From a223669fed8a2f6fdc9fd74dc690d18a5983beba Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Tue, 27 Jun 2023 19:35:56 +0800 Subject: [PATCH] =?UTF-8?q?=E4=B8=BA3498=E9=81=93=E9=A2=98=E7=9B=AE?= =?UTF-8?q?=E5=85=B3=E8=81=94=E4=BA=86=E5=8D=95=E5=85=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 14099 ++++++++++++++++++++++++++++++---------- 1 file changed, 10601 insertions(+), 3498 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 85e25796..2a6cd69a 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -373044,7 +373044,9 @@ "id": "014103", "content": "已知集合$A=\\{1,2\\}$, $B=\\{a, a^2, 3\\}$, 若$A \\cap B=\\{1\\}$, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373064,7 +373066,9 @@ "id": "014104", "content": "若用列举法表示集合$A=\\{x \\in \\mathbf{N} | \\dfrac{8}{6-x} \\in \\mathbf{N}\\}$, 则$A=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373117,7 +373121,9 @@ "id": "014106", "content": "已知点集$M=\\{(x, y) | x^2+y^2<2\\}$, $N=\\{(x, y)|| x|<\\sqrt{2},\\ |y |<\\sqrt{2}\\}$. 若$\\alpha$: 点$P \\in M$, $\\beta$: 点$P \\in N$. 则$\\alpha$是$\\beta$的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373166,7 +373172,9 @@ "id": "014108", "content": "集合$A=\\{x | x^2-5 x-6=0\\}$, $B=\\{x | a x^2-x+6=0,\\ x \\in \\mathbf{R}\\}$, 且$A \\cup B=A$. 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373216,7 +373224,9 @@ "id": "014110", "content": "已知函数$y=f(x)$是$\\mathbf{R}$上的严格增函数, 若$a, b \\in \\mathbf{R}$, 求证: ``$a+b \\geq 0$''是``$f(a)+f(b) \\geq f(-a)+f(-b)$''的充要条件.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373297,7 +373307,9 @@ "id": "014113", "content": "已知集合$A=\\{x | x^2+3 x+2<0\\}, B=\\{x | x^2-4 a x+3 a^2<0\\}$, 且$A \\subset B$, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373317,7 +373329,9 @@ "id": "014114", "content": "若正数$x, y$满足$x y-1=x+y$, 求证: $x, y$均大于$1$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373337,7 +373351,9 @@ "id": "014115", "content": "设集合$A=\\{a+1, a^2-1, a^2-a-1\\}$, 若$1 \\in A$, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373357,7 +373373,9 @@ "id": "014116", "content": "已知集合$A=\\{x | \\dfrac{6}{5-x} \\in \\mathbf{Z}, \\ x \\in \\mathbf{N}\\}$, 试用列举法表示集合$A$, 则$A=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373438,7 +373456,9 @@ "id": "014119", "content": "若不等式$\\dfrac{x-m+1}{x-2 m}<0$成立的一个充分非必要条件是$\\dfrac{1}{3}0$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373644,7 +373676,10 @@ "id": "014128", "content": "设$m \\in \\mathbf{R}$, 若函数$y=\\sqrt{3 x^2-m x+m}$的定义域为$\\mathbf{R}$, 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373664,7 +373699,10 @@ "id": "014129", "content": "若函数$y=(m+1) x^2+m x+(m-1)$的图像都在$x$轴下方 (不含$x$轴), 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373684,7 +373722,9 @@ "id": "014130", "content": "设$a \\in \\mathbf{R}$, 解关于$x$的不等式: $a x^2-(a+1) x+1<0$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373704,7 +373744,9 @@ "id": "014131", "content": "设$a \\in \\mathbf{R}$, 关于$x$的不等式$|x-3|<\\dfrac{x+a}{2}$的解集为$A$.\\\\\n(1) 若$a=2$, 求$A$;\\\\\n(2) 若$A=\\varnothing$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373724,7 +373766,9 @@ "id": "014132", "content": "设$a \\in \\mathbf{R}$, 关于$x$的不等式$|x-3|<\\dfrac{x+a}{2}$的解集为$A$. 若$A \\cap \\mathbf{Z}=\\{3,4\\}$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373802,7 +373846,9 @@ "id": "014135", "content": "设$a \\in \\mathbf{R}$, 若关于$x$的不等式$x^2-a x<0$的解集是区间$(0,1)$的真子集, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373847,7 +373893,9 @@ "id": "014137", "content": "下列不等式中, 解集为$\\{x |-1b>c>d$, 则下列不等式恒成立的是\\bracket{20}.\n\\fourch{$a+d>b+c$}{$a+c>b+d$}{$a c>b d$}{$a d>b c$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -374027,7 +374091,9 @@ "id": "014146", "content": "若$x \\in \\mathbf{R}$, 比较大小: $2 x^2+5 x+3$\\blank{50}$x^2+4 x+2$. (用``$<$''、``$>$''、``$=$''连接)", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -374047,7 +374113,9 @@ "id": "014147", "content": "若正数$x$、$y$满足$\\dfrac{1}{x}+y=4$, 则$\\dfrac{y}{x}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -374067,7 +374135,9 @@ "id": "014148", "content": "若$x<\\dfrac{5}{4}$, 则函数$y=4 x-2+\\dfrac{1}{4 x-5}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -374087,7 +374157,9 @@ "id": "014149", "content": "若正数$x$、$y$满足$\\dfrac{1}{x}+\\dfrac{9}{y}=1$, 则$x+y$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -374107,7 +374179,9 @@ "id": "014150", "content": "已知$\\dfrac{1}{a}<\\dfrac{1}{b}<0$, 有下列不等式: \\textcircled{1} $\\dfrac{1}{a+b}<\\dfrac{1}{a b}$; \\textcircled{2} $|a|+b>0$;\n\\textcircled{3} $a-\\dfrac{1}{a}\\ln b^2$. 其中所有恒成立的不等式的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -374127,7 +374201,9 @@ "id": "014151", "content": "已知$x>0, y>0$, 且$x+y+x y=8$, 求$x+y$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -374147,7 +374223,9 @@ "id": "014152", "content": "已知$x>0, y>0$, 且$x+y+x y=8$, 求$4 x+y$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -374192,7 +374270,9 @@ "id": "014154", "content": "下面四个条件中, 使$a>b$成立的充要条件为\\bracket{20}.\n\\fourch{$a^2>b^2$}{$a^3>b^3$}{$a>b-1$}{$a>b+1$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -374212,7 +374292,9 @@ "id": "014155", "content": "已知$a>0, b>0$, 且$a+b=1$, 有下列不等式: \\textcircled{1} $a^2+b^2 \\geq \\dfrac{1}{2}$, \\textcircled{2} $2^{a-b} \\geq \\dfrac{1}{2}$, \\textcircled{3} $\\log _2 a+\\log _2 b \\geq-2$, \\textcircled{4}$\\sqrt{a}+\\sqrt{b} \\leq \\sqrt{2}$. 其中所有正确的不等式的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -374232,7 +374314,9 @@ "id": "014156", "content": "对任意$x \\in \\mathbf{R}$, 不等式$|x-2|+|x-3| \\geq 2 a^2+a$恒成立, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -374285,7 +374369,9 @@ "id": "014158", "content": "已知$a, b$为非零实数, 则``$a>b$''是``$\\dfrac{1}{a}<\\dfrac{1}{b}$''的\\bracket{20}.\n\\fourch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分也非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -374305,7 +374391,9 @@ "id": "014159", "content": "已知$a>b>0$, 下列不等式中恒成立的是\\bracket{20}.\n\\fourch{$a+b>2 \\sqrt{a b}$}{$a+b<2 \\sqrt{a b}$}{$\\dfrac{a}{2}+2 b>2 \\sqrt{a b}$}{$\\dfrac{a}{2}+2 b<2 \\sqrt{a b}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -374325,7 +374413,9 @@ "id": "014160", "content": "已知实数$a, b$满足$|\\lg a|=|\\lg b|$, 且$a \\neq b$, 那么$\\dfrac{1}{a}+\\dfrac{4}{b}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -374345,7 +374435,9 @@ "id": "014161", "content": "若函数$y=|2 x-a|+2|x+3|$($a>0$)的最小值为$9$, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -374365,7 +374457,9 @@ "id": "014162", "content": "已知$x>0$, $y>0$, 且$x+3 y=x y$, 求$x+y$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -374385,7 +374479,9 @@ "id": "014163", "content": "设$a, b, c \\in \\mathbf{R}$, 比较$a^2+b^2+c^2$与$a b+b c+c a$的值的大小.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -374708,7 +374804,9 @@ "id": "014173", "content": "在天文学中, 天体的明暗程度可以用星等或亮度来描述. 两颗星的星等与亮度满足$m_2-m_1=\\dfrac{5}{2} \\lg \\dfrac{E_1}{E_2}$, 其中星等为$m_k$的星的亮度为$E_k$($k=1,2$). 已知太阳的星等是$-26.7$, 天狼星的星等是$-1.45$, 求太阳与天狼星的亮度的比值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -375078,7 +375176,9 @@ "id": "014184", "content": "我们知道当$a>0$时, $a^{m+n}=a^m \\cdot a^n$对一切$m, n \\in \\mathbf{R}$都成立. 学生小贤在进一步研究指数幂的性质时, 发现有这么一个等式$2^{1+1}=2^1+2^1$, 带着好奇, 他进一步对$2^{m+n}=2^m+2^n$进行深入研究.\\\\\n(1) 当$m=2$时, 求$n$的值;\\\\\n(2) 当$m \\leq 0$时, 求证: $n$是不存在的.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -375098,7 +375198,9 @@ "id": "014185", "content": "下列四组函数中, 同组的两个函数是相同函数的是\\bracket{20}.\n\\twoch{$y=x$与$y=(\\dfrac{1}{x})^{-1}$}{$y=|x|$与$y=\\begin{cases}x, & x>0, \\\\ -x, & x \\leq 0\\end{cases}$}{$y=2 \\ln x$与$y=\\ln x^2$}{$y=x$与$y=\\sqrt[6]{x^6}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -375118,7 +375220,9 @@ "id": "014186", "content": "下列函数是偶函数的为\\bracket{20}.\n\\fourch{$y=\\sin x$}{$y=\\cos x$}{$y=x^3$}{$y=2^x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -375138,7 +375242,9 @@ "id": "014187", "content": "某企业经营一款节能环保产品, 其成本由研发成本与生产成本两部分构成. 研发成本固定为 60 万元, 生产成本为每台 130 元. 根据市场调研, 若该产品产量为$x$万台时, 每万台产品的销售收入为$220-x$($0=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (45:1) node [above] {$A$} coordinate (A);\n\\draw (125:1) node [above] {$B$} coordinate (B);\n\\draw (0,0) circle (1);\n\\draw (O) pic [draw, \"$\\alpha$\", scale = 0.5, angle eccentricity = 1.5] {angle = A--O--B};\n\\draw (A)--(O)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 化简$f(\\alpha)+g(\\alpha)$;\\\\\n(2) 如果$\\dfrac{f(\\alpha)}{g(\\alpha)}=2$, 求$f(\\alpha) \\cdot g(\\alpha)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -375850,7 +375982,9 @@ "id": "014213", "content": "``$\\alpha=\\beta$''是``$\\sin ^2 \\alpha+\\cos ^2 \\beta=1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -375870,7 +376004,9 @@ "id": "014214", "content": "已知点$(-2, y)$在角$\\alpha$的终边上, 若$\\tan (\\pi-\\alpha)=2 \\sqrt{2}$, 则$\\sin \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -375890,7 +376026,9 @@ "id": "014215", "content": "若角$x$满足$2 \\cos (x-\\dfrac{\\pi}{4})=1$, $x \\in(0, \\pi)$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -375910,7 +376048,9 @@ "id": "014216", "content": "若角$\\alpha$的终边与单位圆$x^2+y^2=1$交于点$P(\\dfrac{1}{2}, y)$, 则$\\sin (\\dfrac{\\pi}{2}+\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -375965,7 +376105,9 @@ "id": "014218", "content": "已知$\\alpha \\in(0, \\pi)$, 若$1-2 \\sin 2 \\alpha=\\cos 2 \\alpha$, 则$\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -375985,7 +376127,9 @@ "id": "014219", "content": "已知关于$x$的方程$x^2+3 a x+3 a+1=0$($a>2$)的两实数根分别是$\\tan \\alpha, \\tan \\beta$, 且$\\alpha, \\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 则$\\alpha+\\beta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376005,7 +376149,9 @@ "id": "014220", "content": "已知$\\cos \\theta=-\\dfrac{\\sqrt{2}}{3}$, $\\theta \\in(\\dfrac{\\pi}{2}, \\pi)$, 求$\\dfrac{2}{\\sin 2 \\theta}-\\dfrac{\\cos \\theta}{\\sin \\theta}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376061,7 +376207,9 @@ "id": "014222", "content": "已知$\\tan \\alpha$是关于$x$的方程$x^2+\\dfrac{2 x}{\\cos \\alpha}+1=0$两个实数根中较小的根, 求$\\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376081,7 +376229,9 @@ "id": "014223", "content": "设$\\alpha, \\beta$均为锐角, 且满足$3 \\sin ^2 \\alpha+2 \\sin ^2 \\beta=1$, $2 \\sin 2 \\beta-3 \\sin 2 \\alpha=0$. 求证: $\\alpha+2 \\beta=\\dfrac{\\pi}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376101,7 +376251,9 @@ "id": "014224", "content": "在$\\triangle ABC$中, 若$AB=\\sqrt{2}$, $AC=2$, $A=45^{\\circ}$, 则$\\triangle ABC$的面积$S=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376121,7 +376273,9 @@ "id": "014225", "content": "在$\\triangle ABC$中, 若$AB=4 \\sqrt{3}$, $A=45^{\\circ}$, $C=60^{\\circ}$, 则边$BC=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376211,7 +376365,9 @@ "id": "014228", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$b \\cos C+c \\cos B=a \\sin A$, 则$\\triangle ABC$的形状为\\bracket{20}.\n\\fourch{锐角三角形}{直角三角形}{钝角三角形}{不确定}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -376231,7 +376387,9 @@ "id": "014229", "content": "如图, 某观测站$C$在$A$城的南偏西$20^{\\circ}$方向上, 由$A$城出发有一条公路走向是南偏东$40^{\\circ}$, 测得距$C$点$31$千米的$B$处有一人开车正沿公路向$A$城行驶, 行驶了$20$千米后到达$D$处, 此时$C$、$D$间的距离为$21$千米. 问: 此人还需行驶多少千米才到$A$城?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-110:2.4) node [left] {$C$} coordinate (C);\n\\draw (-50:3.5) node [right] {$B$} coordinate (B);\n\\draw (-50:1.5) node [above right] {$D$} coordinate (D);\n\\draw (A)--(C)--(B)--cycle (C)--(D);\n\\draw [->] (1.7,-1) -- (2.6,-1) node [right] {东};\n\\draw [->] (2,-1.3) -- (2,-0.4) node [above] {北}; \n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376251,7 +376409,9 @@ "id": "014230", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若满足$A=\\dfrac{\\pi}{3}$, $a=4$的$\\triangle ABC$恰有一个, 则$c$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376306,7 +376466,9 @@ "id": "014232", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$a=8$, $b=5$, $c=\\sqrt{153}$则$\\triangle ABC$的面积$S=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376326,7 +376488,9 @@ "id": "014233", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若满足$b \\sin A=a \\cos (B-\\dfrac{\\pi}{6})$. 则角$B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376346,7 +376510,10 @@ "id": "014234", "content": "已知$\\triangle ABC$, 那么``$|\\overrightarrow{AC}|^2+|\\overrightarrow{AB}|^2-|\\overrightarrow{BC}|^2<0$''是``$\\triangle ABC$为钝角三角形''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -376366,7 +376533,9 @@ "id": "014235", "content": "某公司要在$A$、$B$两地连线上的定点$C$处建造广告牌$CD$, 其中$D$为顶端, $AC$长$35$米, $CB$长$80$米, 设$A$、$B$在同一水平面上, 从$A$和$B$看$D$的仰角分别为$\\alpha$和$\\beta$, 现测得$\\alpha=28.12^{\\circ}$, $\\beta=18.45^{\\circ}$, 求$AD$与$CD$的长. (结果精确到$0.01$米)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376386,7 +376555,9 @@ "id": "014236", "content": "在$\\triangle ABC$中, 若$AC=3$, $3 \\sin A=2 \\sin B$, 且$\\cos C=\\dfrac{1}{4}$, 则$AB=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376406,7 +376577,9 @@ "id": "014237", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$. 若$b \\cos A+(a+4 c) \\cos B=0$, 则$B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376426,7 +376599,9 @@ "id": "014238", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 其中$a=4, c=6, \\cos C=\\dfrac{1}{8}$.\\\\\n(1) 求$\\sin A$的值;\\\\\n(2) 求$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376446,7 +376621,9 @@ "id": "014239", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$.\\\\\n(1) 若$\\triangle ABC$的面积$S=\\dfrac{a^2+c^2-b^2}{4}$, 求$B$;\\\\\n(2) 若$a c=\\sqrt{3}$, $\\sin A=\\sqrt{3} \\sin B$, $C=\\dfrac{\\pi}{6}$, 求$c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376466,7 +376643,9 @@ "id": "014240", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$c \\sin C-b \\sin B=a(\\sin A-\\sin B)$.\\\\\n(1) 求角$C$的值;\\\\\n(2) 若$c=3$, 求$\\triangle ABC$周长的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376486,7 +376665,9 @@ "id": "014241", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$b=2$.\\\\\n(1) 若$A+C=\\dfrac{2 \\pi}{3}$, 且$a=2c$, 求$c$的值;\\\\\n(2) 若$A-C=\\dfrac{\\pi}{12}$, $a=\\sqrt{2} c \\sin A$, 求$\\triangle ABC$面积$S$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376506,7 +376687,9 @@ "id": "014242", "content": "在$\\triangle ABC$中, $A=\\dfrac{\\pi}{3}$, 则``$\\sin B<\\dfrac{1}{2}$''是``$\\triangle ABC$是钝角三角形''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -376526,7 +376709,9 @@ "id": "014243", "content": "在临港滴水湖畔拟建造一个四边形的露营基地, 如图$ABCD$所示. 为考虑露营客人娱乐休闲的需求, 在四边形$ABCD$区域中, 将$\\triangle ABD$区域设立成花卉观赏区, $\\triangle BCD$区域设立成烧烤区, 边$AB$、$BC$、$CD$、$DA$修建成观赏步道, 边$BD$修建隔离防护栏. 其中$CD=100$米, $BC=200$米, $\\angle A=\\dfrac{\\pi}3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.7]\n\\draw (0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,0) node [left] {$D$} coordinate (D);\n\\draw (-80:2) node [right] {$B$} coordinate (B);\n\\draw ($(B)!{1/sqrt(3)}!30:(D)$) coordinate (O);\n\\draw ($(O)!1!100:(D)$) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (D) -- (B);\n\\draw (barycentric cs:A=1,B=1,D=1) node {花卉观赏区};\n\\draw (barycentric cs:B=1,C=1,D=1) node {烧烤区};\n\\end{tikzpicture}\n\\end{center}\n(1) 如果烧烤区是一个占地面积为$9600$平方米的钝角三角形, 那么需要修建多长的隔离防护栏(精确到$0.1$米)?\\\\\n(2) 考虑到烧烤区的安全性, 在规划四边形$ABCD$区域时, 首先保证烧烤区的占地面积最大时, 再使得花卉观赏区的面积尽可能大, 则应如何设计观赏步道?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376546,7 +376731,9 @@ "id": "014244", "content": "函数$y=\\tan (\\dfrac{\\pi}{3} x-\\dfrac{\\pi}{5})$的单调区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376634,7 +376821,9 @@ "id": "014247", "content": "在下列函数中, 既在区间$(0, \\dfrac{\\pi}{2})$上是严格增函数, 又是以$\\pi$为最小正周期的偶函数的是\\bracket{20}\n\\fourch{$y=\\sin x$}{$y=\\cos 2 x$}{$y=|\\sin x|$}{$y=\\sin |2 x|$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -376654,7 +376843,9 @@ "id": "014248", "content": "已知$f(x)=2 \\sin x \\cos x+\\sqrt{3} \\cos 2 x$, 若函数$y=f(x)-k$在区间$[0, \\dfrac{\\pi}{4}]$上有两个不同的零点, 则实数$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376721,7 +376912,9 @@ "id": "014250", "content": "求函数$y=2 \\sin x \\cos x+\\sqrt{3} \\cos 2 x$在区间$(0, \\pi)$上的单调减区间和单调增区间;", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376741,7 +376934,9 @@ "id": "014251", "content": "已知$f(x)=\\sin (\\pi x+\\varphi)-\\sqrt{3} \\cos (\\pi x+\\varphi)$($0<\\varphi<\\pi$).\\\\\n(1) 若函数$y=f(x)$是偶函数, 求$\\varphi$的值;\\\\\n(2) 若函数$y=f(x)$的图像关于直线$x=1$对称, 求$\\sin 2 \\varphi$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376761,7 +376956,9 @@ "id": "014252", "content": "已知$f(x)=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$, $|\\varphi|<\\dfrac{\\pi}{2}$), 函数$y=f(x)$的图像与$y$轴的交点为$(0,-\\sqrt{3})$, 它在$y$轴右侧的第一个最高点和第一个最低点的坐标分别为$(x_0, 2)$和$(x_0+\\dfrac{\\pi}{2},-2)$.\\\\\n(1) 求函数$y=f(x)$的表达式及$x_0$的值;\\\\\n(2) 设函数$y=a f(x)+b$, $x\\in [-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$的值域为$[0,3]$, 求实数$a, b$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376781,7 +376978,9 @@ "id": "014253", "content": "设函数$y=-2\\sin(2x-\\dfrac \\pi 3)+5+(-1)^n \\cdot m$($m \\in \\mathbf{R}$, $n$是正整数), 若该函数对任意的$x \\in[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$均有$y>0$恒成立, 求实数$m$取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -376801,7 +377000,9 @@ "id": "014254", "content": "函数$y=\\sin 3 x+|\\sin 3 x|$\\bracket{20}.\n\\twoch{为周期函数, 且最小正周期为$\\dfrac{\\pi}{3}$}{为周期函数, 且最小正周期为$\\dfrac{2 \\pi}{3}$}{为周期函数, 且最小正周期为$2 \\pi$}{不是周期函数}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -376868,7 +377069,9 @@ "id": "014256", "content": "已知$f(x)=\\sin 2 x+\\sin (2 x+\\dfrac{\\pi}{3})$, 若将函数$y=f(x)$的图像向左平移$\\varphi$($\\varphi>0$)个单位长度后得到的图像关于$y$轴对称, 则$\\varphi$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -376888,7 +377091,9 @@ "id": "014257", "content": "若函数$y=\\cos x-\\sin x$在区间$[-a, a]$上是严格减函数, 则$a$的最大值是\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{4}$}{$\\dfrac{\\pi}{2}$}{$\\dfrac{3 \\pi}{4}$}{$\\pi$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -376908,7 +377113,9 @@ "id": "014258", "content": "已知$f(x)=\\sin x+a \\cos x$($a$为常数) 满足$f(x) \\leq f(\\dfrac{\\pi}{6})$. 若函数$y=f(x)$在区间$[x_1, x_2]$上具有单调性, 且满足$f(x_1)+f(x_2)=0$, 求$|x_1+x_2|$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -377009,7 +377216,9 @@ "id": "014261", "content": "已知函数$y=\\cos (\\omega x+\\varphi)$($\\omega>0$, $|\\varphi|<\\dfrac{\\pi}{2}$), 若该函数图像的对称中心到其对称轴$x=\\dfrac{\\pi}{6}$的距离的最小值为$\\dfrac{\\pi}{8}$, 则该函数的表达式为$y=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377111,7 +377320,9 @@ "id": "014264", "content": "已知$f(x)=\\sin x \\cos x-\\sin ^2 x$.\\\\\n(1) 求函数$y=f(x)$的最小值, 并求出此时相应的$x$的值;\\\\\n(2) 写出函数$y=f(x)$, $x \\in[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$的单调区间, 并求该函数的值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -377199,7 +377410,9 @@ "id": "014267", "content": "若角$\\alpha$的终边经过点$P(6,-8)$, 则$\\sin (\\dfrac{3 \\pi}{2}+\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377219,7 +377432,9 @@ "id": "014268", "content": "已知$\\triangle ABC$的面积是$\\dfrac{1}{2}$, $AB=1$, $BC=\\sqrt{2}$, 则$AC=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377333,7 +377548,9 @@ "id": "014271", "content": "函数$y=-\\sin ^2 x-4 \\cos x+6$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377353,7 +377570,10 @@ "id": "014272", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 且$\\triangle ABC$的面积$S$满足$S=\\dfrac{\\sqrt{3}}{2} \\overrightarrow{BA} \\cdot \\overrightarrow{BC}$, $b=2$, 求$a+c$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -377420,7 +377640,9 @@ "id": "014274", "content": "已知$f(x)=2 \\sin (x+\\dfrac{\\pi}{3})$.\\\\\n(1) 若满足$f(x-\\dfrac{\\pi}{3})+f(x)=a$的锐角$x$有两个, 求实数$a$的取值范围;\\\\\n(2) 若满足$f(x-\\dfrac{\\pi}{3})+f(x)=a$的锐角$x$只有一个, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -377440,7 +377662,9 @@ "id": "014275", "content": "如图, 某污水处理厂要在一个矩形污水处理池$ABCD$的池底水平铺设污水净化管道$EH$、$FH$、$EF$来处理污水, 管道越长, 污水净化效果越好. 要求管道的接口$H$是$AB$的中点, $F$分别落在线段$BC$、$AD$上(含线段两端点), $FH \\perp HE$. 已知$AB=40$米, $AD=20 \\sqrt{3}$米, 记$\\angle BHE=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0) node [right] {$B$} coordinate (B);\n\\draw (4,{2*sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (0,{2*sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (A) rectangle (C);\n\\draw ($(A)!0.5!(B)$) node [below] {$H$} coordinate (H);\n\\draw [dashed] (H) --++ (-2,{4/3}) node [left] {$F$} coordinate (F);\n\\draw [dashed] (H) --++ (2,3) node [right] {$E$} coordinate (E);\n\\draw [dashed] (E)--(F);\n\\draw (H) pic [draw, \"$\\theta$\", angle eccentricity = 1.5] {angle = B--H--E};\n\\draw (H) pic [draw, scale = 0.5] {right angle = E--H--F};\n\\end{tikzpicture}\n\\end{center}\n(1) 试将污水净化管道的总长度$L$(即$\\triangle FHE$的周长)表示为$\\theta$的函数, 并求出其定义域;\\\\\n(2) 问$\\theta$取何值时, 污水净化效果最好? 并求出此时管逭的总长度$L$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -377460,7 +377684,9 @@ "id": "014276", "content": "若角$\\alpha$的终边经过点$P(4,-3)$, 则$\\sin (\\dfrac{\\pi}{2}+2 \\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377480,7 +377706,9 @@ "id": "014277", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$A=60^{\\circ}, a=\\sqrt{6}, b=2$, 则$\\cos C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377500,7 +377728,9 @@ "id": "014278", "content": "若对任意实数$x$, 不等式$m+\\cos ^2 x<3+2 \\sin x$恒成立, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377520,7 +377750,9 @@ "id": "014279", "content": "已知向量$\\overrightarrow {m}=(\\dfrac{1}{2}, \\dfrac{1}{2} \\sin 2 x+\\dfrac{\\sqrt{3}}{2} \\cos 2 x)$, $\\overrightarrow {n}=(f(x),-1)$, 且$\\overrightarrow {m} \\perp \\overrightarrow {n}$.\\\\\n(1) 求函数$y=f(x), x \\in[0, \\pi]$的单调增区间;\\\\\n(2) 在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$f(A-\\dfrac{\\pi}{12})=1$, $BC=\\sqrt{3}$, 求$\\triangle ABC$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -377540,7 +377772,9 @@ "id": "014280", "content": "若角$\\alpha$的终边经过点$P(-x,-6)$, 且$\\cos \\alpha=-\\dfrac{5}{13}$, 则$\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377560,7 +377794,10 @@ "id": "014281", "content": "已知等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_{12}=7 \\pi$, 则$\\cos (a_6+a_7)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377580,7 +377817,9 @@ "id": "014282", "content": "已知函数$y=6 \\cos ^2 \\omega x+\\sqrt{3} \\sin 2 \\omega x-3$(其中常数$\\omega>0$) 的最小正周期为$8$, 则函数的单调减区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377647,7 +377886,9 @@ "id": "014284", "content": "已知$f(x)=2 \\sin \\omega x \\cos \\omega x-2 \\sqrt{3} \\cos ^2 \\omega x+\\sqrt{3}$(其中常数$\\omega>0$), 函数$y=f(x)$图像的两条相邻的对称轴之间的距离为$\\dfrac{\\pi}{2}$.\\\\\n(1) 求函数$y=f(x)$的表达式;\\\\\n(2) 在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 角$C$为锐角, 且$f(C)=\\sqrt{3}$, $c=3$, $\\sin B=2 \\sin A$, 求$\\triangle ABC$的周长.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -377748,7 +377989,9 @@ "id": "014287", "content": "在锐角$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$A=2B$.\\\\\n(1) 求$B$的取值范围;\\\\\n(2) 求$\\dfrac{a}{b}+\\dfrac{b}{a}$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -377803,7 +378046,9 @@ "id": "014289", "content": "已知$a$、$b$、$\\alpha$、$\\beta$为实数, $aa$成立, 且$f(0)=-a^2$. 证明: 对任意常数$a$, 关于$x$的方程$f(x+a)=a x$有且仅有$1$个解.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -377938,7 +378191,9 @@ "id": "014295", "content": "函数$y=x^2-3 x-\\dfrac{2}{x}$的零点的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377958,7 +378213,9 @@ "id": "014296", "content": "已知$a \\in \\mathbf{R}$, 若关于$x$的方程$|3^x-1|-2 a=0$有唯一解, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377978,7 +378235,9 @@ "id": "014297", "content": "已知$t \\in \\mathbf{R}$, 若关于$x$的方程$(x-1)^2+2 t^2+6 t=2 t \\cos (x-1)+16$有且仅有一个实数解, 则$t$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -377998,7 +378257,10 @@ "id": "014298", "content": "设$\\theta \\in \\mathbf{R}$. 若$f(x)=3 \\sin x+2$满足: 对任意$x_1 \\in[0, \\dfrac{\\pi}{2}]$, 都存在$x_2 \\in[0, \\dfrac{\\pi}{2}]$, 使得$f(x_1)+2 f(x_2+\\theta)=3$, 则$\\theta$可以是\\bracket{20}.\n\\fourch{$\\dfrac{3 \\pi}{5}$}{$\\dfrac{4 \\pi}{5}$}{$\\dfrac{6 \\pi}{5}$}{$\\dfrac{7 \\pi}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378018,7 +378280,9 @@ "id": "014299", "content": "已知$f(x)=(\\dfrac{1}{2})^x-\\log _3 x$, $x_0$是函数$y=f(x)$的零点, 实数$a$、$b$、$c$满足$0c$. 其中一定不成立的不等式的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378038,7 +378302,10 @@ "id": "014300", "content": "已知$k \\in \\mathbf{R}$, 当$0 \\leq x \\leq 1$时, 不等式$\\sin \\dfrac{\\pi x}{2} \\geq k x$成立, 则$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378058,7 +378325,9 @@ "id": "014301", "content": "已知$f(x)=x-[x]$, 其中$[x]$表示不大于$x$的最大整数, 则方程$5 f(x)-x-2=0$的解的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378078,7 +378347,9 @@ "id": "014302", "content": "设$a, b \\in \\mathbf{Z}$, 若对任意$x \\leq 0$, 都有$(a x+2)(x^2+2 b) \\leq 0$, 则$a+b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378098,7 +378369,9 @@ "id": "014303", "content": "已知$a \\in \\mathbf{R}, y=f(x)$是定义在区间$[-2,2]$上的奇函数, 在区间$[0,2]$上是严格增函数, 若$f(a^2+2 a-3)+f(2-2 a^2)<0$, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378118,7 +378391,9 @@ "id": "014304", "content": "已知$f(x)=\\dfrac{2-x}{x+1}$.\\\\\n(1) 证明: 函数$y=f(x)$在区间$(-1,+\\infty)$上为严格减函数;\\\\\n(2) 是否存在负数$x_0$, 使得$f(x_0)=2^{\\circ}$成立, 若存在求出$x_0$: 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378138,7 +378413,9 @@ "id": "014305", "content": "方程$\\ln x-1=\\dfrac{1-x}{\\mathrm{e}^x}$是否有整数解? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378158,7 +378435,9 @@ "id": "014306", "content": "已知$a \\in \\mathbf{R}$, 若对任意$x_1, x_2 \\in[1,+\\infty)$, 当$x_1a$对任意的$x \\in[0, \\dfrac{1}{2}]$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378198,7 +378479,9 @@ "id": "014308", "content": "若关于$x$的不等式$x^2+x>a$对任意的$x \\in[0, \\dfrac{1}{2}]$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378218,7 +378501,9 @@ "id": "014309", "content": "若关于$x$的不等式$x^2+x>a$对任意的$x \\in(0, \\dfrac{1}{2})$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378238,7 +378523,9 @@ "id": "014310", "content": "若关于$x$的方程$x^2+x=a$在区间$(0, \\dfrac{1}{2})$上有解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378258,7 +378545,9 @@ "id": "014311", "content": "若关于$x$的不等式$x^2+a x+1>0$对任意的$x \\in[0, \\dfrac{1}{2}]$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378278,7 +378567,9 @@ "id": "014312", "content": "若关于$x$的方程$x^2+a x+1=0$在区间$(0, \\dfrac{1}{2})$上有解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378298,7 +378589,9 @@ "id": "014313", "content": "已知定义在区间$[0,2]$上的两个函数$y=f(x)$和$y=g(x)$, 其中$f(x)=x^2-a x+4$($a \\geq 2$), $g(x)=\\dfrac{x^2}{x+1}$, 若对于任意的$x_1, x_2 \\in[0,2]$, $f(x_1) \\geq g(x_2)$恒成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378318,7 +378611,9 @@ "id": "014314", "content": "已知$f(x)=a x+\\dfrac{1}{x+1}$, $a \\in \\mathbf{R}$, 若函数$y=f(x)$区间$[1,2]$上有零点, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378338,7 +378633,9 @@ "id": "014315", "content": "已知定义在区间$[0,2]$上的两个函数$y=f(x)$和$y=g(x)$, 其中$f(x)=x^2-a x+4$($a \\geq 2$), $g(x)=\\dfrac{x^2}{x+1}$, 若对于任意的$x_1 \\in[0,2]$, 总存在$x_2 \\in[0,2]$, 使得$f(x_1) \\geq g(x_2)$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378358,7 +378655,9 @@ "id": "014316", "content": "设$a \\in \\mathbf{R}$, $f(x)=\\dfrac{2^x+a}{2^x+1}$, 若$f(x)<\\dfrac{a+2}{2}$对任意的$x \\in \\mathbf{R}$成立, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378378,7 +378677,9 @@ "id": "014317", "content": "若关于$x$的不等式$x^2-3>a x-a$对任意的$x \\in[3,4]$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378398,7 +378699,9 @@ "id": "014318", "content": "若关于$x$的不等式$2^x-\\dfrac{2}{x}-a>0$在区间$[1,2]$上有实数解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378418,7 +378721,9 @@ "id": "014319", "content": "设$f(x)=\\begin{cases}|x^2+2 x-1|,& x \\leq 0, \\\\ 2^{x-1}+a & , x>0.\\end{cases}$ 若函数$y=f(x)$有两个不同的零点, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378438,7 +378743,10 @@ "id": "014320", "content": "设$f(x)=\\dfrac{-4 x+5}{x+1}$, $g(x)=a \\sin (\\dfrac{\\pi}{3} x)+2 a$, $a>0$, 若对任意$x_1 \\in[0,2]$, 总存在$x_2 \\in[0,2]$, 使得$g(x_1)=f(x_2)$成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378458,7 +378766,9 @@ "id": "014321", "content": "设$f(x)=|2 x+1|$, 若不等式$| f(x)-2 f(\\dfrac{x}{2})| \\leq k$对任意的$x \\in \\mathbf{R}$成立, 则实数$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378478,7 +378788,9 @@ "id": "014322", "content": "设$f(x)=\\begin{cases}x^2-x+3, & x \\leq 1, \\\\ x+\\dfrac{2}{x}, & x>1.\\end{cases}$若关于$x$的不等式$f(x)>|\\dfrac{x}{2}+a|$对任意的$x \\in \\mathbf{R}$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378498,7 +378810,9 @@ "id": "014323", "content": "已知$f(x)=4-\\dfrac{1}{x}$, 若存在区间$[a, b] \\subseteq(\\dfrac{1}{3},+\\infty)$, 使得$\\{y | y=f(x),\\ x \\in[a, b]\\}=[m a, m b]$, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378518,7 +378832,9 @@ "id": "014324", "content": "设$f(x)=\\begin{cases}|x+2|, & x<0, \\\\ x^2-4 x+2, & x \\geq 0,\\end{cases}$ $g(x)=k x+1$. 若函数$y=f(x)-g(x)$的图像经过四个象限, 则实数$k$的取值范围是\\bracket{20}.\n\\fourch{$(-2, \\dfrac{1}{2})$}{$(-6, \\dfrac{1}{2})$}{$(-2,+\\infty)$}{$(-\\infty,-6) \\cup(\\dfrac{1}{2},+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378538,7 +378854,9 @@ "id": "014325", "content": "已知$\\overrightarrow {a}, \\overrightarrow {b}$为两个单位向量, 下列结论中正确的是\\bracket{20}.\n\\twoch{$\\overrightarrow {a}=\\overrightarrow {b}$}{如果$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 那么$\\overrightarrow {a}=\\overrightarrow {b}$}{$\\overrightarrow {a}\\parallel \\overrightarrow {b}$}{如果$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 那么$\\overrightarrow {a}=\\overrightarrow {b}$或$\\overrightarrow {a}=-\\overrightarrow {b}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378558,7 +378876,9 @@ "id": "014326", "content": "已知在平行四边形$ABCD$中, $|\\overrightarrow{AB}|=2$, $|\\overrightarrow{AD}|=3$, $M$为边$CD$的中点, \n若$\\overrightarrow{AM} \\cdot \\overrightarrow{AD}=\\dfrac{21}{2}$, 则向量$\\overrightarrow{AB}$与$\\overrightarrow{AD}$的夹角大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378578,7 +378898,9 @@ "id": "014327", "content": "已知$\\overrightarrow {b}=(1,2)$, 若向量$\\overrightarrow {a}$满足$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 且$|\\overrightarrow {a}|=5$, 则$\\overrightarrow {a}$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378598,7 +378920,9 @@ "id": "014328", "content": "在$\\triangle ABC$中, 若$|\\overrightarrow{AB}+\\overrightarrow{AC}|=|\\overrightarrow{AB}-\\overrightarrow{AC}|$, 则$\\triangle ABC$的形状是\\bracket{20}.\n\\fourch{等腰三角形}{直角三角形}{等边三角形}{等腰直角三角形}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378618,7 +378942,9 @@ "id": "014329", "content": "已知向量$\\overrightarrow {a}=(-1,2)$, $\\overrightarrow {b}=(1,1)$.\\\\\n(1) 已知$k \\in \\mathbf{R}$, 若$\\overrightarrow {c}=k \\overrightarrow {a}+(1-k) \\overrightarrow {b}$, 且$\\overrightarrow {b} \\perp \\overrightarrow {c}$, 求$k$的值;\\\\\n(2) 若$\\overrightarrow{AB}=\\overrightarrow {a}+\\overrightarrow {b}$, $\\overrightarrow{BC}=\\overrightarrow {a}-2 \\overrightarrow {b}$, $\\overrightarrow{CD}=4 \\overrightarrow {a}-2 \\overrightarrow {b}$, 求证: $A$、$C$、$D$三点共线.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378708,7 +379034,9 @@ "id": "014332", "content": "如图, 在正方形$ABCD$中, $E$为$AB$的中点, $F$为$CE$的中点, 则$\\overrightarrow{BF}$等于\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,2) node [above] {$C$} coordinate (C);\n\\draw (0,2) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(E)$) node [left] {$F$} coordinate (F);\n\\draw (A) rectangle (C) --(E) (B)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{3}{4} \\overrightarrow{AB}+\\dfrac{1}{4} \\overrightarrow{AD}$}{$-\\dfrac{1}{4} \\overrightarrow{AB}+\\dfrac{1}{2} \\overrightarrow{AD}$}{$\\dfrac{1}{2} \\overrightarrow{AB}+\\overrightarrow{AD}$}{$\\dfrac{1}{4} \\overrightarrow{AB}+\\dfrac{3}{4} \\overrightarrow{AD}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378728,7 +379056,9 @@ "id": "014333", "content": "已知向量$\\overrightarrow {a}=(1,0), \\overrightarrow {b}=(1,1)$, 当$k$为何实数时:\\\\\n(1) $\\overrightarrow {a}+k \\overrightarrow {b}$与$\\overrightarrow {a}+\\overrightarrow {b}$垂直;\\\\\n(2) $\\overrightarrow {a}+k \\overrightarrow {b}$与$\\overrightarrow {a}+\\overrightarrow {b}$平行.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378748,7 +379078,9 @@ "id": "014334", "content": "在矩形$ABCD$中, $|AB|=\\sqrt{2}$, $|BC|=2$, 点$E$为边$BC$的中点, 点$F$在边$CD$上, $\\overrightarrow{AB} \\cdot \\overrightarrow{AF}=\\sqrt{2}$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{AF}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378768,7 +379100,9 @@ "id": "014335", "content": "已知四边形$ABCD$是边长为$1$的正方形, 则$|\\overrightarrow{AB}+\\overrightarrow{CA}-\\overrightarrow{DC}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378788,7 +379122,9 @@ "id": "014336", "content": "已知点$A(2,3)$, $B(6,-3)$, 且$\\overrightarrow{AB}=3 \\overrightarrow{AP}$, 则点$P$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378808,7 +379144,9 @@ "id": "014337", "content": "设向量$\\overrightarrow {a}=(1,-1)$, $\\overrightarrow {b}=(3,5)$, 求$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的数量投影与投影的坐标.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378828,7 +379166,9 @@ "id": "014338", "content": "已知向量$\\overrightarrow{OA}=(4,1)$, $\\overrightarrow{OB}=(-1,3)$且$\\overrightarrow{OC} \\perp \\overrightarrow{OB}$, $\\overrightarrow{BC}\\parallel \\overrightarrow{OA}$, 求向量$\\overrightarrow{OC}$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378882,7 +379222,9 @@ "id": "014340", "content": "如图, 两块斜边长为$\\sqrt{2}$的直角三角板拼在一起, 求$\\overrightarrow{OD} \\cdot \\overrightarrow{AB}$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (0,2) node [left] {$B$} coordinate (B);\n\\draw (A) ++ (45:{sqrt(6)}) node [right] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [below left] {$C$} coordinate (C);\n\\draw (B)--(O)--(A)--(D)--(C)(A)--(B);\n\\draw pic [draw,scale = 0.5] {right angle = A--O--B};\n\\draw pic [draw,scale = 0.5,\"$45^\\circ$\", angle eccentricity = 2.5] {angle = B--A--O};\n\\draw pic [draw,scale = 0.5,\"$60^\\circ$\", angle eccentricity = 2.5] {angle = A--C--D};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378902,7 +379244,9 @@ "id": "014341", "content": "在$\\triangle ABC$中, $|AB|=5$, $|AC|=6$, $\\cos A=\\dfrac{1}{5}$, $O$是$\\triangle ABC$的外心, 若$\\overrightarrow{OP}=x \\overrightarrow{OB}+y \\overrightarrow{OC}$, 其中$x, y \\in[0,1]$, 则动点$P$的轨迹所覆盖图形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378922,7 +379266,9 @@ "id": "014342", "content": "如图, 已知$\\triangle ABC$的三边长$|AB|=8$, $|BC|=7$, $|AC|=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:3) node [below] {$C$} coordinate (C);\n\\draw (-120:8) node [left] {$B$} coordinate (B);\n\\draw (10:3) node [right] {$Q$} coordinate (Q);\n\\draw (190:3) node [left] {$P$} coordinate (P);\n\\draw (A) circle (3);\n\\draw (P)--(Q)--(C)--(B)--cycle (B)--(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$;\\\\\n(2) 圆$A$的半径为$3$, 设$PQ$是圆$A$的一条直径, 求$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378942,7 +379288,9 @@ "id": "014343", "content": "如图, 点$N$为正方形$ABCD$的中心, $\\triangle ECD$为正三角形, $M$是线段$ED$上任一点, 则直线$BM$、$EN$的位置关系是\\bracket{20}. (填``相交''、``平行''、或``异面'')\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$D$} coordinate (D);\n\\draw (2,0,2) node [right] {$A$} coordinate (A);\n\\draw ($(C)!0.5!(D)$) ++ (0,{sqrt(3)},0) node [above] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(E)$) node [left] {$M$} coordinate (M);\n\\draw ($(A)!0.5!(C)$) node [right] {$N$} coordinate (N);\n\\draw (A)--(B)--(C)--(D)--cycle (B)--(M)(E)--(N)(D)--(E)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378962,7 +379310,9 @@ "id": "014344", "content": "已知等边$\\triangle ABC$的面积为$\\dfrac{9 \\sqrt{3}}{4}$, 其顶点都在球$O$的表面上, 若球心$O$到平面$ABC$的距离为$1$, 则球$O$的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378982,7 +379332,9 @@ "id": "014345", "content": "如图所示, 在正三棱柱$ABC-A_1B_1C_1$中, $AB=3$, $AA_1=4$, $M$为$AA_1$的中点, $P$是$BC$上一点, 且由$P$沿棱柱侧面经过棱$CC_1$到$M$的最短距离为$\\sqrt{29}$, 则$PC$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (1.5,0,{1.5*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,4,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,4,0) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,4,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$M$} coordinate (M);\n\\draw ($(C)!0.2!(C_1)$) coordinate (N);\n\\draw ($(B)!{1/3}!(C)$) node [below right] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle (A_1)--(B_1)--(C_1) (B)--(B_1) (P)--(N);\n\\draw [dashed] (M)--(N)(A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379037,7 +379389,9 @@ "id": "014347", "content": "如图, 已知圆柱$OO_1$的底面半径为$1$, 正$\\triangle ABC$内接于圆柱的下底面圆$O$, 点$O_1$是圆柱的上底面的圆心, 线段$AA_1$是圆柱的母线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [below] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2) node [left] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O_1) ellipse (2 and 0.5);\n\\draw (O)++(-2,0) arc (180:360:2 and 0.5);\n\\draw [dashed] (O)++(-2,0) arc (180:0:2 and 0.5);\n\\draw (2,0) -- (2,2) (-2,0) -- (-2,2);\n\\draw (20:2 and 0.5) node [right] {$A$} coordinate (A);\n\\draw (A) ++ (0,2) node [right] {$A_1$} coordinate (A_1);\n\\draw [dashed] (A)--(A_1);\n\\draw (140:2 and 0.5) node [above] {$C$} coordinate (C);\n\\draw (265:2 and 0.5) node [below] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [below right] {$M$} coordinate (M);\n\\draw [dashed] (A)--(B)--(C)--cycle(A)--(B)(A_1)--(B);\n\\draw [dashed] (C)--(M)(B)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$C$到平面$A_1AB$的距离;\\\\\n(2) 在劣弧$\\overset\\frown{BC}$上存在一点$D$, 满足$\\angle BOD=\\dfrac{\\pi}{6}$, 证明$O_1D\\parallel$平面$A_1AB$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -379126,7 +379480,9 @@ "id": "014350", "content": "若一个圆锥的侧面展开图为半径为$2$的半圆形, 则该圆锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379146,7 +379502,9 @@ "id": "014351", "content": "如图, 已知正方体$ABCD-A_1B_1C_1D_1$, $M$, $N$分别是$A_1D$, $D_1B$的中点, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(D)$) node [below left] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(D1)$) node [right] {$N$} coordinate (N);\n\\draw (B1)--(D1);\n\\draw [dashed] (A1)--(D1)(B)--(D)(A1)--(D)(D1)--(B)(M)--(N);\n\\end{tikzpicture}\n\\end{center}\n\\onech{直线$A_1D$与直线$D_1B$垂直, 直线$MN\\parallel$平面$ABCD$}{直线$A_1D$与直线$D_1B$平行, 直线$MN\\parallel$平面$BDD_1B_1$}{直线$A_1D$与直线$D_1B$相交, 直线$MN\\parallel$平面$ABCD$}{直线$A_1D$与直线$D_1B$异面, 直线$MN\\parallel$平面$BDD_1B_1$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -379201,7 +379559,9 @@ "id": "014353", "content": "在长方体$ABCD-A_1B_1C_1D_1$的各条棱所在直线中, 与直线$AB$异面且垂直的直线有\\blank{50}条.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379256,7 +379616,9 @@ "id": "014355", "content": "若三棱锥$S-ABC$的所有的顶点都在球$O$的球面上, 且$SA \\perp$平面$ABC, AB=2$, $SA=AC=4$, $\\angle BAC=\\dfrac{\\pi}{3}$, 则球$O$的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379276,7 +379638,9 @@ "id": "014356", "content": "如图, 在矩形$ABCD$中, $AB=4$, $AD=2$, $E$为$AB$边的中点, 将$\\triangle ADE$沿$DE$翻折, 得到四棱锥$A_1-DEBC$. 设线段$A_1C$的中点为$M$, 在翻折过程中, 是否总有$BM\\parallel$平面$A_1DE$? 如果有, 证明你的结论.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (4,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(E)$) ++ (0,{sqrt(2)},0) coordinate (T);\n\\draw ($0.3*(A)+sqrt(0.91)*(T)$) node [above] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)!0.5!(C)$) node [above] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(A_1)--(D)--(A)(D)--(E)--(A_1)--(B)--(M);\n\\draw [dashed] (D)--(C)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -379296,7 +379660,9 @@ "id": "014357", "content": "如图, $AB$是圆$O$的直径, 点$C$是圆$O$上异于$A, B$的点, $PO$垂直于圆$O$所在的平面, 且$PO=OB=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above right] {$O$} coordinate (O);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (0,2) node [above] {$P$} coordinate (P);\n\\draw (A) arc (180:360:2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw (-50:2 and 0.5) node [below] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(P)--(B)(P)--(C);\n\\draw [dashed] (A)--(B)(P)--(O)(A)--(C)--(B)(P)--(D)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$D$为线段$AC$的中点, 求证: $AC \\perp$平面$PDO$;\\\\\n(2) 求三棱锥$P-ABC$体积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -379316,7 +379682,9 @@ "id": "014358", "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, 底面$ABC$是以$AC$为斜边的等腰直角三角形, 侧面$AA_1C_1C$为菱形, 点$A_1$在底面上的投影为$AC$的中点$D$, 且$AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (1,{sqrt(3)},0) node [above] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (2,0,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (1,0,0) node [above right] {$D$} coordinate (D);\n\\draw ($(A_1)+(B)-(A)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)!0.4!(B_1)$) node [above right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(C_1)(B_1)--(B);\n\\draw [dashed] (A_1)--(D)--(E)(A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BD \\perp CC_1$;\\\\\n(2) 求点$C$到侧面$AA_1B_1B$的距离;\\\\\n(3) 在线段$A_1B_1$上是否存在点$E$, 使得直线$DE$与侧面$AA_1B_1B$所成角的正弦值为$\\dfrac{\\sqrt{6}}{7}$? 若存在, 请求出$A_1E$的长; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -379406,7 +379774,9 @@ "id": "014361", "content": "已知直线$l$经过点$A(-3, \\sqrt{3})$、$B(\\sqrt{3},-1)$.\\\\\n(1) 直线$l$的斜率是\\blank{50};\\\\\n(2) 直线$l$的倾斜角是\\blank{50};\\\\\n(3) 直线$l$的点斜式方程是\\blank{50};\\\\\n(4) 直线$l$的一个法向量是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379426,7 +379796,9 @@ "id": "014362", "content": "圆$C: x^2+y^2-4 x-6 y+9=0$的圆心到直线$l: 3 x-4 y+2=0$的距离$d=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379446,7 +379818,9 @@ "id": "014363", "content": "若直线$x=-1$与曲线$(x-\\dfrac{m^2}{4})^2+(y-m)^2=1$仅有一个公共点, 则实数$m$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379501,7 +379875,9 @@ "id": "014365", "content": "分别求满足下列条件的圆的方程:\\\\\n(1) 经过点$P(2,2)$, $Q(5,3)$, $R(3,-1)$;\\\\\n(2) 经过点$A(-3,0)$与$B(0,-\\sqrt{3})$, 且圆心在直线$x+y+1=0$上.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -379593,7 +379969,9 @@ "id": "014368", "content": "若直线$l$过点$P(1,2)$且垂直于直线$5 x-2 y+3=0$, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379648,7 +380026,9 @@ "id": "014370", "content": "已知圆$C_1$的半径为$3$, 圆$C_2$的半径为$7$, 若两圆相交, 则两圆的圆心距可能是\\bracket{20}.\n\\fourch{$0$}{$4$}{$8$}{$12$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -379668,7 +380048,9 @@ "id": "014371", "content": "已知圆$C$的一般方程为$x^2+2 x+y^2=0$, 则圆$C$的半径为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379688,7 +380070,9 @@ "id": "014372", "content": "已知直线$l_1$与$l_2: 4 x+3 y+5=0$有相同的法向量, 且直线$l_1$在$x$轴上的截距为$-2$, 则直线$l_1$的点法式方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379743,7 +380127,9 @@ "id": "014374", "content": "若圆$\\mathrm{C}_1: x^2+y^2=4$与圆$C_2: (x-3)^2+(y+m)^2=25$外切, 则实数$m$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379763,7 +380149,9 @@ "id": "014375", "content": "已知圆$C: x^2+y^2-2 y-4=0$, 直线$l: 3 x+y-6=0$, 则直线$l$被圆$C$所截得的弦长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -379783,7 +380171,9 @@ "id": "014376", "content": "已知$\\triangle ABC$的三个顶点的坐标分别是$A(0,1)$, $B(-2,-1)$, $C(5,3)$.\\\\\n(1) 求边$AB$上的中线所在直线的方程;\\\\\n(2) 求$\\triangle ABC$的面积.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -379838,7 +380228,9 @@ "id": "014378", "content": "已知圆$M$与直线$x=2$相切, 圆心$M$在直线$x+y=0$上, 且直线$x-y-2=0$被圆$M$截得的弦长为$2 \\sqrt{2}$.\\\\\n(1) 求圆$M$的标准方程, 并判断圆$M$与圆$N: x^2+y^2-6 x+8 y+15=0$的位置关系;\\\\\n(2) 若不与坐标轴垂直的直线$l$在$x$轴上的截距为$1$, 且直线$l$与圆$M$交于$A$、$B$两点, 在$x$轴上是否存在定点$Q$, 使得直线$AQ$的倾斜角与直线$BQ$的倾斜角互补, 若存在, 求出点$Q$的坐标; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -379858,7 +380250,9 @@ "id": "014379", "content": "计算$\\mathrm{i}+\\mathrm{i}^2+\\mathrm{i}^3+\\mathrm{i}^4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -380014,7 +380408,9 @@ "id": "014384", "content": "若复数$z$满足: $z \\overline{z}+(z+\\overline {z}) \\mathrm{i}=\\dfrac{3-\\mathrm{i}}{2+\\mathrm{i}}$, 求复数$z$及$z+z+z^2$的值, 并证明$\\dfrac{1+z}{1-z}$是纯虚数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380034,7 +380430,9 @@ "id": "014385", "content": "已知$\\alpha$、$\\beta$是实系数一元二次方程$x^2-2 x+m=0$, $m \\in \\mathbf{R}$的两根, 求: $|\\alpha|+|\\beta|$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380088,7 +380486,9 @@ "id": "014387", "content": "已知复数$z$满足$|z-3 \\mathrm{i}|+|z+3 \\mathrm{i}|=10$, 求$|z-\\mathrm{i}|$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380143,7 +380543,9 @@ "id": "014389", "content": "已知$z_1$、$z_2$是方程$x^2+2 x+3=0$在复数范围内的两个根, 则$|z_1-z_2|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -380163,7 +380565,9 @@ "id": "014390", "content": "已知复平面上平行四边形$ABCD$的顶点$A$、$B$、$C$的坐标分别为$(-2,-1)$、$(7,3)$、$(12,9)$, 求向量$\\overrightarrow{AD}$所对应的复数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380183,7 +380587,9 @@ "id": "014391", "content": "已知$z_1$、$z_2 \\in \\mathbf{C}$, 且$z_1 z_2=0$, 求证: $z_1=0$或$z_2=0$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380308,7 +380714,9 @@ "id": "014395", "content": "设复数$z_1=1-\\mathrm{i}$, $z_2=\\cos \\theta+\\mathrm{i} \\sin \\theta$, 其中$\\theta \\in[0, \\pi]$.\\\\\n(1) 若复数$z=\\overline{z_1} \\cdot z_2$为实数, 求$\\theta$的值;\\\\\n(2) 求$|3 z_1+z_2|$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380328,7 +380736,9 @@ "id": "014396", "content": "已知复数$z_1=2-5 \\mathrm{i}$, $z_2=1+(2 \\cos \\theta) \\mathrm{i}$.\\\\\n(1) 求$z_1 \\cdot \\overline{z_1}$;\\\\\n(2) 复数$z_1$、$z_2$对应的向量分别是$\\overrightarrow{OZ_1}$, $\\overrightarrow{OZ_2}$, 其中$O$为坐标原点, 当$\\theta=\\dfrac{\\pi}{3}$时, 求$\\overrightarrow{OZ_1} \\cdot \\overrightarrow{OZ_2}$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380348,7 +380758,9 @@ "id": "014397", "content": "已知$z$是虚数, $z+\\dfrac{1}{z}$是实数.\\\\\n(1) 求$z$为何值时, $|z+2-\\mathrm{i}|$有最小值, 并求出$|z+2-\\mathrm{i}|$的最小值;\\\\\n(2) 设$\\mu=\\dfrac{1-z}{1+z}$, 求证: $\\mu$为纯虚数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380368,7 +380780,9 @@ "id": "014398", "content": "已知$z_1$、$z_2$是实系数一元二次方程的两个虚根, 且$z_1^2=z_2$, 求$z_1$、$z_2$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380388,7 +380802,9 @@ "id": "014399", "content": "关于复数$z$的方程$z^2-(a+\\mathrm{i}) z-\\mathrm{i}-2=0$($a \\in \\mathbf{R}$).\\\\\n(1) 若此方程有实数解, 求$a$的值;\\\\\n(2) 用反证法证明: 对任意的实数$a$, 原方程不可能有纯虚数根.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380408,7 +380824,9 @@ "id": "014400", "content": "判断下列命题的真假.\\\\\n(1) 空间中的任意两条直线都可以确定一个平面;\\\\\n(2) 空间中, 垂直于同一直线的两条直线相互平行;\\\\\n(3) 若直线$a \\perp b$, 直线$a$与平面$\\beta$平行, 则$b \\perp \\beta$;\\\\\n(4) 若直线$a\\parallel$平面$\\alpha$, 直线$a\\parallel$平面$\\beta$, 则平面$\\alpha\\parallel$平面$\\beta$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380428,7 +380846,9 @@ "id": "014401", "content": "如图, 在空间四边形$ABCD$中, 已知$AB$、$BC$、$CD$的中点分别是$P$、$Q$、$R$, 且$PQ=3$、$QR=5$、$PR=7$, 求异面直线$AC$和$BD$所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,1.1,0) node [above] {$A$} coordinate (A);\n\\draw (0.9,-0.7,0) node [below] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [below left] {$Q$} coordinate (Q);\n\\draw ($(C)!0.5!(D)$) node [below right] {$R$} coordinate (R);\n\\draw ($(A)!0.5!(B)$) node [above left] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(D)--cycle (A)--(C)(P)--(Q);\n\\draw [dashed] (B)--(D)(P)--(R)--(Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380448,7 +380868,9 @@ "id": "014402", "content": "如图, 已知$PA \\perp$平面$ABC$, 且$\\triangle ABC$是直角三角形, 其中$\\angle ACB=90^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (-2,0,0) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,1.5,0) node [left] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle (P)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp PC$;\\\\\n(2) 求证: 平面$PAC \\perp$平面$PBC$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380468,7 +380890,9 @@ "id": "014403", "content": "如图, 四棱锥$S-ABCD$的底面为正方形, $SD \\perp$平面$ABCD$, 底面正方形对角线交于点$O$, $G$为$SC$边上的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (D) ++ (0,1.5,0) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(C)$) node [above right] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw (S)--(A)--(B)--(C)--cycle (S)--(B)--(G);\n\\draw [dashed] (S)--(D)--(A) (D)--(C) (D)--(G) (O)--(G) (A)--(C) (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $SA\\parallel$平面$BDG$;\\\\\n(2) 若平面$BDG \\cap$平面$ADS=m$, 求证: $m\\parallel OG$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380488,7 +380912,9 @@ "id": "014404", "content": "如图, 四棱锥$S-ABCD$的底面为正方形, $SD \\perp$平面$ABCD$, 底面正方形对角线交于点$O$, $G$为$SC$边上的中点, 其中$AD=SD=2$, 求直线$OG$与平面$BCS$所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (D) ++ (0,2,0) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(C)$) node [above right] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw (S)--(A)--(B)--(C)--cycle (S)--(B);\n\\draw [dashed] (S)--(D)--(A) (D)--(C) (O)--(G) (A)--(C) (B)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380508,7 +380934,9 @@ "id": "014405", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 与直线$A_1B$互为异面直线的棱有\\blank{50}条.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -380528,7 +380956,9 @@ "id": "014406", "content": "已知$m, n$是两条不同直线, $\\alpha, \\beta$是两个不同平面, 则下列命题错误的是\\bracket{20}.\n\\onech{若$\\alpha, \\beta$不平行, 则在$\\alpha$内不存在与$\\beta$平行的直线}{若$m, n$平行于同一平面, 则$m$与$n$可能异面}{若$m, n$不平行, 则$m$与$n$不可能垂直于同一平面}{若$\\alpha, \\beta$垂直于同一平面, 则$\\alpha$与$\\beta$可能相交}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -380548,7 +380978,9 @@ "id": "014407", "content": "如图, 对于直四棱柱$ABCD-A_1B_1C_1D_1$, 要使$A_1C \\perp B_1D_1$, 则在四边形$ABCD$中, 满足的条件可以是\\blank{50}.(只需写出一个正确的条件)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (A1)--(C);\n\\draw (B1)--(D1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -380568,7 +381000,9 @@ "id": "014408", "content": "工人师傅在检查工作的相邻两个面是否垂直时, 只要用曲尺的短边紧靠在工件的一个面上, 长边在工件的另一个面上转动, 观察短边是否和这个面密合就可以了, 你能说明其中的原理吗?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380588,7 +381022,9 @@ "id": "014409", "content": "如图, 在三棱台$ABC-A_1B_1C_1$的$9$条棱所在直线中, 与直线$A_1B$是异面直线的共有\\blank{50}条.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (-1,0,0) node [left] {$A$} coordinate (A);\n\\draw ({1/2},0,{sqrt(3)/2}) node [below] {$B$} coordinate (B);\n\\draw ({1/2},0,{-sqrt(3)/2}) node [right] {$C$} coordinate (C);\n\\path (0,{sqrt(2)},0) coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(B)!0.5!(P)$) node [right] {$B_1$} coordinate (B_1);\n\\draw ($(C)!0.5!(P)$) node [right] {$C_1$} coordinate (C_1);\n\\draw (A_1)--(A) (B_1)--(B) (C_1)--(C);\n\\draw (A)--(B)--(C)(A_1)--(B_1)--(C_1)--cycle;\n\\draw (A_1)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -380643,7 +381079,9 @@ "id": "014411", "content": "已知直线$l$与平面$\\alpha$相交, 则下列命题中, 正确的个数为\\bracket{20}.\\\\\n\\textcircled{1} 平面$\\alpha$内的所有直线均与直线$l$异面;\\\\\n\\textcircled{2} 平面$\\alpha$内存在与直线$l$垂直的直线;\\\\\n\\textcircled{3} 平面$\\alpha$内不存在直线与直线$l$平行;\\\\\n\\textcircled{4} 平面$\\alpha$内所有直线均与直线$l$相交.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -380698,7 +381136,9 @@ "id": "014413", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, 已知底面$ABCD$是正方形, 点$P$是侧棱$CC_1$上的一点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1.5}\n\\def\\m{1.5}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(C)!0.5!(C1)$) node [right] {$P$} coordinate (P);\n\\draw (B)--(P);\n\\draw [dashed] (B)--(D)--(P)--(A1)(A)--(C1);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$AC_1\\parallel$平面$PBD$, 求$\\dfrac{PC_1}{PC}$的值;\\\\\n(2) 求证: $BD \\perp A_1P$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380718,7 +381158,9 @@ "id": "014414", "content": "已知$PA \\perp$平面$ABC$, $PA=AB=3$, $AC=4$, $M$为$BC$中点, 过点$M$分别作平行于平面$PAB$的直线交$AC$、$PC$于点$E$、$F$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,3) node [left] {$B$} coordinate (B);\n\\draw (0,3,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw ($(A)!0.5!(C)$) node [above right] {$E$} coordinate (E);\n\\draw ($(P)!0.5!(C)$) node [above] {$F$} coordinate (F);\n\\draw (P)--(B)--(C)--cycle(P)--(M)--(F);\n\\draw [dashed] (P)--(A)--(B)(A)--(C)(M)--(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$PM$与平面$ABC$所成的角的大小;\\\\\n(2) 证明: 平面$MEF\\parallel$平面$PAB$, 并求直线$ME$到平面$PA$距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380773,7 +381215,9 @@ "id": "014416", "content": "如图, 在三棱锥$D-ABC$中, 平而$ACD \\perp$平面$ABC$, $AD \\perp AC$, $AB \\perp BC$, $E$、$F$分别为棱$BC$, $CD$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2.5,0) node [left] {$D$} coordinate (D);\n\\draw ({1.5+1.5*cos(80)},0,{1.5*sin(80)}) node [below] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D)$) node [above] {$F$} coordinate (F);\n\\draw (D)--(A)--(B)--(C)--cycle(B)--(D)(E)--(F);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线$EF\\parallel$平面$ABD$;\\\\\n(2) 求证: 直线$BC \\perp$平面$ABD$;\\\\\n(3) 若直线$CD$与平面$ABC$所成的角的大小为$45^{\\circ}$, 直线$CD$与平面$ABD$所成角的大小为$30^{\\circ}$, 求二面角$B-AD-C$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380793,7 +381237,9 @@ "id": "014417", "content": "判断下列命题是否正确, 并说明理由:\\\\\n(1) 有两个面平行, 其余各面都是四边形的几何体叫棱柱;\\\\\n(2) 各个面都是三角形的几何体是三棱锥;\\\\\n(3) 圆柱、圆锥、圆台的底面都是圆面.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380813,7 +381259,9 @@ "id": "014418", "content": "已知一个圆柱的高为定值, 若将其体积扩大为原来的$4$倍, 则它的侧面积扩大为原来的\\blank{50}倍.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -380868,7 +381316,9 @@ "id": "014420", "content": "我国南北朝时期的数学家祖暅提出了计算几何体体积的原理: ``幂势既同, 则积不容异''. 意思是: 两个等高的几何体, 若在任意给定的等高处的截面积相等, 则体积相等. 现有等高的三棱锥和圆锥, 若它们满足祖暅原理的条件, 且圆锥的侧面展开图是一个半径为$3$且圆心角为$\\dfrac{2 \\pi}{3}$的扇形, 则三棱锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -380888,7 +381338,9 @@ "id": "014421", "content": "如图, 已知$BC$为圆锥的底面圆直径, 圆锥的侧面展开图是一个半径为$4$的半圆, $P$是母线$AB$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above right] {$O$} coordinate (O);\n\\draw (0,{sqrt(3)}) node [above right] {$A$} coordinate (A);\n\\draw (-1,0) node [left] {$C$} coordinate (C);\n\\draw (1,0) node [below right] {$B$} coordinate (B);\n\\draw (C)--(A)--(B);\n\\draw (C) arc (180:360:1 and 0.25);\n\\draw [dashed] (C) arc (180:0:1 and 0.25);\n\\draw [dashed] (C)--(B)(O)--(A);\n\\filldraw ($(A)!0.5!(B)$) node [right] {$P$} coordinate (P) circle (0.03);\n\\filldraw (C) circle (0.03);\n\\draw (A) -- ($(A)!-1!(B)$) arc (120:-60:2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求该圆锥的表面积和体积;\\\\\n(2) 动点$M$沿圆锥侧面从点$C$运动到点$P$, 求$M$运动的最短距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380908,7 +381360,9 @@ "id": "014422", "content": "如图正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$\\sqrt{3}$, 体积为$27$, 设$E$是棱$B_1B$上的任意点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.7!(B1)$) node [right] {$E$} coordinate (E);\n\\draw (A)--(E);\n\\draw [dashed] (D)--(E)(D)--(A1)(A1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求棱$A_1A$的长;\\\\\n(2) 求三棱锥$A_1-AED$的体积;\\\\\n(3) 设$F$是棱$C_1C$上的点, 满足$EF\\parallel BC$, 求四棱锥$A_1-AEFD$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -380963,7 +381417,9 @@ "id": "014424", "content": "如图, 已知正三棱柱$ABC-A_1B_1C_1$的底面边长为$2 \\text{cm}$, 高为$5 \\text{cm}$, 一动点从点$A$出发, 沿着三棱柱侧面绕行两周后到达点$A_1$的最短距离是\\blank{50}$\\text{cm}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) -- (B) -- (C);\n\\draw (A) --++ (0,5) node [left] {$A_1$} coordinate (A1);\n\\draw (B) --++ (0,5) node [below right] {$B_1$} coordinate (B1);\n\\draw (C) --++ (0,5) node [right] {$C_1$} coordinate (C1);\n\\draw (A1) -- (B1) -- (C1) -- cycle;\n\\draw [dashed] (A) -- (C);\n\\draw (A) -- ($(B)!{1/6}!(B1)$) -- ($(C)!{1/3}!(C1)$) ($(A)!0.5!(A1)$) -- ($(B)!{2/3}!(B1)$) -- ($(C)!{5/6}!(C1)$);\n\\draw [dashed] ($(C)!{1/3}!(C1)$) -- ($(A)!0.5!(A1)$) ($(C)!{5/6}!(C1)$) -- (A1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -380983,7 +381439,9 @@ "id": "014425", "content": "已知直三棱柱$ABC-A_1B_1C_1$的$6$个顶点都在球$O$的球面上, 若$AB=3$, $AC=4$, $AB \\perp AC$, $AA_1=12$, 则球$O$的半径为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381073,7 +381531,9 @@ "id": "014428", "content": "若圆锥的母线与底面半径之比为$\\sqrt{2}: 1$, 则它的底面积与侧面积之比是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381093,7 +381553,9 @@ "id": "014429", "content": "如图, 正方形$ABCD-A_1B_1C_1D_1$的棱长为$1$, 线段$B_1D_1$有两个动点$E$、$F$, 且$EF=\\dfrac{\\sqrt{2}}{2}$, 则下列结论中错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B1)!0.7!(D1)$) node [right] {$E$} coordinate (E);\n\\draw ($(B1)!0.2!(D1)$) node [above right] {$F$} coordinate (F);\n\\draw [dashed] (A)--(C)(B)--(D)(A)--(E)--(B)(A)--(F)--(B);\n\\draw (B1) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n\\onech{$AC \\perp BE$}{异面直线$AE$、$BF$的所成角为定值}{直线$AB$与平面$BEF$的所成角为定值}{以$A$、$B$、$E$、$F$为顶点的四面体, 其体积不随$E$、$F$位置的变化而变化}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -381113,7 +381575,9 @@ "id": "014430", "content": "已知一个空间几何体的所有棱长均为$1 \\text{cm}$, 其平面展开图如图所示, 则该空间几何体的体积$V=$\\blank{50}$\\text{cm}^3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) rectangle (4,1);\n\\draw (0,0) -- (0,-1) -- (1,-1) -- (1,1) (2,0) -- (2,1) (3,0) -- (3,1);\n\\draw (0,1) --++ (60:1) --++ (-60:1)--++ (60:1) --++ (-60:1)--++ (60:1) --++ (-60:1)--++ (60:1) --++ (-60:1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381133,7 +381597,9 @@ "id": "014431", "content": "已知圆锥的侧面展开图是一个半径为$4$的半圆, 将该圆锥倒置, 放入一个半径为$1$的铁球并注入水, 使水面与球正好相切, 然后将球取出, 这时容器中水的深度为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381188,7 +381654,9 @@ "id": "014433", "content": "如图, 已知圆柱$OO_1$的底面半径为$1$, 点$O_1$是圆柱上底面的圆心. $\\triangle ABC$内接于圆柱的下底面圆$O$, 线段$AA_1$是圆柱的母线, 长度为$2$, 线段$AB$的长为$\\sqrt{3}, C$是优弧$\\overset\\frown{AB}$上异于点$A$、$B$的任意点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2) node [left] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (1,0) node [right] {$A$} coordinate (A) --++ (0,2) node [right] {$A_1$} coordinate (A_1);\n\\draw (-1,0) -- (-1,2);\n\\draw (O_1) ellipse (1 and 0.3);\n\\draw (A) arc (0:-180:1 and 0.3);\n\\draw [dashed] (A) arc (0:180:1 and 0.3);\n\\draw (135:1 and 0.3) node [above] {$C$} coordinate (C);\n\\draw (-105:1 and 0.3) node [below] {$B$} coordinate (B);\n\\draw [dashed] (A)--(B)--(C)--cycle;\n\\draw [dashed] (B)--(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\triangle ABC$为正三角形时, 求点$C$到平面$A_1AB$的距离;\\\\\n(2) 求三棱锥$A_1-ABC$体积的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -381208,7 +381676,9 @@ "id": "014434", "content": "若圆锥的侧面展开图是半径为$5$, 面积为$20 \\pi$的扇形, 则由它的两条母线所确定的该圆锥的截面的面积最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381471,7 +381941,9 @@ "id": "014442", "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $AB=AC=AA_1=2$, $\\angle BAC=90^{\\circ}$, $E$、$F$分别为$CC_1$、$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (A)++(0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw ($(C)-(A)+(A_1)$) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(B)-(A)+(A_1)$) node [left] {$B_1$} coordinate (B_1);\n\\draw (A_1)--(C_1)--(C)--(B)--(B_1)--cycle(B_1)--(C_1);\n\\draw [dashed] (A)--(A_1)(A)--(B)(A)--(C);\n\\draw ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw (E)--(F);\n\\draw [dashed] (A_1)--(B)(F)--(A)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$A_1B$与平面$AEF$所成角的大小;\\\\\n(2) 求点$C$到平面$AEF$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -381526,7 +381998,9 @@ "id": "014444", "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$均为空间单位向量, 它们的夹角为$60^{\\circ}$, 则$|\\overrightarrow {a}+3 \\overrightarrow {b}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381546,7 +382020,9 @@ "id": "014445", "content": "已知$O$为平面$ABCD$外一点, $A$、$B$、$C$、$D$四点中任意三点不共线. 若$\\overrightarrow{OA}=2 x \\overrightarrow{BO}+3 y \\overrightarrow{CO}+4 z \\overrightarrow{DO}$, 则$2 x+3 y+4 z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381566,7 +382042,9 @@ "id": "014446", "content": "如图, $ABC-A_1B_1C_1$是直三棱柱, $\\angle BCA=90^{\\circ}$, 点$D_1$、$F_1$分别是$A_1B_1$、$A_1C_1$的中点, 若$BC=CA=CC_1$, 则$BD_1$与$AF_1$所成角的余弦值是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(2)},0,0) node [left] {$B$} coordinate (B);\n\\draw ({sqrt(2)},0,0) node [right] {$A$} coordinate (A);\n\\draw (0,0,{sqrt(2)}) node [below] {$C$} coordinate (C);\n\\draw (A)++(0,2,0) node [right] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)-(A)+(B)$) node [left] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)-(A)+(C)$) node [below right] {$C_1$} coordinate (C_1);\n\\draw (B)--(C)--(A)--(A_1)--(B_1)--cycle(B_1)--(C_1)--(A_1)(C)--(C_1);\n\\draw [dashed] (B)--(A);\n\\draw ($(A_1)!0.5!(C_1)$) node [below right] {$F_1$} coordinate (F_1);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$D_1$} coordinate (D_1);\n\\draw (F_1)--(A);\n\\draw [dashed] (D_1)--(B);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{\\sqrt{30}}{10}$}{$\\dfrac{1}{2} ;$}{$\\dfrac{\\sqrt{30}}{15}$}{$\\dfrac{\\sqrt{15}}{10}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -381586,7 +382064,9 @@ "id": "014447", "content": "如图, 在四棱锥$P-ABCD$中, 已知$PA \\perp$底面$ABCD$, 底面$ABCD$是正方形, $PA=AB$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,\\l,0) node [above left] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (P)--(A)--(B)(D)--(A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线$BD \\perp$底面$PAC$;\\\\\n(2) 求直线$PC$与平面$PBD$所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -381640,7 +382120,9 @@ "id": "014449", "content": "已知空间向量$\\overrightarrow {a}$、$\\overrightarrow {b}$, 且$\\overrightarrow{AB}=\\overrightarrow {a}+2 \\overrightarrow {b}$, $\\overrightarrow{BC}=-5 \\overrightarrow {a}+6 \\overrightarrow {b}$, $\\overrightarrow{CD}=7 \\overrightarrow {a}-2 \\overrightarrow {b}$, 则一定共线的三点是\\bracket{20}.\n\\fourch{$A$、$B$、$C$}{$B$、$C$、$D$}{$A$、$B$、$D$}{$A$、$C$、$D$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -381660,7 +382142,9 @@ "id": "014450", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=2$, $AA_1=1$, 则$BC_1$与平面$BB_1D_1D$所成角的正弦值为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B)--(C1)(B1)--(D1);\n\\draw [dashed] (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{\\sqrt{6}}{3}$}{$\\dfrac{2 \\sqrt{6}}{5}$}{$\\dfrac{\\sqrt{15}}{5}$}{$\\dfrac{\\sqrt{10}}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -381758,7 +382242,9 @@ "id": "014453", "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $AB=AC=AA_1=2$, $\\angle ABC=90^{\\circ}$, $E$、$F$分别为$CC_1$、$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (A)++(0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw ($(C)-(A)+(A_1)$) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(B)-(A)+(A_1)$) node [left] {$B_1$} coordinate (B_1);\n\\draw (A_1)--(C_1)--(C)--(B)--(B_1)--cycle(B_1)--(C_1);\n\\draw [dashed] (A)--(A_1)(A)--(B)(A)--(C);\n\\draw ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw (E)--(F);\n\\draw [dashed] (A_1)--(B)(F)--(A)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$A_1B$与$EF$所成角的大小;\\\\\n(2) 求平面$ABA_1$与平面$AEF$所成的锐二面角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -381847,7 +382333,9 @@ "id": "014456", "content": "已知椭圆$\\Gamma: 2 x^2+3 y^2=6$的两个焦点为$F_1$、$F_2, P$为$\\Gamma$上一点, 若$|PF_1|=1$, 则$|PF_2|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381867,7 +382355,9 @@ "id": "014457", "content": "若椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的离心率为$\\dfrac{\\sqrt{3}}{2}$, $b=2$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381887,7 +382377,9 @@ "id": "014458", "content": "若方程$\\dfrac{x^2}{4-k}+\\dfrac{y^2}{6+k}=1$表示焦点在$y$轴上的椭圆, 则实数$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381907,7 +382399,9 @@ "id": "014459", "content": "双曲线$\\dfrac{x^2}{3}-y^2=1$的两条渐近线的夹角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}3$", "solution": "", @@ -381940,7 +382434,9 @@ "id": "014460", "content": "抛物线$x=-8 y^2$的准线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -381960,7 +382456,9 @@ "id": "014461", "content": "求以直线$3 x \\pm 4 y=0$为渐近线, 且过点$M(-2,3)$的双曲线的标准方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -381980,7 +382478,9 @@ "id": "014462", "content": "已知抛物线的顶点在原点, 且以某坐标轴为其对称轴, 该抛物线的准线过椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的一个焦点, 且与该椭圆的一个交点为$P(\\dfrac{2}{3},\\dfrac{2 \\sqrt{6}}{3})$, 求此抛物线及椭圆的标准方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382000,7 +382500,9 @@ "id": "014463", "content": "在平面直角坐标系$xOy$中, 已知双曲线$C: 2 x^2-y^2=1$. 设斜率为$1$的直线$l$交双曲线$C$于$P$、$Q$两点, 若直线$l$与圆$x^2+y^2=1$相切, 求证: $OP \\perp OQ$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382054,7 +382556,9 @@ "id": "014465", "content": "已知$F_1(-5,0)$, $F_2(5,0)$两点, 点$M$满足$|MF_1|-|MF_2|=8$, 则点$M$的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382074,7 +382578,9 @@ "id": "014466", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的两个焦点分别为$F_1$、$F_2$, 离心率为$\\dfrac{1}{2}$, 过点$F_2$的直线与$\\Gamma$交于$A$、$B$两点, 若$\\triangle F_1AB$的周长为$8$, 则$\\Gamma$的标准方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382094,7 +382600,9 @@ "id": "014467", "content": "已知抛物线$C: y^2=4 x$的焦点为$F$, $A(x_1,y_1)$, $B(x_2,y_2)$为$C$上两点, 若$y_2^2-2 y_1^2=4$, 则$\\dfrac{|AF|}{|BF|}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382114,7 +382622,9 @@ "id": "014468", "content": "以椭圆$\\dfrac{y^2}{4}+\\dfrac{x^2}{3}=1$的焦点为顶点, 以该椭圆长轴的端点为焦点的双曲线的标准方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382134,7 +382644,9 @@ "id": "014469", "content": "双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的渐近线是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382154,7 +382666,9 @@ "id": "014470", "content": "若椭圆$\\dfrac{x^2}{m^2}+\\dfrac{y^2}{4}=1$经过点$M(-2,\\sqrt{3})$, 则它的焦距为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382174,7 +382688,9 @@ "id": "014471", "content": "以双曲线$\\dfrac{x^2}{4}-\\dfrac{y^2}{5}=1$的中心为顶点, 且以该双曲线的右焦点为焦点的抛物线的标准方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382194,7 +382710,9 @@ "id": "014472", "content": "抛物线$y^2=x$上到焦点的距离等于$2$的点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382214,7 +382732,9 @@ "id": "014473", "content": "在平面直角坐标系$x O y$中, 点$F$是椭圆$x^2+\\dfrac{y^2}{b^2}=1$ ($00$, $b>0$)的左、右焦点分别为$F_1,F_2$, 过$F_2$且斜率为$-\\dfrac{\\sqrt{5}}{2}$的直线与双曲线$C$的左支交于点$A$. 若$(\\overrightarrow{F_1F_2}+\\overrightarrow{F_1A}) \\cdot \\overrightarrow{F_2A}=0$, 则双曲线$C$的渐近线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382274,7 +382798,9 @@ "id": "014476", "content": "在平面直角坐标系$x O y$中, 已知椭圆$\\Gamma: \\dfrac{x^2}{2}+y^2=1$, 过右焦点$F$作两条互相垂直的弦$AB$、$CD$, 设$AB$、$CD$中点分别为$M$、$N$.\\\\\n(1) 证明: 直线$MN$必过定点, 并求出此定点坐标;\\\\\n(2) 若弦$AB$、$CD$的斜率均存在, 求$\\triangle FMN$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382294,7 +382820,9 @@ "id": "014477", "content": "已知直线$y=x-1$与椭圆$\\dfrac{x^2}{2}+y^2=1$交于$A$、$B$两点, $O$为坐标原点, 则$\\triangle OAB$的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382348,7 +382876,9 @@ "id": "014479", "content": "若$P(x, y)$是抛物线$y^2=x$上一动点, 则点$P$到直线$y=x+3$的距离的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$\\dfrac{11\\sqrt{2}}8$", "solution": "", @@ -382381,7 +382911,9 @@ "id": "014480", "content": "已知$t \\in \\mathbf{R}$, 直线$y=m x+1$($m>0$)与双曲线$x^2-y^2=1$相交于$A$、$B$两点, 若线段$AB$的垂直平分线与$x$轴交于点$(t, 0)$, 则$t$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382401,7 +382933,9 @@ "id": "014481", "content": "设$k \\in \\mathbf{R}$, 已知直线$l: y=k x+1$, 双曲线$C: 3x^2-y^2=1$, 若直线$l$与双曲线$C$有两个公共点$A$、$B$.\\\\\n(1) 若$A$、$B$均在双曲线$C$的右支上, 求$k$的取值范围;\\\\\n(2) 若以线段$AB$为直径的圆过坐标原点, 求$k$的值;\\\\\n(3) 若$\\angle AOB$为钝角 (其中$O$为坐标原点), 求$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382421,7 +382955,9 @@ "id": "014482", "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{a^2}-y^2=1$($a>0$), 双曲线$\\Gamma$右支上的任意两点$P_1$、$P_2$的坐标分别为$(x_1, y_1) 、(x_2, y_2)$, 且满足$x_1 x_2-y_1 y_2>0$恒成立, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382475,7 +383011,9 @@ "id": "014484", "content": "已知$F$是椭圆$C_1: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$与抛物线$C_2: y^2=2 p x$($p>0$)的一个共同焦点, $C_1$与$C_2$相交于$A$、$B$两点, 则线段$AB$的长等于\\bracket{20}.\n\\fourch{$\\dfrac{2}{3} \\sqrt{6}$}{$\\dfrac{4}{3} \\sqrt{6}$}{$\\dfrac{5}{3}$}{$\\dfrac{10}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -382495,7 +383033,9 @@ "id": "014485", "content": "若经过点$F_2(2,0)$的直线$l$与双曲线$x^2-\\dfrac{y^2}{3}=1$相交于$A$、$B$两点, 且$|AB|=6$, 则满足条件的直线$l$共有\\blank{50}条.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -382528,7 +383068,9 @@ "id": "014486", "content": "设$t \\in \\mathbf{R}$, 已知双曲线$C: x^2-y^2=1$, 过点$T(t, 0)$作直线$l$和双曲线$C$交于$A$、$B$两点.\\\\\n(1) 求双曲线$C$的焦点和它的渐近线方程;\\\\\n(2) 若$t=0$, 点$A$在第一象限, $AH \\perp x$轴, 垂足为$H$, 求直线$BH$斜率的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382548,7 +383090,9 @@ "id": "014487", "content": "椭圆$x^2+4 y^2=4$的长轴的一个端点为$A$, 以$A$为直角顶点作一个内接于此椭圆的等腰直角三角形, 则此三角形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382568,7 +383112,9 @@ "id": "014488", "content": "已知抛物线$y^2=8 x$的动弦$AB$的长为$12$, 则弦$AB$的中点$M$到$y$轴的最短距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382588,7 +383134,9 @@ "id": "014489", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$的左、右焦点分别为$F_1$、$F_2$, 设点$P$是椭圆$\\Gamma$上一点, 且位于$x$轴的上方, 若$\\triangle PF_1F_2$是等腰三角形, 则点$P$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382608,7 +383156,9 @@ "id": "014490", "content": "已知椭圆$x^2+\\dfrac{y^2}{b^2}=1$($02$. 在平面直角坐标系$x O y$中, 已知点$F(2,0)$, 直线$l: x=t$, 曲线$\\Gamma: y^2=8 x$($0 \\leq x \\leq t$, $y \\geq 0$). 直线$l$与$x$轴交于点$A$、与$\\Gamma$交于点$B$. $P$、$Q$分别是曲线$\\Gamma$与线段$AB$上的动点.\\\\\n(1) 用$t$表示点$B$到点$F$的距离;\\\\\n(2) 设$t=3$, $|FQ|=2$, 线段$OQ$的中点在直线$FP$上, 求$\\triangle AQP$的面积;\\\\\n(3) 设$t=8$, 是否存在以$FP$、$FQ$为邻边的矩形$FPEQ$, 使得点$E$在$\\Gamma$上? 若存在, 求点$P$的坐标; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382681,7 +383235,9 @@ "id": "014493", "content": "已知点$P(0,1)$, 椭圆$\\dfrac{x^2}{4}+y^2=m$($m>1$)上两点$A, B$满足$\\overrightarrow{AP}=2 \\overrightarrow{PB}$, 则当$m=$\\blank{50}时, 点$B$横坐标的绝对值最大.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382701,7 +383257,9 @@ "id": "014494", "content": "设直线$x-3 y+m=0$($m \\neq 0$)与双曲线$\\dfrac{x^2}{4}-\\dfrac{y^2}{b}=1$($b>0$)的两条渐近线分别交于$A$、$B$两点. 若点$P(m, 0)$满足$|PA|=|PB|$, 则实数$b$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382721,7 +383279,9 @@ "id": "014495", "content": "某科研团队在基地$O$点西侧、东侧$20$千米处设有$A$、$B$两站点, 经测量发现点$P$满足$|PA|-|PB|=20$千米, 且点$P$在$O$点北偏东$60^{\\circ}$处, 则$O$、$P$之间的距离为\\blank{50}千米.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382741,7 +383301,9 @@ "id": "014496", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{4}+y^2=1$, $Q(0,1)$为$\\Gamma$的上顶点, 过原点的直线$l$与$\\Gamma$交于不同的两点$A$、$B$, 则$\\triangle ABQ$面积的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382761,7 +383323,9 @@ "id": "014497", "content": "过抛物线$y^2=2 x$上一点$P(2,2)$作两条直线分别交抛物线于$A(x_1, y_1)$、$B(x_2, y_2)$两点.若直线$PA$与$PB$的倾斜角互补, 则$\\dfrac{y_1+y_2}{2}$的值为\\bracket{20}.\n\\fourch{$-\\dfrac{1}{2}$}{$-2$}{$2$}{无法确定}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -382781,7 +383345,9 @@ "id": "014498", "content": "某科研团队在基地$O$点西侧、东侧$20$千米处设有$A$、$B$两站点, 南侧、北侧$15$千米处设有$C$、$D$两站点, 测量距离发现点$Q$满足$|QA|+|QB|=60$千米, $|QC|-|QD|=10$千米, 求$O$、$Q$之间的距离和$Q$点位置(结果精确到$1$千米, $1^{\\circ}$).", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382801,7 +383367,9 @@ "id": "014499", "content": "设$b>0$. 如图, 在平面直角坐标系$xOy$中, $A(x_A, y_A)$是双曲线$\\Gamma_1: \\dfrac{x^2}{4}-\\dfrac{y^2}{b^2}=1$和圆$\\Gamma_2: x^2+y^2=4+b^2$在第一象限内的交点, 曲线$\\Gamma$由$\\Gamma_1$中满足$|x|>x_A$的部分和$\\Gamma_2$中满足$|x|>x_A$的部分构成.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (0,0) circle ({2*sqrt(2)});\n\\draw [domain = -4:4,dashed] plot ({sqrt(\\x*\\x+4)},\\x);\n\\draw [domain = -4:4,dashed] plot ({-sqrt(\\x*\\x+4)},\\x);\n\\draw [domain = {sqrt(2)}:4, thick] plot ({-sqrt(\\x*\\x+4)},\\x);\n\\draw [domain = {sqrt(2)}:4, thick] plot ({sqrt(\\x*\\x+4)},\\x);\n\\draw [domain = {sqrt(2)}:4, thick] plot ({-sqrt(\\x*\\x+4)},-\\x);\n\\draw [domain = {sqrt(2)}:4, thick] plot ({sqrt(\\x*\\x+4)},-\\x);\n\\draw [thick] ({sqrt(6)},{sqrt(2)}) coordinate (A1) arc (30:-30:{2*sqrt(2)}) coordinate (A2);\n\\draw [thick] ({-sqrt(6)},{-sqrt(2)}) coordinate (A3) arc (210:150:{2*sqrt(2)}) coordinate (A4);\n\\foreach \\i in {1,2,3,4}\n{\\filldraw [fill = white] (A\\i) circle (0.1);};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$x_A=\\sqrt{6}$, 求$b$的值;\\\\\n(2) 设$b=\\sqrt{5}$, $F_1$、$F_2$分别为$\\Gamma$与$x$轴的左、右两个交点. 第一象限内的点$P$也在$\\Gamma$上, 且$|PF_1|=8$, 求$\\angle F_1PF_2$的大小.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382821,7 +383389,9 @@ "id": "014500", "content": "太平洋上有$A$、$B$两个岛屿, $B$岛在$A$岛正东$40$海里处. 经多年观察研究发现, 某种鱼群洄游的路线像一个椭圆, 其焦点恰好是$A$、$B$两岛. 曾有渔船在距$A$岛正西$20$海里处发现过鱼群. 某日, 研究人员在$A$、$B$两岛同时用声纳探测仪发出不同频率的探测信号(传播速度相同), $A$、$B$两岛收到鱼群反射信号的时间比为$5: 3$, 则鱼群此时与$A$岛的距离为\\blank{50}海里, 与$B$岛的距离为\\blank{50}海里.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382841,7 +383411,9 @@ "id": "014501", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{9}+\\dfrac{y^2}{5}=1$的左、右焦点分别为$F_1$、$F_2$, 直线$y=k_1 x$($k_1 \\neq 0$)与$\\Gamma$相交于$A$、$B$两点.记$d$为$A$到直线$2 x+9=0$的距离, 当$k_1$变化时, 求证: $\\dfrac{|AF_1|}{d}$为定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382861,7 +383433,9 @@ "id": "014502", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), $P(1,3)$、$Q(3,1)$、$M(-3,1)$、$N(0,2)$这四点中恰有三点在椭圆$C$上.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 点$E$是椭圆$C$上的一个动点, 求$\\triangle EMN$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -382915,7 +383489,9 @@ "id": "014504", "content": "在椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$上任意一点$P, Q$与$P$关于$x$轴对称, 若$\\overrightarrow{F_1P} \\cdot \\overrightarrow{F_2P} \\leq 1$, 则$\\langle\\overrightarrow{F_1P}, \\overrightarrow{F_2Q}\\rangle$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -382978,7 +383554,9 @@ "id": "014506", "content": "已知椭圆$\\dfrac{x^2}{2}+y^2=1$, 作垂直于$x$轴的垂线交椭圆于$A$、$B$两点, 作垂直于$y$轴的垂线交椭圆于$C$、$D$两点, 且$AB=CD$, 两垂线相交于点$P$, 则点$P$的轨迹是\\bracket{20}的一部分.\n\\fourch{椭圆}{双曲线}{圆}{抛物线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -383011,7 +383589,9 @@ "id": "014507", "content": "在相距$1800 \\text{m}$的两个观察站$A$、$B$先后听到远处传来的爆炸声, 已知$A$站听到的时间比$B$站早$5$秒, 声速是$340 \\text{m} / \\text{s}$. 建立适当的平面直角坐标系, 判断爆炸点$P$可能分布在什么样的轨迹上, 并求该轨迹的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -383031,7 +383611,9 @@ "id": "014508", "content": "已知抛物线$y^2=x$上的动点$M(x_0, y_0)$, 过$M$分别作两条直线交抛物线于$P$、$Q$两点, 交直线$x=-1$于$A$、$B$两点.\\\\\n(1) 若点$M$纵坐标为$\\sqrt{2}$, 求$M$点与焦点的距离;\\\\\n(2) 若$P(1,1)$, $Q(1,-1)$, 求证: $y_A \\cdot y_B$为常数.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -383051,7 +383633,9 @@ "id": "014509", "content": "已知$F_1, F_2$是双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a, b>0$)的左、右焦点, 过$F_2$的直线交双曲线的右支于$A, B$两点, 且$|AF_1|=2|AF_2|$, $\\angle AF_1F_2=\\angle F_1BF_2$, 则在下列结论中, 正确结论的序号为\\blank{50}.\\\\\n\\textcircled{1} 双曲线$C$的离心率为$2$;\\\\\n\\textcircled{2} 双曲线$C$的一条渐近线的斜率为$\\sqrt{2}$;\\\\\n\\textcircled{3} 线段$AB$的长为$6 a$;\\\\\n\\textcircled{4} $\\triangle AF_1F_2$的面积为$\\sqrt{15} a^2$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{4}", "solution": "", @@ -383084,7 +383668,9 @@ "id": "014510", "content": "已知抛物线$y^2=x$.\\\\\n(1) 过抛物线焦点$F$的直线交抛物线于$A$、$B$两点, 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值(其中$O$为坐标原点);\\\\\n(2) 过抛物线上的一点$C(x_0, y_0)$, 分别作两条直线交抛物线于另外两点$P(x_P, y_P)$、$Q(x_Q, y_Q)$, 交直线$x=-1$于$A_1(-1,1)$、$B_1(-1,-1)$两点, 求证: $y_P \\cdot y_Q$为常数;\\\\\n(3) 已知点$D(1,1)$, 在抛物线上是否存在异于点$D$的两个不同点$M$、$N$, 使得$DM \\perp MN$? 若存在, 求$N$点纵坐标的取值范围; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -383882,7 +384468,9 @@ "id": "014535", "content": "在等差数列$\\{a_n\\}$中, 公差为常数$d$, 若$a_1<0$, $11 a_5=5 a_8$, 则该数列前$n$项和$S_n$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$-2d$", "solution": "", @@ -383915,7 +384503,9 @@ "id": "014536", "content": "已知等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$S_n=n-5 a_n-85$.\\\\\n(1) 证明: $\\{a_n-1\\}$是等比数列;\\\\\n(2) 试问: 数列$\\{S_n\\}$是否有最小项? 若有, 指出第几项最小; 若没有, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) 有最小项, 第$15$项最小", "solution": "", @@ -383948,7 +384538,9 @@ "id": "014537", "content": "已知数列$\\{a_n\\}$成等差数列, 其前$n$项和为$S_n$, 且$S_{10}=20$, $S_{20}=50$, 求$S_{30}$, 并在等比数列中, 写出相应的问题并完成求解.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384003,7 +384595,9 @@ "id": "014539", "content": "在数列$\\{a_n\\}$中, 若$a_1=2$, 且$a_n=a_{n-1}+\\lg \\dfrac{n}{n-1}$($n \\geq 2$), 则$a_{100}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384058,7 +384652,9 @@ "id": "014541", "content": "已知等差数列$\\{a_n\\}$的公差$d \\neq 0$, 其前$n$项和为$S_n$, 若$S_{10}=0$, 则$S_i$($i=1,2,3, \\cdots, 2023$)中不同的数值有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384078,7 +384674,9 @@ "id": "014542", "content": "(1) 在等差数列$\\{a_n\\}$中, 公差为$d$, 其前$n$项和为$S_n$, 根据要求完成下列表格, (其中$m$为正整数).\n\\begin{center}\n\\begin{tabular}{|c|l|}\n\\hline 用$S_m$表示$S_{2 m}$&$S_{2 m}=2S_m+m^2 d$\\\\\n\\hline 用$S_m$表示$S_{3 m}$&$S_{3 m}=$\\\\\n\\hline 用$S_m$表示$S_{n m}$&$S_{N m}=$\\\\\n\\hline\n\\end{tabular} \n\\end{center}\n(2) 在等比数列$\\{b_n\\}$中, 公比为$q$, 类比问题(1), 请写出相应的结论.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384098,7 +384696,9 @@ "id": "014543", "content": "等差数列$\\{a_n\\}$的首项为$1$, 公差不为$0$. 若$a_2, a_3, a_6$成等比数列, 则$\\{a_n\\}$前$6$项的和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384118,7 +384718,9 @@ "id": "014544", "content": "已知数列$\\{a_n\\}$, 其前$n$项和为$S_n$, 若$S_n=1+\\dfrac{1}{4} a_n$, 则$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384138,7 +384740,9 @@ "id": "014545", "content": "若将数列$\\{2 n-1\\}$与$\\{3 n-2\\}$的公共项从小到大排列得到数列$\\{a_n\\}$, 则$\\{a_n\\}$的前$n$项和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384193,7 +384797,9 @@ "id": "014547", "content": "在等差数列$\\{a_n\\}$中, 前$n$项和为$S_n$, 有以下结论: $S_n=a_1+a_2+\\cdots+a_n=\\dfrac{n(a_1+a_n)}{2}=n a_1+\\dfrac{n(n-1)}{2} d$, 类比上述结论, 写出等比数列$\\{b_n\\}$的前$n$项积$T_n$的结论\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384213,7 +384819,9 @@ "id": "014548", "content": "随着人们生活水平的提高, 很多家庭都购买了家用汽车. 使用汽车共需支出三笔费用: 购置费、燃油费、养护保险费. 某种型号汽车, 购置费共$20$万元; 购买后第$1$年燃油费共$2$万元, 以后每一年都比前一年增加$0.2$万元.\\\\\n(1) 若每年养护保险费均为$1$万元. 设购买该种型号汽车$n$($n \\in \\mathbf{N}$, $n \\geq 1$)年后共支出费用为$S_n$万元, 求$S_n$的表达式;\\\\\n(2) 若购买汽车后的前$6$年, 每年养护保险费均为$1$万元, 由于部件老化和事故多发, 第$7$年起, 每一年的养护保险费都比前一年增加$10 \\%$. 设使用$n$($n \\in \\mathbf{N}$, $n \\geq 1$)年后年平均费用为$c_n$, 当$n=n_0$时, $c_n$最小. 请你列出$n>6$时$c_n$的表达式, 并利用计算器确定$n_0$的值(只需写出$n_0$的值).", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384233,7 +384841,9 @@ "id": "014549", "content": "给定无穷数列$\\{a_n\\}$, 若无穷数列$\\{b_n\\}$满足: 对任意正整数$n$, 都有$|b_n-a_n| \\leq 1$, 则称$\\{b_n\\}$与$\\{a_n\\}$``接近''.\\\\\n(1) 设$\\{a_n\\}$是首项为$1$, 公比为$\\dfrac{1}{2}$的等比数列, $b_n=a_{n+1}+1$, $n \\in \\mathbf{N}$, $n \\geq 1$, 判断数列$\\{b_n\\}$是否与$\\{a_n\\}$接近, 并说明理由;\\\\\n(2) 设数列$\\{a_n\\}$的前四项为: $a_1=1, a_2=2, a_3=4, a_4=8$, $\\{b_n\\}$是一个与$\\{a_n\\}$接近的数列, 记集合$M=\\{x | x=b_i, i=1,2,3,4\\}$, 求$M$中元素的个数$m$的所有可能值;\\\\\n(3) 已知$\\{a_n\\}$是公差为$d$的等差数列, 若存在数列$\\{b_n\\}$满足: $\\{b_n\\}$与$\\{a_n\\}$接近, 且在$b_2-b_1, b_3-b_2, \\cdots, b_{201}-b_{200}$中正数的个数不小于$100$, 求$d$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384253,7 +384863,9 @@ "id": "014550", "content": "设$\\lambda \\in \\mathbf{R}$, 数列$\\{a_n\\}$的通项公式为$a_n=n^2-\\lambda n+8$, 若数列$\\{a_n\\}$为严格增数列, 则$\\lambda$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$(-\\infty,3)$", "solution": "", @@ -384282,7 +384894,9 @@ "id": "014551", "content": "若数列$\\{a_n\\}$的通项公式为$a_n=\\dfrac{n-3.5}{n-4.5}$, 则数列$\\{a_n\\}$的最大项是\\blank{50}; 最小项是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384302,7 +384916,9 @@ "id": "014552", "content": "若数列$\\{a_n\\}$满足$a_1=4$, $a_n=4 a_{n-1}-6$($n \\in \\mathbf{N}$, $n \\geq 2$).\\\\\n(1) 设$b_n=a_n-2$, 求证数列$\\{b_n\\}$是等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384322,7 +384938,9 @@ "id": "014553", "content": "已知数列$\\{a_n\\}$满足$a_1=2$, $a_{n+1}=\\dfrac{1+a_n}{1-a_n}$, 则该数列的前$2023$项的乘积$a_1 a_2 a_3 \\cdots a_{2023}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -384353,7 +384971,9 @@ "id": "014554", "content": "若数列$\\{a_n\\}$满足$a_1=0$, $2 a_{n+1}-a_n a_{n+1}=1$, 则$a_{2023}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384373,7 +384993,9 @@ "id": "014555", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=(n-5)(\\dfrac{1}{2})^n$, 试问: 该数列是否有最大项、最小项? 若有, 分别指出第几项最大、最小; 若没有, 试说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384393,7 +385015,9 @@ "id": "014556", "content": "已知数列$\\{a_n\\}$, $\\{b_n\\}$满足$a_{n+1}-a_n=2(b_{n+1}-b_n)$.\\\\ \n(1) 设$\\{a_n\\}$的第$n_0$项是数列$\\{a_n\\}$的最大项, 即$a_{n_0} \\geq a_n$对一切正整数$n$恒成立, 求证: $\\{b_n\\}$的第$n_0$项是数列$\\{b_n\\}$的最大项;\\\\\n(2) 设$\\lambda \\in \\mathbf{R}$, $a_1=3 \\lambda<0$, $b_n=\\lambda^n$, 求$\\lambda$的取值范围, 使得对任意正整数$m$、$n$, $a_n \\neq 0$, $\\dfrac{a_m}{a_n} \\in(\\dfrac{1}{6}, 6)$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384413,7 +385037,9 @@ "id": "014557", "content": "对于无穷数列$\\{a_n\\}$与$\\{b_n\\}$, 记$A=\\{x | x=a_n\\}, B=\\{x | x=b_n\\}$, 若同时满足条件: \\textcircled{1} $\\{a_n\\}$, $\\{b_n\\}$均为严格增数列; \\textcircled{2} $A \\cap B=\\varnothing$且$A \\cup B=\\{x | x \\in \\mathbf{N}, x \\geq 1\\}$, 则称$\\{a_n\\}$与$\\{b_n\\}$是无穷互补数列.\\\\\n(1) 若$a_n=2 n-1$, $b_n=4 n-2$, 判断$\\{a_n\\}$与$\\{b_n\\}$是否为无穷互补数列, 并说明理由;\\\\\n(2) 若$a_n=2^n$且$\\{a_n\\}$与$\\{b_n\\}$是无穷互补数列, 求数列$\\{b_n\\}$的前$2023$项的和;\\\\\n(3) 若$\\{a_n\\}$与$\\{b_n\\}$是无穷互补数列, $\\{a_n\\}$为等差数列且$a_{10}=23$, 求$\\{a_n\\}$与$\\{b_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384433,7 +385059,9 @@ "id": "014558", "content": "定义``等和数列'': 在一个数列中, 如果每一项与它的后一项的和都为同一个常数, 那么这个数列叫做等和数列, 这个常数叫做该数列的公和. 已知数列$\\{a_n\\}$是等和数列, 且$a_1=2$, 公和为$5$, 那么$a_{28}=$\\blank{50}, 这个数列的前$n$项和$S_n$的计算公式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384453,7 +385081,9 @@ "id": "014559", "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 若$a_1=1$, $\\{\\dfrac{S_n}{a_n}\\}$是公差为$\\dfrac{1}{3}$的等差数列, 则数列$\\{a_n\\}$的通项公式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384473,7 +385103,9 @@ "id": "014560", "content": "已知数列$\\{a_n\\}$的前$n$项的和$S_n$满足$S_{n+1}+S_n=n$. 对于以下两个命题: \\textcircled{1} 若$a_1=-1$, 则$S_{2023}=1010$; \\textcircled{2} 数列$\\{a_{n+1}+a_n\\}$是常数列, 下列说法正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}都正确}{\\textcircled{1}正确, \\textcircled{2}不正确}{\\textcircled{1}\\textcircled{2}都不正确}{\\textcircled{1}不正确, \\textcircled{2}正确}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -384504,7 +385136,9 @@ "id": "014561", "content": "已知数列$\\{a_n\\}$, $\\{b_n\\}$满足$a_1=b_1=1$, $a_{n+1}=a_n+b_n+\\sqrt{a_n^2+b_n^2}$, $b_{n+1}=a_n+b_n-\\sqrt{a_n^2+b_n^2}$, 设$c_n=3^n(\\dfrac{1}{a_n}+\\dfrac{1}{b_n})$, 则数列$\\{c_n\\}$的前$2023$项之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384524,7 +385158,9 @@ "id": "014562", "content": "若$a_n=2 n-7$, 下列说法中, 所有正确说法的序号是\\blank{50}.\\\\\n\\textcircled{1} 数列$\\{a_n\\}$是严格增数列;\\\\\n\\textcircled{2} 数列$\\{n a_n\\}$是严格增数列;\\\\\n\\textcircled{3} 数列$\\{\\dfrac{a_n}{n}\\}$是严格增数列;\\\\\n\\textcircled{4} 数列$\\{\\dfrac{1}{a_n}\\}$是严格减数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384544,7 +385180,9 @@ "id": "014563", "content": "已知数列$\\{a_n\\}$中$a_1=1$, $a_2=2$, $a_3=4$, 满足$a_{n+1}=-a_{n-2}$($n \\in \\mathbf{N}$, $n \\geq 3$), 则$a_{12}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384564,7 +385202,9 @@ "id": "014564", "content": "已知数列$\\{a_n\\}$的通项公式$a_n=\\dfrac{1}{(n+1)^2}$, 记$f(n)=(1-a_1)(1-a_2) \\cdots(1-a_n)$, 则$f(n)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384584,7 +385224,9 @@ "id": "014565", "content": "已知数列$\\{a_n\\}$满足$a_1=1, n a_{n+1}=(n+1) a_n+1$, 设$t \\in \\mathbf{R}$, 若对于任意的$a \\in[-2,2]$, 不等式$\\dfrac{a_{n+1}}{n+1}<3-a \\cdot 2^t$恒成立, 则$t$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384604,7 +385246,9 @@ "id": "014566", "content": "下图是某神奇``黄金分割草''的生长图, 第$1$阶段生长为坚直向上长为$1$米的枝干, 第$2$阶段在枝头生长出两根新的枝干, 新枝干的长度是原来的$\\dfrac{\\sqrt{5}-1}{2}$, 且与旧枝成$120^\\circ$角, 第$3$阶段又在每个枝头各长出两根新的枝干, 新枝干的长度是原来的$\\dfrac{\\sqrt{5}-1}{2}$, 且与旧枝成$120^\\circ$角, $\\cdots$, 依次生长, 直到永远.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{0.618}\n\\draw (0,0) -- (0,1);\n\\draw (3,0) -- (3,1) --++ (30:\\l) (3,1)--++(150:\\l);\n\\draw (6,0) -- (6,1) --++ (30:\\l) coordinate (A1) (6,1)--++(150:\\l) coordinate (A2) (A2)--++ (210:{\\l*\\l}) (A2) --++ (90:{\\l*\\l}) (A1) --++ (-30:{\\l*\\l}) (A1) --++ (90:{\\l*\\l});\n\\draw (9,0) -- (9,1) --++ (30:\\l) coordinate (A1) (9,1)--++(150:\\l) coordinate (A2) (A2)--++ (210:{\\l*\\l}) coordinate (B1) (A2) --++ (90:{\\l*\\l}) coordinate (B2) (A1) --++ (-30:{\\l*\\l}) coordinate (B4) (A1) --++ (90:{\\l*\\l}) coordinate (B3) (B1) --++ (-90:{\\l*\\l*\\l}) (B1) --++ (150:{\\l*\\l*\\l}) (B2) --++ (150:{\\l*\\l*\\l}) (B2) --++ (30:{\\l*\\l*\\l}) (B3) --++ (150:{\\l*\\l*\\l}) (B3) --++ (30:{\\l*\\l*\\l}) (B4) --++ (30:{\\l*\\l*\\l}) (B4) --++ (-90:{\\l*\\l*\\l});\n\\draw (12,1) node {$\\cdots$};\n\\draw (0,0) node [below] {第$1$阶段};\n\\draw (3,0) node [below] {第$2$阶段};\n\\draw (6,0) node [below] {第$3$阶段};\n\\draw (9,0) node [below] {第$4$阶段};\n\\draw (12,0) node [below] {$\\cdots\\cdots$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求第$3$阶段``黄金分割草''的高度;\\\\\n(2) 求第$13$阶段``黄金分割草''的所有枝干的长度之和;(结果精确到$0.01$米)", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384624,7 +385268,9 @@ "id": "014567", "content": "已知数列$\\{a_n\\}$中, $a_1=a$($a \\in \\mathbf{R}$, $a \\neq-\\dfrac{1}{2}$), $a_n=2 a_{n-1}+\\dfrac{\\mathbf{1}}{n}+\\dfrac{\\mathbf{1}}{n(n+1)}$($n \\geq 2$, $n \\in \\mathbf{N}$), 又数列$\\{b_n\\}$满足: $b_n=a_n+\\dfrac{1}{n+1}$.\\\\\n(1) 求证: 数列$\\{b_n\\}$是等比数列;\\\\\n(2) 若数列$\\{a_n\\}$是严格增数列, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384644,7 +385290,9 @@ "id": "014568", "content": "若数列$\\{a_n\\}$满足$a_1>0$, $a_{n+1} a_n-a_n^2=1$, 且$\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\cdots+\\dfrac{1}{a_{2023}}=2023$, 则\\bracket{20}.\n\\twoch{$20230$). 数列$\\{b_n\\}$定义如下: 对于正整数$m$, $b_m$是使得不等式$a_n \\geq m$成立的所有$n$中的最小值.\\\\\n(1) 若$p=\\dfrac{1}{2}$, $q=-\\dfrac{1}{3}$, 求$b_3$;\\\\\n(2) 若$p=2$, $q=-1$, 求数列$\\{b_n\\}$的前$2m$项和;\\\\\n(3) 是否存在$p$、$q$, 使得$b_m=3 m+2$? 如果存在, 求出所有满足条件的$p$、$q$的值; 如果不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -384684,7 +385334,9 @@ "id": "014570", "content": "书架上层放有$6$本不同的数学书, 下层放有$5$本不同的语文书.\\\\\n(1) 若从中任取一本书, 则共有\\blank{50}种不同的取法;\\\\\n(2) 若从中任取数学书与语文书各一本, 则共有\\blank{50}种不同的取法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "(1) $11$; (2) $30$", "solution": "", @@ -384715,7 +385367,9 @@ "id": "014571", "content": "将$5$名北京冬奥会志愿者分配到花样滑冰、短道速滑、冰球和冰壶$4$个项目进行培训, 若每名志愿者只分配到$1$个项目, 且每个项目至少分配$1$名志愿者, 则不同的分配方案共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384735,7 +385389,9 @@ "id": "014572", "content": "已知$n$是大于等于$3$的正整数, 且$\\mathrm{C}_n^2+\\mathrm{C}_n^3=\\mathrm{P}_n^2$, 则$n$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384755,7 +385411,9 @@ "id": "014573", "content": "设$a \\in \\mathbf{R}$, 在$(2 x+\\dfrac{a}{x})^7$的二项展开式中, 若$x^{-3}$项的系数是$84$, 则$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384775,7 +385433,9 @@ "id": "014574", "content": "用$0,1,2,3,4$这$5$个数字, 组成四位数.\\\\\n(1) 共可以组成\\blank{50}个没有重复数字的四位数;\\\\\n(2) 共可以组成个\\blank{50}没有重复数字的四位偶数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "(1) $96$; (2) $60$", "solution": "", @@ -384808,7 +385468,9 @@ "id": "014575", "content": "某社团共有$10$人, 其中男生$6$人, 女生$4$人.\\\\\n(1) 这$10$名同学站成一排拍照, 若女生都不相邻, 则共有\\blank{50}种不同的排法;\\\\\n(2) 现从这$10$名同学中选出$4$名同学安排到$4$个小区参加志愿者活动, 每个小区一人, 若选出的同学中男生女生都要有, 则共有\\blank{50}种不同的安排方法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "(1) $604800$; (2) $4656$", "solution": "", @@ -384841,7 +385503,9 @@ "id": "014576", "content": "某社团共有$10$人, 其中男生$6$人, 女生$4$人, 并且男、女生中各有$1$人是队长. 这$10$名同学站成一排拍照, 若男生队长不站排头, 女生队长不站排尾, 则共有\\blank{50}种不同的排法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -384861,7 +385525,9 @@ "id": "014577", "content": "某社团共有$10$人, 其中男生$6$人, 女生$4$人, 并且男、女生中各有$1$人是队长. 现从这$10$名同学中选出$4$名同学去参加志愿者活动, 既要有队长, 又要有女生的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{13}{21}$", "solution": "", @@ -384892,7 +385558,9 @@ "id": "014578", "content": "设$(2 x+1)^n=a_0+a_1 x+a_2 x^2+\\cdots+a_n x^n$.\\\\\n(1) 若$a_0+a_1+a_2+\\cdots+a_n=6561$, 求$a_3$的值;\\\\\n(2) 若$n=8$, 求$(a_0+a_2+\\cdots+a_8)^2-(a_1+a_3+\\cdots+a_7)^2$的值;\\\\\n(3) 若$n=15$, 求$a_0, a_1, \\cdots, a_n$中的最大项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "(1) $448$; (2) $6561$; (3) 最大项为$a_10$($=3075072$)", "solution": "", @@ -384923,7 +385591,9 @@ "id": "014579", "content": "在一次运动会上有四项比赛的冠军分别在甲、乙、丙三人中产生, 那么不同的夺冠情况共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$81$", "solution": "", @@ -384954,7 +385624,9 @@ "id": "014580", "content": "已知有$4$名男生, $6$名女生, 若从这$10$人中任选$3$人, 则恰有$1$名男生和$2$名女生的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 12$", "solution": "", @@ -384984,7 +385656,9 @@ "id": "014581", "content": "若在$(x+\\dfrac{1}{x})^{10}$的二项展开式中第$k$项的系数最大, 则$(2 x+\\dfrac{1}{\\sqrt{x}})^k$的二项展开式中, 常数项是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$60$", "solution": "", @@ -385017,7 +385691,9 @@ "id": "014582", "content": "甲乙丙丁戊$5$名同学站成一排参加文艺汇演, 甲不站在两端, 丙和丁相邻的不同排列方式共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$24$", "solution": "", @@ -385050,7 +385726,9 @@ "id": "014583", "content": "已知$x$是小于等于$7$的正整数, 若$\\mathrm{C}_{13}^{2 x-1}=\\mathrm{C}_{13}^{x+2}$, 则$x$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$3$或$4$", "solution": "", @@ -385081,7 +385759,9 @@ "id": "014584", "content": "从$7$个人中选$4$人负责元旦三天假期的值班工作, 若第一天安排$2$人, 第二天和第三天均安排$1$人, 且人员不重复, 则共有\\blank{50}种不同的安排方式. (结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$420$", "solution": "", @@ -385114,7 +385794,9 @@ "id": "014585", "content": "第$14$届国际数学教育大会(ICME-14)于$2021$年$7$月$12$日至$18$日在上海举办, 已知张老师和李老师都在$7$天中随机选择了连续的$3$天参会, 则两位老师所选的日期恰好都不相同的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{6}{25}$", "solution": "", @@ -385145,7 +385827,9 @@ "id": "014586", "content": "设$(x-1)(x+1)^5=a_0+a_1 x+a_2 x^2+a_3 x^3+\\cdots+a_6 x^6$, 则$a_3=$\\blank{50}.(结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$0$", "solution": "", @@ -385178,7 +385862,9 @@ "id": "014587", "content": "若$O$是正六边形$A_1A_2A_3A_4A_5A_6$的中心, $B=\\{\\overrightarrow{OA_i} | i=1,2,3,4,5,6\\}$, $\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c} \\in B$, 且$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$互不相等, 要使得$(\\overrightarrow {a}+\\overrightarrow {b}) \\cdot \\overrightarrow {c}=0$, 则有序向量组$(\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c})$的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$48$", "solution": "", @@ -385211,7 +385897,9 @@ "id": "014588", "content": "在高三一班元旦晩会上, 有$6$个演唱节目, $4$个舞蹈节目.\\\\\n(1) 若$4$个舞蹈节目排在一起, 则不同的节目安排顺序共有多少种?\\\\\n(2) 若要求每$2$个舞蹈节目之间至少安排$1$个演唱节目, 则不同的节目安排顺序共有多少种?\\\\\n(3) 若已定好节目单, 后来情况有变, 需加上诗歌朗诵和快板$2$个节目, 但不能改变原来节目的相对顺序, 则共有多少种不同的节目演出顺序?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "(1) $120960$; (2) $604800$; (3) $132$", "solution": "", @@ -385242,7 +385930,10 @@ "id": "014589", "content": "已知数列$\\{a_n\\}$共有$11$项, $a_1=0$, $a_{11}=4$, 且$|a_{k+1}-a_k|=1$($k=1,2,3, \\cdots, 10$), 满足这样条件的不同数列的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元", + "第八单元" + ], "genre": "填空题", "ans": "$120$", "solution": "", @@ -385273,7 +385964,9 @@ "id": "014590", "content": "规定$\\mathrm{C}_x^m=\\dfrac{x(x-1) \\cdots(x-m+1)}{m !}$, 其中$x \\in \\mathbf{R}$, $m$是正整数, 且$\\mathrm{C}_x^0=1$, 这是组合数$\\mathrm{C}_n^m$($n, m$是正整数, 且$m \\leq n$)的一种推广.\\\\\n(1) 求$\\mathrm{C}_{-15}^5$的值;\\\\\n(2) 组合数的两个性质: \\textcircled{1} $\\mathrm{C}_n^m=\\mathrm{C}_n^{n-m}$; \\textcircled{2} $\\mathrm{C}_n^m+\\mathrm{C}_n^{m-1}=\\mathrm{C}_{n+1}^m$是否都能推广到$\\mathrm{C}_x^m$($x \\in \\mathbf{R}$, $m$是正整数)的情形? 若能推广, 则写出推广的形式并给出证明; 若不能, 则说明理由;\\\\\n(3) 已知组合数$\\mathrm{C}_n^m$是正整数, 证明: 当$x \\in \\mathbf{Z}$, $m$是正整数时, $\\mathrm{C}_x^m \\in \\mathbf{Z}$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -385293,7 +385986,9 @@ "id": "014591", "content": "从甲、乙等$5$名同学中随机选$3$名参加社区服务工作, 则甲、乙两人中至少有一人入选的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{9}{10}$", "solution": "", @@ -385324,7 +386019,9 @@ "id": "014592", "content": "$100$件产品中有$5$件次品, 不放回地抽取$2$次, 每次抽出$1$件, 已知第一次抽出的是次品, 则第二次抽出正品的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{95}{99}$", "solution": "", @@ -385354,7 +386051,9 @@ "id": "014593", "content": "袋中有大小与质地均相同的$12$个小球, 分别为红球、黑球、黄球、绿球, 从中任取$1$个球, 若得到红球的概率是$\\dfrac{1}{4}$, 得到黑球或黄球的概率$\\dfrac{5}{12}$, 得到黄球或绿球的概率是$\\dfrac{1}{2}$, 则得到绿球的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 13$", "solution": "", @@ -385385,7 +386084,9 @@ "id": "014594", "content": "已知随机变量$X$服从正态分布$N(2, \\sigma^2)$, 若$P(22.5)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$0.14$", "solution": "", @@ -385416,7 +386117,9 @@ "id": "014595", "content": "实力相当的甲、乙两人参加乒乓球比赛, 规定$5$局$3$胜制(即$5$局内谁先赢$3$局就算胜出并停止比赛).\\\\\n(1) 试求甲打完$5$局才获胜的概率;\\\\\n(2) 按比赛规则甲获胜的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 3{16}$; (2) $\\dfrac 12$", "solution": "", @@ -385446,7 +386149,9 @@ "id": "014596", "content": "某工厂有三个车间生产同一产品, 第一车间的次品率为$0.05$, 第一车间的次品率为$0.03$, 第一车间的次品率为$0.01$, 各车间的产品数量分别为$1500$件、$2000$件、$1500$件, 出厂时, 三个车间的产品完全混合, 现从中任取$1$件产品, 求该产品是次品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "$0.03$", "solution": "", @@ -385476,7 +386181,9 @@ "id": "014597", "content": "甲、乙两个学校进行体育比赛, 比赛共设三个项目, 每个项目胜方得$10$分, 负方得$0$分, 没有平局. 三个项目比赛结束后, 总得分高的学校获得冠军. 已知甲学校在三个项目中获胜的概率分别为$0.5$、$0.4$、$0.8$, 各项目的比赛结果相互独立.\\\\\n(1) 求甲学校获得冠军的概率;\\\\\n(2) 用$X$表示乙学校的总得分, 求$X$的分布与期望.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "(1) $0.6$; (2) $X\\sim \\begin{pmatrix}0 & 10 & 20 & 30 \\\\ 0.16 & 0.44 & 0.34 & 0.06\\end{pmatrix}$, $E[X]=13$", "solution": "", @@ -385507,7 +386214,9 @@ "id": "014598", "content": "已知随机变量$X$服从二项分布$B(12,0.25)$, 且$E[a X-3]=3$($a \\in \\mathbf{R}$), 则$D[a X-3]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$9$", "solution": "", @@ -385538,7 +386247,9 @@ "id": "014599", "content": "有若干个大小与质地均相同的红球和白球. 已知甲袋中有$6$个红球, $4$个白球, 乙袋中有$8$个红球, $6$个白球, 若随机取一个袋子, 再从该袋中随机取一个球, 则该球是红球的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{41}{70}$", "solution": "", @@ -385569,7 +386280,9 @@ "id": "014600", "content": "袋中装有大小与质地均相同的$2$个白球和$3$个黑球.\\\\\n(1) 从中有放回地摸两次, 每次摸$1$个球, 求两球颜色不同的概率;\\\\\n(2) 从中不放回地摸两次, 每次摸$1$个球, 记$X$为摸出两球中白球的个数, 求$X$的期望和方差.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{12}{25}$; (2) $\\dfrac 9{25}$", "solution": "", @@ -385599,7 +386312,9 @@ "id": "014601", "content": "哥德巴赫猜想是指``每个大于$2$的偶数都可以表示为两个素数的和'', 例如$10=7+3$, $16=13+3$. 在不超过$32$的所有素数中, 随机选取两个不同的数, 其和等于$32$的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 2{55}$", "solution": "", @@ -385630,7 +386345,9 @@ "id": "014602", "content": "袋中有大小与质地均相同的$5$个红球, $4$个白球, 现随机地从中取出一个球, 记录颜色后, 将其放回袋中, 并随之放入$2$个与之颜色相同的球, 再从袋中第二次取出一球, 则第二次取出的是白球的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 49$", "solution": "", @@ -385660,7 +386377,9 @@ "id": "014603", "content": "某实验测试的规则如下: 每位学生最多可做$3$次实验, 一旦实验成功, 则停止实验, 否则做完$3$次为止. 设某学生每次实验成功的概率为$p$($01.39$, 则$p$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$(0,0.7)$", "solution": "", @@ -385691,7 +386410,9 @@ "id": "014604", "content": "已知两个随机变量$X, Y$, 其中$X \\sim B(4, \\dfrac{1}{4})$, $Y \\sim N(\\mu, \\sigma^2)$($\\sigma>0$), 若$E[X]=E[Y]$, 且$P(|Y|<1)=0.4$, 则$P(Y>3)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$0.1$", "solution": "", @@ -385721,7 +386442,9 @@ "id": "014605", "content": "某射击小组共有$25$名射手, 其中一级射手$5$人, 二级射手$10$人, 三级射手$10$人, 若一、二、三级射手能通过选拔进入比赛的概率分别是$0.9$, $0.8$, $0.4$, 则从中任选一名射手能通过选拔进入比赛的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$0.66$", "solution": "", @@ -385751,7 +386474,9 @@ "id": "014606", "content": "在四次独立重复试验中, 事件$A$在每次试验中发生的概率相同, 若事件$A$至少发生一次的概率为$\\dfrac{65}{81}$, 则事件$A$发生次数$X$的期望是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 43$", "solution": "", @@ -385781,7 +386506,9 @@ "id": "014607", "content": "某批零件(个数非常多)的尺寸$X$服从正态分布$N(10, \\sigma^2)$, 且满足$p(X<9)=\\dfrac{1}{6}$, 零件的尺寸与$10$的误差的绝对值不超过$1$即合格, 从这批产品中随机抽取$n$件, 若要保证抽取的合格零件不少于$2$件的概率不低于$0.9$, 则$n$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -385811,7 +386538,9 @@ "id": "014608", "content": "甲、乙两人各拿两颗骰子做抛掷游戏, 规则如下: 若掷出的点数之和为$3$的倍数, 原掷骰子的人再继续掷; 若掷出的点数之和不是$3$的倍数, 就由对方接着掷. 第一次由甲开始掷, 求第$n$次由甲掷的概率$P_n$(用含$n$的式子表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "$P_n=\\dfrac 12(1+(-\\dfrac 13)^{n-1})$", "solution": "", @@ -385842,7 +386571,9 @@ "id": "014609", "content": "如图, $\\triangle ABC$中, $AD \\perp BC$于$D$, $\\angle BAC=45^{\\circ}$, $BD=2$, $CD=1$, 则$\\triangle ABC$的面积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (-2,0) node [left] {$B$} coordinate (B);\n\\draw (1,0) node [right] {$C$} coordinate (C);\n\\draw (0,0) node [below] {$D$} coordinate (D);\n\\draw (0,{(sqrt(17)+3)/2}) node [above] {$A$} coordinate (A);\n\\draw (B)--(A)--(C)--cycle(A)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -385862,7 +386593,9 @@ "id": "014610", "content": "已知实数$a>1$, 则方程$a^x+x-2=0$的根$x_0$满足\\bracket{20}.\n\\fourch{$x_0<-1$}{$-11$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -385882,7 +386615,9 @@ "id": "014611", "content": "已知不等式$2 x-1>m(x^2-1)$对满足$|m| \\leq 2$的一切实数$m$恒成立, 则$x$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -385902,7 +386637,9 @@ "id": "014612", "content": "已知点$A$、$B$、$C$是圆$O$上不同的三点, 线段$OC$与线段$AB$交于点$D$(圆心$O$与点$D$不重合), 若$\\overrightarrow{OC}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$($\\lambda, \\mu \\in \\mathbf{R}$), 则$\\lambda+\\mu$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -385922,7 +386659,9 @@ "id": "014613", "content": "已知$\\{a_n\\}$是首项为$a$, 公差为$1$的等差数列, $b_n=\\dfrac{1+a_n}{a_n}$, 若对于任意正整数$n$, 都有$b_n \\geq b_8$成立, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$(-8,-7)$", "solution": "", @@ -385948,7 +386687,9 @@ "id": "014614", "content": "已知$a \\in \\mathbf{R}$, 若关于$x$的方程$\\lg a x=2 \\lg (x-1)$有解, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -385968,7 +386709,10 @@ "id": "014615", "content": "已知$a \\neq 0$, $a \\in \\mathbf{R}$, 若关于$x$的不等式$(a x-1)(x^2-a x-1) \\geq 0$对于任意$x>0$都成立, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{2}}2$", "solution": "", @@ -385994,7 +386738,9 @@ "id": "014616", "content": "已知$m \\in \\mathbf{R}$, $f(x)=\\begin{cases}2 x^2-x,& x \\leq 0, \\\\ -x^2+x,& x>0,\\end{cases}$ $g(x)=f(x)-m$, 若函数$y=g(x)$恰有三个零点$x_1, x_2, x_3$, 则$x_1 x_2 x_3$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386014,7 +386760,9 @@ "id": "014617", "content": "已知抛物线$C: y^2=x$的焦点为$F$, $A(x_0, y_0)$是抛物线$C$上一点, 若$|AF|=|\\dfrac{5}{4} x_0|$, 则$x_0=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386034,7 +386782,9 @@ "id": "014618", "content": "已知$k \\in \\mathbf{R}$, 若不等式$x^2-k x+k-1>0$对任意$x \\in(1,2)$恒成立, 则$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386054,7 +386804,9 @@ "id": "014619", "content": "在矩形$ABCD$中, 边$AB, AD$的长分别为$2,1$. 若点$M, N$分别是边$BC, CD$上的点, 且满足$\\dfrac{|\\overrightarrow{BM}|}{|\\overrightarrow{BC}|}=\\dfrac{|\\overrightarrow{CN}|}{|\\overrightarrow{CD}|}$, 则$\\overrightarrow{AM} \\cdot \\overrightarrow{AN}$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386074,7 +386826,9 @@ "id": "014620", "content": "已知$a \\in \\mathbf{R}$, $A=\\{x | x^2-4 x+3<0\\}$, $B=\\{x | x^2-6 x+8<0\\}$, $C=\\{x | 2 x^2-9 x+a<0\\}$, 若$A \\cap B \\subseteq C$, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386094,7 +386848,10 @@ "id": "014621", "content": "关于$x$的方程$(x^2-1)^2-|x^2-1|+k=0$, 给出下列四个命题:\\\\\n\\textcircled{1} 存在实数$k$, 使得方程恰有$2$个不同的实根;\\\\\n\\textcircled{2} 存在实数$k$, 使得方程恰有$4$个不同的实根;\\\\\n\\textcircled{3} 存在实数$k$, 使得方程恰有$5$个不同的实根;\\\\\n\\textcircled{4} 存在实数$k$, 使得方程恰有$8$个不同的实根, 其中所有真命题的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386114,7 +386871,10 @@ "id": "014622", "content": "方程$x^2+2 x-1=0$的解可视为函数$y=x+2$的图像与函数$y=\\dfrac{1}{x}$的图像交点的横坐标. 若方程$x^4+a x-4=0$的各个实根$x_1, x_2, \\cdots, x_k$($k \\leq 4$)所对应的点$(x_i, \\dfrac{4}{x_i})$($i=1,2,3, \\cdots, k$)均在直线$y=x$的同侧, 则$a$取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386134,7 +386894,9 @@ "id": "014623", "content": "已知$a \\in \\mathbf{R}$, $f(x)=\\dfrac{1}{3} x^3-\\dfrac{1}{2} a x^2+x$, 若函数$y=f(x)$在区间$(\\dfrac{1}{2}, 3)$内既有极大值点又有极小值点, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386154,7 +386916,9 @@ "id": "014624", "content": "设数列$\\{a_n\\}$的首项$a_1$为常数, 且$a_1 \\neq \\dfrac{3}{5}$, 又$a_{n+1}=3^n-2 a_n$. 若$\\{a_n\\}$是严格递增数列, 求$a_1$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "$(0,\\dfrac 35)\\cup (\\dfrac 35,1)$", "solution": "", @@ -386181,7 +386945,9 @@ "id": "014625", "content": "设$y=f(x)$是定义在$\\mathbf{R}$上的偶函数, 其图像关于直线$x=1$成轴对称, 对任意的$x_1, x_2 \\in[0, \\dfrac{1}{2}]$, 都有$f(x_1+x_2)=f(x_1) \\cdot f(x_2)$且$f(1)=a>0$. 设数列$\\{a_n\\}$满足$a_n=f(2 n+\\dfrac{1}{2 n})$($n \\geq 1$, $n \\in \\mathbf{N}$), 求$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -386201,7 +386967,9 @@ "id": "014626", "content": "设偶函数$y=f(x)$是定义在$\\mathbf{R}$上的可导函数, 且$f(1)=0$. 当$x<0$时, 有$x f'(x)-f(x)>0$恒成立, 则不等式$f(x)>0$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386221,7 +386989,9 @@ "id": "014627", "content": "设$f(x)=\\dfrac{2}{x}+a \\ln x-2$($a>0$), 设函数$y=f(x)$.\\\\\n(1) 若对于任意$x \\in(0,+\\infty)$都有$f(x)>2(a-1)$成立, 求实数$a$的取值范围;\\\\\n(2) 设$b \\in \\mathbf{R}$, 记$g(x)=f(x)+x-b$. 当$a=1$时, 函数$y=g(x)$在区间$[\\mathrm{e}^{-1}, \\mathrm{e}]$上有两个零点, 求$b$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -386241,7 +387011,9 @@ "id": "014628", "content": "某校为了解高三年级学生体重情况, 从该年级$1000$名学生中抽取$125$名学生测量他们的体重进行分析. 在这项调查中, 抽取的$125$名学生的体重是\\bracket{20}.\n\\fourch{总体}{样本}{总体的容量}{样本容量}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -386261,7 +387033,9 @@ "id": "014629", "content": "某校小卖部为研究学生对饮料的喜好情况, 从该校$450$名同学中用随机数法抽取$30$人参加这一项调查. 将这$450$名同学编号为$001,002, \\ldots, 449,450$, 在以下随机数表中从任意一个随机数开始读出三位数组, 假设从第$2$行第$7$列的数字开始, 则第$5$个被抽到的同学的编号为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{llllll}\n16227794 & 39495443 & 54821737 & 93237887 & 35209643 & 84263491 \\\\ 64844217 & 55721754 & 55068331 & 04744767 & 21763350 & 25839212 \\\\ 06766301 & 63785916 & 95556719 & 98105071 & 75128673 & 58074439\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$447$", "solution": "", @@ -386292,7 +387066,9 @@ "id": "014630", "content": "某书店新进了一批书籍, 下表是某月中连续$6$天的销售情况记录:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 日期 & 6 日 & 7 日 & 8 日 & 9 日 & 10 日 & 11 日 \\\\\n\\hline 当日销售量/本 & 30 & 40 & 28 & 44 & 38 & 42 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n根据上表估计该书店该月(按$31$天计算)的销售总量约是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386312,7 +387088,9 @@ "id": "014631", "content": "某校课题小组为了研究高一年级学生甲、乙两门学科成绩的线性相关关系, 在高一第二学期期末考试后随机抽取了$5$名同学 (记为$A$、$B$、$C$、$D$、$E$) 的甲、乙两门学科成绩(满分均为$100$分), 如图. 后来发现$D$同学数据记录有误, 那么去掉数据$D(82,88)$后, 下列说法错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.25]\n\\draw [->] (80,75) -- (95,75) node [below] {学科甲成绩$x$};\n\\draw [->] (80,75) -- (80,90) node [left] {学科乙成绩$y$};\n\\filldraw (81,78) circle (0.1) node [right] {$A(81,78)$};\n\\filldraw (83,81) circle (0.1) node [right] {$E(83,81)$};\n\\filldraw (87,84) circle (0.1) node [right] {$B(87,84)$};\n\\filldraw (89,87) circle (0.1) node [right] {$C(89,87)$};\n\\filldraw (82,88) circle (0.1) node [right] {$D(82,88)$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{样本线性相关系数$r$变大}{最小二乘拟合误差变大}{变量$x$、$y$的相关程度变高}{线性相关系数$r$越接近于$1$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -386343,7 +387121,9 @@ "id": "014632", "content": "甲、乙两城之间的长途客车均由$A$和$B$两家公司运营, 为了解这两家公司长途客车的运行情况, 随机调查了甲、乙两城之间的$500$个班次, 得到下面的$2 \\times 2$列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 准点班次数 & 未准点班次数 \\\\\n\\hline$A$& 240 & 20 \\\\\n\\hline$B$& 210 & 30 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n根据上表数据, 判断甲、乙两城之间的长途客车准点情况与客车所属公司\\blank{50}. (填写``有关''或``无关'')\\\\\n($\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 显著水平取$0.05$, $P(\\chi^2 \\geq 3.841) \\approx 0.05$).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "无关", "solution": "", @@ -386374,7 +387154,9 @@ "id": "014633", "content": "深入实施科教兴国战略是中华人民伟大复兴的必由之路. $2020$年第七次全国人口普查中(未包括中国香港、澳门特别行政区和台湾省的人口数据), 我国$31$个省级行政区$15$岁及以上男性和女性的文盲人口比重($\\%$)情况, 经统计得到如下的茎叶图(其中$a$是$0$至$9$中的一个数字).\n\\begin{center}\n\\begin{tabular}{cccccccc|c|cccccccccccc}\n&&&&&&&女&&男\\\\\n&&&&&&&&0&4&6&7&7&7&8&9&9&9\\\\\n&&&&&&5&4&1&0&0&0&1&2&3&3&5&5&8&8&9\\\\\n&&9&9&2&1&1&1&2&1&6&7&7&7\\\\\n&&&&&2&0&0&3&3\\\\\n9&9&9&8&4&3&3&0&4&1&7\\\\\n&&&&&&7&0&5\\\\\n&&&&&8&5&2&6&2\\\\\n&&&&&&&6&7\\\\\n&&&&&&5&5&8\\\\\n&&&&&&&1&12\\\\\n&&&&&&&5&13\\\\\n&&&&&&&1&14\\\\\n&&&&&&&&20&4\\\\\n&&&&&&&$a$&36\n\\end{tabular}\n\\end{center}\n(1) 根据茎叶图判断男性样本数据和女性样本数据的离散程度, 并求离散程度较小的样本数据的第$80$百分位数;\\\\\n(2) 若女性样本数据的极差为$35.3(\\%)$, 求该样本数据的的平均数与方差(结果精确到$0.1$);\\\\\n(3) 为了调查今年某地区$15$岁及以上男性和女性文盲人口情况, 研究小组准备采用分层随机抽样方法抽取$5000$人进行调查. 已知该地区$15$岁及以上的男性约有$4.2$百万人, 女性约有$3.8$百万人. 分别求出抽取的男性人数和女性人数.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) 女性样本离散程度较高, 男性样本数据的第$80$百分位数为$2.7$($\\%$); (2) $a=8$, 平均数约为$6.3$, 方差约为$41.3$; (3) 男性$2625$人, 女性$2375$人", "solution": "", @@ -386405,7 +387187,9 @@ "id": "014634", "content": "某研究小组通过部分直辖市的统计年鉴, 整理了各区$15$岁及以上人口中大学专科、 本科和研究生学历的人口比重($\\%$)如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 地区 & \\makecell{专科\\\\学历\\\\比重} & \\makecell{本科\\\\学历\\\\比重} & \\makecell{研究生\\\\学历\\\\比重} & 地区 & \\makecell{专科\\\\学历\\\\比重} & \\makecell{本科\\\\学历\\\\比重} & \\makecell{研究生\\\\学历\\\\比重} & 地区 & \\makecell{专科\\\\学历\\\\比重} & \\makecell{本科\\\\学历\\\\比重} & \\makecell{研究生\\\\学历\\\\比重} \\\\\n\\hline 1 区 & 12.37 & 18.14 & 4.84 & 12 区 & 10.74 & 10.22 & 1.01 & 23 区 & 16.37 & 17.21 & 3.15 \\\\\n\\hline 2 区 & 14.69 & 26.5 & 9.74 & 13 区 & 13.27 & 17.92 & 3.12 & 24 区 & 15.07 & 17.1 & 3.21 \\\\\n\\hline 3 区 & 14.63 & 26.74 & 8.28 & 14 区 & 11.35 & 12.1 & 1.56 & 25 区 & 16.65 & 20.04 & 3.29 \\\\\n\\hline 4 区 & 15.67 & 23.65 & 5.38 & 15 区 & 10.46 & 11.86 & 1.13 & 26 区 & 13.62 & 17.3 & 2.72 \\\\\n\\hline 5 区 & 15.89 & 23.47 & 5.77 & 16 区 & 7.03 & 6.41 & 0.42 & 27 区 & 16.7 & 26.34 & 6.46 \\\\\n\\hline 6 区 & 15.33 & 23.92 & 5.43 & 17 区 & 15.82 & 28.71 & 8.44 & 28 区 & 16.05 & 19.51 & 4.23 \\\\\n\\hline 7 区 & 14.52 & 23.55 & 8.01 & 18 区 & 15.01 & 29.25 & 12.28 & 29 区 & 12.74 & 12.62 & 3.84 \\\\\n\\hline 8 区 & 14.68 & 22.21 & 5.89 & 19 区 & 15.28 & 30.76 & 9.56 & 30 区 & 13.7 & 11.41 & 0.98 \\\\\n\\hline 9 区 & 15.45 & 17.82 & 2.83 & 20 区 & 16.29 & 26.37 & 7.27 & 31 区 & 13 & 11.73 & 1.14 \\\\\n\\hline 10 区 & 13.01 & 14.93 & 3.03 & 21 区 & 16.97 & 28.17 & 8.21 & 32 区 & 12.79 & 12.51 & 1.26 \\\\\n\\hline 11 区 & 13.91 & 19.98 & 6.14 & 22 区 & 13.08 & 33.14 & 17.81 &&&&\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n请从以下几个方面对上表中的数据进行统计分析:\\\\\n(1) 各类学历的频率分布直方图;\\\\\n(2) 各类学历的集中趋势和离散程度;\\\\\n(3) 相关指标的相关性分析.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) 图略; (2) \\begin{tabular}{|c||c|c||c|c|c|}\n\\hline\n学历 & 均值 & 中位数 & 极差 & 方差 & 标准差 \\\\ \\hline\n专科 & $14.13$ & $14.66$ & $9.94$ & $4.52$ & $2.13$ \\\\ \\hline\n本科 & $20.05$ & $19.75$ & $26.73$ & $45.84$ & $6.77$ \\\\ \\hline\n研究生 & $5.20$ & $4.54$ & $17.39$ & $14.03$ & $3.75$ \\\\ \\hline\n\\end{tabular} (3) 专科与本科的相关系数约为$0.68$, 有较强的正相关性; 专科与研究生的相关系数约为$0.41$, 有较弱的正相关性; 本科与研究生的相关性约为$0.90$, 有很强的正相关性", "solution": "", @@ -386436,7 +387220,9 @@ "id": "014635", "content": "某同学要调查一个地区博物馆的展品数量, 有以下说法:\n\\textcircled{1} 可以通过调查获取相关数据;\n\\textcircled{2} 可以通过统计报表获取相关数据;\n\\textcircled{3} 可以通过实验获取相关数据;\n\\textcircled{4} 可以通过互联网获取相关数据. 其中所有正确说法的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{4}", "solution": "", @@ -386467,7 +387253,9 @@ "id": "014636", "content": "分别统计了甲、乙两位同学$16$周的各周课外体育运动时长(单位: $\\text{h}$), 得如下茎叶图: 则下列结论中错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{cccc|c|cccccccc}\n&&&甲&&乙\\\\\n&&6&1&5&\\\\\n8&5&3&0&6&3\\\\\n7&5&3&2&7&4&6\\\\\n6&4&2&1&8&1&2&2&5&6&6&6&6\\\\\n&&4&2&9&0&2&3&8\\\\\n&&&&10&1\n\\end{tabular}\n\\end{center}\n\\onech{甲同学周课外体育运动时长的样本中位数为$7.4$}{乙同学周课外体育运动时长的样本平均数大于$8$}{甲同学周课外体育运动时长大于$8$的概率的估计值大于$0.4$}{乙同学周课外体育运动时长大于$8$的概率的估计值大于$0.6$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -386498,7 +387286,9 @@ "id": "014637", "content": "为了解甲、乙两种离子在小鼠体内的残留程度, 进行如下试验: 将$200$只小鼠随机分成$A$、$B$两组, 每组$100$只, 其中给$A$组小鼠服甲离子溶液, 给$B$组小鼠服乙离子溶液, 给每只小鼠服的溶液体积相同、摩尔浓度相同. 经过一段时间后, 用某种科学方法测算出残留在小鼠体内离子的百分比. 根据试验数据分别得到如下直方图:\n记$C$为事件: ``乙离子残留在体内的百分比不低于$5.5$'', 根据直方图得到概率$P(C)$的估计值为$0.70$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {6.5/0.05/0.05,5.5/0.1/0.1,1.5/0.15/0.15,4.5/0.2/0.2,3.5/0.3/0.3}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {1.5/0.15/0.15,2.5/0.2/0.2,3.5/0.3/0.3,4.5/0.2/0.2,5.5/0.1/0.1,6.5/0.05/0.05}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$7.5$};\n\\draw (4.5,-0.1) node {甲离子残留百分比直方图};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/b,7.5/0.15/0.15,6.5/0.2/0.2,5.5/0.35/a}\n{\\draw [dashed] ({\\i-1},\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/0.1,4.5/0.15/0.15,5.5/0.35/0.35,6.5/0.2/0.2,7.5/0.15/0.15}\n{\\draw ({\\i-1},0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$8.5$};\n\\draw (4.5,-0.1) node {乙离子残留百分比直方图};\n\\end{tikzpicture}\n\\end{center}\n(1) 求乙离子残留百分比直方图中$a$、$b$的值;\\\\\n(2) 分别估计甲、乙离子残留百分比的平均值(同一组中的数据用该组区间的中点值为代表).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) $a=0.3$, $b=0.1$; (2) 甲离子残留百分比的平均值约为$4.05$($\\%$), 乙离子残留百分比的平均值约为$6$($\\%$)", "solution": "", @@ -386529,7 +387319,9 @@ "id": "014638", "content": "通过随机抽样, 我们获得某种产品加工前含水率与加工后含水率的一组测试数据, 如下表所示.\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}\n\\hline 加工前含水率 ($\\%$) & 16.7 & 18.2 & 18.2 & 17.9 & 17.4 & 16.6 & 17.2 & 17.7 & 15.7 & 17.1 \\\\\n\\hline 加工后含水率($\\%$)& 17.1 & 18.4 & 18.6 & 18.5 & 18.2 & 17.1 & 18.0 & 18.2 & 16.0 & 17.5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 请绘制上述数据的散点图, 并依据散点图观察两组数据的相关性;\\\\\n(2) 计算产品加工前含水率与加工后含水率之间的相关系数, 并判断两个变量的相关程度;\\\\\n(3) 依据表中给出的某种产品加工前含水率与加工后含水率的一组测试数据, 试预测当产品加工前含水率为$19 \\%$时, 加工后含水率的数值.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) 图略, 有明显的正相关关系; (2) $r\\approx 0.97$, 相关性明显, 为正相关; (3) 拟合直线为$y=1.0173x+0.1909$, 当产品加工前含水率为$19\\%$时, 加工后含水率估计为$19.5\\%$", "solution": "", @@ -386560,7 +387352,9 @@ "id": "014639", "content": "一支田径队有男运动员$48$人, 女运动员$36$人, 若用分层抽样的方法从全体运动员中抽取一个容量为$21$的样本, 则抽取男运动员的人数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386580,7 +387374,9 @@ "id": "014640", "content": "甲、乙两城市某月初连续$7$天的日均气温数据如图所示, 则在这$7$天中, 有以下说法:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\foreach \\i in {1,2,...,7} {\n\\draw [gray] (\\i,0) -- (\\i,7) (0,\\i) -- (7,\\i); \n\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.2,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw [->] (0,0) -- (8,0) node [below] {日期};\n\\draw [->] (0,0) -- (0,8) node [left] {气温$^\\circ\\text{C}$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,5) -- (2,3) -- (3,6) -- (4,3) -- (5,7) -- (6,5) -- (7,6);\n\\draw [dashed] (1,5) -- (2,4) -- (3,6) -- (4,5) -- (5,5) -- (6,4) -- (7,6);\n\\draw (7.5,5.5) -- (9.5,5.5) node [right] {甲};\n\\draw [dashed] (7.5,4) -- (9.5,4) node [right] {乙};\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 甲城市日均气温的平均数与中位数相等;\\\\\n\\textcircled{2} 甲城市的日均气温比乙城市的日均气温稳定;\\\\\n\\textcircled{3} 乙城市日均气温的极差为$3^{\\circ} \\text{C}$;\\\\\n\\textcircled{4} 乙城市日均气温的众数为$5^{\\circ} \\text{C}$.\n其中所有正确说法的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{4}", "solution": "", @@ -386611,7 +387407,9 @@ "id": "014641", "content": "某校抽取$100$名学生测身高, 其中身高最大值为$186 \\text{cm}$, 最小值为$154 \\text{cm}$, 根据身高数据绘制频率组距分布直方图, 组距为$5$, 且第一组下限为$153.5$, 则组数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$7$", "solution": "", @@ -386642,7 +387440,9 @@ "id": "014642", "content": "翠冠梨是盛产于江南夏季的一种丰水蜜梨, 是盛夏解渴消暑的时令佳品. 某产区标准化的果园每亩梨树$42$株. 为调查该产区$8000$亩翠冠梨的产量, 在收获期从绿港村$600$亩梨园中随机抽取了$10$棵梨树, 测得其产量 (单位: $\\text{kg}$) 分别为$72$、$75$、$84$、$94$、$60$、$78$、$99$、$78$、$91$、$90$. 预估该产区翠冠梨的总产量为\\blank{50}吨.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$27.6\\times 10^3$", "solution": "", @@ -386673,7 +387473,9 @@ "id": "014643", "content": "某国家足球队$26$名球员的年龄分布茎叶图如图所示:\n\\begin{center}\n\\begin{tabular}{c|ccccccccccccccccc}\n1 & 8 & 9 \\\\\n2 & 1 & 2 & 3 & 3 & 4 & 5 & 5 & 5 & 6 & 6 & 7 & 8 & 8 & 8 & 9 & 9 & 9 \\\\ \n3 & 0 & 1 & 2 & 2 & 2 & 3 & 4 \n\\end{tabular}\n\\end{center}\n该国家足球队$25$岁的球员共有\\blank{50}位, 该国家足球队球员年龄的第$75$百分位数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$3$; $30$", "solution": "", @@ -386704,7 +387506,9 @@ "id": "014644", "content": "某地经过多年的环境治理, 已将荒山改造成了绿水青山. 为估计一林区某种树木的总材积量, 随机选取了$10$棵这种树木, 测量每棵树的根部横截面积 (单位: $\\text{m}^2$) 和材积量 (单位: $\\text{m}^3$), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 样本号$i$& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 总和 \\\\\\hline\n\\makecell{根部横截\\\\面积$x_i$} & 0.04 & 0.06 & 0.04 & 0.08 & 0.08 & 0.05 & 0.05 & 0.07 & 0.07 & 0.06 & 0.6 \\\\\\hline\n材积量$y_i$ & 0.25 & 0.40 & 0.22 & 0.54 & 0.51 & 0.34 & 0.36 & 0.46 & 0.42 & 0.40 & 3.9 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 估计该林区这种树木平均一棵的根部横截面积与平均一棵的材积量;\\\\\n(2) 求该林区这种树木的根部横截面积与材积量的样本相关系数(精确到$0.01$);\\\\\n(3) 现测量了该林区所有这种树木的根部横截面积, 并得到所有这种树木的根部横截面积总和为$186 \\text{m}^2$. 已知树木的材积量与其根部横截面积近似成正比. 利用以上数据给出该林区这种树木的总材积量的估计值.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) 横截面积的平均值约为$0.06\\text{m}^2$; 材积量的平均值约为$0.39\\text{m}^3$; (2) $r\\approx 0.97$; (3) 比例系数的估计值$\\hat{k}=\\dfrac{\\displaystyle\\sum_{i=1}^{10}x_iy_i}{\\displaystyle\\sum_{i=1}^{10}x_i^2}$, 约为$1210.9\\text{m}^3$", "solution": "", @@ -386735,7 +387539,10 @@ "id": "014645", "content": "一医疗团队为研究某地的一种地方性疾病与当地居民的卫生习惯(卫生习惯分为良好和不够良好两类) 的关系, 在已患该疾病的病例中随机调查了$100$例 (称为病例组), 同时在未患该疾病的人群中随机调查了$100$人 (称为对照组), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 不够良好 & 良好 \\\\\n\\hline 病例组 & 40 & 60 \\\\\n\\hline 对照组 & 10 & 90 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 判断患该疾病群体与未患该疾病群体的卫生习惯是否有关?\\\\\n(2) 从该地的人群中任选一人, $A$表示事件``选到的人卫生习惯不够良好'', $B$表示事件``选到的人患有该疾病''. $\\dfrac{P(B | A)}{P(\\overline {B} | A)}$与$\\dfrac{P(B | \\overline {A})}{P(\\overline {B} | \\overline {A})}$的比值是卫生习惯不够良好对患该疾病风险程度的一项度量指标, 记该指标为$R$.\\\\\n(i) 证明: $R=\\dfrac{P(A | B)}{P(\\overline {A} | B)} \\cdot \\dfrac{P(\\overline {A} | \\overline {B})}{P(A | \\overline {B})}$;\\\\\n(ii) 利用该调查数据, 给出$P(A | B), P(A | \\overline {B})$的估计值, 并利用 (i) 的结果给出$R$的估计值.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "(1) $\\chi^2=24$, 认为有关; (2) 证明略; (3) $R$的估计值为$6$", "solution": "", @@ -386766,7 +387573,10 @@ "id": "014646", "content": "本市某区对全区高中生的身高(单位: 厘米) 进行统计, 得到如下的频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 120]\n\\draw [->] (0,0) -- (0,0.035) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.002) -- (0.4,0.002) -- (0.6,-0.002) -- (0.8,0) -- (10,0) node [below] {身高/厘米};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {2/0.022/0.022,3/0.027/0.027,4/0.025/0.025,5/0.015/x,6/0.01/0.01,7/0.001/0.001}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};}; \n\\foreach \\i/\\j/\\k in {2/0.022/150,3/0.027/160,4/0.025/170,5/0.015/180,6/0.01/190,7/0.001/200}\n{\\draw (\\i,0) node [below] {$\\k$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);};\n\\draw (8,0) node [below] {$210$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若数据分布均匀, 记随机变量$X$为各区间中点所代表的身高, 写出$X$的分布及期望;\\\\\n(2) 已知本市身高在区间$[180,210]$的市民人数约占全市总人数的$10 \\%$, 且全市高中生约占全市总人数的$1.2 \\%$. 现在要以该区本次统计数据估算全市高中生身高情况, 从本市市民中任取$1$人, 若此人的身高位于区间$[180,210]$, 试估计此人是高中生的概率;\\\\\n(3) 现从身高在区间$[170,190)$的高中生中分层抽样抽取一个$80$人的样本. 若身高在$[170,180)$中样本的均值为$176$厘米, 方差为$10$; 身高在区间$[180,190)$中样本的均值为$184$厘米, 方差为$16$, 试求这$80$人身高的方差.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "(1) $X\\sim \\begin{pmatrix} 155 & 165 & 175 & 185 & 195 & 205 \\\\ 0.22 & 0.27 & 0.25 & 0.15 & 0.1 & 0.01\\end{pmatrix}$, $E[X]=171.7$($\\text{cm}$); (2) $0.0312$; (3) $27.25$", "solution": "", @@ -386797,7 +387607,9 @@ "id": "014647", "content": "已知集合$A=\\{-1,3,2 m-1\\}$, $B=\\{3, m^2\\}$. 若$B \\subseteq A$, 则实数$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386817,7 +387629,9 @@ "id": "014648", "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{3,4,5\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386839,7 +387653,9 @@ "id": "014649", "content": "若全集$U=\\mathbf{R}$, 集合$A=\\{x | x \\geq 1\\} \\cup\\{x | x \\leq 0\\}$, 则$\\overline {A}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386859,7 +387675,9 @@ "id": "014650", "content": "已知$a\\in \\mathbf{R}$, 那么``$a>1$''是``$\\dfrac 1a<1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -386879,7 +387697,9 @@ "id": "014651", "content": "设$a$、$b$是非零实数, 若$a0$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386919,7 +387741,9 @@ "id": "014653", "content": "设$x \\in \\mathbf{R}$, 不等式$|x-3|<1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386939,7 +387763,9 @@ "id": "014654", "content": "若实数$x$、$y$满足$x y=1$, 则$x^2+2 y^2$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386959,7 +387785,9 @@ "id": "014655", "content": "若正实数$x$、$y$满足$x+4 y=1$, 则$x y$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386979,7 +387807,9 @@ "id": "014656", "content": "已知$\\alpha \\in\\{-2,-1,-\\dfrac{1}{2}, \\dfrac{1}{2} 1,2,3\\}$. 若幂函数$y=x^\\alpha$为奇函数, 且在区间$(0,+\\infty)$上为严格减函数, 则$\\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -386999,7 +387829,9 @@ "id": "014657", "content": "已知$f(x)=\\mathrm{e}^{|x-a|}$($a$为常数). 若$y=f(x)$在区间$[1,+\\infty)$上是严格增函数, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387019,7 +387851,9 @@ "id": "014658", "content": "设函数$y=f(x)$是定义在$\\mathbf{R}$上的奇函数, 当$x \\in(0,+\\infty)$时, $f(x)=\\lg x$, 则满足$f(x)>0$的$x$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387039,7 +387873,9 @@ "id": "014659", "content": "已知函数$y=f(x)$的图像是折线段$ABC$, 其中$A(0,0)$、$B(\\dfrac{1}{2}, 5)$、$C(1,0)$. 函数$y=x f(x)$($0 \\leq x \\leq 1$)的图像与$x$轴围成的图形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387059,7 +387895,9 @@ "id": "014660", "content": "设常数$a \\geq 0$, 根据$a$的不同取值, 讨论函数$y=\\dfrac{2^x+a}{2^x-a}$的奇偶性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -387079,7 +387917,9 @@ "id": "014661", "content": "设常数$a \\in \\mathbf{R}$, 根据$a$的不同取值, 讨论函数$y=x^2+\\dfrac{a}{x}$($x \\neq 0$)的奇偶性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -387099,7 +387939,9 @@ "id": "014662", "content": "对于定义在$\\mathbf{R}$上的函数$y=f(x)$, 考察以下性质, $p$: 存在非零实数$a$, 使得$f(x+a)0.\\end{cases}$若$f(0)$是函数$y=f(x)$的最小值, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$[-1,2]$}{$[-1,0]$}{$[1,2]$}{$[0,2]$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -387139,7 +387983,9 @@ "id": "014664", "content": "如图, $A, B, C$三地有直道相通, $AB=5$千米, $AC=3$千米, $BC=4$千米. 现甲、乙两警员同时从$A$地出发匀速前往$B$地, 经过$t$小时, 他们之间的距离为$f(t)$(单位: 千米). 甲的路线是$AB$, 速度为$5$千米/小时, 乙的路线是$ACB$, 速度为$8$千米/小时. 乙到达$B$地后在原地等待. 设$t=t_1$时, 乙到达$C$地. 已知警员的对讲机的有效通话距离是$3$千米. 当$t_1 \\leq t \\leq 1$时, 求$f(t)$的表达式, 并判断$f(t)$在区间$[t_1, 1]$上的最大值是否超过$3$? 说明理由.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (5,0) node [right] {$B$} coordinate (B);\n\\draw (1.8,2.4) node [above] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -387159,7 +388005,9 @@ "id": "014665", "content": "已知$f(x)=\\begin{cases}|\\lg | x-1 \\|, & x \\neq 1, \\\\ 0, & x=1.\\end{cases}$关于$x$的方程$f^2(x)+b f(x)+c=0$有$7$个不同实数解的一个充要条件是\\bracket{20}.\n\\fourch{$b<0$且$c>0$}{$b>0$且$c<0$}{$b<0$且$c=0$}{$b \\geq 0$且$b<0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -387179,7 +388027,10 @@ "id": "014666", "content": "设$a \\in \\mathbf{R}$. 若存在唯一的$m\\in \\mathbf{Z}$使得关于$x$的不等式组$\\dfrac{1}{2} x^2-\\dfrac{1}{2}0, f(x)=2 \\sin (\\omega x)$, 若$y=f(x)$在区间$[-\\dfrac{\\pi}{4}, \\dfrac{2 \\pi}{3}]$上是严格增函数, 则$\\omega$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387400,7 +388272,9 @@ "id": "014677", "content": "已知等差数列$\\{a_n\\}$的公差不为零, $S_n$为其前$n$项和. 若$S_5=0$, 则$S_1, S_2, \\cdots, S_{100}$这$100$个数中所有不同数值的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387420,7 +388294,9 @@ "id": "014678", "content": "已知数列$\\{a_n\\}$前$n$项和为$S_n$, 若$S_n+a_n=2$, 则$S_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387440,7 +388316,9 @@ "id": "014679", "content": "某公司积极投入本市``五个新城''建设, 将总部迁入其中一个新城. 该公司$2021$年第一季度的营业收入为$1.1$亿元, 利润为$0.16$亿元. 预测显示: 在以$2021$年为第一年的未来十年(每年$4$个季度, 共$40$个季度)内, 该公司每一季度的营业收入比上一季度增加$0.05$亿元, 利润比上一季度增长$4 \\%$. 据此预测, 解答以下问题:\\\\\n(1) 求$2021$年至$2025$年, 该公司五年共$20$个季度营业收入共计多少亿元?\\\\\n(2) 该公司在哪年哪季度的利润将首次超过该季度的营业收入的$18 \\%$?", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -387460,7 +388338,9 @@ "id": "014680", "content": "已知$f(x)=x^3-x$, $g(x)=x^2+a$, 若曲线$y=f(x)$在点$(-1, f(-1))$处的切线也是曲线$y=g(x)$的切线, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387480,7 +388360,9 @@ "id": "014681", "content": "若$x=-2$是函数$y=(x^2+a x-1) \\mathrm{e}^{x-1}$的极值点, 则该函数的极小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387500,7 +388382,9 @@ "id": "014682", "content": "已知$k$为常数, 若$(x-k)^2 \\mathrm{e}^{\\frac{x}{k}} \\leq \\dfrac{1}{\\mathrm{e}}$对任意的$x \\in(0,+\\infty)$成立, 求$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -387520,7 +388404,9 @@ "id": "014683", "content": "在正三角形$ABC$中, $D$是$BC$上的点, 若$|AB|=3$, $|BD|=1$, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AD}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387540,7 +388426,9 @@ "id": "014684", "content": "直角坐标系$xOy$中, $\\overrightarrow {i}$、$\\overrightarrow {j}$分别是与$x$、$y$轴正方向同向的单位向量. 在直角$\\triangle ABC$中, 若$\\overrightarrow{AB}=2 \\overrightarrow {i}+\\overrightarrow {j}$, $\\overrightarrow{AC}=3 \\overrightarrow {i}+k \\overrightarrow {j}$, 则$k$的可能值的个数是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -387560,7 +388448,9 @@ "id": "014685", "content": "已知圆心为$O$、半径为$1$的圆上有三点$A$、$B$、$C$. 若$7 \\overrightarrow{OA}+5 \\overrightarrow{OB}+8 \\overrightarrow{OC}=\\overrightarrow{0}$, 则$|\\overrightarrow{BC}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387580,7 +388470,9 @@ "id": "014686", "content": "设$m \\in \\mathbf{R}$, $m^2+m-2+(m^2-1) \\mathrm{i}$是纯虚数, 其中$\\mathrm{i}$是虚数单位, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387600,7 +388492,9 @@ "id": "014687", "content": "若复数$z=1+2 \\mathrm{i}$, 其中$\\mathrm{i}$是虚数单位, 则$(z+\\dfrac{1}{\\overline {z}}) \\cdot \\overline {z}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387620,7 +388514,9 @@ "id": "014688", "content": "已知$a$、$b \\in \\mathbf{R}$, 若$2+a \\mathrm{i}, b+\\mathrm{i}$($\\mathrm{i}$是虚数单位)是实系数一元二次方程$x^2+p x+q=0$的两个根, 那么$p$、$q$的值分别是 \\bracket{20}.\n\\fourch{$p=-4$, $q=5$}{$p=-4$, $q=3$}{$p=4$, $q=5$}{$p=4$, $q=3$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -387640,7 +388536,9 @@ "id": "014689", "content": "在如图所示的棱长为$10$的正方体$ABCD-A_1B_1C_1D_1$中, 一条平行于$A_1C$的直线与正方体的表面交于$P$、$Q$两点, 其中点$P$在侧面$ADD_1A_1$上, 且到$A_1D_1$的距离为$3$, 到$AA_1$的距离为$2$, 则点$Q$所在的面是 \\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$ABCD$}{$ABB_1A_1$}{$BCC_1B_1$}{$CDD_1C_1$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -387660,7 +388558,9 @@ "id": "014690", "content": "《九章算术》中, 称底面为矩形而有一侧棱垂直于底面的四棱锥为阳马. 设$AA_1$是正六棱柱的一条侧棱, 如图, 若阳马以该六棱柱的顶点为顶点, 以$AA_1$为底面矩形的一边, 则这样的阳马的个数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}\n\\coordinate (A) at (0,0,0) node [below] {$A$};\n\\path (A) --++ (45:{sqrt(3)/2}) --++ (1,0) coordinate (B);\n\\path (A) --++ (45:{sqrt(3)}) coordinate (C);\n\\path (C) --++ (-2,0) coordinate (D);\n\\path (B) --++ (-4,0) coordinate (E);\n\\coordinate (F) at (-2,0);\n\\draw (E) -- (F) -- (A) -- (B);\n\\draw [dashed] (B) -- (C) -- (D) -- (E) -- (F);\n\\foreach \\i in {(A),(B),(E),(F)}{\\draw \\i --++ (0,2);};\n\\foreach \\i in {(D),(C)}{\\draw [dashed] \\i --++ (0,2);};\n\\path (A) --++ (0,2) coordinate (A1) node [above] {$A_1$};\n\\path (B) --++ (0,2) coordinate (B1);\n\\path (C) --++ (0,2) coordinate (C1);\n\\path (D) --++ (0,2) coordinate (D1);\n\\path (E) --++ (0,2) coordinate (E1);\n\\path (F) --++ (0,2) coordinate (F1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- (E1) -- (F1) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$4$}{$8$}{$12$}{$16$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -387680,7 +388580,9 @@ "id": "014691", "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$的棱所在的直线中, 与直线$BC_1$异面的直线的条数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{1.5}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw (B) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -387700,7 +388602,9 @@ "id": "014692", "content": "若圆柱的高为$4$, 底面积为$9 \\pi$, 则该圆柱的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387720,7 +388624,9 @@ "id": "014693", "content": "若正三棱柱的所有棱长均为$a$, 且其体积为$16 \\sqrt{3}$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387740,7 +388646,9 @@ "id": "014694", "content": "若一个圆锥的侧面展开图是面积为$2 \\pi$的半圆面, 则该圆锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387760,7 +388668,9 @@ "id": "014695", "content": "若三个球的半径$R_1, R_2, R_3$满足$R_1+2R_2=3R_3$, 则它们的表面积$S_1, S_2, S_3$, 满足的等量关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387780,7 +388690,9 @@ "id": "014696", "content": "在四面体$ABCD$中, 已知$AB \\perp CD, BC \\perp DA$. 求证: $BD \\perp AC$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -387800,7 +388712,9 @@ "id": "014697", "content": "如图, 在正四棱柱$ABCD-A_1B_1C_1D_1$中, 底面$ABCD$的边长为$3$, $BD_1$与底面所成角的大小为$\\arctan \\dfrac{2}{3}$, 则该正四棱柱的高等于\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (B)--(D1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387821,7 +388735,9 @@ "id": "014698", "content": "若$\\overrightarrow {n}=(-2,1)$是直线$l$的一个法向量, 则$l$的倾斜角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387841,7 +388757,9 @@ "id": "014699", "content": "若直线$2 x+m y+1=0$与直线$y=3 x-1$平行, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387861,7 +388779,9 @@ "id": "014700", "content": "已知$P_1(a_1, b_1)$与$P_2(a_2, b_2)$是直线$y=k x+1$($k$为常数) 上两个不同的点, 则关于$x$和$y$的方程组$\\begin{cases}a_1 x+b_1 y=1, \\\\ a_2 x+b_2 y=1\\end{cases}$的解的情况是\\bracket{20} .\n\\twoch{无论$k$、$P_1$、$P_2$如何, 总是无解}{无论$k$、$P_1$、$P_2$如何, 总有唯一解}{存在$k$、$P_1$、$P_2$, 使之恰有两解}{存在$k$、$P_1$、$P_2$, 使之有无穷多解}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -387881,7 +388801,9 @@ "id": "014701", "content": "已知平行直线$l_1: 2 x+y-1=0$, $l_2: 2 x+y+1=0$, 则$l_1$与$l_2$的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387901,7 +388823,9 @@ "id": "014702", "content": "已知椭圆的中心在原点, 一个焦点为$F(-2 \\sqrt{3}, 0)$, 且长轴长是短轴长的$2$倍, 则该椭圆的标准方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387921,7 +388845,9 @@ "id": "014703", "content": "设$m$为常数, 若点$F(0,5)$是双曲线$\\dfrac{y^2}{m}-\\dfrac{x^2}{9}=1$的一个焦点, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387951,7 +388877,9 @@ "id": "014704", "content": "抛物线$y^2=2 p x$($p>0$)上的动点$Q$到焦点的距离的最小值为$1$, 则$p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -387971,7 +388899,9 @@ "id": "014705", "content": "双曲线$x^2-\\dfrac{y^2}{b^2}=1(b>0)$的左、右焦点分别为$F_1$、$F_2$, 直线$l$过$F_2$且与双曲线交于$A$、$B$两点.\\\\\n(1) 若$l$的倾斜角为$\\dfrac{\\pi}{2}$, $\\triangle F_1AB$是等边三角形, 求双曲线的渐近线方程;\\\\\n(2) 设$b=\\sqrt{3}$. 若$l$的斜率存在, 且$(\\overrightarrow{F_1A}+\\overrightarrow{F_1B}) \\cdot \\overrightarrow{AB}=0$, 求$l$的斜率.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -387991,7 +388921,9 @@ "id": "014706", "content": "在报名的$3$名男教师和$6$名女教师中, 选取$5$人参加义务献血, 若要求男、女教师都有, 则不同的选取方式的种数为\\blank{50}.(结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388011,7 +388943,9 @@ "id": "014707", "content": "若排列数$\\mathrm{P}_6^m=6 \\times 5 \\times 4$, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388033,7 +388967,9 @@ "id": "014708", "content": "组合数$\\mathrm{C}_n^r$($n>r \\geq 1$, $n$、$r \\in \\mathbf{Z})$恒等于\\bracket{20}.\n\\fourch{$\\dfrac{r+1}{n+1} \\mathrm{C}_{n-1}^{r-1}$}{$(n+1)(r+1) \\mathrm{C}_{n-1}^{r-1}$}{$n r \\mathrm{C}_{n-1}^{r-1}$}{$\\dfrac{n}{r} \\mathrm{C}_{n-1}^{r-1}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -388053,7 +388989,9 @@ "id": "014709", "content": "若$(1-2 x)^4=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4$, 则$a_0+a_4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388073,7 +389011,9 @@ "id": "014710", "content": "从一副混合后的扑克牌($52$张)中随机抽取$1$张, 设$A$表示事件``抽得红桃K'', $B$表示事件``抽得为黑桃'', 则$P(A \\cup B)=$\\blank{50}. (结果用最简分数表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388093,7 +389033,9 @@ "id": "014711", "content": "已知事件$A$, 其对立事件记为$\\overline {A}$, 若$P(A)=0.5$, 则$P(\\overline {A})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388113,7 +389055,9 @@ "id": "014712", "content": "若事件$E$与$F$相互独立, 且$P(E)=P(F)=\\dfrac{1}{4}$, 则$P(E \\cap F)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388133,7 +389077,9 @@ "id": "014713", "content": "从$1,2,3,4,5$中任取$2$个不同的数, 事件$A$表示``取到的$2$个数之和为偶数'', 事件$B$表示``取到的$2$个数均为偶数'', 则$P(B | A)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388153,7 +389099,9 @@ "id": "014714", "content": "设$A$、$B$为两个事件, 若$P(B)=0.4$, $P(A)=0.5$, $P(B | A)=0.3$, 则$P(B | \\overline {A})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388173,7 +389121,9 @@ "id": "014715", "content": "一批产品的二等品率为$0.02$, 从这批产品中每次随机取一件, 有放回地抽取$100$次, 若用$X$表示抽到的二等品件数, 则$D[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388193,7 +389143,9 @@ "id": "014716", "content": "袋中有大小与质地均相同的$4$个红球、$m$个黄球、$n$个绿球, 从该袋内随机取出两个球, 记取出的红球数为$X$. 若取出的两个球都是红球的概率为$\\dfrac{1}{6}$, 一个红球和一个黄球的概率为$\\dfrac{1}{3}$, 则$E[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388213,7 +389165,9 @@ "id": "014717", "content": "若随机变量$X$服从正态分布$N(2, \\sigma^2)$, 且$P(X<4)=0.8$, 则$P(0|PF_2|$, 求$\\dfrac{|PF_1|}{|PF_2|}$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388460,7 +389436,9 @@ "id": "014729", "content": "从集合$\\{a, b, c\\}$的所有子集中依次随机取出$2$个不同的子集, 记为$M$和$N$, 求$M \\subseteq N$的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388480,7 +389458,9 @@ "id": "014730", "content": "从集合$\\{a, b, c\\}$的所有子集中随机取出$3$个不同的集合, 求满足``任取其中两个集合$M$和$N$, 总成立$M \\subseteq N$或$N \\subseteq M$''的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388500,7 +389480,9 @@ "id": "014731", "content": "设$a \\in \\mathbf{R}$, 求函数$y=x^2+|x-a|+1$, $x \\in \\mathbf{R}$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388520,7 +389502,9 @@ "id": "014732", "content": "已知圆$O: x^2+y^2=1$, 点$A(1,0)$绕圆心$O$旋转$\\dfrac{\\pi}{6}$至点$B$, 点$C(0,1)$, 则$|BC|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388540,7 +389524,9 @@ "id": "014733", "content": "设$a \\in \\mathbf{R}$, 求函数$y=x^3-a x$的单调增区间.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "当$a>0$时, 单调增区间为$(-\\infty,-\\dfrac{\\sqrt{3a}}3]$和$[\\dfrac{\\sqrt{3a}}3,+\\infty)$; 当$a\\le 0$时, 单调增区间为$(-\\infty,+\\infty)$", "solution": "", @@ -388568,7 +389554,9 @@ "id": "014734", "content": "已知数列$\\{a_n\\}$, $a_n=2^n+1$, 试判断$a_n$是否可能为$7$的倍数, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388588,7 +389576,9 @@ "id": "014735", "content": "已知三个两两不重合的平面$\\alpha$、$\\beta$和$\\gamma$, 三者之间的交线共有\\blank{50}条.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -388608,7 +389598,10 @@ "id": "014736", "content": "已知$a \\in \\mathbf{R}$, 记函数$y=\\sqrt{2-\\dfrac{x+3}{x+1}}$的定义域为$A$, 函数$y=\\lg [(x-a-1)(2 a-x)]$的定义域为$B$. 若$B \\subseteq A$, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "填空题", "ans": "$(-\\infty,-2]\\cup [\\dfrac 12,1)\\cup (1,+\\infty)$", "solution": "", @@ -388635,7 +389628,9 @@ "id": "014737", "content": "已知$a$、$b \\in \\mathbf{R}$, 讨论函数$y=a \\sin x+b$的奇偶性.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388655,7 +389650,9 @@ "id": "014738", "content": "设$a \\in \\mathbf{R}$, 方程$\\dfrac{1}{\\sqrt{\\dfrac{1}{x}+a}}=\\dfrac{1}{\\sqrt{(a-3) x+2 a-4}}$的解集中恰有一个元素, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388675,7 +389672,9 @@ "id": "014739", "content": "已知数列$\\{a_n\\}$是等比数列, $a_1=1, a_2=a$, 数列$\\{b_n\\}$满足$b_n=a_n \\cdot a_{n+1}$, 求数列$\\{b_n\\}$的前$n$项和$S_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388695,7 +389694,10 @@ "id": "014740", "content": "已知$a \\in \\mathbf{R}$, 设$x_1$、$x_2$是方程$2 x^2+3 a x+a^2-a=0$的两根, 试将$|x_1|+|x_2|$表示为$a$的函数.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388715,7 +389717,9 @@ "id": "014741", "content": "若四面体$ABCD$各棱的长为$1$或$2$, 且该四面体不是正四面体, 求四面体体积$V$的所有可能的值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -388735,7 +389739,9 @@ "id": "014742", "content": "已知命题$A$: 若$3 \\leq ab^a$, 请据此证明: 若$a$、$b$是正整数, $a0$且$x+2 y=1$, 则$\\dfrac{1}{x}+\\dfrac{2}{y}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$9$", "solution": "", @@ -389019,7 +390041,9 @@ "id": "014751", "content": "如图所示的茎叶图记录了甲、乙两组各$5$名工人某日的产量数据. 若这两组数据的中位数相等, 且平均值也相等, 则$x+y=$\\blank{50}.\n\\begin{center}\n\\begin{tabular}{cc|c|ccc}\n\\multicolumn{2}{c|}{甲组} & & \\multicolumn{3}{c}{乙组}\\\\ \\hline\n& 6 & 5 & 9 \\\\ \n2 & 5 & 6 & 1 & 7 & $y$ \\\\\n$x$ & 4 & 7 & 8\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$8$", "solution": "", @@ -389052,7 +390076,9 @@ "id": "014752", "content": "对于两个均不等于$1$的正数$m$、$n$, 定义: $m * n=\\begin{cases}\\log _m n,& m \\geq n, \\\\ \\log _n m, & m=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A) arc (120:180:2) node [below left] {$B$} coordinate (B) arc (240:300:2) node [below right] {$C$} coordinate (C) arc (0:60:2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$10-4\\sqrt{7}$", "solution": "", @@ -389151,7 +390181,9 @@ "id": "014755", "content": "在下列条件下, 能确定一个平面的是\\bracket{20}.\n\\twoch{空间的任意三点}{空间的任意一条直线和任意一点}{空间的任意两条直线}{梯形的两条腰所在的直线}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -389184,7 +390216,9 @@ "id": "014756", "content": "已知集合$A=\\{x|| x-1 |>2\\}$,$B=\\{x | x^2+p x+q \\leq 0\\}$, 若$A \\cup B=\\mathbf{R}$, 且$A \\cap B=[-2,-1)$, 则$p$、$q$的值分别为\\bracket{20}.\n\\fourch{$-1$、$-6$}{$1$、$-6$}{$3$、$2$}{$-3$、$2$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -389217,7 +390251,9 @@ "id": "014757", "content": "已知函数$f(x)=3^x-(\\dfrac{1}{3})^x+2$, 若$f(a^2)+f(a-2)>4$, 则实数$a$的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty, 1)$}{$(-\\infty,-2) \\cup(1,+\\infty)$}{$(-2,1)$}{$(-1,2)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -389250,7 +390286,9 @@ "id": "014758", "content": "数学家们在探寻自然对数底$\\mathrm{e} \\approx 2.71828$与圆周率$\\pi$之间的联系时, 发现了以下公式($\\mathrm{i}$为虚数单位):\\\\\n(I) $\\mathrm{e}^x=1+\\dfrac{x}{1 !}+\\dfrac{x^2}{2 !}+\\dfrac{x^3}{3 !}+\\cdots+\\dfrac{x^n}{n !}+\\cdots$;\\\\\n(II) $\\sin x=\\dfrac{x}{1 !}-\\dfrac{x^3}{3 !}+\\dfrac{x^5}{5 !}-\\dfrac{x^7}{7 !}+\\cdots+(-1)^{n-1} \\dfrac{x^{2 n-1}}{(2 n-1) !}+\\cdots$;\\\\\n(III) $\\cos x=1-\\dfrac{x^2}{2 !}+\\dfrac{x^4}{4 !}-\\dfrac{x^6}{6 !}+\\cdots+(-1)^{n-1} \\dfrac{x^{2 n-2}}{(2 n-2) !}+\\cdots$.\\\\\n上述公式中, $x \\in \\mathbf{C}$, $n \\in \\mathbf{N}$, $n\\ge 1$, 据此判断, 当$x\\in \\mathbf{C}$时, 以下命题\n\\textcircled{1} $\\mathrm{e}^{\\mathrm{i}x}=\\cos x+\\mathrm{i} \\sin x$; \n\\textcircled{2} $\\mathrm{e}^{\\mathrm{i}x}=\\sin x+\\mathrm{i} \\cos x$; \n\\textcircled{3} $\\mathrm{e}^{\\mathrm{i} \\pi}+1=0$; \n\\textcircled{4} $\\mathrm{e}^{\\mathrm{i} \\pi}+\\mathrm{i}=0$; \n\\textcircled{5} $|\\mathrm{e}^{\\mathrm{i}x}+\\mathrm{e}^{-\\mathrm{i}x}| \\leq 2$中, 正确的个数是\\bracket{20}.\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -389284,7 +390322,9 @@ "id": "014759", "content": "锐角$\\triangle ABC$中, 角$A$、$B$、$C$的对边分别为$a$、$b$、$c$. 已知$\\sin ^2B+\\sin ^2C=\\sin ^2A+\\sin B \\sin C$.\\\\\n(1) 求$A$;\\\\\n(2) 若$a=3$, 求$b+c$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac\\pi 3$; (2) 最大值为$6$", "solution": "", @@ -389317,7 +390357,9 @@ "id": "014760", "content": "如图, 在多面体$EFG-ABCD$中, 四边形$ABCD$、$CFGD$、$ADGE$均是边长为$1$的正方形, 点$H$在棱$EF$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$E$} coordinate (E);\n\\draw (D) ++ (0,2,0) node [above] {$G$} coordinate (G);\n\\draw (C) ++ (0,2,0) node [right] {$F$} coordinate (F);\n\\draw ($(E)!0.6!(F)$) node [below left] {$H$} coordinate (H);\n\\draw (A)--(B)--(C)--(F)--(G)--(E)--cycle;\n\\draw (B)--(F)--(E)--cycle(B)--(H)--(G);\n\\draw [dashed] (G)--(D)--(C)(D)--(B)(D)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求该几何体的体积;\\\\\n(2) 证明: 存在点$H$, 使得$DH \\perp BF$;\\\\\n(3) 求$BD$与平面$BEF$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 56$; (2) 证明略($H$和点$E$重合); (3) $\\arcsin\\dfrac{\\sqrt{6}}3$", "solution": "", @@ -389350,7 +390392,9 @@ "id": "014761", "content": "高斯是德国著名的数学家, 近代数学奠基者之一, 享有``数学王子''的称号. 以他的名字定义的函数称为高斯函数$f(x)=[x]$, 其中$[x]$表示不超过$x$的最大整数. 已知数列$\\{a_n\\}$满足$a_1=2$, $a_2=6$, $a_{n+2}+5 a_n=6 a_{n+1}$, 若$b_n=[\\log _5 a_{n+1}]$, $S_n$为数列$\\{\\dfrac{1000}{b_n b_{n+1}}\\}$的前$n$项和.\\\\\n(1) 证明: 数列$\\{a_{n+1}-a_n\\}$是等比数列, 并求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求$[S_{2023}]$的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) 证明略; $a_n=5^{n-1}+1$; (2) $999$", "solution": "", @@ -389383,7 +390427,9 @@ "id": "014762", "content": "已知椭圆$E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 依次连接椭圆$E$的四个顶点构成的四边形面积为$4 \\sqrt{3}$.\\\\\n(1) 若$a=2$, 求椭圆$E$的标准方程;\\\\\n(2) 以椭圆$E$的右顶点为焦点、以原点为顶点的抛物线$G$, 若$G$上动点$M$到点$H(10,0)$的最短距离为$4 \\sqrt{6}$, 求$a$的值;\\\\\n(3) 当$a=2$时, 设点$F$为椭圆$E$的右焦点, $A(-2,0)$, 直线$l$交$E$于$P$、$Q$(均不与点$A$重合) 两点, 直线$l$、$AP$、$AQ$的斜率分别为$k$、$k_1$、$k_2$, 若$k k_1+k k_2+3=0$, 求$\\triangle FPQ$的周长.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}4+\\dfrac{y^2}3=1$; (2) $4$; (3) $8$", "solution": "", @@ -389416,7 +390462,9 @@ "id": "014763", "content": "已知函数$f(x)=a x^3-b x^2+c$, 其中实数$a>0$, $b \\in \\mathbf{R}$, $c \\in \\mathbf{R}$.\\\\\n(1) $b=3 a$时, 求函数$y=f(x)$的极值点;\\\\\n(2) $a=1$时, $x^2 \\ln x \\geq f(x)-2 x-c$在$[3,4]$上恒成立, 求$b$的取值范围;\\\\\n(3) 证明: 当$b=3 a$, 且$5 a=latex]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.5:4.5, samples = 100] plot (\\x,{-\\x*(\\x-2)*(\\x-4)/6});\n\\foreach \\i in {1,2,3,4}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\draw (1,1) node {$y=f'(x)$};\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 函数$y=f(x)$在区间$(-\\infty, 1)$上严格减;\\\\\n\\textcircled{2} 函数$y=f(x)$在区间$(2,4)$上严格增;\\\\\n\\textcircled{3} 函数$y=f(x)$在$x=0$处取得极小值;\\\\\n\\textcircled{4} 函数$y=f(x)$在$x=4$处取得极大值.\\\\", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389529,7 +390585,9 @@ "id": "014768", "content": "已知函数$y=f(x)$, 其导函数$y=f'(x)$的图像如图所示. 以下四个选项中, 可能表示函数$y=f(x)$的图像是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1, samples = 100] plot (\\x,{1.5*(1-\\x*\\x)});\n\\foreach \\i in {-1,1}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\draw (1,2) node {$y=f'(x)$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1, samples = 100] plot (\\x,{pow(\\x,3)});\n\\foreach \\i in {-1,1}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\end{tikzpicture}\n}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1, samples = 100] plot (\\x,{1.5*\\x-0.5*pow(\\x,3)});\n\\foreach \\i in {-1,1}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1, samples = 100] plot (\\x,{0.5*\\x*\\x+\\x-0.5});\n\\foreach \\i in {-1,1}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1, samples = 100] plot (\\x,{-0.5*\\x*\\x+\\x+0.5});\n\\foreach \\i in {-1,1}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -389549,7 +390607,10 @@ "id": "014769", "content": "现有一张半径为$2$米的圆形铁皮, 从中裁剪出一块扇形铁皮 (如图$1$), 并卷成一个深度为$h$米的圆锥筒 (如图$2$) 的容器. 当圆锥筒容器的深度$h$为多少米时, 其容积最大? 容积的最大值为多少立方米? (铁皮厚度忽略不计)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle (2);\n\\draw (0,0) --++ (30:2) (0,0) --++ (150:2);\n\\draw (0,-2) node [below] {图$1$};\n\\draw (4,-2) --++ (1.5,3) node [midway, below right] {$2$} (4,-2) --++ (-1.5,3);\n\\draw (4,1) ellipse (1.5 and 0.5);\n\\draw (4,-2) -- (4,1) node [midway, left] {$h$} --++ (1.5,0) node [midway, above] {$r$};\n\\draw (4,-2) node [below] {图$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -389569,7 +390630,9 @@ "id": "014770", "content": "已知$f(x)=2 x^4-x$.\\\\\n(1) 求曲线$y=f(x)$在点$A(1,1)$处的切线$l$的方程;\\\\\n(2) 证明: 曲线$y=f(x)$上除切点$A(1,1)$外, 其余点都在(1)中切线$l$的上方.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -389589,7 +390652,9 @@ "id": "014771", "content": "已知$a \\in \\mathbf{R}$, $f(x)=a x-(2 a+1) \\ln x-\\dfrac{2}{x}$.\\\\\n(1) 若$x=1$是函数$y=f(x)$的驻点, 求$a$的值;\\\\\n(2) 当$a>0$时, 求函数$y=f(x)$的单调区间.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -389609,7 +390674,9 @@ "id": "014772", "content": "设曲线$y=\\ln x+2 x$的斜率为$3$的切线为$l$, 则$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389629,7 +390696,9 @@ "id": "014773", "content": "已知$f(x)=\\dfrac{x}{x^2+1}$, 则函数$y=f(x)$的导数$f'(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389649,7 +390718,9 @@ "id": "014774", "content": "已知定义在区间$(-3,3)$上的奇函数$y=f(x)$的导函数是$y=f'(x)$, 当$x \\geq 0$时, $y=f(x)$的图像如图所示, 则关于$x$的不等式$\\dfrac{f'(x)}{x}>0$的解集为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3] plot (\\x,{\\x*(2-\\x)});\n\\draw [dashed] (1,1) -- (1,0) node [below] {$1$};\n\\draw [dashed] (3,-3) -- (3,0) node [above] {$3$};\n\\draw (2,0) node [below left] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389669,7 +390740,9 @@ "id": "014775", "content": "已知$f(x)=\\dfrac{1}{2} \\sin 2 x+\\cos x$, 求函数$y=f(x), x \\in[0,2 \\pi]$的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -389689,7 +390762,9 @@ "id": "014776", "content": "已知$f(x)=x^2+x$, 则曲线$y=f(x)$在点$(0, f(0))$处的切线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389709,7 +390784,9 @@ "id": "014777", "content": "函数$y=\\dfrac{\\ln x}{x}$的驻点为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389729,7 +390806,9 @@ "id": "014778", "content": "已知函数$y=f(x)$, 其导函数$y=f'(x)$的图像如图所示, 则下列所有真命题的序号为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-2,-1,1,2,3,4}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\draw (-2,-2) .. controls ++ (0.2,0.6) and ++ (-0.1,-0.25) .. (-1,0) .. controls ++ (0.1,0.25) and ++ (-0.6,0) .. (1,2) .. controls ++ (0.6,0) and ++ (-0.1,0.4) .. (2,0) .. controls ++ (0.3,-1.2) and ++ (-0.5,0) .. (4,-2);\n\\draw [dashed] (-2,-2) -- (-2,0) (1,2) -- (1,0) (4,-2) -- (4,0);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $y=f(x)$在区间$[-2,-1]$上严格增;\\\\\n\\textcircled{2} $x=-1$是$y=f(x)$的极小值点;\\\\\n\\textcircled{3} $y=f(x)$在区间$[-1,2]$上严格增, 在区间$[2,4]$上严格减;\\\\\n\\textcircled{4} $x=2$是$y=f(x)$的极小值点.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389749,7 +390828,10 @@ "id": "014779", "content": "已知$0<\\varphi<\\pi$, $f(x)=x \\cos (x+\\varphi)-\\cos x$, 函数$y=f(x)$, $x \\in(-\\pi, \\pi)$是偶函数, 则$y=f(x)$的单调减区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389769,7 +390851,9 @@ "id": "014780", "content": "用总长$148 \\text{cm}$的钢条制作一个长方体容器的框架, 若容器底面的长比宽多$5 \\text{cm}$, 要使它的容积最大, 则容器底面的宽为\\blank{50}$\\text{cm}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389789,7 +390873,9 @@ "id": "014781", "content": "已知函数$y=f(x)$与它的导函数$y=f'(x)$的定义域均为$\\mathbf{R}$, 现有下述两个命题:\\\\\n\\textcircled{1} ``$y=f(x)$为奇函数''是``$y=f'(x)$为偶函数''的充分非必要条件;\\\\\n\\textcircled{2} ``$y=f(x)$为严格增函数''是``$y=f'(x)$为严格增函数''的必要非充分条件. 则说法正确的选项是\\bracket{20}.\n\\twoch{命题\\textcircled{1}和\\textcircled{2}均为真命题}{命题\\textcircled{1}为真命题, 命题\\textcircled{2}为假命题}{命题\\textcircled{1}为假命题, 命题\\textcircled{2}为真命题}{命题\\textcircled{1}和\\textcircled{2}均为假命题}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -389809,7 +390895,10 @@ "id": "014782", "content": "已知$a \\in \\mathbf{R}$, $f(x)=\\sin ^2 x-a \\sin x$. 若函数$y=f(x)-f(\\dfrac{\\pi}{2}-x)$在区间$[0, \\dfrac{\\pi}{2}]$上是严格增函数, 则$a$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -389829,7 +390918,9 @@ "id": "014783", "content": "设$f(x)=a x^3-(a+1) x^2+x$, $g(x)=k x+m$, 其中$a \\geq 0$, $k$、$m \\in \\mathbf{R}$. 若对任意$x \\in[0,1]$, 均有$f(x) \\leq g(x)$, 则称函数$y=g(x)$是函数$y=f(x)$的``控制函数'', 且记所有的``控制函数''$y=g(x)$在$x=x_0$($0 \\leq x_0 \\leq 1$)处的最小值为$\\overline {f}(x_0)$.\\\\\n(1) 若$a=2, g(x)=x$, 问: 函数$y=g(x)$是否为函数$y=f(x)$的``控制函数''? 请说明理由;\\\\\n(2) 若$a=0$, 且直线$y=h(x)$是曲线$y=f(x)$在点$(\\dfrac{1}{4}, f(\\dfrac{1}{4}))$处的切线, 证明: 函数$y=h(x)$是函数$y=f(x)$的``控制函数'', 并求$\\overline {f}(\\dfrac{1}{4})$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -391786,7 +392877,9 @@ "id": "014847", "content": "已知$n$是正整数, 设抛物线: $y=n(n+1) x^2-(2 n+1) x+1$的图像在$x$轴上截得的线段的长度为$a_n$, 求$\\displaystyle\\lim_{n\\to\\infty}(a_1+a_2+\\cdots+a_n)$的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -391806,7 +392899,10 @@ "id": "014848", "content": "设函数$f(x)=x(\\dfrac{1}{2})^x+\\dfrac{1}{x+1}$, $O$为坐标原点, $A_n$为函数$y=f(x)$图象上横坐标$n$($n \\in \\mathbf{N}$, $n\\ge 1$)的点, 向量$\\overrightarrow{OA_n}$与向量$\\overrightarrow {i}=(1,0)$的夹角为$\\theta_n$, 求满足$\\tan \\theta_1+\\tan \\theta_2+\\cdots+\\tan \\theta_n<\\dfrac{5}{3}$的最大整数$n$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -391826,7 +392922,10 @@ "id": "014849", "content": "设$g(k)$是关于$x$不等式$\\log _2 x+\\log _2(3 \\sqrt{2^{2 k+2}}-x) \\geq 2 k+3$($k \\in \\mathbf{N}$, $k\\ge 1$)的整数解的个数, 求$g(k)$.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -391846,7 +392945,10 @@ "id": "014850", "content": "设二次函数$f(x)=(k-4) x^2+k x$, 且对任意实数$x$, 都有$f(x) \\leq 6 x+2$恒成立, 数列$\\{a_n\\}$满足$a_{n+1}=f(a_n)$.\\\\\n(1) 求函数$f(x)$的解析式和值域;\\\\\n(2) 试写出一个区间$(a, b)$, 使得当$a_1 \\in(a, b)$时, 数列$\\{a_n\\}$是递增数列, 并说明理由;\\\\\n(3) 已知$a_1=\\dfrac{1}{3}$, 是否存在非零整数$\\lambda$, 使得对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 都有$\\log _3(\\dfrac{1}{\\frac{1}{2}-a_1})+\\log _3(\\dfrac{1}{\\frac{1}{2}-a_2})+\\cdots+\\log _3(\\dfrac{1}{\\frac{1}{2}-a_n})>-1+(-1)^{n-1}\\cdot 2 \\lambda+n \\log _32$恒成立, 若存在, 求之; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -391866,7 +392968,10 @@ "id": "014851", "content": "已知函数$f(x)=\\dfrac{4 x-2}{x+1}$($x \\neq-1$, $x \\in \\mathbf{R}$), 数列$\\{a_n\\}$满足$a_1=a$($a \\neq-1$, $a \\in \\mathbf{R}$), $a_{n+1}=f(a_n)$($n\\in \\mathbf{N}$, $n\\ge 1$). 若数列$\\{a_n\\}$是常数列, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -391886,7 +392991,9 @@ "id": "014852", "content": "函数$f(x)$是定义在$[0,1]$上的增函数, 满足$f(x)=2 f(\\dfrac{x}{2})$且$f(1)=1$, 在每个区间$(\\dfrac{1}{2^i}, \\dfrac{1}{2^{i-1}}]$($i=1,2, \\cdots$)上, $y=f(x)$的图像都是斜率为同一常数$k$的直线的一部分. 求$f(0)$及$f(\\dfrac{1}{2})$, $f(\\dfrac{1}{4})$的值, 并归纳出$f(\\dfrac{1}{2^i})$($i=1,2, \\cdots$)的表达式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -391906,7 +393013,9 @@ "id": "014853", "content": "数列$\\{a_n\\}$的通项公式为$a_n=n+\\dfrac{c}{n}$(其中$c$为实常数), 若数列$\\{a_n\\}$是递增数列, 求$c$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -391926,7 +393035,9 @@ "id": "014854", "content": "函数$f(x)=x^2+m$, 其中$m$为实常数, 定义数列$\\{a_n\\}$如下: $a_1=0$, $a_{n+1}=f(a_n)$, $n \\in \\mathbf{N}$, $n\\ge 1$.\\\\\n(1) 当$m=1$时, 求$a_2, a_3, a_4$的值;\\\\\n(2) 是否存在实数$m$, 使$a_2, a_3, a_4$成等比数列? 若存在, 请求出实数$m$的值, 并求出等比数列的公比; 若不存在, 请说明理由;\\\\\n(3) 设$m=-1$, $f^{-1}(x)$为$f(x)$在$x \\in[0,+\\infty)$的反函数, 数列$\\{b_n\\}$满足: $b_1=1$, $b_{n+1}=f^{-1}(b_n^2)$($n \\in \\mathbf{N}$, $n\\ge 1$), 记$S_n=b_1^2+b_2^2+\\cdots+b_n^2$, 求使$S_n>2020$成立的最小正整数$n$的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -391946,7 +393057,9 @@ "id": "014855", "content": "设数列$a_n=-n^2+10 n+11$($n \\in \\mathbf{N}$, $n\\ge 1$)的前$n$项和为$S_n$, 则当$S_n$取得最大值时, $n$的值为\\bracket{20}.\n\\fourch{$10$}{$11$}{$10$或$11$}{$12$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -391966,7 +393079,9 @@ "id": "014856", "content": "根据市场调查结果, 预测某种家用商品从年初开始的$n$个月内累积的需求量$S_n$(万件) 近似地满足关系式$S_n=\\dfrac{n}{90}(21 n-n^2-5)$($n=1,2, \\cdots, 12$), 按此预测, 在本年度内, 需求量超过$1.5$万件的月份是\\bracket{20}.\n\\fourch{$5$、$6$月}{$6$、$7$月}{$7$、$8$月}{$8$、$9$月}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -391986,7 +393101,9 @@ "id": "014857", "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_4 \\geq 10$, $S_5 \\leq 15$, 则$a_4$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392006,7 +393123,9 @@ "id": "014858", "content": "设$\\{a_n\\}$是公比为$q$的等比数列, 且满足条件$a_1>1$, $a_{2006} a_{2007}-1>0$, $\\dfrac{a_{2006}-1}{a_{2007}-1}<0$. 设$T_n=a_1 a_2 a_3 \\cdots a_n$, 则使$T_n<1$成立的最小正整数$n$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392026,7 +393145,9 @@ "id": "014859", "content": "数列$\\{a_n\\}$中, $a_1=8$, $a_4=2$, 且满足$a_{n+2}=2 a_{n+1}-a_n$($n \\in \\mathbf{N}$, $n\\ge 1$).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$b_n=\\dfrac{1}{n(12-a_n)}$($n \\in \\mathbf{N}$, $n\\ge 1$), 其前$n$项和为$S_n$, 问是否存在最大的整数$m$, 使得对任意正整数$n$, 恒有$S_n>\\dfrac{m}{32}$成立? 若存在, 求$m$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392046,7 +393167,9 @@ "id": "014860", "content": "对于任意$n \\in \\mathbf{N}$且$n>1$, 求证: $(1+\\dfrac{1}{3})(1+\\dfrac{1}{5}) \\cdots(1+\\dfrac{1}{2 n-1})>\\dfrac{\\sqrt{2 n+1}}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392066,7 +393189,9 @@ "id": "014861", "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 已知$S_5S_8$. 则下列结论中错误的是\\bracket{20}.\n\\twoch{$d<0$}{$a_7=0$}{$S_9>S_5$}{$S_6$, $S_7$均为$S_n$的最大值}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -392086,7 +393211,9 @@ "id": "014862", "content": "某纯净水制造厂在净化水过程中, 每增加一次过滤可减少水中杂质$20 \\%$, 要使水中杂质减少到原来的$5 \\%$以下, 则至少需过滤的次数为\\bracket{20}.\n\\fourch{$11$}{$12$}{$13$}{$14$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -392106,7 +393233,9 @@ "id": "014863", "content": "已知等比数列$\\{a_n\\}$中$a_2=1$, 则其前$3$项的和$S_3$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392126,7 +393255,9 @@ "id": "014864", "content": "已知$a_n=9 n-8$($n \\in \\mathbf{N}$, $n\\ge 1$), 且对任意$m \\in \\mathbf{N}$, $m\\ge 1$, 数列$\\{a_n\\}$中落入区间$(9^m, 9^{2 m})$内的项的个数为$b_m$, 则数列$\\{b_m\\}$的前$m$项和$S_m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392146,7 +393277,9 @@ "id": "014865", "content": "已知等比数列$\\{a_n\\}$满足$a_n>a_{n+1}$, 且$a_3+a_6=18$, $a_4 \\cdot a_5=32$. 求数列$\\{a_n\\}$中所有小于$1$的项的各项和$S$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392166,7 +393299,9 @@ "id": "014866", "content": "设数列$\\{a_n\\}$的各项都是正数, 其前$n$项和为$S_n$, 且$a_n^2=2S_n-a_n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$b_n=3^n+(-1)^{n-1} \\lambda \\cdot 2^{a_n}$($\\lambda$为非零整数, $n$为正整数), 试确定$\\lambda$的值, 使得对任意正整数$n$, 恒有$b_{n+1}>b_n$成立.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392186,7 +393321,9 @@ "id": "014867", "content": "已知$F_1, F_2$分别是椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{12}=1$的左、右焦点, 点$P$是椭圆上的任意一点, 则$\\dfrac{|PF_1|-|PF_2|}{|PF_1|}$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392206,7 +393343,9 @@ "id": "014868", "content": "若当$P(m, n)$为圆$x^2+(y-1)^2=1$上任意一点时, 不等式$m+n+c \\geq 0$恒成立, 则$c$的取值范围是\\bracket{20}.\n\\fourch{$-1-\\sqrt{2} \\leq c \\leq \\sqrt{2}-1$}{$\\sqrt{2}-1 \\leq c \\leq \\sqrt{2}+1$}{$c \\leq-\\sqrt{2}-1$}{$c \\geq \\sqrt{2}-1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -392226,7 +393365,9 @@ "id": "014869", "content": "在平面直角坐标系$x O y$中, $A(-1,0)$, $B(1,0)$, $C(0,1)$, 经过原点的直线$l$将$\\triangle ABC$分成左、右两部分, 记左、右两部分的面积分别为$S_1$、$S_2$, 求$\\dfrac{(1+S_1)^2}{1-S_2^2}$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392246,7 +393387,9 @@ "id": "014870", "content": "已知曲线$C: \\dfrac{x^2}{3}+\\dfrac{y^2}{4}=1$. 设曲线$C$与$y$轴交于$D, E$两点, 点$Q(0, m)$在线段$DE$上, 点$P$在曲线$C$上运动. 若当点$P$的坐标为$(0,2)$时, $|\\overrightarrow{QP}|$取得最小值, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392266,7 +393409,9 @@ "id": "014871", "content": "已知抛物线$y^2=4 x$的焦点为$F$, 过$F$作互相垂直的两条直线$l_1, l_2$, $l_1$与抛物线交于$A$、$B$两点, $l_2$与抛物线交于$C$、$D$两点, $M$、$N$分别是线段$AB$、$CD$的中点, 求$\\triangle FMN$面积的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392286,7 +393431,9 @@ "id": "014872", "content": "已知平面上的曲线$C$及点$P$, 在$C$上任取一点$Q$, 线段$PQ$长度的最小值称为点$P$到曲线$C$的距离, 记作$d(P, C)$. 则点$P(0,3)$到曲线$C: x^2-y^2=1$的距离$d(P, C)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392306,7 +393453,9 @@ "id": "014873", "content": "设$F_1$是椭圆$\\dfrac{x^2}{4}+y^2=1$的左焦点, $O$为坐标原点, 点$P$在椭圆上, 则$\\overrightarrow{PF_1} \\cdot \\overrightarrow{PO}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392326,7 +393475,9 @@ "id": "014874", "content": "平面直角坐标系$xOy$中, 过椭圆$M: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右焦点$F$作直线$x+y-\\sqrt{3}=0$交$M$于$A, B$两点, $P$为$AB$的中点, 且$OP$的斜率为$\\dfrac{1}{2}$.\\\\\n(1) 求$M$的方程;\\\\\n(2)$C, D$为$M$上的两点, 若四边形$ABCD$的对角线$CD \\perp AB$, 求四边形$ABCD$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392346,7 +393497,9 @@ "id": "014875", "content": "已知椭圆$C: \\dfrac{x^2}{3}+y^2=1$. $O$为原点, 直线$l: y=k x+t$($k \\neq 0$)与椭圆$C$交于$A, B$两点, 若存在点$P(0,-\\dfrac{1}{2})$, 使得$\\overrightarrow{BA} \\cdot \\overrightarrow{PA}=\\overrightarrow{AB} \\cdot \\overrightarrow{PB}$.\\\\\n(1) 求证: $3 k^2+1=4 t$;\\\\\n(2) 求$\\triangle AOB$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392366,7 +393519,9 @@ "id": "014876", "content": "已知函数$f(x)=a+(1-a)(\\cos x+\\sin x)$, 当$x \\in[0, \\dfrac{\\pi}{2}]$时, $-2 \\leq f(x) \\leq 2$恒成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392386,7 +393541,9 @@ "id": "014877", "content": "给定常数$c>0$, 定义函数$f(x)=2|x+c+4|-|x+c|$, 数列$a_1, a_2, a_3, \\cdots$满足$a_{n+1}=f(a_n)$, $n$是正整数.\\\\\n(1) 若$a_1=-c-2$, 求$a_2$及$a_3$;\\\\\n(2) 求证: 对任意正整数$n$, $a_{n+1}-a_n \\geq c$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392406,7 +393563,9 @@ "id": "014878", "content": "给定常数$a$, 已知函数$f(x)=\\sqrt{x}(x-a)$的单调性如下:\n当$a \\leq 0$时, 函数$f(x)$在区间$[0,+\\infty)$上单调递增; 当$a>0$时, 函数$f(x)$在区间$[0, \\dfrac{a}{3}]$上单调递减, 在区间$[\\dfrac{a}{3},+\\infty)$上单调递增. 设$g(a)$为$f(x)$在区间$[0,2]$上的最小值.\\\\\n(1) 写出$g(a)$的表达式, 并写出$g(a)$的单调区间(不要求证明);\\\\\n(2) 求实数$a$的取值范围, 使得$-6 \\sqrt{3} \\leq g(a) \\leq-2$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392426,7 +393585,9 @@ "id": "014879", "content": "有两个相同的直三棱柱, 高为$\\dfrac{1}{a}$, 底面三角形的三边长分别为$3 a, 4 a, 5 a$($a>0$). 用它们拼成一个三棱柱或四棱柱, 在所有可能的情形中, 全面积最小的是一个四棱柱, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392446,7 +393607,9 @@ "id": "014880", "content": "已知椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$上的动点$P(x, y)$与定点$M(m, 0)$($00$), 数列$\\{a_n a_{n+1}\\}$是公比为$q$($q>0$)的等比数列, $b_n=a_{2 n-1}+a_{2 n}$, 记$S_n=b_1+b_2+\\cdots+b_n$. 求$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{1}{S_n}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392486,7 +393651,9 @@ "id": "014882", "content": "正数列$\\{a_n\\}$的前$n$项和$S_n$满足: $r S_n=a_n a_{n+1}-1$, $a_1=a>0$, 常数$r \\in \\mathbf{N}$.\\\\\n(1) 求证: $a_{n+2}-a_n$是一个定值;\\\\\n(2) 若数列$\\{a_n\\}$是一个周期数列, 求该数列的周期;\\\\\n(3) 若数列$\\{a_n\\}$是一个有理数等差数列, 求$S_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) 定值为$r$, 证明略; (2) 当$a=1$时, 周期为$1$; 当$a\\in (0,1)\\cup (1,+\\infty)$时, 周期为$2$; (3) $S_n=n$($r=0$时)或$S_n=\\dfrac 34n^2+\\dfrac 54n$($r=3$时)", "solution": "", @@ -392513,7 +393680,9 @@ "id": "014883", "content": "已知点$A(7,4)$, $B(-8,2)$, 在$x$轴上求点$C$, 使经过点$C$, 且以$A, B$为焦点的椭圆的长轴长最短, 则$C$点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392533,7 +393702,9 @@ "id": "014884", "content": "函数$y=\\cos x+2|\\cos x|$($x \\in[0,2 \\pi]$)的图像与直线$y=k$有且仅有两个不同的公共点, 则实数$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{0\\}\\cup (1,3]$", "solution": "", @@ -392560,7 +393731,9 @@ "id": "014885", "content": "已知集合$A=\\{x |(x-5)(x+1)<0\\}$, 集合$B=\\{x | x^2-p x-10<0\\}$, 且$A \\subseteq B$, 则实数$p$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392580,7 +393753,10 @@ "id": "014886", "content": "函数$y=\\dfrac{1}{1-x}$的图像与函数$y=2 \\sin \\pi x$($-2 \\leq x \\leq 4$)的图像所有公共点的横坐标之和等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392600,7 +393776,9 @@ "id": "014887", "content": "设有函数$f(x)=a+\\sqrt{-x^2-4 x}$和$g(x)=\\dfrac{4}{3} x+1$, 已知$x \\in[-4,0]$时恒有$f(x) \\leq g(x)$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-\\infty,-5]$", "solution": "", @@ -392627,7 +393805,9 @@ "id": "014888", "content": "若关于$x$的方程$m=x-x^2$在区间$[-2,2]$上只有一个实数根, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392647,7 +393827,9 @@ "id": "014889", "content": "若关于$x$的方程$2 m=\\sqrt{x}-x$有两个不同的实数根, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392667,7 +393849,9 @@ "id": "014890", "content": "若关于$x$的方程$(2 m-1) x=x^2+1$在区间$[-2,1]$上有两个不同的实数根, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392687,7 +393871,9 @@ "id": "014891", "content": "已知函数$f(x)$的定义域是$\\mathbf{R}$, 满足对任意$x\\in \\mathbf{R}$, 都成立$f(x+1)=\\dfrac{1-f(x)}{1+f(x)}$.\\\\\n(1) 证明: $2$是函数$f(x)$的一个周期;\\\\\n(2) 当$x \\in[0,1)$时, $f(x)=x$, 求$f(x)$在$[-1,0)$上的解析式;\\\\\n(3) 设$a>0$, 对于 (2) 中的函数$f(x)$, 关于$x$的方程$f(x)=a x$恰有$100$个根, 求正实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $f(x)=-\\dfrac x{2+x}$; (3) $(\\dfrac{1}{101},\\dfrac{1}{99})$", "solution": "", @@ -392715,7 +393901,9 @@ "id": "014892", "content": "已知点$A(1,1)$, 点$F$是抛物线$y=\\dfrac{1}{9} x^2$的焦点, 点$P$是抛物线上的一个动点, 当$|PA|+|PF|$最小时, $P$点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392735,7 +393923,9 @@ "id": "014893", "content": "不等式$3 x^2<\\log _a x$在区间$(0, \\dfrac{1}{3})$上恒成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392755,7 +393945,10 @@ "id": "014894", "content": "由方程$x|x|+y|y|=1$确定的函数$y=f(x)$在$(-\\infty,+\\infty)$上是\\bracket{20}.\n\\fourch{增函数}{减函数}{先增后减}{先减后增}", "objs": [], - "tags": [], + "tags": [ + "第七单元", + "第二单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -392782,7 +393975,9 @@ "id": "014895", "content": "关于$x$的方程$(x^2-1)^2-|x^2-1|+k=0$, 给出下列四个命题, 其中是假命题的是\\blank{50}.\\\\\n\\textcircled{1} 存在实数$k$, 使得方程恰有$3$个不同的实根;\\\\\n\\textcircled{2} 存在实数$k$, 使得方程恰有$4$个不同的实根;\\\\\n\\textcircled{3} 存在实数$k$, 使得方程恰有$5$个不同的实根;\\\\\n\\textcircled{4} 存在实数$k$, 使得方程恰有$8$个不同的实根.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392802,7 +393997,9 @@ "id": "014896", "content": "设平面点集$A=\\{(x, y) |(y-x)(y-\\dfrac{1}{x}) \\geq 0\\}$, $B=\\{(x, y) |(x-1)^2+(y-1)^2 \\leq 1\\}$, 则$A \\cap B$所表示的平面图形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -392822,7 +394019,9 @@ "id": "014897", "content": "若关于$x$的方程$\\dfrac{|x|}{x-3}=k x^2$有四个不同的实数根, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "$(-\\infty,-\\dfrac 49)$", "solution": "", @@ -392849,7 +394048,9 @@ "id": "014898", "content": "若关于$x$的方程$a=x^2-3|x|+2$有四个实数根, 求实数$a$的取值范围;", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392869,7 +394070,9 @@ "id": "014899", "content": "若关于$x$的方程$x^2+a x+2=0$在$(0,2)$内只有一个实数根, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392889,7 +394092,9 @@ "id": "014900", "content": "对于定义域为$\\mathbf{R}$的函数$f(x)$, 若存在非零实数$x_0$, 使函数$f(x)$在$(-\\infty, x_0)$和$(x_0,+\\infty)$上均有零点, 则称$x_0$为函数$f(x)$的一个``界点''. 下列四个函数中, 不存在``界点''的是\\bracket{20}.\n\\fourch{$f(x)=x^2+\\pi x$}{$f(x)=2^x-x^2$}{$f(x)=2-|x-1|$}{$f(x)=x-\\sin x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -392909,7 +394114,9 @@ "id": "014901", "content": "关于$x$的方程$x^2-1-|x^2-1|+k=0$, 给出下列四个命题, 其中假命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 存在实数$k$, 使得方程恰有$1$个不同的实根;\\\\\n\\textcircled{2} 存在实数$k$, 使得方程恰有$2$个不同的实根;\\\\\n\\textcircled{3} 存在实数$k$, 使得方程恰有$4$个不同的实根;\\\\\n\\textcircled{4} 存在实数$k$, 使得方程有无数个实根.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "\\textcircled{3}", "solution": "", @@ -392935,7 +394142,9 @@ "id": "014902", "content": "已知数列$a_n=n+1$, $b_n=\\begin{cases}1,& n=1, \\\\ (-\\dfrac{1}{10})(\\dfrac{9}{10})^{n-2}, & n \\geq 2,\\end{cases}$ $c_n=-a_n \\cdot b_n$($n \\in \\mathbf{N}$, $n\\ge 1$). 在数列$\\{c_n\\}$中, 是否存在正整数$k$, 使得对于任意的正整数$n$, 都有$c_n \\leq c_k$成立? 若存在, 写出所有满足条件的正整数$k$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392955,7 +394164,9 @@ "id": "014903", "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$), 设$P$是双曲线$C$上任意一点, $O$为坐标原点, 设$F$为双曲线右焦点过右焦点$F$的动直线$l$交双曲线于$A$、$B$两点, 是否存在这样的$a, b$的值, 使得$\\triangle OAB$为等边三角形? 若存在, 求出所有满足条件的$a, b$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -392975,7 +394186,10 @@ "id": "014904", "content": "已知函数$f(x)=2 x+1$, $x \\in \\mathbf{N}$, $x\\ge 1$. 若存在正整数$x_0$, $n$, 使$f(x_0)+f(x_0+1)+\\cdots+f(x_0+n)=63$成立, 则称$(x_0, n)$为函数$f(x)$的一个``生成点''. 函数$f(x)$的``生成点''共有\\bracket{20}.\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -392995,7 +394209,9 @@ "id": "014905", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 点$(n, \\dfrac{S_n}{n})$在直线$y=\\dfrac{1}{2} x+\\dfrac{11}{2}$上. 数列$\\{b_n\\}$满足$b_{n+2}-2 b_{n+1}+b_n=0$($n \\in \\mathbf{N}$, $n\\ge 1$)且$b_3=11$, 前$9$项和为$153$.\\\\\n(1) 求数列$\\{a_n\\}$、$\\{b_n\\}$的通项公式;\\\\\n(2) 设$f(n)=\\begin{cases}a_n,& n=2 l-1, \\\\ b_n, &n=2 l\\end{cases}$($l \\in \\mathbf{N}$, $l\\ge 1$), 问是否存在正整数$m$, 使得$f(m+15)=5 f(m)$成立? 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) $a_n=n+5$, $b_n=3n+2$; (2) 存在, $m=11$", "solution": "", @@ -393022,7 +394238,9 @@ "id": "014906", "content": "在平面直角坐标系$xOy$中, 原点为$O$, 抛物线$C$的方程为$x^2=4 y$, 线段$AB$是抛物线$C$的一条动弦. 当$|AB|=8$时, 设圆$D: x^2+(y-1)^2=r^2$($r>0$), 若存在且仅存在两条动弦$AB$, 满足直线$AB$与圆$D$相切, 求半径$r$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393042,7 +394260,9 @@ "id": "014907", "content": "定义函数$y=f(x)$, $x \\in D$($D$为定义域)图像上的点到坐标原点的距离为函数的$y=f(x)$, $x \\in D$的模. 若模存在最大值, 则称之为函数$y=f(x)$, $x \\in D$的长距; 若模存在最小值, 则称之为函数$y=f(x)$, $x \\in D$的短距.\\\\\n(1) 分别判断函数$f_1(x)=\\dfrac{1}{x}$与$f_2(x)=\\sqrt{-x^2-4 x+5}$是否存在长距与短距;\\\\\n(2) 对于任意$x \\in[1,2]$, 是否存在实数$a$, 使得函数$f(x)=\\sqrt{2 x|x-a|}$的短距不小于$2$且长距不大于$4$? 若存在, 求出$a$的取值范围; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $f_1(x)$存在短距, 不存在长距, $f_2(x)$存在短距, 也存在长距; (2) 存在满足条件的实数$a$, $a$的范围为$[-1,-\\dfrac 12]\\cup [\\dfrac 52,5]$", "solution": "", @@ -393069,7 +394289,9 @@ "id": "014908", "content": "若干个能唯一确定一个数列的量称为该数列的``基本量''. 设$\\{a_n\\}$是公比为$q$的无穷\n等比数列, 下列$\\{a_n\\}$的四组量中, 一定能成为该数列``基本量''的是第\\blank{50}组.(其中$n$为大于$1$的整数, $S_n$为$\\{a_n\\}$的前$n$项和)\n\\textcircled{1} $S_1$与$S_2$; \\textcircled{2} $a_2$与$S_3$; \\textcircled{3} $a_1$与$a_n$; \\textcircled{4} $q$与$a_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -393089,7 +394311,9 @@ "id": "014909", "content": "设$g(x)$是定义在$\\mathbf{R}$上, 以$1$为周期的函数. 若函数$f(x)=x+g(x)$在区间$[3,4]$上的值域为$[-2,5]$. 则\\\\\n(1) $f(x)$在区间$[4,5]$上的值域为\\blank{50};\\\\\n(2) $f(x)$在区间$[-8,8]$上的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "(1) $[-1,6]$; (2) $[-13,9]$", "solution": "", @@ -393115,7 +394339,9 @@ "id": "014910", "content": "在平面直角坐标系$xOy$中, 对于直线$l: a x+b y+c=0$和点$P_1(x_1, y_1)$, $P_2(x_2, y_2)$, 记$\\eta=(a x_1+b y_1+c)(a x_2+b y_2+c)$. 若$\\eta<0$, 则称点$P_1, P_2$被直线$l$分割. 若曲线$C$与直线$l$没有公共点, 且曲线$C$上存在点$P_1, P_2$被直线$l$分割, 则称直线$l$为曲线$C$的一条分割线.\\\\\n(1) 设点$A$的坐标为$(2,2)$, 直线$l: x+y-1=0$.\\\\\n(i) 求证: 点$A$、$B(-1,0)$被直线$l$分割;\\\\\n(ii) 求证: 存在一点$C$, 点$C$、$A$不被直线$l$分割;\\\\\n(2) 设曲线$C_1: x y=1$是否存在分割线? 若存在, 写出一条分割线, 并证明其为曲线$C_1$的分割线; 若不存在, 说明理由;\\\\\n(3) 求证: 曲线$C_2: |x y|=1$恰存在两条分割线.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) (i) $A$与$B$被直线$l$分割; (ii) $A$与$C$不被直线$l$分割; (2) 如$x+y=0$, 理由略; (3) 证明略", "solution": "", @@ -393140,7 +394366,9 @@ "id": "014911", "content": "若实数$x, y, m$满足$\\lg (x-m)>\\lg (y-m)$, 则称$x$比$y$``更真''于$m$. 若$4 x-x^2-1$比$x-1$``更真''于$1$, 则$x$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(2,3)$", "solution": "", @@ -393166,7 +394394,9 @@ "id": "014912", "content": "方程$x^2+\\sqrt{2} x-1=0$的解可视为函数$y=x+\\sqrt{2}$的图像与函数$y=\\dfrac{1}{x}$的图像交点的横坐标. 若方程$x^4+a x-4=0$的各个实根$x_1, x_2, \\cdots, x_k$($k \\leq 4$)所对应的点$(x_i, \\dfrac{4}{x_i})$($i=1,2, \\cdots, k$)均在直线$y=x$的同侧, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -393188,7 +394418,9 @@ "id": "014913", "content": "已知平面上的线段$l$及点$P$. 任取$l$上一点$Q$, 线段$PQ$长度的最小值称为点$P$到线段$l$的距离, 记作$d(P, l)$. 设线段$l_1: y=2$($-2 \\leq x \\leq 2)$.\\\\\n(1) 分别求点$P_1(0,3)$、$P_2(4,3)$到线段$l_1$的距离$d(P_1, l_1)$、$d(P_2, l_1)$;\\\\\n(2) 求点的集合$D=\\{P | d(P, l_1)=1\\}$;\\\\\n(3) 设$A(0,2)$, 写出$\\Omega=\\{P|d(P, l_1)=| PA |\\}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $d(P_1,l_1)=1$, $d(P_2,l_1)=\\sqrt{5}$;\\\\\n(2) $D=\\left\\{(x,y)|\\begin{cases}x\\ge 2, \\\\ (x-2)^2+(y-2)^2=1,\\end{cases}\\text{ 或 } \\begin{cases} x\\le =2, \\\\ (x+2)^2+(y-2)^2=1, \\end{cases}\\text{ 或 }\\begin{cases} -2b_n$, $n=1,2, \\cdots$, 其中$\\overrightarrow {j}$为方向与$y$轴正方向相同的单位向量, 则称$\\{A_n\\}$为$T$点列.\\\\\n(1) 判断$A_1(1,1)$, $A_2(2, \\dfrac{1}{2})$, $A_3(3, \\dfrac{1}{3})$, $\\cdots$, $A_n(n, \\dfrac{1}{n})$, $\\cdots$是否为$T$点列, 并说明理由;\\\\\n(2) 若$\\{A_n\\}$为$T$点列, 且点$A_2$在点$A_1$的右上方. 任取其中连续三点$A_k, A_{k+1}, A_{k+2}$, 判断$\\triangle A_k A_{k+1} A_{k+2}$的形状(锐角三角形、直角三角形、钝角三角形), 并予以证明;\\\\\n(3) 若$\\{A_n\\}$为$T$点列, 从小到大排列的四个不同的正整数$m,n,p,q$满足$m+q=n+p$, 求证: $\\overrightarrow{A_n A_q} \\cdot \\overrightarrow {j}>\\overrightarrow{A_m A_p} \\cdot \\overrightarrow {j}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) 是$T$点列, 理由略; (2) 是钝角三角形, 证明略; (3) 证明略", "solution": "", @@ -393241,7 +394475,9 @@ "id": "014915", "content": "已知$a, b, c$均为正实数, 求证: $\\dfrac{1}{a}+\\dfrac{1}{b}+\\dfrac{1}{c} \\geq \\dfrac{1}{\\sqrt{a b}}+\\dfrac{1}{\\sqrt{b c}}+\\dfrac{1}{\\sqrt{c a}}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "证明略", "solution": "", @@ -393268,7 +394504,9 @@ "id": "014916", "content": "已知函数$f(x)$、$g(x)$的定义域均为$\\mathbf{R}$, $x_1$、$x_2$是在$\\mathbf{R}$上任意选取的两个实数. 若$f(x)$是奇函数且不等式$|f(x_1)+f(x_2)| \\geq|g(x_1)+g(x_2)|$恒成立, 问$g(x)$是否也为奇函数? 证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "是奇函数, 证明略", "solution": "", @@ -393294,7 +394532,9 @@ "id": "014917", "content": "已知函数$f(x)$、$g(x)$的定义域均为$\\mathbf{R}$, $x_1$、$x_2$是在$\\mathbf{R}$上任意选取的两个实数. 若$f(x)$是周期函数且不等式$|f(x_1)-f(x_2)| \\geq|g(x_1)-g(x_2)|$恒成立, 问$g(x)$是否为周期函数? 证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "是周期函数, 证明略", "solution": "", @@ -393321,7 +394561,9 @@ "id": "014918", "content": "若$a, b\\in \\mathbf{R}$, 满足$2 a+b+2 \\leq 0$.\\\\\n(1) 求证: 关于$t$的方程$t^2+a t+b-2=0$有实数解;\\\\\n(2) 求证: 关于$x$的方程$x^2+\\dfrac{1}{x^2}+a(x+\\dfrac{1}{x})+b=0$有正实数解.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", "solution": "", @@ -393348,7 +394590,9 @@ "id": "014919", "content": "对于无穷数列$\\{a_n\\}$, 若存在正常数$M$, 使得对任意正整数$n$, 总成立$|a_n|0$且$a \\neq b$, 由$a, b, \\dfrac{a+b}{2}, \\sqrt{a b}$按一定顺序构成的数列\\bracket{20}.\n\\twoch{可能是等差数列, 也可能是等比数列}{可能是等差数列, 但不可能是等比数列}{不可能是等差数列, 但可能是等比数列}{不可能是等差数列, 也不可能是等比数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -393542,7 +394800,9 @@ "id": "014927", "content": "设函数$f(x)=a x^2-2 x+2$, 对于满足$10$, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393562,7 +394822,9 @@ "id": "014928", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=-\\dfrac{3}{2} n^2+\\dfrac{27}{2} n$, 设$T_n=|a_1|+|a_2|+\\cdots+|a_n|$, 求$T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393582,7 +394844,9 @@ "id": "014929", "content": "若函数$f(x)=\\begin{cases}2^x-a, & x \\leq 1, \\\\ (x-a)(x-2 a), & x>1\\end{cases}$恰有两个零点, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -393602,7 +394866,9 @@ "id": "014930", "content": "我们称点$P$到图形$C$上任意一点距离的最小值为点$P$到图形$C$的距离, 那么平面内到定圆$C$的距离与到定点$A$的距离相等的点的轨迹不可能是\\bracket{20}.\n\\fourch{圆}{椭圆}{双曲线的一支}{直线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -393629,7 +394895,9 @@ "id": "014931", "content": "在集合$U=\\{a, b, c, d\\}$的子集中选出$4$个不同的子集, 需同时满足以下两个条件: \\textcircled{1} $\\varnothing, U$都要选出; \\textcircled{2} 对选出的任意两个子集$A$和$B$, 必有$A \\subseteq B$或$B \\subseteq A$. 那么共有\\blank{50}种不同的选法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$36$", "solution": "", @@ -393656,7 +394924,9 @@ "id": "014932", "content": "给定常数$c>0$, 定义函数$f(x)=2|x+c+4|-|x+c|$, 数列$a_1, a_2, a_3, \\cdots$满足$a_{n+1}=f(a_n)$, $n \\in \\mathbf{N}$, $n \\ge 1$.\\\\\n(1) 求证: 对任意正整数$n$, $a_{n+1}-a_n \\geq c$;\\\\ \n(2) 是否存在$a_1$, 使得$a_1, a_2, a_3, \\cdots, a_n, \\cdots$成等差数列? 若存在, 求出所有这样的$a_1$; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393676,7 +394946,9 @@ "id": "014933", "content": "是否存在第一象限的角$\\alpha$和第三象限的角$\\beta$, 使得$\\tan \\alpha \\tan \\beta=\\tan (\\alpha-\\beta)$? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393700,7 +394972,9 @@ "id": "014934", "content": "是否存在第二象限的角$\\alpha$和第四象限的角$\\beta$, 使得$\\tan \\alpha \\tan \\beta=\\tan (\\alpha-\\beta)$? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393724,7 +394998,9 @@ "id": "014935", "content": "是否存在第一象限的角$\\alpha$和第三象限的角$\\beta$, 使得$\\sin \\alpha \\sin \\beta=\\sin (\\alpha-\\beta)$? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393748,7 +395024,10 @@ "id": "014936", "content": "设$n \\in \\mathbf{N}$, $n \\ge 1$, 数列$\\{a_n\\}$、$\\{b_n\\}$满足: $a_n$为$(x+4)^n-(x+1)^n$的展开式中各项系数之和, $b_n=[\\dfrac{a_1}{5}]+[\\dfrac{2 a_2}{5^2}]+\\cdots+[\\dfrac{n a_n}{5^n}]$($[x]$表示不超过实数$x$的最大整数), 则当$n$变化时, $b_n-5 n$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -393768,7 +395047,9 @@ "id": "014937", "content": "如图, 棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $E$为棱$CC_1$的中点, 点$P$、$Q$分别为面$A_1B_1C_1D_1$和线段$B_1C$上的动点, 则$\\triangle PEQ$周长的最小值为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(C)!0.5!(C1)$) node [right] {$E$} coordinate (E);\n\\draw ($(B1)!0.3!(C)$) node [below] {$Q$} coordinate (Q);\n\\draw ($1/3*(A1)+1/3*(B1)+1/3*(C1)$) node [left] {$P$} coordinate (P);\n\\draw (B1)--(C)(E)--(Q);;\n\\draw [dashed] (Q)--(P)--(E);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$2 \\sqrt{2}$}{$\\sqrt{10}$}{$\\sqrt{11}$}{$2 \\sqrt{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -393790,7 +395071,9 @@ "id": "014938", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{2}+y^2=1$的右焦点为$F_2$, 过$F_2$作两条不重合的动直线$l_1$、$l_2$, 其中$l_1$与$\\Gamma$交于$A$、$B$两点, $l_2$与$\\Gamma$交于$C$、$D$两点, $M$、$N$分别是线段$AB$、$CD$的中点. 若直线$MN$过定点$(\\dfrac{2}{3}, 0)$, 试问$l_1$与$l_2$的夹角是否为定值? 如果是, 求出该定值; 如果不是, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393810,7 +395093,9 @@ "id": "014939", "content": "如图, 已知椭圆$\\Gamma: \\dfrac{x^2}{2}+y^2=1$的右焦点为$F_2$. 过点$F_2$作互相垂直且与$x$轴不重合的两条直线, 分别交$\\Gamma$于$A$、$B$两点和$C$、$D$两点, $M$、$N$分别是线段$AB$、$CD$的中点. 求证: 直线$MN$过定点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [draw, name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\path [domain = 0:2, name path = AB] plot (\\x,{2*(\\x-1)});\n\\path [domain = -2:2, name path = CD] plot (\\x,{-0.5*(\\x-1)});\n\\path [name intersections = {of = AB and elli, by = {A,B}}];\n\\path [name intersections = {of = CD and elli, by = {D,C}}];\n\\draw (A) node [below] {$A$}-- (B) node [above] {$B$};\n\\draw (C) node [below] {$C$} -- (D) node [above] {$D$};\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(D)$) node [above] {$N$} coordinate (N);\n\\draw (M)--(N);\n\\draw (1,0) node [above] {$F_2$} coordinate (F_2);\n\\foreach \\i in {A,B,C,D,M,N,F_2}\n{\\filldraw (\\i) circle (0.03);};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393830,7 +395115,9 @@ "id": "014940", "content": "设定义在$\\mathbf{R}$上的函数$f(x)$满足: 对于任意的$x_1$、$x_2 \\in \\mathbf{R}$, 当$x_1 \\neq x_2$时, 都有$\\dfrac{f(x_1)-f(x_2)}{x_1-x_2} \\geq 0$, 且存在$u \\neq v$, 使得$f(u) \\neq f(v)$. 求证: $f(x)$不是周期函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393850,7 +395137,9 @@ "id": "014941", "content": "已知函数$f(x)=2^x+x$的反函数为$y=f^{-1}(x)$, 若$a, b, c$依次成公差大于零的等差数列, 且各项均在函数$f(x)$的值域中, 求证: $2 f^{-1}(b)>f^{-1}(a)+f^{-1}(c)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393872,7 +395161,9 @@ "id": "014942", "content": "已知函数$f(x)=3 \\cos x$, 则函数$y=f(2 x)-f(x)$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -393892,7 +395183,9 @@ "id": "014943", "content": "若关于$x$的不等式$|2^x-m|-\\dfrac{1}{2^x}<0$在区间$[0,1]$内恒成立, 则实数$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(\\dfrac 32,2)$", "solution": "", @@ -393918,7 +395211,9 @@ "id": "014944", "content": "在$xOy$平面上, 将曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$($x>0$)、直线$y=\\dfrac{4}{3} x$、直线$y=0$和直线$y=4$围成的封闭图形记为$D$, 记$D$绕$y$轴旋转一周所得的几何体为$\\Omega$, 利用祖暅原理得出$\\Omega$的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$36\\pi$", "solution": "", @@ -393943,7 +395238,9 @@ "id": "014945", "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_2=3$, 若$|a_{n+1}-a_n|=2^n$($n \\in \\mathbf{N}$, $n\\ge 1$), 且$\\{a_{2 n-1}\\}$是递增数列、$\\{a_{2 n}\\}$是递减数列, 求$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{a_{2 n-1}}{a_{2 n}}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -393963,7 +395260,9 @@ "id": "014946", "content": "已知曲线$C_1: \\dfrac{x^2}{4}-\\dfrac{y^2}{3}=1$和曲线$C_2: |y|=|x|+2$, 求证: 不存在过点$(1,0)$的直线, 同时与$C_1, C_2$都有公共点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "证明略", "solution": "", @@ -393989,7 +395288,9 @@ "id": "014947", "content": "若关于$x$的方程$|x^2-2 a x-3|=4 a$有且仅有两个实数解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$\\{0\\}\\cup (1,3)$", "solution": "", @@ -394016,7 +395317,9 @@ "id": "014948", "content": "设$a_1, d$为实数, 且首项为$a_1$、公差为$d$的等差数列$\\{a_n\\}$的前$n$项和$S_n$满足$S_5S_6+15=0$, 则$d$的取值范围是\\blank{50}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394036,7 +395339,9 @@ "id": "014949", "content": "若关于$x$的方程$\\cos 2 x-2 \\cos x+m=0$有实数解, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394056,7 +395361,9 @@ "id": "014950", "content": "若关于$x$的方程$\\cos 2 x-2 \\cos x+m=0$在$[0, \\pi]$内有且仅有两个解, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[1,\\dfrac 32)$", "solution": "", @@ -394083,7 +395390,9 @@ "id": "014951", "content": "若关于$x$的不等式$\\cos 2 x-2 \\cos x+m>0$恒成立, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394103,7 +395412,9 @@ "id": "014952", "content": "若$\\{x | \\cos 2 x-2 \\cos x+m>0, \\ x \\in \\mathbf{R}\\} \\neq \\varnothing$, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394123,7 +395434,9 @@ "id": "014953", "content": "已知函数$f(x)=\\dfrac{4 x+a}{x^2+1}$的值域为$[-1, b]$, 求实数$a, b$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "$a=3$, $b=4$", "solution": "", @@ -394149,7 +395462,9 @@ "id": "014954", "content": "已知函数$f(x)=\\sqrt{x^2-1}-a x+1$有且仅有两个零点, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394169,7 +395484,9 @@ "id": "014955", "content": "在正四棱柱$ABCD-A_1B_1C_1D_1$中, 顶点$B_1$到对角线$BD_1$、平面$A_1BCD_1$的距离分别为$h$、$d$, 则下列命题中正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1)--(B);\n\\draw [dashed] (B)--(D1)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\onech{若侧棱的长小于底面的边长, 则$\\dfrac{h}{d}$的取值范围为$(0,1)$}{若侧棱的长小于底面的边长, 则$\\dfrac{h}{d}$的取值范围为$(\\dfrac{\\sqrt{2}}{2}, \\dfrac{2 \\sqrt{3}}{3})$}{若侧棱的长大于底面的边长, 则$\\dfrac{h}{d}$的取值范围为$(\\dfrac{2 \\sqrt{3}}{3}, \\sqrt{2})$}{若侧棱的长大于底面的边长, 则$\\dfrac{h}{d}$的取值范围为$(\\dfrac{2 \\sqrt{3}}{3},+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -394189,7 +395506,9 @@ "id": "014956", "content": "已知点$P(0,1)$, 椭圆$\\dfrac{x^2}{4}+y^2=m$($m>1$)上两点$A, B$满足$\\overrightarrow{AP}=2 \\overrightarrow{PB}$, 则当$m=$\\blank{50}时, 点$B$横坐标的绝对值最大.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -394216,7 +395535,9 @@ "id": "014957", "content": "已知实数$a, b$满足$\\dfrac{8}{(a+1)^3}+\\dfrac{10}{a+1}-b^3-5 b>0$, 对于命题: \\textcircled{1} 若$a, b$两数中有一个大于$1$, 则另一个必小于$1$; \\textcircled{2} 若$a \\in(-2,-1)$, 则$a>b$, 下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}均为真命题}{\\textcircled{1}和\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \n\\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -394243,7 +395564,9 @@ "id": "014958", "content": "若关于$x$的方程$m+\\sqrt{1-x}=x$有实数解, 则实数$m$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394263,7 +395586,9 @@ "id": "014959", "content": "设常数$a$使方程$\\sin x+\\sqrt{3} \\cos x=a$在闭区间$[0,2 \\pi]$上恰有三个解$x_1, x_2, x_3$, 则$x_1+x_2+x_3=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394283,7 +395608,9 @@ "id": "014960", "content": "已知数列$\\{a_n\\}$满足: $a_{n+1}(a_n+1)=1$($n \\in \\mathbf{N}$, $n\\ge 1$), 且$a_1=1$, 若$\\displaystyle\\lim_{n\\to\\infty} a_n=A$, 则$A=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394303,7 +395630,9 @@ "id": "014961", "content": "已知函数$f(x)=\\dfrac{(1-t) x-t^2}{x}$($t \\in \\mathbf{R}$)的定义域为$D$, 若存在区间$[a, b] \\subseteq D$, 使得当$x \\in[a, b]$时, $f(x)$的取值范围也是$[a, b]$, 则当$t$变化时, $b-a$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$\\dfrac 23\\sqrt{3}$", "solution": "", @@ -394329,7 +395658,10 @@ "id": "014962", "content": "(1) 求证: 关于$x$的方程$x^n+x-1=0$($n \\in \\mathbf{N}$, $n \\geq 2$)在区间$(\\dfrac{1}{2}, 1)$内存在唯一解;\\\\\n(2) 设(1) 中方程的唯一解为$x_n$, 判断数列$\\{x_{n+1}\\}$的单调性并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第四单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", "solution": "", @@ -394355,7 +395687,9 @@ "id": "014963", "content": "泳池长$90$米, 甲乙两人分别在泳池两边, 同时相向游泳, 甲的速度是$3$米/秒, 乙的速度是$2$米/秒, 若不计转向时间, $3$分钟后他们共相遇多少次?", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "解答题", "ans": "", "solution": "", @@ -394375,7 +395709,9 @@ "id": "014964", "content": "函数$y=f(x)$的图像如图所示, 在区间$[a, b]$上可找到$n$($n \\geq 2$, $n \\in \\mathbf{N}$)个不同的实数$x_1,x_2, \\cdots, x_n$, 使得$\\dfrac{f(x_1)}{x_1}=\\dfrac{f(x_2)}{x_2}=\\cdots=\\dfrac{f(x_n)}{x_n}$, 则$n$的取值集合为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0.5,0) node [below] {$a$} -- (1,1.5) -- (1.5,0.2) -- (2,1) -- (2.5,0) node [below] {$b$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\{2,3\\}$}{$\\{3,4\\}$}{$\\{2,3,4\\}$}{$\\{3,4,5\\}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -394395,7 +395731,10 @@ "id": "014965", "content": "对任意$s \\in(-\\infty, 0) \\cup(0,+\\infty)$和$t \\in[-1,1]$, 不等式$s^2+\\dfrac{16}{s^2}-2 s t-\\dfrac{8}{s} \\sqrt{1-t^2}-a \\geq 0$恒成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第七单元" + ], "genre": "填空题", "ans": "$(-\\infty,8-4\\sqrt{2}]$", "solution": "", @@ -394422,7 +395761,10 @@ "id": "014966", "content": "若四面体$S-ABC$的一条棱长为$x$, 其余棱长均为$1$, 它的体积是$V(x)$, 则函数$V(x)$在其定义域上\\bracket{20}.\n\\twoch{是增函数但无最大值}{是增函数且有最大值}{不是增函数且无最大值}{不是增函数但有最大值}", "objs": [], - "tags": [], + "tags": [ + "第六单元", + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -394449,7 +395791,9 @@ "id": "014967", "content": "若两函数$y=x+a$与$y=\\sqrt{1-2 x^2}$的图像有两个公共点$A$、$B$, $O$是坐标原点, $\\triangle OAB$是锐角三角形, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394469,7 +395813,9 @@ "id": "014968", "content": "如图, 在平面直角坐标系$xOy$中, 菱形$ABCD$的边长为$4$, 且$|OB|=|OD|=6$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (5,2) node [right] {$C$} coordinate (C);\n\\path [draw, name path = arc] (2,-2) arc (-90:90:2);\n\\path [name path = OC] (O) -- (C);\n\\path [name intersections = {of = OC and arc, by = A}];\n\\draw (A) node [below left] {$A$} coordinate (A);\n\\path [name path = circle1] (A) circle (4);\n\\path [name path = circle2] (C) circle (4);\n\\path [name intersections = {of = circle1 and circle2, by = {B,D}}];\n\\draw (A)--(B)--(C)--(D)--cycle(B) node [above] {$B$} --(O)--(D) node [below] {$D$};\n\\draw [dashed] (2,-2)--(2,2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $|OA| \\cdot|OC|$为定值;\\\\\n(2) 当点$A$在半圆$(x-2)^2+y^2=4$($2 \\leq x \\leq 4$)上运动时, 求点$C$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) 定值为$20$, 证明略; (2) $x=5$且$-5\\le y\\le 5$", "solution": "", @@ -394496,7 +395842,9 @@ "id": "014969", "content": "已知圆$O$的半径为$1$, $PA$、$PB$为该圆的两条切线, $A$、$B$为两个切点, 则$\\overrightarrow{PA} \\cdot \\overrightarrow{PB}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$2\\sqrt{2}-3$", "solution": "", @@ -394523,7 +395871,9 @@ "id": "014970", "content": "已知$f(x)$是定义在$\\mathbf{R}$上的偶函数, 且$f(x)$在$[0,+\\infty)$上是严格增函数, 如果对于任意$x \\in[1,2]$, 不等式$f(a x+1) \\leq f(x-3)$恒成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$[-1,0]$", "solution": "", @@ -394551,7 +395901,9 @@ "id": "014971", "content": "已知$f(x)$是定义在$[1,+\\infty)$上的函数, 且$f(x)=\\begin{cases}1-|2 x-3|,& 1 \\leq x<2, \\\\ \\dfrac{1}{2} f(\\dfrac{1}{2} x), & x \\geq 2,\\end{cases}$则函数$y=2 x f(x)-3$在区间$(1,2022)$上的零点个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$11$", "solution": "", @@ -394578,7 +395930,9 @@ "id": "014972", "content": "设$\\theta \\in(0, \\dfrac{\\pi}{2}]$, 函数$f(x)=5 \\sin (2 x-\\theta)$, $x \\in[0,5 \\pi]$, 若函数$F(x)=f(x)-3$的所有零点依次记为$x_1, x_2, x_3, \\cdots, x_n$, 且$x_1=latex]\n\\fill [pattern = north east lines] (0,0.2) rectangle (4,0);\n\\draw (0,0) -- (4,0);\n\\draw (0.4,0) -- (0.4,-1) -- (3.6,-1) -- (3.6,0) (2,0)--(2,-1);\n\\draw (0.4,-1.1) -- (0.4,-1.7) (3.6,-1.1)-- (3.6,-1.7);\n\\draw [<->] (0.4,-1.4) -- (3.6,-1.4) node [midway, fill = white] {$30-3x$};\n\\draw [<->] (3.9,-1) -- (3.9,0) node [midway, fill = white, rotate = 90] {$x$};\n\\draw (3.7,-1) -- (4,-1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -394652,7 +396010,9 @@ "id": "014975", "content": "动物园要建造一面靠墙(该墙的长度为$12$米)的$2$间面积相同的长方形熊猫居室. 如果可供建造围墙的材料长是$30$米, 那么宽$x$为多少米时才能使所建造的熊猫居室面积最大? 熊猫居室的最大面积是多少平方米?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [pattern = north east lines] (0,0.2) rectangle (4,0);\n\\draw (0,0) -- (4,0);\n\\draw (0.4,0) -- (0.4,-1) -- (3.6,-1) -- (3.6,0) (2,0)--(2,-1);\n\\draw (0.4,-1.1) -- (0.4,-1.7) (3.6,-1.1)-- (3.6,-1.7);\n\\draw [<->] (0.4,-1.4) -- (3.6,-1.4) node [midway, fill = white] {$30-3x$};\n\\draw [<->] (3.9,-1) -- (3.9,0) node [midway, fill = white, rotate = 90] {$x$};\n\\draw (3.7,-1) -- (4,-1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -394672,7 +396032,9 @@ "id": "014976", "content": "北京天坛的圜丘坛为古代祭天的场所, 分上、中、下三层. 上层中心有一块圆形石板(称为天心石), 环绕天心石砌$9$块扇面形石板构成第一环, 向外每环依次增加$9$块. 中层起, 每层的第一环比上一层的最后一环多$9$块, 向外每环依次也增加$9$块. 已知每层环数相同, 且下层比中层多$729$块, 则三层共有扇面形石板(不含天心石)\\blank{50}块.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394692,7 +396054,9 @@ "id": "014977", "content": "小明将上周每天骑车上学路上的情况用图像表示. 很遗憾图像的先后次序不小心被打乱了. 还好小明同时用文字进行了记录:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (2,0) node [below] {时间};\n\\draw [->] (0,0) -- (0,2) node [right] {离开家的距离};\n\\draw (0,0) node [below left] {$O$};\n\\draw [thick] (0,0) -- (0.8,0.8) -- (1.2,0.8) -- (2,1.6);\n\\draw (1,-0.5) node {A};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (2,0) node [below] {时间};\n\\draw [->] (0,0) -- (0,2) node [right] {离开家的距离};\n\\draw (0,0) node [below left] {$O$};\n\\draw [thick] (0,0) -- (0.5,0.5) -- (0.7,0) -- (1.2,0) -- (2,1.6);\n\\draw (1,-0.5) node {B};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (2,0) node [below] {时间};\n\\draw [->] (0,0) -- (0,2) node [right] {离开家的距离};\n\\draw (0,0) node [below left] {$O$};\n\\draw [thick,domain = 0:1.6, samples = 50] plot (\\x,{\\x*\\x*\\x/1.6/1.6}); \n\\draw (1,-0.5) node {C};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (2,0) node [below] {时间};\n\\draw [->] (0,0) -- (0,2) node [right] {离开家的距离};\n\\draw (0,0) node [below left] {$O$};\n\\draw [thick] (0,0) -- (1,1.3) -- (1.3,1.3) -- (2,1.6);\n\\draw (1,-0.5) node {D};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (2,0) node [below] {时间};\n\\draw [->] (0,0) -- (0,2) node [right] {离开家的距离};\n\\draw (0,0) node [below left] {$O$};\n\\draw [thick] (0,0) -- (1.6,1.6);\n\\draw (1,-0.5) node {E};\n\\end{tikzpicture}\n\\end{center}\n周一: 匀速骑车前进;\\\\\n周二: 匀速骑车前进, 中间遇到红灯停了一次, 然后用与之前同样的速度匀速前进;\\\\\n周三: 骑车出门晩了, 越骑越快;\\\\\n周四: 匀速骑车出门后一会儿想起忘带东西又回去拿, 然后再赶回学校;\\\\\n周五: $\\cdots$.\\\\\n请将图像的编号填入表格中对应日期的下方:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 日期 & 周一 & 周二 & 周三 & 周四 & 周五 \\\\\n\\hline 图像编号 &&&&&\\\\\n\\hline\n\\end{tabular} \n\\end{center}\n并描述周五小明上学途中可能发生的情况, 填在下面的空格中:\\\\\n周五: \\blank{200}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394712,7 +396076,9 @@ "id": "014978", "content": "为防止和严惩酒后驾车, 国家质量监督检验检疫局于$2004$年$5$月$31$日发布了《车辆驾驶人员血液、呼气酒精含量阈值与检验》国家标准. 标准规定, 车辆驾驶人员血液中的酒精含量大于或等于$20$毫克/百毫升, 小于$80$毫克/百毫升为饮酒驾车, 血液中的酒精含量大于或等于$80$毫克/百毫升为醉酒驾车, 经过大样本容量的试验分析得到一个函数模型$y=f(x)$近似地表达$y$与$x$的关系: $f(x)=\\begin{cases}a(x-1.3)^2+46.83, & 0=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (120:2) node [left] {$B$} coordinate (B);\n\\draw ($(O)!0.5!(B)$) node [left] {$Q$} coordinate (Q);\n\\path [name path = QP] (Q) --++ (2.5,0);\n\\path [name path = arc,draw] (A) arc (0:120:2);\n\\path [name intersections = {of = QP and arc, by = P}];\n\\draw (Q) -- (P) node [right] {$P$};\n\\draw (B)--(O)--(A)(O)--(P);\n\\draw ($(Q)!0.4!(P)$) ++ (0,0.5) node {桃花区};\n\\draw ($(Q)!0.4!(P)$) ++ (0,-0.2) node {郁金香区};\n\\draw ($(O)!0.7!(A)$) ++ (0,0.3) node {海棠区};\n\\end{tikzpicture}\n\\end{center}\n(1) 当$Q$是$OB$的中点时, 求$PQ$的长; (结果精确到$1$米)\\\\\n(2) 郁金香因其花朵饱满、颜色鲜㓞而受到游客欢迎. 请规划观赏区划分方案, 使郁金香种植区$\\triangle OPQ$的面积尽可能地大.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -394752,7 +396120,9 @@ "id": "014980", "content": "某项目组决定开发一款``猫捉老鼠''的游戏. 如图, 两个信号源$A$、$B$相距$10$米, $O$是$AB$的中点, 过$O$点的直线$l$与直线$AB$的夹角为$45^{\\circ}$. 机器猫在直线$l$上运动, 机器鼠的运动轨迹始终满足: 接收到$A$点的信号比接收到$B$点的信号晩$\\dfrac{8}{v_0}$秒 (注: 信号每秒传播$v_0$米). 在时刻$t_0$时, 测得机器鼠距离$O$点为$4$米. 以$O$为原点, 直线$AB$为$x$轴建立平面直角坐标系.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\filldraw (-1,0) circle (0.03) node [below] {$A$} coordinate (A);\n\\filldraw (1,0) circle (0.03) node [below] {$B$} coordinate (B);\n\\draw (-1.6,-1.6) -- (1.6,1.6) node [right] {$l$};\n\\draw (0.8,0.8) node [fill = white] {\\rotatebox{45}{猫}};\n\\draw ({4/3},0.8) node {鼠};\n\\end{tikzpicture}\n\\end{center}\n(1) 求时刻$t_0$时机器鼠所在位置的坐标;\\\\\n(2) 游戏设定: 机器鼠在距离直线$l$不超过$1.5$米的区域运动时, 有``被抓''的风险. 如果机器鼠保持目前的运动轨迹不变, 是否有``被抓''的风险?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -394772,7 +396142,9 @@ "id": "014981", "content": "有一块正方形菜地$EFGH$, $EH$所在直线是一条小河, 收获的蔬菜可以送到$F$点或河边运走. 于是, 菜地分为两个区域$S_1$和$S_2$, 其中$S_1$中的蔬菜运到河边较近, $S_2$中的蔬菜运到$F$点较近, 而菜地内$S_1$和$S_2$的分界线$C$上的点到河边与到$F$点的距离相等. 现建立平面直角坐标系, 如图, 其中原点$O$为线段$EF$的中点, 点$F$的坐标为$(1,0)$, 则菜地内分界线$C$的方程为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,0) node [below] {$E$} coordinate (E) rectangle (2,4) node [above] {$G$} coordinate (G);\n\\draw (2,0) node [below] {$F$} coordinate (F);\n\\draw (-2,4) node [above] {$H$} coordinate (H);\n\\draw (-1,2.5) node {$S_1$};\n\\draw (1,1) node {$S_2$};\n\\draw [very thick, domain = 0:4, samples = 100] plot ({pow(\\x,2)/8},\\x);\n\\filldraw ({1/2},2) circle (0.05) node [right] {$M$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -394792,7 +396164,9 @@ "id": "014982", "content": "产能利用率是指实际产出与生产能力的比率, 工业产能利用率是衡量工业生产经营状况的重要指标. 下图为某地区统计局发布的$2015$年至$2018$年第$2$季度该地区工业产能利用率的折线图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, yscale = 0.5]\n\\draw (1,4.2) coordinate (P);\n\\foreach \\i/\\j in {1/74.2,2/74.3,3/74.0,4/74.6,5/72.9,6/73.1,7/73.2,8/73.8,9/75.8,10/76.8,11/76.8,12/78.0,13/76.5,14/76.8}\n{\\filldraw (\\i,{\\j-70}) circle (0.05 and 0.1) coordinate (Q) node [above = 0.2] {$\\j$};\n\\draw (P)--(Q);\n\\draw (\\i,{\\j-70}) coordinate (P);};\n\\draw (0,0) rectangle (15,10);\n\\draw (7.5,10) node [above] {分季度工业产能利用率};\n\\foreach \\i in {70,72,74,76,78,80}\n{\\draw (0,{\\i-70}) node [left] {$\\i$};};\n\\draw (0,10) node [above left] {$(\\%)$};\n\\foreach \\i/\\j in {1/1,2/2,3/3,4/4,5/1,6/2,7/3,8/4,9/1,10/2,11/3,12/4,13/1,14/2}\n{\\draw (\\i,0) node [below] {\\small$\\j$季度};};\n\\draw (0.6,-1) -- (4.4,-1) node [below, midway] {$2015$年};\n\\draw (4.6,-1) -- (8.4,-1) node [below, midway] {$2016$年};\n\\draw (8.6,-1) -- (12.4,-1) node [below, midway] {$2017$年};\n\\draw (12.6,-1) -- (14.4,-1) node [below, midway] {$2018$年};\n\\end{tikzpicture}\n\\end{center}\n在统计学中, 同比是指本期统计数据与上一年同期统计数据相比较, 例如$2016$年第二季度与$2015$年第二季度相比较; 环比是指本期统计数据与上期统计数据相比较, 例如$2015$年第二季度与$2015$年第一季度相比较.\n根据上述信息, 下列结论中正确的是\\bracket{20}.\n\\twoch{2015 年第三季度环比有所提高}{2016 年第一季度同比有所提高}{2017 年第三季度同比有所提高}{2018 年第一季度环比有所提高}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -394812,7 +396186,9 @@ "id": "014983", "content": "某居民小区为缓解业主停车难的问题, 拟对小区内一块扇形空地$AOB$进行改造. 如图所示, 平行四边形$OMPN$区域为停车场, 其余部分建成绿地, 点$P$在围墙$AB$弧上, 点$M$和点$N$分别在道路$OA$和道路$OB$上, 且$OA=90$米, $\\angle AOB=\\dfrac{\\pi}{3}$, 设$\\angle POB=\\theta$, 停车场$OMPN$的面积为$S$.\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (0,0) node [left] {$B$} arc (180:140:3) node [left] {$P$} coordinate (P) arc (140:120:3) node [above] {$A$} -- (3,0) node [right] {$O$}-- cycle;\n\\draw [dashed] (3,0) -- (P);\n\\draw (P) --++ ({3/sin(60)*sin(20)},0) node [right] {$M$};\n\\draw (P) --++ (-60:{3/sin(60)*sin(40)}) node [below] {$N$};\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\theta=\\dfrac{\\pi}{12}$时, 求$S$的值;(结果精确到$0.1$平方米)\\\\\n(2) 写出$S$关于$\\theta$的函数关系式, 并求当$\\theta$为何值时, $S$取到最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -394832,7 +396208,9 @@ "id": "014984", "content": "据相关数据统计, 至$2021$年底全国已开通5G基站$140$万个, 部分省市的政府工作报告将``推进5G通信网络建设''列入$2022$年的重点工作, $2022$年一月份全国开通5G基站$4$万个.\\\\\n(1) 如果从$2022$年$2$月份起, 每个月比上一个月多开通$2000$个, 那么, 到$2022$年底全国共开通5G基站多少万个;(结果精确到$0.1$万个)\\\\\n(2) 如果$2022$年计划开通5G基站$60$万个, 并且自$2023$年起每年新开通的基站数量比上一年增加$x \\%$, 若到$2024$年底全国开通的5G基站总数至少达到$500$万个, 求$x$的最小值. (结果精确到$0.01$)", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -394852,7 +396230,9 @@ "id": "014985", "content": "某民营企业开发出了一种新产品, 预计能获得$50$万元到$1500$万元的经济收益. 企业财务部门研究对开发该新产品的团队进行奖励, 并讨论了一个奖励方案: 奖金金额$y$(单位: 万元) 随经济收益$x$(单位: 万元) 的增加而严格增加, 且$y>0$, 奖金金额不超过$20$万元.\\\\\n(1) 请你为该企业构建一个$y$关于$x$的函数模型, 并说明你的函数模型符合企业奖励要求的理由;\\\\\n(2) 若该企业采用函数$y=\\begin{cases}\\dfrac{1}{50} x+1, & 50 \\leq x \\leq 500, \\\\ 19+\\dfrac{1-a}{x}, & 500=latex]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw [domain = -2:2, pattern = horizontal lines] plot ({sqrt(\\x*\\x+1)},\\x) -- (2,2) -- (0,0) -- (2,-2) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$4\\pi$", "solution": "", @@ -394918,7 +396303,10 @@ "id": "014988", "content": "我国南宋时期的数学家杨辉, 在他$1261$年所著的《详解九章算法》一书中, 以如下图所示的三角形图表的方式呈现了一些二项展开式中的二项式系数, 此图称为``杨辉三角'', 也称为``贾宪三角''. 在此图中, 从第三行开始, 首尾两数为$1$, 其他各数均为它``肩上''两数之和.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\foreach \\i in {0,1,...,6}\n{\\draw (\\i,0) --++ (60:{6-\\i});\n\\draw (\\i,0) --++ (120:\\i);\n\\draw (60:\\i) --++ ({6-\\i},0);};\n\\foreach \\i/\\j in {0/1,1/6,2/15,3/20,4/15,5/6,6/1}\n{\\draw [fill=white] (\\i,0) circle (0.4) node {\\tiny\\zhnumber{\\j}};};\n\\foreach \\i/\\j in {0/1,1/5,2/10,3/10,4/5,5/1}\n{\\draw [fill=white] (60:1) ++ (\\i,0) circle (0.4) node {\\tiny\\zhnumber{\\j}};};\n\\foreach \\i/\\j in {0/1,1/4,2/6,3/4,4/1}\n{\\draw [fill=white] (60:2) ++ (\\i,0) circle (0.4) node {\\tiny\\zhnumber{\\j}};};\n\\foreach \\i/\\j in {0/1,1/3,2/3,3/1}\n{\\draw [fill=white] (60:3) ++ (\\i,0) circle (0.4) node {\\tiny\\zhnumber{\\j}};};\n\\foreach \\i/\\j in {0/1,1/2,2/1}\n{\\draw [fill=white] (60:4) ++ (\\i,0) circle (0.4) node {\\tiny\\zhnumber{\\j}};};\n\\foreach \\i/\\j in {0/1,1/1}\n{\\draw [fill=white] (60:5) ++ (\\i,0) circle (0.4) node {\\tiny\\zhnumber{\\j}};};\n\\foreach \\i/\\j in {0/1}\n{\\draw [fill=white] (60:6) ++ (\\i,0) circle (0.4) node {\\tiny\\zhnumber{\\j}};};\n\\end{tikzpicture}\n\\end{center}\n(1) 如图, 以``杨辉三角''第三行第一个数为首项, 从左上至右下, 将第三斜列各数取出按原来的顺序排列得一数列: $1,3,6,10,15, \\cdots$, 记作数列$\\{a_n\\}$, 写出$a_n$与$a_{n-1}$($n \\in \\mathbf{N}$, $n \\geq 2$)的递推关系, 并求其通项公式;\\\\\n(2) 对于 (1) 中的数列$\\{a_n\\}$, 若数列$\\{b_n\\}$满足$b_1+\\dfrac{1}{2} b_2+\\dfrac{1}{3} b_3+\\cdots+\\dfrac{1}{n} b_n=2 a_n$($n \\in \\mathbf{N}$, $n\\ge 1$), 求数列$\\{b_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -394938,7 +396326,9 @@ "id": "014989", "content": "数学中有许多形状优美的曲线, 如星形线. 设定大圆的半径是小圆半径的$4$倍, 让小圆在大圆内部沿着大圆的圆周滚动, 小圆圆周上的任一点形成的轨迹即为星形线. 已知小圆圆周上某一点$P$形成的星形线$C$的方程为$x^{\\frac{2}{3}}+y^{\\frac{2}{3}}=a^{\\frac{2}{3}}$($a>0$), 有如下结论:\\\\\n\\textcircled{1} 曲线$D: |x|+|y|=a$的周长大于星形线$C$的周长;\\\\\n\\textcircled{2} 星形线$C$上任意两点间距离的最大值为$2 a$;\\\\\n\\textcircled{3} 星形线$C$与圆$E: x^2+y^2=\\dfrac{a^2}{4}$有且仅有$4$个公共点.\\\\\n其中所有正确结论的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}", "solution": "", @@ -394963,7 +396353,9 @@ "id": "014990", "content": "对于函数$f(x)$和实数$t$, 若在其定义域内存在实数$x_0$, 使得$f(x_0+t)=f(x_0)+f(t)$成立, 则称$f(x)$是``$t$跃点''函数, 并称$x_0$是函数$f(x)$的``$t$跃点''.\\\\\n(1) 求证: 函数$f(x)=2^x+3 x^2$, $x \\in[0,3]$是``$2$跃点''函数;\\\\\n(2) 若函数$g(x)=x^3-\\dfrac{a}{2} x+3 a$, $x \\in(-3,+\\infty)$是``$1$跃点''函数, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $[-\\dfrac 14,+\\infty)$", "solution": "", @@ -394989,7 +396381,10 @@ "id": "014991", "content": "若实系数一元二次方程$a x^2+b x+c=0$在复数集$\\mathbf{C}$内的根为$x_1$、$x_2$, 则有$a(x-x_1)(x-x_2)=a x^2-a(x_1+x_2) x+a x_1 x_2=0$, 所以$x_1+x_2=-\\dfrac{b}{a}$, $x_1 x_2=\\dfrac{c}{a}$(韦达定理), 类比此方法求解如下问题: 设实系数一元三次方程$a x^3+b x^2+c x+d=0$在复数集$\\mathbf{C}$内的根为$x_1$、$x_2$、$x_3$, 则$\\dfrac{1}{x_1}+\\dfrac{1}{x_2}+\\dfrac{1}{x_3}$的值为\\bracket{20}.\n\\fourch{$-\\dfrac{c}{d}$}{$\\dfrac{b}{d}$}{$\\dfrac{c}{a}$}{$\\dfrac{d}{a}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第五单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -395015,7 +396410,9 @@ "id": "014992", "content": "``阿基米德多面体''也称为半正多面体, 是由边数不全相同的正多边形为面围成的多面体, 它体现了数学的对称美. 如图, 将正方体沿交于一顶点的三条棱的中点截去一个三棱锥, 共可截去八个三棱锥, 得到八个面为正三角形、六个面为正方形的``阿基米德多面体'', 则异面直线$AB$与$CD$所成角的大小是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(220:0.5cm)}]\n\\draw (-1,0,0) coordinate (A);\n\\draw (0,0,1) node [below] {$C$} coordinate (B);\n\\draw (1,0,0) coordinate (C);\n\\draw (0,0,-1) coordinate (D);\n\\draw (-1,1,1) coordinate (E);\n\\draw (1,1,1) node [left] {$D$} coordinate (F);\n\\draw (1,1,-1) coordinate (G);\n\\draw (-1,1,-1) coordinate (H);\n\\draw (-1,2,0) node [left] {$A$} coordinate (M);\n\\draw (0,2,1) coordinate (N);\n\\draw (1,2,0) coordinate (P);\n\\draw (0,2,-1) node [above] {$B$} coordinate (Q);\n\\draw (A)--(B)--(E)--cycle (B)--(C)--(F)--cycle (C)--(G) (G)--(P) (F)--(P)--(N) --cycle (M)--(N)--(E)--cycle (M)--(Q)--(P);\n\\draw [dashed] (A)--(D)--(H)--cycle (C)--(D)--(G) (M)--(H)--(Q)--(G);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$30^{\\circ}$}{$45^{\\circ}$}{$60^{\\circ}$}{$120^{\\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -395041,7 +396438,9 @@ "id": "014993", "content": "对于非零向量$\\overrightarrow {a}$、$\\overrightarrow {b}$, 定义一种向量的运算: $\\overrightarrow {a} \\otimes \\overrightarrow {b}=\\dfrac{\\overrightarrow {a} \\cdot \\overrightarrow {b}}{\\overrightarrow {b} \\cdot \\overrightarrow {b}}$. 设集合$P=\\{\\dfrac{n}{2} | n \\in \\mathbf{N}\\}$, 若非零向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$\\overrightarrow {a} \\otimes \\overrightarrow {b} \\in P$, $\\overrightarrow {b} \\otimes \\overrightarrow {a} \\in P$, 且其夹角$\\theta \\in(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$, 则$\\overrightarrow {a} \\otimes \\overrightarrow {b}$的所有可能的值组成的集合为\\bracket{20}.\n\\fourch{$\\{\\dfrac{3}{2}, \\dfrac{5}{2}\\}$}{$\\{\\dfrac{1}{2}, \\dfrac{3}{2}\\}$}{$\\{1\\}$}{$\\{\\dfrac{1}{2}\\}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -395063,7 +396462,9 @@ "id": "014994", "content": "在平面直角坐标系$xOy$中, 已知任意角$\\theta$的顶点与原点$O$重合, 始边与$x$轴的正半轴重合. 若角$\\theta$的终边经过点$P(x_0, y_0)$且$|OP|=r$($r>0$), 定义$\\mathrm{sicos} \\theta=\\dfrac{x_0+y_0}{r}$, 称``函数$y=\\mathrm{sicos} x$, $x \\in \\mathbf{R}$''为``正余弦函数''. 对于正余弦函数, 有如下结论:\\\\\n\\textcircled{1} 该函数是偶函数;\\\\\n\\textcircled{2} 该函数图像的一个对称中心是点$(\\dfrac{3 \\pi}{4}, 0)$;\\\\\n\\textcircled{3} 该函数的单调递减区间是$[2 k \\pi-\\dfrac{3 \\pi}{4}, 2 k \\pi+\\dfrac{\\pi}{4}]$($k \\in \\mathbf{Z}$);\\\\\n\\textcircled{4} 该函数的图像与直线$y=\\dfrac{3}{2}$没有公共点.\\\\\n其中所有正确结论的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{4}", "solution": "", @@ -395089,7 +396490,9 @@ "id": "014995", "content": "如果有穷数列$a_1, a_2, a_3, \\cdots, a_n$($n$为正整数) 满足条件$a_1=a_n$, $a_2=a_{n-1}$, $\\cdots$, $a_n=a_1$, 即$a_i=a_{n-i+1}$($i=1,2, \\cdots, n$), 我们称其为``对称数列''. 例如, 由组合数组成的数列$\\mathrm{C}_m^0, \\mathrm{C}_m^1, \\cdots, \\mathrm{C}_m^m$就是``对称数列''.\\\\\n(1) 设$\\{b_n\\}$是项数为$7$的``对称数列'', 其中$b_1, b_2, b_3, b_4$是等差数列, 且$b_1=2$, $b_4=11$. 依次写出$\\{b_n\\}$的每一项;\\\\\n(2) 设$\\{c_n\\}$是项数为$2 k-1$的``对称数列''($k$是大于$1$的正整数), 其中$c_k, c_{k+1}, \\cdots, c_{2 k-1}$是首项为$50$, 公差为$-4$的等差数列. 记$\\{c_n\\}$各项的和为$S_{2 k-1}$. 当$k$为何值时, $S_{2 k-1}$取得最大值? 并求出$S_{2 k-1}$的最大值;\\\\\n(3) 对于确定的$m$($m$是大于$1$的正整数), 写出所有项数不超过$2 m$的``对称数列'', 使得$1,2,2^2, \\cdots, 2^{m-1}$依次是该数列中连续的项; 当$m>1500$时, 求其中一个``对称数列''前$2022$项的和$S_{2022}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) $2,5,8,11,8,5,2$;\\\\\n(2) $k=13$时$S_{2k-1}$取到最大, 最大值为$626$;\\\\\n(3) 共有四种满足要求的数列:\\\\\n第一种: $1,2,\\cdots,2^{m-2},2^{m-1},2^{m-1},2^{m-2},\\cdots,2,1$($2m$项), $S_{2022}=\\begin{cases}2^{m+1}-2^{2m-2022}-1, & 15000$恒成立, 若记$y=f(x)$的零点个数为$n$, 则$n$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$17$", "solution": "", @@ -402715,7 +404140,9 @@ "id": "015259", "content": "在空间中, $O$是一个定点, $\\overrightarrow{OA}$、$\\overrightarrow{OB}$、$\\overrightarrow{OC}$是给定的三个不共面的向量, 且它们两两之间的夹角都是锐角. 若向量$\\overrightarrow{OP}$满足$|\\overrightarrow{OA} \\cdot \\overrightarrow{OP}|=|\\overrightarrow{OA}|$, $|\\overrightarrow{OB} \\cdot \\overrightarrow{OP}|=2|\\overrightarrow{OB}|$, $|\\overrightarrow{OC} \\cdot \\overrightarrow{OP}|=3|\\overrightarrow{OC}|$, 则满足题意的点$P$的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$8$", "solution": "", @@ -402748,7 +404175,9 @@ "id": "015260", "content": "在$(a+b)^{10}$的二项展开式中, 第$3$项为\\bracket{20}.\n\\fourch{$\\mathrm{C}_{10}^2 a^8 b^2$}{$\\mathrm{C}_{10}^2 a^2 b^8$}{$\\mathrm{C}_{10}^3 a^7 b^3$}{$\\mathrm{C}_{10}^3 a^3 b^7$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -402781,7 +404210,9 @@ "id": "015261", "content": "下列以$t$为参数的参数方程中, 能够表示方程$x y=1$的是\\bracket{20}.\n\\fourch{$\\begin{cases}x=t^{\\frac{1}{2}}, \\\\ y=t^{-\\frac{1}{2}}\\end{cases}$}{$\\begin{cases}x=\\sin t, \\\\ y=\\dfrac{1}{\\sin t}\\end{cases}$}{$\\begin{cases}x=\\cos t, \\\\ y=\\dfrac{1}{\\cos t}\\end{cases}$}{$\\begin{cases}x=\\tan t, \\\\ y=\\dfrac{1}{\\tan t}\\end{cases}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -402814,7 +404245,9 @@ "id": "015262", "content": "计算: $\\displaystyle\\lim _{h \\to 0} \\dfrac{\\sin 2(x+h)-\\sin (2 x)}{h}=$\\bracket{20}.\n\\fourch{$0$}{$\\cos 2 x$}{$2 \\cos x$}{$2 \\cos 2 x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -402847,7 +404280,9 @@ "id": "015263", "content": "设函数$y=f(x)$在定义域$D$上的导数值均存在, 其导函数为$y=f'(x)$, 关于这两个函数的图像, 有如下两个命题:\\\\\n命题$p$: 若$y=f'(x)$的图像关于直线$x=x_0$对称, 则$y=f(x)$的图像也关于直线$x=x_0$对称;\\\\\n命题$q$: 若$y=f'(x)$是减函数, 且其图像向右方无限延伸时会与$x$轴无限趋近, 则函数$y=f(x)$是增函数, 且其图像向右方无限延伸时也会存在一条平行或重合于$x$轴的直线$l$, 使得$y=f(x)$的图像与$l$无限趋近.\\\\\n下列判断正确的是\\bracket{20}.\n\\twoch{$p$和$q$都是真命题}{$p$和$q$都是假命题}{$p$是真命题, $q$是假命题}{$p$是假命题, $q$是真命题}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -402880,7 +404315,9 @@ "id": "015264", "content": "已知函数$f(x)=\\mathrm{e}^x-x$, $x \\in \\mathbf{R}$.\\\\\n(1) 求$f'(0)$的值, 并写出该函数在点$(0, f(0))$处的切线方程;\\\\\n(2) 求函数$y=f(x)$在区间$[-1,1]$上的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $y=1$; (2) 最大值为$\\mathrm{e}-1$, 最小值为$1$", "solution": "", @@ -402913,7 +404350,9 @@ "id": "015265", "content": "如图, 在三棱锥$P-ABC$中, $PA \\perp$平面$ABC$, $\\angle BAC=90^{\\circ}$, $|AB|=|AP|=|AC|=2$, $M$、$N$分别为$PA$、$PC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(215:0.5cm)}]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(P)$) node [above right] {$M$} coordinate (M);\n\\draw ($(P)!0.5!(C)$) node [right] {$N$} coordinate (N);\n\\draw (P)--(B)--(C)--cycle(B)--(N);\n\\draw [dashed] (B)--(M)--(N)(P)--(A)--(B)(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$BN$与平面$ABC$所成角的大小;\\\\\n(2) 求平面$MNB$与平面$ABC$所成二面角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\arcsin\\dfrac{\\sqrt{6}}6$; (2) $\\arccos\\dfrac{2\\sqrt{5}}5$或$\\pi-\\arccos\\dfrac{2\\sqrt{5}}5$", "solution": "", @@ -402946,7 +404385,9 @@ "id": "015266", "content": "设$a \\in \\mathbf{R}$. 函数$f(x)=a x^2+\\dfrac{6}{x}$, $x>0$.\\\\\n(1) 当$a=3$时, 求函数$y=f(x)$的单调区间;\\\\\n(2) 设常数$c>0$. 当$a=0$时, 关于$x$的不等式$f(x)+2 x^3 \\geq 13 c$在$[c,+\\infty)$恒成立, 求$c$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 在$(0,1]$上是严格减函数, 在$[1,+\\infty)$上是严格增函数; (2) $(0,\\dfrac{8}{13}]\\cup [\\sqrt{6},+\\infty)$", "solution": "", @@ -402979,7 +404420,9 @@ "id": "015267", "content": "设实数$m \\neq 0$. 对任意给定的实数$x$, 都有$(3+m x)^{99}=a_0+a_1 x+a_2 x^2+\\ldots+a_{99} x^{99}$.\\\\\n(1) 当$m=1$时, 求$a_{97}+a_{98}$的值;\\\\\n(2) 若$m$是整数, 且满足$6<\\dfrac{a_5}{a_4}<7$成立, 求$a_0+a_1+a_2+\\cdots+a_{99}$的值;\\\\\n(3) 当$m \\in(12,13)$时, 根据$m$的取值, 讨论$(3+m x)^{99}$的二项展开式中系数最大的项是第几项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "(1) $43956$; (2) $4^{99}$; (3) 当$m\\in (12,\\dfrac{243}{19})$时, 最大项为第$81$项; 当$m=\\dfrac{243}{19}$时, 最大项为第$81$项与第$82$项; 当$m\\in (\\dfrac{243}{19},13)$时, 最大项为第$82$项", "solution": "", @@ -403012,7 +404455,9 @@ "id": "015268", "content": "设常数$\\lambda \\in(0,1)$. 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, 点$Q$满足$\\overrightarrow{D_1Q}=\\lambda \\overrightarrow{D_1C_1}$, 点$M$、$N$分别为棱$AD$、$AB$上的动点 (均不与顶点重合), 且满足$|\\overrightarrow{AN}|=\\lambda|\\overrightarrow{DM}|$, 记$|\\overrightarrow{DM}|=a$. 以$A$为原点, 分别以$\\overrightarrow{AB}$、$\\overrightarrow{AD}$与$\\overrightarrow{AA_1}$的方向为$x$、$y$与$z$轴的正方向, 建立如图空间直角坐标系.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [above right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [above right] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D1);\n\\draw (A) ++ (0,\\l,0) node [above left] {$A_1$} coordinate (A1);\n\\draw (B1) -- (C1) -- (D1) -- (A1) -- cycle;\n\\draw (B) -- (B1) (C) -- (C1) (D) -- (D1);\n\\draw [dashed] (A) -- (A1);\n\\draw [->] (B) -- ($(A)!1.4!(B)$) node [right] {$x$};\n\\draw [->] (D) -- ($(A)!1.4!(D)$) node [below] {$y$};\n\\draw [->] (A1) -- ($(A)!1.4!(A1)$) node [right] {$z$};\n\\filldraw ($(C1)!0.6!(D1)$) node [below] {$Q$} coordinate (Q) circle (0.03);\n\\filldraw ($(A)!0.6!(D)$) node [below left] {$M$} coordinate (M) circle (0.03);\n\\filldraw ($(A)!0.24!(B)$) node [left] {$N$} coordinate (N) circle (0.03);\n\\draw [dashed] (A1)--(M)--(N)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 用$\\lambda$和$a$表示点$M$、$N$、$Q$的坐标;\\\\\n(2) 设$a=\\dfrac{1}{2}$, 若$\\angle MA_1N=\\angle AMN$, 求常数$\\lambda$的值;\\\\\n(3) 记$Q$到平面$MA_1N$的距离为$h(a)$. 求证: 若关于$a$的方程$h(a)=\\dfrac{\\sqrt{5}}{2} \\lambda$在$(0,1)$上恰有两个不同的解, 则这两个解中至少有一个大于$\\dfrac{1}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $M(0,1-a,0)$, $N(\\lambda a,0,0)$, $Q(\\lambda,1,1)$; (2) $\\lambda = \\dfrac{2\\sqrt{11}}{11}$; (3) 证明略", "solution": "", @@ -403045,7 +404490,9 @@ "id": "015269", "content": "函数$y=\\log _2 x$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(0,+\\infty)$", "solution": "", @@ -403078,7 +404525,9 @@ "id": "015270", "content": "函数$y=2 \\sin 3 x+4$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$\\dfrac{2\\pi}{3}$", "solution": "", @@ -403111,7 +404560,9 @@ "id": "015271", "content": "已知集合$A=\\{1\\}$, $B=\\{x | x^2+2 x+a=0,\\ x \\in \\mathbf{R}\\}$, 且$A \\subset B$, 则实数$a$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$-3$", "solution": "", @@ -403144,7 +404595,9 @@ "id": "015272", "content": "扇形$OAB$所在圆的半径长为$1$ , $\\overset{\\frown}{AB}$所对的圆心角$\\angle AOB$大小为$\\dfrac{\\pi}{3}$, 则扇形$OAB$的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{6}$", "solution": "", @@ -403177,7 +404630,9 @@ "id": "015273", "content": "指数函数$y=(a-1)^x$在区间$[0,2]$上的最大值为$4$, 则实数$a$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -403210,7 +404665,9 @@ "id": "015274", "content": "函数$y=\\sin (x+\\dfrac{\\pi}{2})$的单调减区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[2k\\pi,2k\\pi+\\pi]$, $k\\in \\mathbf{Z}$", "solution": "", @@ -403243,7 +404700,9 @@ "id": "015275", "content": "已知$y=f(x)$是定义域为$\\mathbf{R}$的奇函数, 当$x>0$时, $f(x)=1-2 x$, 则当$x<0$时, $y=f(x)$的表达式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$f(x)=-1-2x$", "solution": "", @@ -403276,7 +404735,9 @@ "id": "015276", "content": "方程$|x-3|+|x-4|=1$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$[3,4]$", "solution": "", @@ -403309,7 +404770,9 @@ "id": "015277", "content": "对任意实数$x$, 定义$[x]$表示小于等于$x$的最大整数, 例如$[1.8]=1$, $[-1.8]=-2$, 则方程$x^2-[x]-1=0$的解的个数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -403342,7 +404805,9 @@ "id": "015278", "content": "某河道水上游览航线一经开放就受到公众喜爱, 其中有一条航线是: 从码头A出发顺流而下到码头B, 然后不做停留原路返回到码头A(不计调头时间). 假设游船在静水中的船速恒定不变, 且整个航程中途不做停靠, 以下结论正确的是\\blank{50}(填序号).\\\\\n\\textcircled{1} 水流速度越大整个航程所需时间越长;\\\\\n\\textcircled{2} 水流速度越大整个航程所需时间越短;\\\\\n\\textcircled{3} 水流速度大小不会影响整个航程所需时间.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "\\textcircled{1}", "solution": "", @@ -403375,7 +404840,10 @@ "id": "015279", "content": "已知函数$y=f(x)$的表达式是$f(x)=x^4+4^{|x|}$, 若$\\alpha \\in(0, \\pi)$, 且$f(\\sin \\alpha)b$''是``$a^2>b^2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -403474,7 +404946,9 @@ "id": "015282", "content": "函数$y=\\dfrac{\\ln |x|}{x}$的大致图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-3.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = 0.33:3.5, samples = 100] plot (\\x,{ln(\\x)/\\x});\n\\draw [domain = 0.33:3.5, samples = 100] plot (-\\x,{ln(\\x)/(-\\x)});\n\\foreach \\i in {1,2,3}\n{\\draw (\\i,0.2) -- (\\i,0) (-\\i,0.2) -- (-\\i,0);\n\\draw (0.2,\\i) -- (0,\\i) (0.2,-\\i) -- (0,-\\i);};\n\\draw (1,0) node [below] {$1$} (-1,0) node [below] {$-1$};\n\\draw (0,1) node [left] {$1$} (0,-1) node [left] {$-1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-3.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = 0.33:3.5, samples = 100] plot (\\x,{ln(\\x)/\\x});\n\\draw [domain = 0.33:3.5, samples = 100] plot (-\\x,{-ln(\\x)/(-\\x)});\n\\foreach \\i in {1,2,3}\n{\\draw (\\i,0.2) -- (\\i,0) (-\\i,0.2) -- (-\\i,0);\n\\draw (0.2,\\i) -- (0,\\i) (0.2,-\\i) -- (0,-\\i);};\n\\draw (1,0) node [below] {$1$} (-1,0) node [below] {$-1$};\n\\draw (0,1) node [left] {$1$} (0,-1) node [left] {$-1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-3.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = 0.33:3.5, samples = 100] plot (\\x,{-ln(\\x)/\\x});\n\\draw [domain = 0.33:3.5, samples = 100] plot (-\\x,{ln(\\x)/(-\\x)});\n\\foreach \\i in {1,2,3}\n{\\draw (\\i,0.2) -- (\\i,0) (-\\i,0.2) -- (-\\i,0);\n\\draw (0.2,\\i) -- (0,\\i) (0.2,-\\i) -- (0,-\\i);};\n\\draw (1,0) node [below] {$1$} (-1,0) node [below] {$-1$};\n\\draw (0,1) node [left] {$1$} (0,-1) node [left] {$-1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-3.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = 0.33:3.5, samples = 100] plot (\\x,{-ln(\\x)/\\x});\n\\draw [domain = 0.33:3.5, samples = 100] plot (-\\x,{-ln(\\x)/(-\\x)});\n\\foreach \\i in {1,2,3}\n{\\draw (\\i,0.2) -- (\\i,0) (-\\i,0.2) -- (-\\i,0);\n\\draw (0.2,\\i) -- (0,\\i) (0.2,-\\i) -- (0,-\\i);};\n\\draw (1,0) node [below] {$1$} (-1,0) node [below] {$-1$};\n\\draw (0,1) node [left] {$1$} (0,-1) node [left] {$-1$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -403507,7 +404981,9 @@ "id": "015283", "content": "已知$\\triangle ABC$的三个内角$A$、$B$、$C$满足$\\sin C+\\sin (B-A)=\\sin 2A$, 则$\\triangle ABC$的形状是\\bracket{20}.\n\\twoch{等腰三角形}{直角三角形}{等腰直角三角形}{等腰三角形或直角三角形}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -403540,7 +405016,9 @@ "id": "015284", "content": "已知对任意正数$a$、$b$、$c$, 当$a+b+c=1$时, 都有$2^a+2^b+2^c=latex]\n\\draw (0,0) coordinate (O);\n\\draw (O) ++ (-1,{2/sqrt(3)}) node [above] {$A$} coordinate (A);\n\\draw (O) ++ (1,{2/sqrt(3)}) node [above] {$B$} coordinate (B);\n\\draw (B) ++ (-60:1) node [right] {$C$} coordinate (C);\n\\draw (O) ++ (-100:{sqrt(7)/sqrt(3)}) node [below] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求线段$AC$的长;\\\\\n(2) 求四边形$ABCD$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\sqrt{7}$; (2) $\\dfrac{9\\sqrt{3}}{4}$", "solution": "", @@ -403672,7 +405156,9 @@ "id": "015288", "content": "在月亮和太阳的引力作用下, 海水水面发生的周期性涨落现象叫做潮汐. 一般早潮叫潮, 晩潮叫汐. 受潮汐影响, 港口的水深也会相应发生变化. 下图记录了某港口某一天整点时刻的水深$y$(单位: 米) 与时间$x$(单位: 时)的大致关系:\n假设$4$月份的每一天水深与时间的关系都符合图中所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (25,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,13) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {1,2,...,24} \n{\\draw [dotted] (\\i,-1) -- (\\i,13);}\n\\foreach \\i in {1,2,...,48}\n{\\draw ({\\i/2},0.2) -- ({\\i/2},0);};\n\\foreach \\i in {2,4,...,24}\n{\\draw (\\i,0) node [below] {$\\i$};};\n\\foreach \\i in {1,2,...,12}\n{\\draw [dotted] (-1,\\i) -- (25,\\i);\n\\draw (0,\\i) node [left] {$\\i$};};\n\\foreach \\i in {1,2,...,24}\n{\\draw (0.2,{\\i/2}) -- (0,{\\i/2});};\n\\foreach \\i in {0,1,...,24}\n{\\filldraw (\\i,{8+3*sin((\\i-11)*30)}) circle (0.08);};\n\\end{tikzpicture}\n\\end{center}\n(1) 请运用函数模型$y=A \\sin (\\omega x+\\varphi)+h$($A>0$, $\\omega>0$,$-\\dfrac{\\pi}{2}<\\varphi<\\dfrac{\\pi}{2}$, $h \\in \\mathbf{R}$), 根据以上数据写出水深$y$与时间$x$的函数的近似表达式;\\\\\n(2) 根据该港口的安全条例, 要求船底与水底的距离必须不小于$3.5$米, 否则该船必须立即离港. 一艘船满载货物, 吃水 (即船底到水面的距离) $6$米, 计划明天进港卸货.\\\\\n(i) 求该船可以进港的时间段;\\\\\n(ii) 该船今天会到达港口附近, 明天$0$点可以及时进港并立即开始卸货. 已知卸货时吃水深度以每小时$0.3$米的速度匀速减少, 卸完货后空船吃水$3$米. 请设计一个卸货方案, 在保证严格遵守该港口安全条例的前提下, 使该船明天尽早完成卸货(不计停靠码头和驶离码头所需时间).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $y=3\\sin(\\dfrac{\\pi}{6}x+\\dfrac{\\pi}{6})+8$, $x\\in [0,24]$; (2) 可以进港的时间段为0点至6点, 以及12点至16点; 在0点进港开始卸货, 5点暂时驶离港口, 11点返回港口继续卸货, 16点完成卸货任务", "solution": "", @@ -403705,7 +405191,9 @@ "id": "015289", "content": "已知函数$y=F(x)$与$y=f(x)$的定义域为$\\mathbf{R}$, 若对任意区间$[u, v] \\subseteq \\mathbf{R}$, 存在$p \\in[u, v]$且$q \\in[u, v]$, 使$f(p) \\leq \\dfrac{F(u)-F(v)}{u-v} \\leq f(q)$, 则称$y=f(x)$是$y=F(x)$的生成函数.\\\\\n(1) 求证: $f(x)=2 x$是$F(x)=x^2-3$的生成函数;\\\\\n(2) 若$f(x)=x^2+2$是$y=F(x)$的生成函数, 判断并证明$y=F(x)$的单调性;\\\\\n(3) 若$y=f(x)$是$y=F(x)$的生成函数, 实数$a \\neq 0$, 求$y=F(a x+b)$的一个生成函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $y=F(x)$是$(-\\infty,+\\infty)$上的严格增函数, 证明略; (3) $y=af(ax+b)$", "solution": "", @@ -404898,7 +406386,9 @@ "id": "015332", "content": "若关于$x$的方程$(\\dfrac{1}{2})^x+m=\\sqrt{x+1}$在实数范围内有解, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404918,7 +406408,9 @@ "id": "015333", "content": "设$a>0$, 已知第一象限的两个点$P_1(x_1, y_1)$, $P_2(x_2, y_2)$分别在双曲线$\\Gamma_1: \\dfrac{x^2}{a^2}-y^2=1$和$\\Gamma_2: a^2 x^2-y^2=1$的右\n支上, 则$\\dfrac{x_1 x_2}{y_1 y_2}$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404938,7 +406430,9 @@ "id": "015334", "content": "设复数$z_1, z_2$满足: $z_1=\\mathrm{i} \\cdot \\overline{z_2}$, 且$|z_1-1|=1$, 其中$\\mathrm{i}$是虚数单位, 则$|z_1-z_2|$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404958,7 +406452,9 @@ "id": "015335", "content": "已知定义域为区间$D$的函数$y=f(x)$, 其导函数为$y=f'(x)$, 满足: 对任意$x \\in D$, 都有$f'(x)<1$. 证明: 方程$f(x)-x=0$至多只有一个实数解.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404978,7 +406474,9 @@ "id": "015336", "content": "设$S_n$是无穷数列$\\{a_n\\}$的前$n$项和, 若对任意正整数$n$, 不等式$\\dfrac{S_n}{n}<\\dfrac{S_{n+1}}{n+1}$恒成立, 则称数列$\\{a_n\\}$为和谐数列. 给出下列$3$个命题:\\\\\n\\textcircled{1} 若对任意正整数$n$, 均有$a_n=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(C)!0.5!(C1)$) node [right] {$E$} coordinate (E);\n\\draw ($(B1)!0.3!(C)$) node [below] {$Q$} coordinate (Q);\n\\draw ($1/3*(A1)+1/3*(B1)+1/3*(C1)$) node [left] {$P$} coordinate (P);\n\\draw (B1)--(C)(E)--(Q);;\n\\draw [dashed] (Q)--(P)--(E);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$2 \\sqrt{2}$}{$\\sqrt{10}$}{$\\sqrt{11}$}{$2 \\sqrt{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -405018,7 +406518,9 @@ "id": "015338", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{2}+y^2=1$的右焦点为$F_2$, 过$F_2$作两条不重合的动直线$l_1$、$l_2$, 其中$l_1$与$\\Gamma$交于$A$、$B$两点, $l_2$与$\\Gamma$交于$C$、$D$两点, $M$、$N$分别是线段$AB$、$CD$的中点. 若直线$MN$过定点$(\\dfrac{2}{3}, 0)$, 试问$l_1$与$l_2$的夹角是否为定值? 如果是, 求出该定值; 如果不是, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405038,7 +406540,9 @@ "id": "015339", "content": "设定义域为$\\mathbf{R}$的函数$y=f(x)$是增函数, 且存在实数$u, v$, 使得$u\\ln x$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405158,7 +406672,9 @@ "id": "015345", "content": "已知$y=f(x)$是定义域为$\\mathbf{R}$的奇函数, 且$x \\leq 0$时, $f(x)=\\mathrm{e}^x-1$, 其中$\\mathrm{e}$为自然对数的底数, 则$y=f(x)$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405178,7 +406694,10 @@ "id": "015346", "content": "如图, 在$\\triangle ABC$中, 点$D$、$E$是线段$BC$上两个动点, 且$\\overrightarrow{AD}+\\overrightarrow{AE}=x \\overrightarrow{AB}+y \\overrightarrow{AC}$, 则$\\dfrac{1}{x}+\\dfrac{9}{y}$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0) node [right] {$C$} coordinate (C);\n\\draw ($(B)!0.3!(C)$) node [below] {$D$} coordinate (D);\n\\draw ($(B)!0.65!(C)$) node [below] {$E$} coordinate (E);\n\\draw (1,2) node [above] {$A$} coordinate (A);\n\\draw [->] (A)--(B);\n\\draw [->] (A)--(C);\n\\draw [->] (A)--(D);\n\\draw [->] (A)--(E);\n\\draw (B)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405198,7 +406717,9 @@ "id": "015347", "content": "如图, 棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$的内切球为球$O, E$、$F$分别是棱$AB$和棱$CC_1$的中点, $G$在棱$BC$上移动, 则下列命题正确的个数是\\bracket{20}.\\\\\n\\textcircled{1} 存在点$G$, 使$OD$垂直于平面$EFG$;\\\\\n\\textcircled{2} 对于任意点$G, OA$平行于平面$EFG$;\\\\\n\\textcircled{3} 直线$EF$被球$O$截得的弦长为$\\sqrt{2}$;\\\\\n\\textcircled{4} 过直线$EF$的平面截球$O$所得的所有截面圆中, 半径最小的圆的面积为$\\dfrac{\\pi}{2}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [above right] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (A) ++ (0,\\l,0) node [above] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [right] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (D) -- (D1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (A) -- (A1);\n\\filldraw ($(A1)!0.5!(C)$) node [right] {$O$} coordinate (O) circle (0.03);\n\\draw [dashed] (O) circle (1) ellipse (1 and 0.25);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(C1)$) node [right] {$F$} coordinate (F);\n\\draw ($(B)!0.8!(C)$) node [below] {$G$} coordinate (G);\n\\draw (F)--(G);\n\\draw [dashed] (F)--(E)--(G);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{0}{1}{2}{3}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -405218,7 +406739,9 @@ "id": "015348", "content": "设$a \\in \\mathbf{R}$, 若集合$A=\\{(x, y) |(x+y)^2+x+y-2 \\leq 0\\}$, $B=\\{(x, y) |(x-a)^2+(y-2 a-1)^2 \\leq a^2-1\\}$, \n且$A \\cap B \\neq \\varnothing$, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405238,7 +406761,9 @@ "id": "015349", "content": "已知$a$、$b$、$\\alpha$、$\\beta \\in \\mathbf{R}$, 满足$\\sin \\alpha+\\cos \\beta=a$, $\\cos \\alpha+\\sin \\beta=b$, $00$), 若$E[X]=E[Y]$, 且$P(|Y|<1)=0.4$, 则$P(Y>3)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$0.1$", "solution": "", @@ -446757,7 +448304,9 @@ "id": "017232", "content": "已知甲袋中有$3$个白球和$2$个红球, 乙袋中有$2$个白球和$4$个红球. 若先随机取一只袋, 再从该袋中先后随机取$2$个球, 则在第一次取出的球是红球的前提下, 第二次取出的球是白球的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{17}{32}$", "solution": "", @@ -446790,7 +448339,10 @@ "id": "017233", "content": "函数$y=(1+\\cos x)^{2023}+(1-\\cos x)^{2023}$, $x \\in[-\\dfrac{2 \\pi}{3}, \\dfrac{2 \\pi}{3}]$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第三单元" + ], "genre": "填空题", "ans": "$[2,2^{2023}]$", "solution": "", @@ -446823,7 +448375,9 @@ "id": "017234", "content": "已知平面向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$满足$|\\overrightarrow {a}|=1$, $\\langle\\overrightarrow {a}, \\overrightarrow {b}\\rangle=\\langle 7 \\overrightarrow {a}-\\overrightarrow {c}, 9 \\overrightarrow {a}-\\overrightarrow {c}\\rangle=\\dfrac{\\pi}{6}$, 则$|\\overrightarrow {b}-\\overrightarrow {c}|$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$[\\dfrac{1}{2},+\\infty)$", "solution": "", @@ -446856,7 +448410,9 @@ "id": "017235", "content": "若$a<0\\dfrac{1}{b}$}{$-a>b$}{$a^2>b^2$}{$a^3a_m$''是``$\\{a_n\\}$是严格递增数列''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -446955,7 +448515,9 @@ "id": "017238", "content": "已知函数$f(x)$是定义在$\\mathbf{R}$上的连续可导函数, 其导函数为$f'(x)$, 且对任意$x \\in \\mathbf{R}$均有$f(x)-f(-x)=2 x$. 若当$x<0$时, $f'(x)>1$恒成立, 且$f(a-2)-f(1-2 a)>3 a-3$, 则实数$a$的取值范围是\\bracket{20}.\n\\fourch{$(-1,1)$}{$(1,+\\infty)$}{$(-\\infty,-1)$}{$(-\\infty,-1) \\cup(1,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -446988,7 +448550,9 @@ "id": "017239", "content": "直三棱柱$ABC-A_1B_1C_1$中, 底面$ABC$为等腰直角三角形, $AB \\perp AC$, $AB=AC=2$, $AA_1=4, M$是侧棱$CC_1$上一点, 设$MC=h$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (A) ++ (0,3,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,3,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,3,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(C)!0.3!(C_1)$) node [right] {$M$} coordinate (M);\n\\draw (B)--(C)--(C_1)--(B_1)--cycle (B_1)--(A_1)--(C_1)(B)--(M);\n\\draw [dashed] (B)--(A)--(C)(B)--(A_1)--(C)(A)--(M)(A)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$BM \\perp A_1C$, 求$h$的值;\\\\\n(2) 若$h=2$, 求直线$BA_1$与平面$ABM$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $h=1$; (2) $\\arcsin\\dfrac{\\sqrt{10}}{5}$", "solution": "(1) 由底面$ABC$为等腰直角三角形且 $AB \\perp AC$知$AB\\perp$平面$ACC_1A_1$, \\\\\n从而$BM$在平面$ACC_1A_1$上的投影为$AM$,\n故由$BM \\perp A_1C$ 知 $AM\\perp A_1C $, \\\\\n结合$AC=2$, $AA_1=4$ 得 $MC=1$, 即$h=1$.\\\\\n(2) 如图建系：以$A$为原点，分别以$\\overrightarrow{AB}$、$\\overrightarrow{AC}$、$\\overrightarrow{AA_1}$方向为$x$轴、 $y$轴、$z$ 轴正方向建立平面直角坐标系.\\\\\n$A(0,0,0),B(2,0,0),C(0,2,0),A_1(0,0,4),M(0,2,2)$,\n$\\overrightarrow{BA_1}=(-2,0,4),\\overrightarrow{AB}=(2,0,0),\\overrightarrow{AM}=(0,2,2),$\\\\\n设平面$ABM$的一个法向量为$\\overrightarrow{n}=(x,y,z)$, 则$\\begin{cases}\n2x=0,\\\\\n2y+2z=0.\n\\end{cases}$ 取$\\overrightarrow{n}=(0,1,-1),$设直线$BA_1$与平面$ABM$所成的角为$\\theta$, 则$\\sin\\theta=|\\cos \\langle\\overrightarrow{BA_1},\\overrightarrow{n}\\rangle|=\\dfrac{|\\overrightarrow{BA_1}\\cdot\\overrightarrow{n}|}{|\\overrightarrow{BA_1}|\\cdot|\\overrightarrow{n}|}=\\dfrac{\\sqrt{10}}{5}$, 故直线$BA_1$与平面$ABM$所成的角为$\\arcsin\\dfrac{\\sqrt{10}}{5}$.", @@ -447021,7 +448585,9 @@ "id": "017240", "content": "已知$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$.\\\\\n(1) 若$B=\\dfrac{\\pi}{3}$, $b=\\sqrt{5}$, $\\triangle ABC$的面积$S=\\sqrt{3}$, 求$a-c$的值;\\\\\n(2) 若$2 \\cos C(\\overrightarrow{BA} \\cdot \\overrightarrow{BC}+\\overrightarrow{AB} \\cdot \\overrightarrow{AC})=c^2$, 求角$C$的大小.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $a-c=\\pm 1$; (2) $C=\\dfrac{\\pi}{3}$", "solution": "(1) 由 $S=\\dfrac{1}{2}ac\\sin B=\\dfrac{\\sqrt{3}}{4}ac=\\sqrt{3}$知$ac=4$，\\\\\n由$a^2+c^2-b^2=2ac\\cos B$知$a^2+c^2=9$.\\\\\n结合两式得$(a-c)^2=1$, 故$a-c=\\pm 1$\\\\\n(2) 由$2\\cos C (ac\\cos B+cb\\cos A)=c^2$知$2a\\cos B\\cos C+2b \\cos A\\cos C=c$,\\\\\n又由正弦定理知 $2\\cos C(\\sin A\\cos B+\\sin B\\cos A)=\\sin C$，\\\\\n$2\\cos C \\sin (A+B)=2\\cos C \\sin C=\\sin C$,其中$C\\in (0,\\pi), \\sin C>0$,\\\\\n故$\\cos C=\\dfrac{1}{2}$, $C\\in(0,\\pi)$, $C=\\dfrac{\\pi}{3}$.", @@ -447054,7 +448620,9 @@ "id": "017241", "content": "疫苗在上市前必须经过严格的检测, 并通过临床实验获得相关数据, 以保证疫苗使用的安全和有效.某生物制品研究所将某一型号疫苗用在动物小白鼠身上进行科研和临床实验, 得到统计数据如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 未感染病毒 & 感染病毒 & 总计 \\\\\n\\hline 未注射疫苗 & 40 &$p$&$x$\\\\\n\\hline 注射疫苗 & 60 &$q$&$y$\\\\\n\\hline 总计 & 100 & 100 & 200 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n现从未注射疫苗的小白鼠中任取$1$只, 取到``感染病毒''的小白鼠的概率为$\\dfrac{3}{5}$.\\\\\n(1) 求$2 \\times 2$列联表中的数据$p, q, x, y$的值;\\\\\n(2) 是否有$95 \\%$的把握认为注射此种疫苗有效? 说明理由;\\\\\n(3) 在感染病毒的小白鼠中, 按未注射疫苗和注射疫苗的比例抽取$10$只进行病例分析, 然后从这$10$只小白鼠中随机抽取$4$只对注射疫苗情况进行核实, 记$X$为$4$只中未注射疫苗的小白鼠的只数, 求$X$的分布与期望$E[X]$.\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中$n=a+b+c+d$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$P(\\chi^2 \\geq k)$& 0.10 & 0.05 & 0.01 & 0.005 & 0.001 \\\\\n\\hline$k$& 2.706 & 3.841 & 6.635 & 7.879 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) $p=60$, $q=40$, $x=100$, $y=100$; (2) $\\chi^2=8$, 有$95 \\%$的把握认为注射此种疫苗有效; (3) 分布为$\\begin{pmatrix}0&1&2&3&4\\\\\\dfrac{1}{210}&\\dfrac{4}{35}&\\dfrac{3}{7}&\\dfrac{8}{21}&\\dfrac{1}{14}\\end{pmatrix}$, $E[X]=\\dfrac{12}{5}$", "solution": "(1) $\\dfrac{p}{p+40}=\\dfrac{3}{5}$得$p=60$, $q=40$, $x=100$, $y=100$.\\\\\n(2) 原假设$H_0:$ 是否注射此种疫苗与是否感染病毒无关.\\\\\n$\\chi^2=\\dfrac{200\\times (40\\times 40-60\\times 60)^2}{100\\times 100\\times 100\\times 100}=8>3.841$\\\\\n故拒绝原假设，即有$95 \\%$的把握认为注射此种疫苗有效.\\\\\n(3) 抽取$6$只未注射疫苗、$4$只注射疫苗的小白鼠.\\\\\n$P(X=0)=\\dfrac{C_6^0C_4^4}{C_{10}^4}=\\dfrac{1}{210}$;\\\\\n$P(X=1)=\\dfrac{C_6^1C_4^3}{C_{10}^4}=\\dfrac{4}{35}$;\\\\\n$P(X=2)=\\dfrac{C_6^2C_4^2}{C_{10}^4}=\\dfrac{3}{7}$;\\\\\n$P(X=3)=\\dfrac{C_6^3C_4^1}{C_{10}^4}=\\dfrac{8}{21}$;\\\\\n$P(X=4)=\\dfrac{C_6^4C_4^0}{C_{10}^4}=\\dfrac{1}{14}$.\\\\\n故$X$的分布为$\\begin{pmatrix}\n0&1&2&3&4\\\\\n\\dfrac{1}{210}&\\dfrac{4}{35}&\\dfrac{3}{7}&\\dfrac{8}{21}&\\dfrac{1}{14}\n\\end{pmatrix},$\\\\\n期望$E[X]=0\\times \\dfrac{1}{210}+1\\times \\dfrac{4}{35}+2\\times \\dfrac{3}{7}+3\\times \\dfrac{8}{21}+4\\times \\dfrac{1}{14}=\\dfrac{12}{5}$.", @@ -447087,7 +448655,9 @@ "id": "017242", "content": "已知椭圆$C: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$.\\\\\n(1) 求该椭圆的离心率;\\\\\n(2) 设点$P(x_0, y_0)$是椭圆$C$上一点, 求证: 过点$P$的椭圆$C$的切线方程为$\\dfrac{x_0 x}{4}+\\dfrac{y_0 y}{3}=1$;\\\\\n(3) 若点$M$为直线$l: x=4$上的动点, 过点$M$作该椭圆的切线$MA, MB$, 切点分别为$A, B$, 求$\\triangle MAB$的面积的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) 证明略; (3) $\\dfrac 92$", "solution": "(1) 椭圆的离心率$e=\\dfrac{1}{2}$;\\\\\n(2) 证明: 当$x_0=2$时，$y_0=0,$ 过点$P$的椭圆$C$的切线方程为 $x=2$,符合$\\dfrac{x_0 x}{4}+\\dfrac{y_0 y}{3}=1$，\\\\\n同理，当$x_0=-2$时，也符合；\\\\\n当$x_0\\neq\\pm 2$时，设过点$P$的椭圆$C$的切线方程为$y-y_0=k(x-x_0)$($k$存在),\\\\\n联立$\\begin{cases}\ny-y_0=k(x-x_0),\\\\\n \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1\n\\end{cases}$得$(3+4k^2)x^2+8k(y-kx_0)x+4(y_0-kx_0)^2-12=0$,\\\\\n$\\Delta=0$得$(x_0^2-4)k^2-2x_0y_0k+y_0^2-3=0$,解得$k=\\dfrac{x_0y_0}{x_0^2-4}=\\dfrac{x_0y_0}{4(1-\\dfrac{y_0^2}{3})-4}=-\\dfrac{3x_0}{4y_0}.$\\\\\n故$y-y_0=-\\dfrac{3x_0}{4y_0}(x-x_0)$,即$\\dfrac{x_0 x}{4}+\\dfrac{y_0 y}{3}=1$.\\\\\n综上，过点$P$的椭圆$C$的切线方程为$\\dfrac{x_0 x}{4}+\\dfrac{y_0 y}{3}=1$.\\\\\n(3) 设$A(x_1,y_1).B(x_2,y_2),x_1\\neq x_2,M(4,t)$.\\\\\n则切线$MA:\\dfrac{x_1x}{4}+\\dfrac{y_1y}{3}=1,$代入$(4,t)$得$x_1+\\dfrac{ty_1}{3}=1$,\\\\\n同理，$x_1+\\dfrac{ty_1}{3}=1$，\\\\\n故$A(x_1,y_1),B(x_2,y_2)$在直线$x+\\dfrac{ty}{3}=1$上，故直线$AB:x=-\\dfrac{ty}{3}+1$.\\\\\n联立$\\begin{cases}\nx=-\\dfrac{ty}{3}+1,\\\\\n\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1\n\\end{cases}$\n得\n$(4+\\dfrac{t^2}{3})y^2-2ty-9=0$, $\\Delta=16t^2+144>0$,\\\\\n$|AB|=\\sqrt{1+\\dfrac{t^2}{9}}\\cdot |y_1-y_2|=\\sqrt{1+\\dfrac{t^2}{9}}\\cdot \\dfrac{\\sqrt{16t^2+144}}{4+\\dfrac{t^2}{3}}$,\\\\\n$M$到直线$AB$的距离$d=\\dfrac{|4+\\dfrac{t^2}{3}-1|}{\\sqrt{1+\\dfrac{t^2}{9}}}=\\dfrac{3+\\dfrac{t^2}{3}}{\\sqrt{1+\\dfrac{t^2}{9}}}$,\\\\\n$\\triangle MAB$的面积$S=\\dfrac{1}{2}|AB|\\cdot d=\\dfrac{1}{2}\\cdot \\sqrt{1+\\dfrac{t^2}{9}}\\cdot \\dfrac{\\sqrt{16t^2+144}}{4+\\dfrac{t^2}{3}}\\cdot \\dfrac{3+\\dfrac{t^2}{3}}{\\sqrt{1+\\dfrac{t^2}{9}}}=\\dfrac{2(t^2+9)\\sqrt{t^2+9}}{t^2+12}$,\\\\\n令$\\lambda=\\sqrt{t^2+9}\\geq3, S=f(\\lambda)=\\dfrac{2\\lambda^3}{\\lambda^2+3},$\n则$f^{'} (\\lambda)=\\dfrac{2\\lambda^4+18\\lambda^2}{(\\lambda^2+3)^2}>0$,故$f(\\lambda)$在$[3,+\\infty)$严格增，$f(\\lambda)_{\\min}=f(3)=\\dfrac{9}{2}$,故$\\triangle MAB$的面积的最小值为$\\dfrac{9}{2},$ 此时$M$的坐标为$(4,0)$.", @@ -447120,7 +448690,9 @@ "id": "017243", "content": "设函数$f(x)=\\ln (x+1)$, $g(x)=\\dfrac{x}{x+1}$.\\\\\n(1) 记$x_1=g(1)$, $x_{n+1}=g(x_n)$, $n \\in \\mathbf{N}$, $n \\geq 1$. 证明: 数列$\\{\\dfrac{1}{x_n}\\}$为等差数列;\\\\\n(2) 设$m \\in \\mathbf{Z}$. 若对任意$x>0$均有$f(x)>m g(x)-1$成立, 求$m$的最大值;\\\\\n(3) 是否存在正整数$t$使得对任意$n \\in \\mathbf{N}$, $n \\geq t$, 都有$\\displaystyle f(n-t)0,\\dfrac{1}{x_{n+1}}=\\dfrac{x_n+1}{x_n}=\\dfrac{1}{x_n}+1$,即$\\dfrac{1}{x_{n+1}}-\\dfrac{1}{x_{n}}=1$,\\\\因此数列$\\{\\dfrac{1}{x_n}\\}$是以$2$为首项，$1$为公差的等差数列.\\\\\n(2) 对任意$x>0$ 均有$f(x)-mg(x)=\\ln (x+1)-\\dfrac{mx}{x+1}+1>0,$\\\\\n令$h(x)=\\ln (x+1)-\\dfrac{mx}{x+1}+1,x>0,$则$h^{'}(x)=\\dfrac{x+1-m}{(x+1)^2}$.\\\\\n当$m\\geq4$时，$h(1)=\\ln 2-\\dfrac{m}{2}+1<2-\\dfrac{m}{2}<0,$这与$h(x)>0$对$x>0$恒成立矛盾；\\\\\n当$m=3$时，$h(x)=\\ln (x+1)-\\dfrac{3x}{x+1}+1,h^{'}(x)\\dfrac{x-2}{(x+1)^2}.h(x)$在$(0,2]$严格减，在$[2,+\\infty)$严格增，故$h(x)_{\\min}=h(2)=\\ln 3-2+1=\\ln 3-1>0$,符合题意.\\\\\n综上，整数$m$的最大值为$3$.\\\\\n(3) 对任意正整数$t$,取$n=100t,$则 \\\\$\\displaystyle f(n-t)=f(99t)=\\ln (1+99t)=\\ln \\dfrac{1+99t}{99t}+\\ln \\dfrac{99t}{99t-1}+\\cdots +\\ln \\dfrac{2}{1}=\\sum_{k=1}^{99t}\\ln (1+\\dfrac{1}{k}).$\\\\\n$\\displaystyle n-\\sum_{k=1}^{100t}g(k)=\\sum_{k=1}^{100t}(1- g(k))=\\sum_{k=1}^{100t} \\dfrac{1}{1+k}=\\sum_{k=1}^{99t} \\dfrac{1}{1+k}+\\sum_{k=99t+1}^{100t} \\dfrac{1}{1+k}.$\\\\\n$\\displaystyle f(n-t)-[ n-\\sum_{k=1}^{100t}g(k)]=\\sum_{k=1}^{99t} (\\ln (1+\\dfrac{1}{k})-\\dfrac{1}{1+k})-\\sum_{k=99t+1}^{100t} \\dfrac{1}{1+k}.$\\\\\n令$H(x)=\\ln (x+1)-\\dfrac{x}{x+1},x>0$,则$H^{'}(x)=\\dfrac{x}{(x+1)^2}>0$恒成立，故$H(x)$在$(0,+\\infty)$严格增，$H(x)>H(0)=0$，故$\\ln (1+x)>\\dfrac{x}{x+1}$对$x>0$恒成立.\\\\\n因此$\\ln (1+\\dfrac{1}{k})>\\dfrac{\\dfrac{1}{k}}{1+\\dfrac{1}{k}}=\\dfrac{1}{1+k}$,\\\\\n$\\displaystyle f(n-t)-[ n-\\sum_{k=1}^{100t}g(k)]=\\sum_{k=1}^{99t} (\\ln (1+\\dfrac{1}{k})-\\dfrac{1}{1+k})-\\sum_{k=99t+1}^{100t} \\dfrac{1}{1+k}$\\\\\n$\\displaystyle>\\sum_{k=2}^{99t} (\\ln (1+\\dfrac{1}{k})-\\dfrac{1}{1+k})+(\\ln 2-\\dfrac{1}{2})- \\dfrac{t}{1+99t+1}>\\ln 2-\\dfrac{1}{2}-\\dfrac{1}{99+\\dfrac{2}{t}}>0.1-\\dfrac{1}{99}>0.$\\\\\n因此不存在正整数$t$使得对任意$n \\in \\mathbf{N}$, $n \\geq t$, 都有$\\displaystyle f(n-t)0$)的最小正周期为$T$. 若$\\dfrac{2}{3} \\pi0$)上, 过点$B(0,-1)$的直线交$C$于$P, Q$两点, 则\\blank{50}.\\\\\n\\textcircled{1} $C$的准线为$y=-1$; \\textcircled{2} 直线$AB$与$C$相切; \\textcircled{3} $|OP| \\cdot|OQ|>|OA|^2$; \\textcircled{4} $|BP| \\cdot|BQ|>|BA|^2$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447373,7 +448968,9 @@ "id": "017255", "content": "已知函数$f(x)$及其导函数$f'(x)$的定义域均为$\\mathbf{R}$, 记$g(x)=f'(x)$. 若$f(\\dfrac{3}{2}-2 x), g(2+x)$均为偶函数, 则\\blank{50}.\\\\\n\\textcircled{1} $f(0)=0$; \\textcircled{2} $g(-\\dfrac{1}{2})=0$; \\textcircled{3} $f(-1)=f(4)$; \\textcircled{4} $g(-1)=g(2)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447393,7 +448990,9 @@ "id": "017256", "content": "$(1-\\dfrac{y}{x})(x+y)^8$的展开式中$x^2 y^6$的系数为\\blank{50}(用数字作答).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447413,7 +449012,9 @@ "id": "017257", "content": "写出与圆$x^2+y^2=1$和$(x-3)^2+(y-4)^2=16$都相切的一条直线的方程: \\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447433,7 +449034,9 @@ "id": "017258", "content": "若曲线$y=(x+a) \\mathrm{e}^x$有两条过坐标原点的切线, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447453,7 +449056,9 @@ "id": "017259", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), $C$的上顶点为$A$, 两个焦点为$F_1, F_2$, 离心率为$\\dfrac{1}{2}$, 过$F_1$且垂直于$AF_2$的直线与$C$交于$D, E$两点, $|DE|=6$, 则$\\triangle ADE$的周长是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447473,7 +449078,9 @@ "id": "017260", "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 已知$a_1=1$, $\\{\\dfrac{S_n}{a_n}\\}$是公差为$\\dfrac{1}{3}$的等差数列.\\\\\n(1) 求$\\{a_n\\}$得通项公式;\\\\\n(2) 证明: $\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\cdots+\\dfrac{1}{a_n}<2$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447493,7 +449100,9 @@ "id": "017261", "content": "记$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$, 已知$\\dfrac{\\cos A}{1+\\sin A}=\\dfrac{\\sin 2B}{1+\\cos 2B}$.\\\\\n(1) 若$C=\\dfrac{2 \\pi}{3}$, 求$B$;\\\\\n(2) 求$\\dfrac{a^2+b^2}{c^2}$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447513,7 +449122,9 @@ "id": "017262", "content": "如图, 直三棱柱$ABC-A_1B_1C_1$的体积为$4$, $\\triangle A_1BC$的面积为$2 \\sqrt{2}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)!0.5!(C)$) node [above] {$D$} coordinate (D);\n\\draw (A)--(A_1)(B)--(B_1)(C)--(C_1)(A)--(B)--(C)(A_1)--(B_1)--(C_1)--cycle(A_1)--(B);\n\\draw [dashed] (A)--(C)--(A_1)(A)--(D)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$A$到平面$A_1BC$的距离;\\\\\n(2) 设$D$为$A_1C$的中点, $AA_1=AB$, 平面$A_1BC \\perp$平面$ABB_1A_1$, 求二面角$A-BD-C$的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447533,7 +449144,10 @@ "id": "017263", "content": "一医疗团队为研究某地的一种地方性疾病与当地居民的卫生习惯 (卫生习惯分为良好和不够良好两类) 的关系, 在已患该疾病的病例中随机调查了$100$例 (称为病例组), 同时在未患该疾病的人群中随机调査了$100$人 (称为对照组), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 不够良好 & 良好 \\\\\n\\hline 病例组 & 40 & 60 \\\\\n\\hline 对照组 & 10 & 90 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 能否有$99 \\%$的把握认为患该疾病群体与未患该疾病群体的卫生习惯有差异?\\\\\n(2) 从该地的人群中任选一人, $A$表示事件``选到的人卫生习惯不够良好'', $B$表示事件``选到的人患有该疾病'', $\\dfrac{P(B | A)}{P(\\overline {B} | A)}$与$\\dfrac{P(B | \\overline {A})}{P(\\overline {B} | \\overline {A})}$的比值是卫生习惯不够良好对患该疾病风险程度的一项度量指标, 记该指标为$R$.\\\\\n(I) 证明: $R=\\dfrac{P(A | B)}{P(\\overline {A} | B)} \\cdot \\dfrac{P(\\overline {A} | \\overline {B})}{P(A | \\overline {B})}$;\\\\\n(II) 利用该调査数据, 给出$P(A | B)$, $P(A | \\overline {B})$的估计值, 并利用(I)的结果给出$R$的估计值.\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline$P(\\chi^2 \\geq k)$& 0.050 & 0.010 & 0.001 \\\\\n\\hline$k$& 3.841 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元", + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447553,7 +449167,9 @@ "id": "017264", "content": "已知点$A(2,1)$在双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{a^2-1}=1$($a>1$)上, 直线$l$交$C$于$P, Q$两点, 直线$AP, AQ$的斜率之和为$0$.\\\\\n(1) 求$l$的斜率;\\\\\n(2) 若$\\tan \\angle PAQ=2 \\sqrt{2}$, 求$\\triangle PAQ$的面积.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447573,7 +449189,9 @@ "id": "017265", "content": "已知函数$f(x)=\\mathrm{e}^x-a x$和$g(x)=a x-\\ln x$有相同的最小值.\\\\\n(1) 求$a$;\\\\\n(2) 证明: 存在直线$y=b$, 其与两条曲线$y=f(x)$和$y=g(x)$共有三个不同的交点, 并且从左到右的三个交点的横坐标成等差数列.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447593,7 +449211,9 @@ "id": "017266", "content": "已知集合$A=\\{-1,1,2,4\\}$, $B=\\{x \\| x-1 | \\leq 1\\}$, 则$A \\cap B=$\\bracket{20}.\n\\fourch{$\\{-1,2\\}$}{$\\{1,2\\}$}{$\\{1,4\\}$}{$\\{-1,4\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -447613,7 +449233,9 @@ "id": "017267", "content": "$(2+2 \\mathrm{i})(1-2 \\mathrm{i})=$\\bracket{20}.\n\\fourch{$-2+4 \\mathrm{i}$}{$-2-4 \\mathrm{i}$}{$6+2 \\mathrm{i}$}{$6-2 \\mathrm{i}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -447633,7 +449255,10 @@ "id": "017268", "content": "中国的古建筑不仅是挡风遮雨的住处, 更是美学和哲学的体现. 如图是某古建筑物的剖面图, 其中$DD_1$, $CC_1$, $BB_1$, $AA_1$是举, $OD_1$, $DC_1$, $CB_1$, $BA_1$是相等的步, 相邻桁的举步之比分别为$\\dfrac{DD_1}{OD_1}=0.5$, $\\dfrac{CC_1}{DC_1}=k_1$, $\\dfrac{BB_1}{CB_1}=k_2$, $\\dfrac{AA_1}{BA_1}=k_3$, 已知$k_1, k_2, k_3$成公差为$0.1$的等差数列, 且直线$OA$的斜率为$0.725$, 则$k_3=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (9.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (1,0) node [below] {$D_1$} coordinate (D_1);\n\\draw (D_1) ++ (0,0.5) node [below right] {$D$} coordinate (D);\n\\draw (D) ++ (1,0) node [below] {$C_1$} coordinate (C_1);\n\\draw (C_1) ++ (0,0.7) node [below right] {$C$} coordinate (C);\n\\draw (C) ++ (1,0) node [below] {$B_1$} coordinate (B_1);\n\\draw (B_1) ++ (0,0.8) node [below right] {$B$} coordinate (B);\n\\draw (B) ++ (1,0) node [below] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (0,0.9) node [above] {$A$} coordinate (A);\n\\draw (D_1)--(D) (C_1)--(C) (B_1)--(B) (A_1) -- (A);\n\\draw (D) --++ (6,0) coordinate (D_2) (C) --++ (4,0) coordinate (C_2) (B) --++ (2,0) coordinate (B_2);\n\\draw (B_2) --++ (0,-0.8) (C_2) --++ (0,-0.7) (D_2) --++ (0,-0.5);\n\\draw (O)--(D)--(C)--(B)--(A)--(B_2)--(C_2)--(D_2)--(8,0);\n\\draw [dashed] (O)--(A);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$0.75$}{$0.8$}{$0.85$}{$0.9$}", "objs": [], - "tags": [], + "tags": [ + "第四单元", + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -447654,7 +449279,9 @@ "id": "017269", "content": "已知$\\overrightarrow {a}=(3,4)$, $\\overrightarrow {b}=(1,0)$, $\\overrightarrow {c}=\\overrightarrow {a}+t \\overrightarrow {b}$, $\\langle\\overrightarrow {a}, \\overrightarrow {c}\\rangle=\\langle \\overrightarrow {b}, \\overrightarrow {c}\\rangle$, 则$t=$\\bracket{20}.\n\\fourch{$-6$}{$-5$}{$5$}{$6$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -447674,7 +449301,9 @@ "id": "017270", "content": "有甲乙丙丁戊$5$名同学站成一排参加文艺汇演, 若甲不站在两端, 丙和丁相邻的不同排列方式有多少种\\bracket{20}.\n\\fourch{$12$种}{$24$种}{$36$种}{$48$种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -447694,7 +449323,9 @@ "id": "017271", "content": "角$\\alpha, \\beta$满足$\\sin (\\alpha+\\beta)+\\cos (\\alpha+\\beta)=2 \\sqrt{2} \\cos (\\alpha+\\dfrac{\\pi}{4}) \\sin \\beta$, 则\\bracket{20}.\n\\fourch{$\\tan (\\alpha+\\beta)=1$}{$\\tan (\\alpha+\\beta)=-1$}{$\\tan (\\alpha-\\beta)=1$}{$\\tan (\\alpha-\\beta)=-1$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -447714,7 +449345,9 @@ "id": "017272", "content": "正三棱台高为$1$, 上下底边长分别是$3 \\sqrt{3}$和$4 \\sqrt{3}$, 所有顶点在同一球面上, 则球的表面积是\\bracket{20}.\n\\fourch{$100 \\pi$}{$128 \\pi$}{$144 \\pi$}{$192 \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -447734,7 +449367,9 @@ "id": "017273", "content": "若函数$f(x)$的定义域为$\\mathbf{R}$, 且$f(x+y)+f(x-y)=f(x) f(y)$, $f(1)=1$, 则$\\displaystyle\\sum_{k=1}^{22} f(k)=$\\bracket{20}.\n\\fourch{$-3$}{$-2$}{$0$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -447754,7 +449389,9 @@ "id": "017274", "content": "函数$f(x)=\\sin (2 x+\\varphi)$($0<\\varphi<\\pi$)的图像关于$(\\dfrac{2 \\pi}{3}, 0)$中心对称, 则\\blank{50}.\\\\\n\\textcircled{1} $y=f(x)$在$(0, \\dfrac{5 \\pi}{12})$单调递减;\\\\ \\textcircled{2} $y=f(x)$在$(-\\dfrac{\\pi}{12}, \\dfrac{11 \\pi}{12})$有$2$个极值点;\\\\\n\\textcircled{3} 直线$x=\\dfrac{7 \\pi}{6}$是一条对称轴;\\\\\n\\textcircled{4} 直线$y=\\dfrac{\\sqrt{3}}{2}-x$是一条切线.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447774,7 +449411,9 @@ "id": "017275", "content": "已知$O$为坐标原点, 过抛物线$C: y^2=2 p x (p>0)$的焦点$F$的直线与$C$交于$A, B$两点, 点$A$在第一象限, 点$M(p, 0)$, 若$|AF|=|AM|$, 则\\blank{50}.\\\\\n\\textcircled{1} 直线$AB$的斜率为$2 \\sqrt{6}$; \\textcircled{2} $|OB|=|OF|$; \\textcircled{3} $|AB|>4|OF|$; \\textcircled{4} $\\angle OAM+\\angle OBM<180^{\\circ}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447794,7 +449433,9 @@ "id": "017276", "content": "如图, 四边形$ABCD$为正方形, $ED \\perp$平面$ABCD$, $FB\\parallel ED$, $AB=ED=2FB$, 记三棱锥$E-ACD$, $F-ABC$, $F-ACE$的体积分别为$V_1$, $V_2$, $V_3$, 则\\blank{50}.\\\\\n\\textcircled{1} $V_3=2V_2$; \\textcircled{2} $V_3=V_1$; \\textcircled{3} $V_3=V_1+V_2$; \\textcircled{4} $2V_3=3V_1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$E$} coordinate (E);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (D) ++ (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (B) ++ (0,1,0) node [left] {$F$} coordinate (F);\n\\draw (E)--(A)--(B)--(C)--cycle;\n\\draw (B)--(F)--(E)(A)--(F)--(C);\n\\draw [dashed] (E)--(D)--(A)(D)--(C)(A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447814,7 +449455,9 @@ "id": "017277", "content": "若实数$x, y$满足$x^2+y^2-x y=1$, 则\\blank{50}.\\\\\n\\textcircled{1} $x+y \\leq 1$; \\textcircled{2} $x+y \\geq-2$; \\textcircled{3} $x^2+y^2 \\leq 2$; \\textcircled{4} $x^2+y^2 \\geq 1$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447834,7 +449477,9 @@ "id": "017278", "content": "已知随机变量$X$服从正态分布$N(2, \\sigma^2)$, 且$P(22.5)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447854,7 +449499,9 @@ "id": "017279", "content": "写出曲线$y=\\ln |x|$过坐标原点的切线方程: \\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447874,7 +449521,9 @@ "id": "017280", "content": "已知点$A(-2,3)$, $B(0, a)$, 若直线$AB$关于$y=a$的对称直线与圆$(x+3)^2+(y+2)^2=1$存在公共点, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447894,7 +449543,9 @@ "id": "017281", "content": "已知椭圆$\\dfrac{x^2}{6}+\\dfrac{y^2}{3}=1$, 直线$l$与椭圆在第一象限交于$A, B$, 与$x$轴, $y$轴分别交于$M, N$, 且$|MA|=|NB|$, $|MN|=2 \\sqrt{3}$, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -447914,7 +449565,9 @@ "id": "017282", "content": "已知$\\{a_n\\}$为等差数列, $\\{b_n\\}$是公比为$2$的等比数列, 且$a_2-b_2=a_3-b_3=b_4-a_4$.\\\\\n(1) 证明: $a_1=b_1$;\\\\\n(2) 求集合$\\{k | b_k=a_m+a_1,\\ 1 \\leq m \\leq 500\\}$中元素的个数.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447934,7 +449587,9 @@ "id": "017283", "content": "记$\\triangle ABC$的三个内角分别为$A$、$B$、$C$, 其对边分别为$a, b, c$, 分别以$a, b, c$为边长的三个正三角形的面积依次为$S_1, S_2, S_3$, 已知$S_1-S_2+S_3=\\dfrac{\\sqrt{3}}{2}$, $\\sin B=\\dfrac{1}{3}$.\\\\\n(1) 求$\\triangle ABC$的面积;\\\\\n(2) 若$\\sin A \\sin C=\\dfrac{\\sqrt{2}}{3}$, 求$b$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447954,7 +449609,10 @@ "id": "017284", "content": "在某地区进行流行病调查, 随机调查了$100$名某种疾病患者的年龄, 得到如下的样本数据频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.06, yscale = 180]\n\\draw [->] (0,0) -- (105,0) node [below] {年龄/岁};\n\\draw [->] (0,0) -- (0,0.03) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0/0.001,10/0.002,20/0.012,30/0.017,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0/0.001,20/0.012,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (90,0) node [below] {$90$};\n\\end{tikzpicture}\n\\end{center}\n(1) 估计该地区这种疾病患者的平均年龄 (同一组中的数据用该组区间的中点值代表);\\\\\n(2) 估计该地区一人患这种疾病年龄在区间$[20,70)$的概率.\\\\\n(3) 已知该地区这种疾病的患病率为$0.1 \\%$, 该地区的年龄位于区间$[40,50)$的人口占该地区总人口的$16 \\%$, 从该地区任选一人, 若此人年龄位于区间$[40,50)$, 求此人患该种疾病的概率. (样本数据中的患者年龄位于各地区的频率作为患者年龄位于该区间的概率, 精确到 $0.0001$)\\\\", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447974,7 +449632,9 @@ "id": "017285", "content": "如图, $PO$是三棱锥$P-ABC$的高, $PA=PB$, $AB \\perp AC$, $E$是$PB$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(135:0.5cm)}, scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({4*sqrt(3)},0,0) node [right] {$B$} coordinate (B);\n\\draw ({2*sqrt(3)},0,2) node [right] {$O$} coordinate (O);\n\\draw (0,0,12) node [left] {$C$} coordinate (C);\n\\draw (O) ++ (0,3,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(B)$) node [right] {$E$} coordinate (E);\n\\draw (C)--(A)--(B)--(P)--cycle(A)--(E)(A)--(P);\n\\draw [dashed] (P)--(O)(C)--(E)(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $OE\\parallel$平面$PAC$;\\\\\n(2) 若$\\angle ABO=\\angle CBO=30^{\\circ}$, $PO=3$, $PA=5$, 求二面角$C-AE-B$的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -447994,7 +449654,9 @@ "id": "017286", "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的右焦点为$F(2,0)$, 渐近线方程为$y= \\pm \\sqrt{3} x$.\\\\\n(1) 求$C$的方程;\\\\\n(2) 过$F$的直线与$C$的两条渐近线分别交于$A, B$两点, 点$P(x_1, y_1), Q(x_2, y_2)$在$C$上, 且$x_1>x_2>0$, $y_1>0$, 过$P$且斜率为$-\\sqrt{3}$的直线与过$Q$且斜率为$\\sqrt{3}$的直线交于点$M$. 请从下面\\textcircled{1}\\textcircled{2}\\textcircled{3}中选取两个作为条件, 证明另外一个条件成立:\n\\textcircled{1}$M$在$AB$上; \\textcircled{2}$PQ\\parallel AB$; \\textcircled{3}$|MA|=|\\mathrm{MB}|$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448014,7 +449676,9 @@ "id": "017287", "content": "已知函数$f(x)=x \\mathrm{e}^{a x}-\\mathrm{e}^x$.\\\\\n(1) 当$a=1$时, 讨论$f(x)$的单调性;\\\\\n(2) 当$x>0$时, $f(x)<-1$, 求$a$的取值范围;\\\\\n(3) 设$n \\in \\mathbf{N}$, $n\\ge 1$, 证明: $\\dfrac{1}{\\sqrt{1^2+1}}+\\dfrac{1}{\\sqrt{2^2+2}}+\\cdots+\\dfrac{1}{\\sqrt{n^2+n}}>\\ln (n+1)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448034,7 +449698,9 @@ "id": "017288", "content": "设集合$A=\\{1,2\\}$, $B=\\{x | 2,4,6\\}$则$A \\cup B=$\\bracket{20}.\n\\fourch{$\\{2\\}$}{$\\{1,2\\}$}{$\\{2,4,6\\}$}{$\\{1,2,4,6\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448054,7 +449720,9 @@ "id": "017289", "content": "已知$a, b \\in \\mathbf{R}$, $a+3 \\mathrm{i}=(b+\\mathrm{i}) \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$a=$\\bracket{20}.\n\\fourch{$a=1$, $b=-3$}{$a=-1$, $b=3$}{$a=-1$, $b=-3$}{$a=1$, $b=3$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448074,7 +449742,9 @@ "id": "017290", "content": "若实数$x, y$满足约束条件$\\begin{cases}x-2 \\geq 0,\\\\ 2 x+y-7 \\leq 0,\\\\ x-y-2 \\leq 0,\\end{cases}$ 则$z=3 x+4 y$的最大值是 \\bracket{20}.\n\\fourch{$20$}{$18$}{$13$}{$6$}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448094,7 +449764,10 @@ "id": "017291", "content": "设$x \\in \\mathbf{R}$, 则``$\\sin x=1$''是``$\\cos x=0$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448114,7 +449787,9 @@ "id": "017292", "content": "某几何体的三视图如图所示(单位: $\\text{cm}$), 则该几何体的体积(单位: $\\text{cm}^3)$是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0) -- (4,0) -- (3,2) -- (1,2) -- cycle (1,2) -- (1,4) arc (180:0:1) -- (3,2) (1,4) -- (3,4);\n\\foreach \\i in {0,1,3,4}\n{\\draw (\\i,-0.3) --++ (0,-0.4);};\n\\draw [->] (-0.5,-0.5) -- (0,-0.5);\n\\draw [->] (4.5,-0.5) -- (4,-0.5);\n\\draw [<->] (1,-0.5) -- (3,-0.5);\n\\draw (0.5,-0.5) node {$1$} (3.5,-0.5) node {$1$};\n\\draw (2,-0.5) node [fill = white] {$2$};\n\\draw (2,-1.5) node {正视图};\n\\draw (6,0) -- (10,0) -- (9,2) -- (7,2) -- cycle (7,2) -- (7,4) arc (180:0:1) -- (9,2) (7,4) -- (9,4);\n\\foreach \\i in {6,7,9,10}\n{\\draw (\\i,-0.3) --++ (0,-0.4);};\n\\draw [->] (5.5,-0.5) -- (6,-0.5);\n\\draw [->] (10.5,-0.5) -- (10,-0.5);\n\\draw [<->] (7,-0.5) -- (9,-0.5);\n\\draw (6.5,-0.5) node {$1$} (9.5,-0.5) node {$1$};\n\\draw (8,-0.5) node [fill = white] {$2$};\n\\draw (8,-1.5) node {侧视图};\n\\foreach \\i in {0,2,4,5}\n{\\draw (4.7,\\i) -- (5.3,\\i);};\n\\draw [<->] (5,0) -- (5,2) node [midway, fill = white] {$2$};\n\\draw [<->] (5,2) -- (5,4) node [midway, fill = white] {$2$};\n\\draw (5,4.5) node {$1$};\n\\draw [->] (5,5.5) -- (5,5);\n\\draw (2,-4) circle (1) circle (2);\n\\draw (2,-6.5) node {俯视图};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$22 \\pi$}{$8 \\pi$}{$\\dfrac{22}{3} \\pi$}{$\\dfrac{16}{3} \\pi$}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448134,7 +449809,9 @@ "id": "017293", "content": "为了得到$y=2 \\sin 3 x$的图像, 只要把函数$y=2 \\sin (3 x+\\dfrac{\\pi}{5})$图像上所有点\\bracket{20}.\n\\twoch{向左平移$\\dfrac{\\pi}{5}$个单位长度}{向右平移$\\dfrac{\\pi}{5}$个单位长度}{向左平移$\\dfrac{\\pi}{15}$个单位长度}{向右平移$\\dfrac{\\pi}{15}$个单位长度}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448154,7 +449831,9 @@ "id": "017294", "content": "已知$2^a=5$, $\\log _83=b$, 则$4^{a-3 b}=$\\bracket{20}.\n\\fourch{$25$}{$5$}{$\\dfrac{25}{9}$}{$\\dfrac{25}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448174,7 +449853,9 @@ "id": "017295", "content": "如图, 已知正三棱柱$ABC-A_1B_1C_1$, $AC=AA_1$, $E, F$分别是棱$BC, A_1C_1$上的点. 记$EF$与$AA_1$所成的角为$\\alpha$, $EF$与平面$ABC$所成的角为$\\beta$, 二面角$F-BC-A$的平面角为$\\gamma$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\draw ($(B)!0.6!(C)$) node [below right] {$E$} coordinate (E);\n\\draw ($(A_1)!0.3!(C_1)$) node [above] {$F$} coordinate (F);\n\\draw [dashed] (E)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\alpha \\leq \\beta \\leq \\gamma$}{$\\beta \\leq \\alpha \\leq \\gamma$}{$\\beta \\leq \\gamma \\leq \\alpha$}{$\\alpha \\leq \\gamma \\leq \\beta$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448194,7 +449875,9 @@ "id": "017296", "content": "已知$a, b \\in \\mathbf{R}$, 若对任意$x \\in \\mathbf{R}$, $a|x-b|+|x-4|-|2 x-5| \\geq 0$, 则\\bracket{20}.\n\\fourch{$a \\leq 1$, $b \\geq 3$}{$a \\leq 1$, $b \\leq 3$}{$a \\geq 1$, $b \\geq 3$}{$a \\geq 1$, $b \\leq 3$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448214,7 +449897,9 @@ "id": "017297", "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=a_n-\\dfrac{1}{3} a_n^2$($n \\in \\mathbf{N}$, $n\\ge 1$), 则\\bracket{20}.\n\\fourch{$2<100 a_{100}<\\dfrac{5}{2}$}{$\\dfrac{5}{2}<100 a_{100}<3$}{$3<100 a_{100}<\\dfrac{7}{2}$}{$\\dfrac{7}{2}<100 a_{100}<4$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448234,7 +449919,9 @@ "id": "017298", "content": "我国南宋著名数学家秦九韶, 发现了从三角形三边求面积的公式, 他把这种方法称为``三斜求积'', 它填补了我国传统数学的一个空白. 如果把这个方法写成公式, 就是$S=$$\\sqrt{\\dfrac{1}{4}[c^2 a^2-(\\dfrac{c^2+a^2-b^2}{2})^2]}$, 其中$a, b, c$是三角形的三边, $S$是三角形的面积. 设某三角形的三边$a=\\sqrt{2}$, $b=\\sqrt{3}$, $c=2$, 则该三角形的面积$S=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -448254,7 +449941,9 @@ "id": "017299", "content": "已知多项式$(x+2)(x-1)^4=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4+a_5 x^5$, 则$a_2=$\\blank{50}, $a_1+a_2+a_3+a_4+a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -448274,7 +449963,9 @@ "id": "017300", "content": "若$3 \\sin \\alpha-\\cos \\beta=\\sqrt{10}$, $\\alpha+\\beta=\\dfrac{\\pi}{2}$, 则$\\sin \\alpha=$\\blank{50}, $\\cos 2 \\beta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -448294,7 +449985,9 @@ "id": "017301", "content": "已知$f(x)=\\begin{cases}-x^2+2, & x \\leq 1, \\\\ x+\\dfrac{1}{x}-1, & x>1,\\end{cases}$则$f(f(\\dfrac{1}{2}))=$\\blank{50}; 若当$x \\in[a, b]$时, $1 \\leq f(x) \\leq 3$, 则$b-a$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -448314,7 +450007,9 @@ "id": "017302", "content": "现有$7$张卡片, 分别写上数字$1,2,2,3,4,5,6$. 从这$7$张卡片中随机抽取$3$张, 记所抽取卡片上数字的最小值为$\\xi$, 则$P(\\xi=2)=$\\blank{50}, $E[\\xi ]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -448334,7 +450029,9 @@ "id": "017303", "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左焦点为$F$, 过$F$且斜率为$\\dfrac{b}{4 a}$的直线交双曲线于点$A(x_1, y_1)$, 交双曲线的渐近线于点$B(x_2, y_2)$且$x_1<0=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (5,0,0) node [below] {$B$} coordinate (B);\n\\draw (5,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (C) ++ (-3,0,0) node [below] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(C)$) node [right] {$N$} coordinate (N);\n\\draw (N) ++ (0,1.5,0) node [above] {$F$} coordinate (F);\n\\draw (F) ++ (-1,0,0) node [above] {$E$} coordinate (E);\n\\draw ($(E)!0.5!(A)$) node [left] {$M$} coordinate (M);\n\\draw (A)--(B)--(F)--(E)--cycle;\n\\draw (B)--(C)--(F)(N)--(F)(B)--(M);\n\\draw [dashed] (C)--(D)--(A)(D)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $FN \\perp AD$;\\\\\n(2) 求直线$BM$与平面$ADE$所成角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448414,7 +450117,9 @@ "id": "017307", "content": "已知等差数列$\\{a_n\\}$的首项$a_1=-1$, 公差$d>1$, 记$\\{a_n\\}$的前$n$项和为$S_n$($n \\in \\mathbf{N}$, $n\\ge 1$).\\\\\n(1) 若$S_4-2 a_2 a_3+6=0$, 求$S_n$;\\\\\n(2) 若对于每个$n \\in \\mathbf{N}$, $n\\ge 1$, 存在实数$c_n$, 使$a_n+c_n, a_{n+1}+4 c_n, a_{n+2}+15 c_n$成等比数列, 求$d$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448434,7 +450139,9 @@ "id": "017308", "content": "如图, 已知椭圆$\\dfrac{x^2}{12}+y^2=1$. 设$A, B$是椭圆上异于$P(0,1)$的两点, 且点$Q(0, \\dfrac{1}{2})$在线段$AB$上, 直线$PA, PB$分别交直线$y=-\\dfrac{1}{2} x+3$于$C, D$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-4,0) -- (6.5,0) node [above] {$x$};\n\\draw [->] (0,-1.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [name path = elli, draw] (0,0) ellipse ({2*sqrt(3)} and 1);\n\\draw (110:{sqrt(12)} and 1) node [above] {$P$} coordinate (P);\n\\draw (0,0.5) node [below right] {$Q$} coordinate (Q);\n\\path [name path = line, draw] ($(6,0)!1.1!(0,3)$) -- ($(6,0)!-0.1!(0,3)$);\n\\draw ({-2*sqrt(3)},0) node [below left] {$A$} coordinate (A);\n\\path [name path = AB] (A) -- ($(A)!2!(Q)$);\n\\path [name intersections = {of = elli and AB, by = B}];\n\\draw (B) node [below] {$B$};\n\\path [name path = AP] (A) -- ($(A)!2.5!(P)$);\n\\path [name path = BP] (P) -- ($(P)!2!(B)$);\n\\path [name intersections = {of = AP and line, by = C}];\n\\path [name intersections = {of = BP and line, by = D}];\n\\draw (A)--(C) node [above] {$C$} (P) -- (D) node [above] {$D$};\n\\draw (A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$P$到椭圆点的距离的最大值;\\\\\n(2) 求$|CD|$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448454,7 +450161,9 @@ "id": "017309", "content": "设函数$f(x)=\\dfrac{\\mathrm{e}}{2 x}+\\ln x$($x>0$).\\\\\n(1) 求$f(x)$的单调区间;\\\\\n(2) 已知$a, b \\in \\mathbf{R}$, 曲线$y=f(x)$上不同的三点$(x_1, f(x_1)),(x_2, f(x_2)),(x_3, f(x_3))$处的切线都经过点$(a, b)$. 证明:\\\\\n(I) 若$a>\\mathrm{e}$, 则$0N_0$时, $a_n>0$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448594,7 +450315,9 @@ "id": "017316", "content": "在北京冬奥会上, 国家速滑馆``冰丝带''使用高效环保的二氧化碳跨临界直制冰技术, 为实现绿色东奥作出了贡献, 如图描述了一定条件下二氧化碳所处的状态与$T$和$\\lg P$的关系, 其中$T$表示温度, 单位是$\\text{K}$, $P$表示压强, 单位是$\\text{bar}$. 下列结论中正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (4.5,0) node [below] {$T$};\n\\draw [->] (0,0) -- (0,4.5) node [left] {$\\lg P$};\n\\foreach \\i/\\j in {0/200,1/250,2/300,3/350,4/400}\n{\\draw (\\i,0) --++ (0,-0.2) node [below] {$\\j$};\n\\draw (0,\\i) --++ (-0.2,0) node [left] {$\\i$};};\n\\draw (4,0) -- (4,4) -- (0,4);\n\\draw [domain = 0:2] plot (\\x,{2*ln(\\x+1.1)/ln(3.1)});\n\\draw (0.4,{2*ln(0.4+1.1)/ln(3.1)}) coordinate (A);\n\\draw (A) -- (0.5,2.5);\n\\draw [domain = 0.5:4] plot (\\x,{1.5/ln(8)*(ln(\\x)-ln(4))+4});\n\\draw [dashed] (2,2) --++ (2,0) (2,2) -- (2,{1.5/ln(8)*(ln(2)-ln(4))+4});\n\\draw (2.5,1) node {气态};\n\\draw (3,3) node {超临界状态};\n\\draw (0.5,3.5) node {固态};\n\\draw (1.2,2.2) node {液态}; \n\\end{tikzpicture}\n\\end{center}\n\\onech{当$T=220$, $P=1026$时, 二氧化碳处于液态}{当$T=270$, $P=128$时, 二氧化碳处于气态}{当$T=300$, $P=9987$时, 二氧化碳处于超临界状态}{当$T=360$, $P=729$时, 二氧化碳处于超临界状态}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448614,7 +450337,9 @@ "id": "017317", "content": "若$(2 x-1)^4=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$, 则$a_0+a_2+a_4=$\\bracket{20}.\n\\fourch{$40$}{$41$}{$-40$}{$-41$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448634,7 +450359,9 @@ "id": "017318", "content": "已知正三棱锥$P-ABC$的$6$条棱长均为$6$, $S$是$\\triangle ABC$及其内部的点构成的集合, 设集合$T=\\{Q \\in S | PQ \\leq 5\\}$, 则$T$表示的区域的面积为\\bracket{20}.\n\\fourch{$\\dfrac{3 \\pi}{4}$}{$\\pi$}{$2 \\pi$}{$3 \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448654,7 +450381,9 @@ "id": "017319", "content": "在$\\triangle ABC$中, $AC=3$, $BC=4$, $\\angle C=90^{\\circ}$. $P$为$\\triangle ABC$所在平面内的动点, 且$PC=1$, 则$\\overrightarrow{PA} \\cdot \\overrightarrow{PB}$的取值范围是\\bracket{20}.\n\\fourch{$[-5,3]$}{$[-3,5]$}{$[-6,4]$}{$[-4,6]$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448674,7 +450403,9 @@ "id": "017320", "content": "函数$f(x)=\\dfrac{1}{x}+\\sqrt{1-x}$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -448694,7 +450425,9 @@ "id": "017321", "content": "已知双曲线$y^2+\\dfrac{x^2}{m}=1$的渐近线方程为$y= \\pm \\dfrac{\\sqrt{3}}{3} x$, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -448714,7 +450447,9 @@ "id": "017322", "content": "若函数$f(x)=A \\sin x-\\sqrt{3} \\cos x$的一个零点为$\\dfrac{\\pi}{3}$, 则$A=$\\blank{50}, $f(\\dfrac{\\pi}{12})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -448734,7 +450469,9 @@ "id": "017323", "content": "设函数$f(x)=\\begin{cases}-a x+1, & x=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (0,0,2) node [below] {$C$} coordinate (C);\n\\draw (1.95,{sqrt(4-1.95*1.95)}) node [right] {$A$} coordinate (A);\n\\draw (A) ++ (0,2,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [left] {$C_1$} coordinate (C_1);\n\\draw ($(A)!0.5!(C)$) node [below] {$N$} coordinate (N);\n\\draw ($(B_1)!0.5!(A_1)$) node [above] {$M$} coordinate (M);\n\\draw (C)--(A)--(A_1)--(B_1)--(C_1)--cycle(A_1)--(C_1);\n\\draw [dashed] (C)--(B)--(A)(B)--(B_1)(B)--(N)--(M)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN\\parallel$平面$BCC_1B_1$;\\\\\n(2) 若$AB \\perp MN$, 求直线$AB$与平面$BMN$所成角的正弦值;\\\\\n(3) 若$BM=MN$, 求直线$AB$与平面$BMN$所成角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448814,7 +450557,9 @@ "id": "017327", "content": "在校运会上, 只有甲、乙、丙三名同学参加铅球比赛, 比赛成绩达到$9.50 \\mathrm{m}$(含$9.50 \\text{m}$)以上的同学获优秀奖. 为预测优秀奖的人数及冠军得主, 收集了甲、乙、丙以往的比赛成绩, 并整理得到如下数据 (单位: $\\text{m}$):\\\\\n甲: $9.80,9.70,9.55,9.54,9.48,9.42,9.40,9.35,9.30,9.25$\\\\\n乙: $9.78,9.56,9.51,9.36,9.32,9.23$\\\\\n丙: $9.85,9,65,9.20,9.16$\\\\\n假设用频率估计概率, 且甲、乙、丙的比赛成绩相互独立.\\\\\n(1) 估计甲在校运动会铅球比赛中获得优秀奖的概率;\\\\\n(2) 设$X$是甲、乙、丙在校运动会铅球比赛中获得优秀奖的总人数, 估计$X$的数学期望$E[X[$;\\\\\n(3) 在校运动会铅球比赛中, 甲、乙、丙谁获得冠军的概率估计值最大? (结论不要求证明)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448834,7 +450579,9 @@ "id": "017328", "content": "已知椭圆$E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的一个顶点为$A(0,1)$, 焦距为$2 \\sqrt{3}$.\\\\\n(1) 求椭圆$E$的方程;\\\\\n(2) 过点$P(-2,1)$作斜率为$k$的直线与椭圆$E$交于不同的两点$B, C$, 直线$AB, AC$分别与$x$轴交于点$M, N$. 当$|MN|=2$时, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448854,7 +450601,9 @@ "id": "017329", "content": "已知函数$f(x)=\\mathrm{e}^x \\ln (1+x)$.\\\\\n(1) 求曲线$y=f(x)$在$(0, f(0))$处的切线方程;\\\\\n(2) 设$g(x)=f'(x)$, 讨论$g(x)$在$[0,+\\infty)$上的单调性;\\\\\n(3) 证明: 对任意的$s, t \\in(0,+\\infty)$, 有$f(s+t)>f(s)+f(t)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448874,7 +450623,9 @@ "id": "017330", "content": "已知$Q: a_1, a_2, \\cdots, a_k$为有穷整数数列. 给定正整数$m$, 若对任意的$n \\in\\{1,2, \\cdots, m\\}$, 在$Q$中存在$a_1, a_{i+1}, a_{i+2}, \\cdots, a_{i+j}$($j \\geq 0$), 使得$a_i+a_{i+1}+a_{i+2}+\\cdots+a_{i+j}=n$, 则称$Q$为$m-$连续可表数列.\\\\\n(1) 判断$Q: 2,1,4$是否为$5-$连续可表数列? 是否为$6-$连续可表数列? 说明理由;\\\\\n(2) 若$Q: a_1, a_2, \\cdots, a_k$为$8-$连续可表数列, 求证: $k$的最小值为 $4$;\\\\\n(3) 若$Q: a_1, a_2, \\cdots, a_k$为$20-$连续可表数列, $a_1+a_2+\\cdots+a_k<20$, 求证: $k \\geq 7$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -448894,7 +450645,9 @@ "id": "017331", "content": "若$z=-1+\\sqrt{3} \\mathrm{i}$, 则$\\dfrac{z}{z \\overline {z}-1}=$\\bracket{20}.\n\\fourch{$-1+\\sqrt{3} \\mathrm{i}$}{$-1-\\sqrt{3} \\mathrm{i}$}{$-\\dfrac{1}{3}+\\dfrac{\\sqrt{3}}{3} \\mathrm{i}$}{$-\\dfrac{1}{3}-\\dfrac{\\sqrt{3}}{3} \\mathrm{i}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448914,7 +450667,9 @@ "id": "017332", "content": "某社区通过公益讲座宣传中国非物质文化遗产保护知识. 为了解讲座效果, 随机抽取$10$位社区居民, 让他们在讲座前和讲座后各回答一份相关知识问卷, 这$10$位社区居民在讲座前和讲座后问卷答题的正确率如下图. 则下列选项正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.7, yscale = 0.08]\n\\draw [->] (0,55) -- (11,55);\n\\draw [->] (0,55) -- (0,56) -- (0.2,57) -- (-0.2,58) -- (0,59) -- (0,105);\n\\draw (0,55) node [below left] {$O$};\n\\foreach \\i in {1,2,...,10}\n{\\draw (\\i,55.5) -- (\\i,55) node [below] {$\\i$};};\n\\foreach \\i in {60,65,...,100}\n{\\draw [dotted] (10.5,\\i) -- (0,\\i) node [left] {$\\i\\%$};};\n\\draw (5.5,45) node {居民编号};\n\\draw (-2,80) node [rotate = 90] {正确率};\n\\filldraw (12,70) circle (0.05 and {7/16}) ++ (1,0) node {讲座后};\n\\filldraw (12,80) node {\\tiny$\\times$} node {\\tiny$+$} ++ (1,0) node {讲座前};\n\\foreach \\i/\\j/\\k in {1/65/90,2/60/85,3/70/80,4/60/90,5/65/85,6/75/85,7/90/95,8/85/100,9/80/85,10/95/100}\n{\\filldraw (\\i,\\j) node {\\tiny$\\times$} node {\\tiny$+$} (\\i,\\k) circle (0.05 and {7/16});};\n\\end{tikzpicture}\n\\end{center}\n\\onech{讲座前问卷答题的正确率的中位数小于$70 \\%$}{讲座后问卷答题的正确率的平均数大于$85 \\%$}{讲座前问卷答题的正确率的标准差小于讲座后正确率的标准差}{讲座后问卷答题的正确率的极差大于讲座前正确率的极差}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448934,7 +450689,9 @@ "id": "017333", "content": "设全集$U=\\{-2,-1,0,1,2,3\\}$, 集合$A=\\{-1,2\\}$, $B=\\{x | x^2-4 x+3=0\\}$, 则$\\overline{A \\cup\nB}=$\\bracket{20}.\n\\fourch{$\\{1,3\\}$}{$\\{0,3\\}$}{$\\{-2,1\\}$}{$\\{-2,0\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448954,7 +450711,9 @@ "id": "017334", "content": "如图, 网格纸上绘制的是一个多面体的三视图, 网格小正方形的边长为$1$, 则该多面体的体积为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\foreach \\i in {0,1,2,3,4,5,6}\n{\\draw [dashed,thin] (\\i,0) -- (\\i,4) (\\i+8,0) -- (\\i+8,4) (\\i,-6) -- (\\i,-2);};\n\\foreach \\i in {0,1,2,3,4}\n{\\draw [dashed,thin] (0,\\i) -- (6,\\i) (8,\\i) -- (14,\\i) (0,\\i-6) -- (6,\\i-6);};\n\\draw [ultra thick] (1,1) -- (5,1) -- (3,3) -- (1,3) -- cycle;\n\\draw [ultra thick] (10,1) rectangle (12,3);\n\\draw [ultra thick] (1,-5) rectangle (5,-3) (3,-5) -- (3,-3);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$8$}{$12$}{$16$}{$20$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448974,7 +450733,10 @@ "id": "017335", "content": "函数$y=(3^x-3^{-x}) \\cos x$在区间$[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}]$的图像大致为\\bracket{20}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{(pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{abs((pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180))});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{-(pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{-abs((pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180))});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -448994,7 +450756,9 @@ "id": "017336", "content": "当$x=1$时, 函数$f(x)=a \\ln x+\\dfrac{b}{x}$取得最大值$-2$, 则$f'(2)=$\\bracket{20}.\n\\fourch{$-1$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449014,7 +450778,9 @@ "id": "017337", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 已知$B_1D$与平面$ABCD$和平面$AA_1B_1B$所成的角均为$30^{\\circ}$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\def\\l{{sqrt(3)}}\n\\def\\m{1}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (B_1)--(D)--(B);\n\\draw (A)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$AB=2AD$}{$AB$与平面$AB_1C_1D$所成的角为$30^{\\circ}$}{$AC=CB_1$}{$B_1D$与平面$BB_1C_1C$所成的角为$45^{\\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449034,7 +450800,9 @@ "id": "017338", "content": "沈括的《梦溪笔谈》是中国古代科技史上的杰作, 其中收录了计算圆弧长度的``会圆术'', 如图, $\\overset\\frown{AB}$是以为$O$圆心, $OA$为半径的圆弧, $C$是$AB$的中点, $D$在$\\overset\\frown{AB}$上, $CD \\perp AB$. ``会圆术''给出$\\overset\\frown{AB}$的弧长的近似值$s$的计算公式: $s=AB+\\dfrac{CD^2}{OA}$. 当$OA=2$, $\\angle AOB=60^{\\circ}$时, $s=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (60:2) node [right] {$B$} coordinate (B);\n\\draw (120:2) node [left] {$A$} coordinate (A);\n\\draw (90:2) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [below] {$C$} coordinate (C);\n\\draw (A)--(O)--(B) arc (60:120:2) -- (B) (C)--(D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{11-3 \\sqrt{3}}{2}$}{$\\dfrac{11-4 \\sqrt{3}}{2}$}{$\\dfrac{9-3 \\sqrt{3}}{2}$}{$\\dfrac{9-4 \\sqrt{3}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449054,7 +450822,9 @@ "id": "017339", "content": "甲、乙两个圆锥的母线长相等, 侧面展开图的圆心角之和为$2\\pi$, 侧面积分别为$S_{\\text{甲}}$和$S_{\\text{乙}}$, 体积分别为$V_{\\text{甲}}$和$V_{\\text{乙}}$, 若$\\dfrac{S_{\\text{甲}}}{{S_{\\text{乙}}}}=2$, 则$\\dfrac{V_{\\text{甲}}}{{V_{\\text{乙}}}}=$\\bracket{20}.\n\\fourch{$\\sqrt{5}$}{$2 \\sqrt{2}$}{$\\sqrt{10}$}{$\\dfrac{5 \\sqrt{10}}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449074,7 +450844,9 @@ "id": "017340", "content": "椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左顶点为$A$, 点$P, Q$均在$C$上, 且关于$y$轴对称. 若直线$AP, AQ$的斜率之积为$\\dfrac{1}{4}$, 则$C$的离心率为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{3}}{2}$}{$\\dfrac{\\sqrt{2}}{2}$}{$\\dfrac{1}{2}$}{$\\dfrac{1}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449094,7 +450866,9 @@ "id": "017341", "content": "已知$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$区间在$(0, \\pi)$上恰有三个极值点, 两个零点, 则$\\omega$的取值范围是\\bracket{20}.\n\\fourch{$[\\dfrac{5}{3}, \\dfrac{13}{6})$}{$[\\dfrac{5}{3}, \\dfrac{19}{6})$}{$(\\dfrac{13}{6}, \\dfrac{8}{3}]$}{$(\\dfrac{13}{6}, \\dfrac{19}{6}]$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449114,7 +450888,9 @@ "id": "017342", "content": "已知$a=\\dfrac{31}{32}$, $b=\\cos \\dfrac{1}{4}$, $c=4 \\sin \\dfrac{1}{4}$, 则\\bracket{20}.\n\\fourch{$c>b>a$}{$b>a>c$}{$a>b>c$}{$a>c>b$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449134,7 +450910,9 @@ "id": "017343", "content": "设向量$\\overrightarrow {a}, \\overrightarrow {b}$的夹角的余弦值为$\\dfrac{1}{3}$, 且$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=3$, 则$(2 \\overrightarrow {a}+\\overrightarrow {b}) \\cdot \\overrightarrow {b}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -449154,7 +450932,9 @@ "id": "017344", "content": "若双曲线$y^2-\\dfrac{x^2}{m^2}=1$($m>0$)的渐近线与圆$x^2+y^2-4 y+3=0$相切, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -449174,7 +450954,9 @@ "id": "017345", "content": "从正方体的$8$个顶点中任选$4$个, 则这$4$个点在同一平面上的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -449194,7 +450976,9 @@ "id": "017346", "content": "已知$\\triangle ABC$中, 点$D$在边$BC$上, $\\angle ADB=120^{\\circ}$, $AD=2$, $CD=2BD$. 当$\\dfrac{AC}{AB}$取得最小值时, $BD=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -449214,7 +450998,9 @@ "id": "017347", "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和. 已知$\\dfrac{2S_n}{n}+n=2 a_n+1$.\\\\\n(1) 证明: $\\{a_n\\}$是等差数列;\\\\\n(2) 若$a_4, a_7, a_9$成等比数列, 求$S_n$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449234,7 +451020,9 @@ "id": "017348", "content": "在四棱锥$P-ABCD$中, $PD \\perp$底面$ABCD$, $CD\\parallel AB$, $AD=DC=CB=1$, $AB=2$, $PD=\\sqrt{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5, z = {(245:0.5cm)}]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (0.5,0,{-0.5*sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (D) ++ (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (D) ++ (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(P)--cycle;\n\\draw [dashed] (A)--(D)--(C)(D)--(P)(D)--(B)(P)--(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $BD \\perp PA$;\\\\\n(2) 求$PD$与平面$PAB$的所成的角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449254,7 +451042,10 @@ "id": "017349", "content": "甲、乙两个学校进行体育比赛, 比赛共设三个项目, 每个项目胜方得$10$分, 负方得$0$分, 没有平局. 三个项目比赛结束后, 总得分高的学校获得冠军. 已知甲学校在三个项目中获胜的概率分别为$0.5,0.4,0.8$, 各项目的比赛结果相互独立.\\\\\n(1) 求甲学校获得冠军的概率;\\\\\n(2) 用$X$表示乙学校的总得分, 求$X$的分布列与期望.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449274,7 +451065,9 @@ "id": "017350", "content": "设抛物线$C: y^2=2 p x$($p>0$)的焦点为$F$, 点$D(p, 0)$, 过$F$的直线交$C$于$M, N$两点, 当直线$MD \\perp x$轴时, $|MF|=3$.\\\\\n(1) 求$C$的方程;\\\\\n(2) 设直线$MD$、$ND$与$C$的另一个交点分别为$A, B$, 记直线$MN$、$AB$的倾斜角分别为$\\alpha$, $\\beta$, 当$\\alpha-\\beta$取得最大值时, 求直线$AB$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449294,7 +451087,9 @@ "id": "017351", "content": "已知函数$f(x)=\\dfrac{\\mathrm{e}^x}{x}-\\ln x+x-a$.\\\\\n(1) 若$f(x) \\geq 0$, 求$a$的取值范围;\\\\\n(2) 证明: 若$f(x)$有两个零点$x_1, x_2$, 则$x_1 x_2<1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449314,7 +451109,9 @@ "id": "017352", "content": "在直角坐标系$xOy$中, 曲线$C_1$的参数方程为$\\begin{cases}x=\\dfrac{2+t}{6},\\\\ y=\\sqrt{t}\\end{cases}$($t$是参数), 曲线$C_2$的参数方程为$\\begin{cases}x=-\\dfrac{2+s}{6}, \\\\ y=-\\sqrt{s}\\end{cases}$($s$是参数).\\\\\n(1) 写出$C_1$的普通方程;\\\\\n(2) 以坐标原点为极点, $x$轴正半轴建立极坐标系, 曲线$C_3$的极坐标方程为$2 \\cos \\theta-\\sin \\theta=0$, 求$C_3$与$C_1$交点的直角坐标, 及$C_3$与$C_2$交点的直角坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449334,7 +451131,9 @@ "id": "017353", "content": "已知实数$a, b, c$均为正数, 满足$a^2+b^2+4 c^2=3$, 证明:\\\\\n(1) $a+b+2 c \\leq 3$;\\\\\n(2) 若$b=2 c$, 则$\\dfrac{1}{a}+\\dfrac{1}{c} \\geq 3$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449354,7 +451153,9 @@ "id": "017354", "content": "设集合$A=\\{-2,-1,0,1,2\\}$, $B=\\{x | 0 \\leq x<\\dfrac{5}{2}\\}$, 则$A \\cap B=$\\bracket{20}.\n\\fourch{$\\{0,1,2\\}$}{$\\{-2,-1,0\\}$}{$\\{0,1\\}$}{$\\{1,2\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449374,7 +451175,9 @@ "id": "017355", "content": "某社区通过公益讲座宣传中国非物质文化遗产保护知识. 为了解讲座效果, 随机抽取$10$位社区居民, 让他们在讲座前和讲座后各回答一份相关知识问卷, 这$10$位社区居民在讲座前和讲座后问卷答题的正确率如下图. 则下列选项正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.7, yscale = 0.08]\n\\draw [->] (0,55) -- (11,55);\n\\draw [->] (0,55) -- (0,56) -- (0.2,57) -- (-0.2,58) -- (0,59) -- (0,105);\n\\draw (0,55) node [below left] {$O$};\n\\foreach \\i in {1,2,...,10}\n{\\draw (\\i,55.5) -- (\\i,55) node [below] {$\\i$};};\n\\foreach \\i in {60,65,...,100}\n{\\draw [dotted] (10.5,\\i) -- (0,\\i) node [left] {$\\i\\%$};};\n\\draw (5.5,45) node {居民编号};\n\\draw (-2,80) node [rotate = 90] {正确率};\n\\filldraw (12,70) circle (0.05 and {7/16}) ++ (1,0) node {讲座后};\n\\filldraw (12,80) node {\\tiny$\\times$} node {\\tiny$+$} ++ (1,0) node {讲座前};\n\\foreach \\i/\\j/\\k in {1/65/90,2/60/85,3/70/80,4/60/90,5/65/85,6/75/85,7/90/95,8/85/100,9/80/85,10/95/100}\n{\\filldraw (\\i,\\j) node {\\tiny$\\times$} node {\\tiny$+$} (\\i,\\k) circle (0.05 and {7/16});};\n\\end{tikzpicture}\n\\end{center}\n\\onech{讲座前问卷答题的正确率的中位数小于$70 \\%$}{讲座后问卷答题的正确率的平均数大于$85 \\%$}{讲座前问卷答题的正确率的标准差小于讲座后正确率的标准差}{讲座后问卷答题的正确率的极差大于讲座前正确率的极差}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449394,7 +451197,9 @@ "id": "017356", "content": "若$z=1+\\mathrm{i}$. 则$|\\mathrm{i} z+3 \\overline {z}|=$\\bracket{20}.\n\\fourch{$4 \\sqrt{5}$}{$4 \\sqrt{2}$}{$2 \\sqrt{5}$}{$2 \\sqrt{2}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449414,7 +451219,9 @@ "id": "017357", "content": "如图, 网格纸上绘制的是一个多面体的三视图, 网格小正方形的边长为$1$, 则该多面体的体积为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\foreach \\i in {0,1,2,3,4,5,6}\n{\\draw [dashed,thin] (\\i,0) -- (\\i,4) (\\i+8,0) -- (\\i+8,4) (\\i,-6) -- (\\i,-2);};\n\\foreach \\i in {0,1,2,3,4}\n{\\draw [dashed,thin] (0,\\i) -- (6,\\i) (8,\\i) -- (14,\\i) (0,\\i-6) -- (6,\\i-6);};\n\\draw [ultra thick] (1,1) -- (5,1) -- (3,3) -- (1,3) -- cycle;\n\\draw [ultra thick] (10,1) rectangle (12,3);\n\\draw [ultra thick] (1,-5) rectangle (5,-3) (3,-5) -- (3,-3);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$8$}{$12$}{$16$}{$20$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449434,7 +451241,9 @@ "id": "017358", "content": "将函数$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$($\\omega>0$)的图像向左平移$\\dfrac{\\pi}{2}$个单位长度后得到曲线$C$, 若$C$关于$y$轴对称, 则$\\omega$的最小值是\\bracket{20}.\n\\fourch{$\\dfrac{1}{6}$}{$\\dfrac{1}{4}$}{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449454,7 +451263,9 @@ "id": "017359", "content": "从分别写有$1,2,3,4,5,6$的$6$张卡片中无放回随机抽取$2$张, 则抽到的$2$张卡片上的数字之积是$4$的倍数的概率为\\bracket{20}\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{1}{3}$}{$\\dfrac{2}{5}$}{$\\dfrac{2}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449474,7 +451285,10 @@ "id": "017360", "content": "函数$y=(3^x-3^{-x}) \\cos x$在区间$[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}]$的图像大致为\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{(pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{abs((pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180))});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{-(pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{-abs((pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180))});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449494,7 +451308,9 @@ "id": "017361", "content": "当$x=1$时, 函数$f(x)=a \\ln x+\\dfrac{b}{x}$取得最大值$-2$, 则$f'(2)=$\\bracket{20}.\n\\fourch{$-1$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449514,7 +451330,9 @@ "id": "017362", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 已知$B_1D$与平面$ABCD$和平面$AA_1B_1B$所成的角均为$30^{\\circ}$, 则\\bracket{20}.\n\\fourch{$AB=2AD$}{$AB$与平面$AB_1C_1D$所成的角为$30^{\\circ}$}{$AC=CB_1$}{$B_1D$与平面$BB_1C_1C$所成的角为$45^{\\circ}$}\n\\fourch{$\\sqrt{5}$}{$2 \\sqrt{2}$}{$\\sqrt{10}$}{$\\dfrac{5 \\sqrt{10}}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449534,7 +451352,9 @@ "id": "017363", "content": "甲、乙两个圆锥的母线长相等, 侧面展开图的圆心角之和为$2\\pi$, 侧面积分别为$S_{\\text{甲}}$和$S_{\\text{乙}}$, 体积分别为$V_{\\text{甲}}$和$V_{\\text{乙}}$, 若$\\dfrac{S_{\\text{甲}}}{{S_{\\text{乙}}}}=2$, 则$\\dfrac{V_{\\text{甲}}}{{V_{\\text{乙}}}}=$\\bracket{20}.\n\\fourch{$\\sqrt{5}$}{$2 \\sqrt{2}$}{$\\sqrt{10}$}{$\\dfrac{5 \\sqrt{10}}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449554,7 +451374,9 @@ "id": "017364", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的离心率为$\\dfrac{1}{3}$, $A_1, A_2$分别为$C$的左、右顶点, $B$为$C$的上顶点. 若$\\overrightarrow{BA_1} \\cdot \\overrightarrow{BA_2}=-1$, 则$C$的方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{18}+\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}{9}+\\dfrac{y^2}{8}=1$}{$\\dfrac{x^2}{3}+\\dfrac{y^2}{2}=1$}{$\\dfrac{x^2}{2}+y^2=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449574,7 +451396,9 @@ "id": "017365", "content": "已知$9^m=10$, $a=10^m-11$, $b=8^m-9$, 则\\bracket{20}.\n\\fourch{$a>0>b$}{$a>b>0$}{$b>a>0$}{$b>0>a$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449594,7 +451418,9 @@ "id": "017366", "content": "已知向量$\\overrightarrow {a}=(m, 3)$, $\\overrightarrow {b}=(1, m+1)$. 若$\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -449614,7 +451440,9 @@ "id": "017367", "content": "设点$M$在直线$2 x+y-1=0$上, 点$(3,0)$和$(0,1)$均在圆$M$上, 则圆$M$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -449634,7 +451462,9 @@ "id": "017368", "content": "记双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的离心率为$e$, 写出满足条件``直线$y=2 x$与$C$无公共点''的$e$的一个值\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -449654,7 +451484,9 @@ "id": "017369", "content": "已知$\\triangle ABC$中, 点$D$在边$BC$上, $\\angle ADB=120^{\\circ}$, $AD=2$, $CD=2BD$. 当$\\dfrac{AC}{AB}$取得最小值时, $BD=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -449674,7 +451506,10 @@ "id": "017370", "content": "甲、乙两城之间的长途客车均由$A$和$B$两家公司运营, 为了解这两家公司长途客车的运行情况, 随机调查了甲、乙两城之间的$500$个班次, 得到下面列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& 准点班次数 & 未准点班次数 \\\\\\hline\n$A$& 240 & 20 \\\\\\hline\n$B$& 210 & 30 \\\\\\hline\n\\end{tabular}\n\\end{center}\n(1) 根据上表, 分别估计这两家公司甲、乙两城之间的长途客车准点的概率;\\\\\n(2) 能否有$90 \\%$的把握认为甲、乙两城之间的长途客车是否准点与客车所属公司有关?\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$,\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$P(\\chi^2 \\geq k)$& 0.100 & 0.050 & 0.010 \\\\\\hline\n$k$& 2.706 & 3.841 & 6.635 \\\\\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449694,7 +451529,9 @@ "id": "017371", "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和. 已知$\\dfrac{2S_n}{n}+n=2 a_n+1$.\\\\\n(1) 证明: $\\{a_n\\}$是等差数列;\\\\\n(2) 若$a_4, a_7, a_9$成等比数列, 求$S_n$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449714,7 +451551,9 @@ "id": "017372", "content": "小明同学参加综合实践活动, 设计了一个封闭的包装盒, 包装盒如图所示: 底面$ABCD$是边长为$8$(单位: $\\text{cm}$)的正方形, $\\triangle EAB$, $\\triangle FBC$, $\\triangle GCD$, $\\triangle HDA$均为正三角形, 且它们所在的平面都与平面$ABCD$垂直.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,-2) node [right] {$C$} coordinate (C);\n\\draw (0,0,-2) node [below] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) ++ (0,{sqrt(3)},0) node [above] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) ++ (0,{sqrt(3)},0) node [right] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(D)$) ++ (0,{sqrt(3)},0) node [above] {$G$} coordinate (G);\n\\draw ($(D)!0.5!(A)$) ++ (0,{sqrt(3)},0) node [left] {$H$} coordinate (H);\n\\draw (A)--(B)--(C)(E)--(F)--(G)--(H)--cycle(H)--(A)--(E)--(B)--(F)--(C);\n\\draw [dashed] (A)--(D)--(C)(H)--(D)--(G)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $EF\\parallel$平面$ABCD$;\\\\\n(2) 求该包装盒的容积 (不计包装盒材料的厚度).", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449734,7 +451573,9 @@ "id": "017373", "content": "已知函数$f(x)=x^3-x$, $g(x)=x^2+a$, 曲线$y=f(x)$在点$(x_1, f(x_1))$处的切线也是曲线$y=g(x)$的切线.\\\\\n(1) 若$x_1=-1$, 求$a$;\\\\\n(2) 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449754,7 +451595,9 @@ "id": "017374", "content": "设抛物线$C: y^2=2 p x$($p>0$)的焦点为$F$, 点$D(p, 0)$, 过$F$的直线交$C$于$M, N$两点. 当直线$MD$垂直于$x$轴时, $|MF|=3$.\\\\\n(1) 求$C$的方程;\\\\\n(2) 设直线$MD, ND$与$C$的另一个交点分别为$A, B$, 记直线$MN, AB$的倾斜角分别为$\\alpha$, $\\beta$. 当$\\alpha-\\beta$取得最大值时, 求直线$AB$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449774,7 +451617,9 @@ "id": "017375", "content": "在直角坐标系$xOy$中, 曲线$C_1$的参数方程为$\\begin{cases}x=\\dfrac{2+t}{6},\\\\ y=\\sqrt{t}\\end{cases}$($t$为参数), 曲线$C_2$的参数方程为$\\begin{cases}x=-\\dfrac{2+s}{6},\\\\ y=-\\sqrt{s}\\end{cases}$($s$为参数).\\\\\n(1) 写出$C_1$的普通方程;\\\\\n(2) 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 曲线$C_3$的极坐标方程为$2 \\cos \\theta-\\sin \\theta=0$, 求$C_3$与$C_1$交点的直角坐标, 及$C_3$与$C_2$交点的直角坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449794,7 +451639,9 @@ "id": "017376", "content": "已知$a, b, c$均为正数, 且$a^2+b^2+4 c^2=3$, 证明:\\\\\n(1) $a+b+2 c \\leq 3$;\\\\\n(2) 若$b=2 c$, 则$\\dfrac{1}{a}+\\dfrac{1}{c} \\geq 3$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -449814,7 +451661,9 @@ "id": "017377", "content": "设全集$U=\\{1,2,3,4,5\\}$, 集合$M$满足$\\overline{M}=\\{1,3\\}$, 则\\bracket{20}.\n\\fourch{$2 \\in M$}{$3 \\in M$}{$4 \\notin M$}{$5 \\notin M$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449834,7 +451683,9 @@ "id": "017378", "content": "已知$z=1-2 \\mathrm{i}$, 且$z+a \\cdot \\overline {z}+b=0$, 其中$a, b$为实数, 则\\bracket{20}.\n\\fourch{$a=1$, $b=-2$}{$a=-1$, $b=2$}{$a=1$, $b=2$}{$a=-1$, $b=-2$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449854,7 +451705,9 @@ "id": "017379", "content": "已知向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=\\sqrt{3}$, $|\\overrightarrow {a}-2 \\overrightarrow {b}|=3$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}=$\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$1$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449874,7 +451727,9 @@ "id": "017380", "content": "嫦娥二号卫星在完成探月任务后, 继续进行深空探测, 成为我国第一颗环绕太阳飞行的人造卫星. 为研究嫦娥二号绕日周期与地球绕日周期的比值, 用到数列$\\{b_n\\}$: $b_1=1+\\dfrac{1}{a_1}$, $b_2=1+\\dfrac{1}{a_1+\\dfrac{1}{a_2}}$, $b_3=1+\\dfrac{1}{a_1+\\dfrac{1}{a_2+\\dfrac{1}{a_3}}}$, $\\cdots$, 以此类推, 其中$a_k$($k=1,2, \\cdots$)均为正整数, 则\\bracket{20}.\n\\fourch{$b_1=latex, node distance = 10pt]\n\\node [draw, rounded corners] (start) {开始};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of start] (step1) {输入$a=1$, $b=1$, $n=1$};\n\\node [draw, below = of step1] (step2) {$b=b+2a$};\n\\node [draw, below = of step2] (step3) {$a=b-a$, $n=n+1$};\n\\node [draw, diamond, aspect = 2, below = of step3] (switch) {$|\\dfrac{b^2}{a^2}-2|<0.01$};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of switch] (step4) {输出$n$};\n\\node [draw, rounded corners, below = of step4] (end) {结束};\n\\coordinate [left = 15pt of switch] (stepx);\n\\foreach \\i/\\j in {start/step1,step1/step2,step2/step3,step3/switch,step4/end}\n{\\draw [->] (\\i)--(\\j);};\n\\draw [->] (switch) -- node [right] {是} (step4);\n\\draw (switch) -- (stepx) node[below, midway] {否};\n\\draw [->] (stepx) -- (stepx|-step1) -> (step1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$3$}{$4$}{$5$}{$6$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449934,7 +451793,9 @@ "id": "017383", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E, F$分别为$AB, BC$的中点, 则\\bracket{20}.\n\\twoch{平面$B_1EF \\perp$平面$BDD_1$}{平面$B_1EF \\perp$平面$A_1BD$}{平面$B_1EF\\parallel$平面$A_1AC$}{平面$B_1EF\\parallel$平面$A_1C_1D$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449954,7 +451815,9 @@ "id": "017384", "content": "已知等比数列$\\{a_n\\}$的前$3$项和为$168$, $a_2-a_5=42$, 则$a_6=$\\bracket{20}.\n\\fourch{$14$}{$12$}{$6$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449974,7 +451837,9 @@ "id": "017385", "content": "已知球$O$的半径为$1$, 四棱锥的顶点为$O$, 底面的四个顶点均在球$O$的球面上, 则当该四棱锥的体积最大时, 其高为\\bracket{20}.\n\\fourch{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{3}$}{$\\dfrac{\\sqrt{2}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -449994,7 +451859,9 @@ "id": "017386", "content": "某棋手与甲、乙、丙三位棋手各比赛一盘, 各盘比赛结果相互独立. 已知该棋手与甲、乙、丙比赛获胜的概率分别为$p_1, p_2, p_3$且$p_3>p_2>p_1>0$. 记该棋手连胜两盘的概率为$p$, 则\\bracket{20}.\n\\twoch{$p$与该棋手和甲, 乙, 丙的比赛次序无关}{该棋手在第二盘与甲比赛, $p$最大}{该棋手在第二盘与乙比赛, $p$最大}{该棋手在第二盘与丙比赛, $p$最大}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -450014,7 +451881,9 @@ "id": "017387", "content": "双曲线$C$的两个焦点$F_1, F_2$, 以$C$的实轴为直径的圆记为$D$, 过$F_1$作$D$的切线与$C$交于$M, N$两点, 且$\\cos \\angle F_1NF_2=\\dfrac{3}{5}$, 则$C$的离心率为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{5}}{2}$}{$\\dfrac{3}{2}$}{$\\dfrac{\\sqrt{13}}{2}$}{$\\dfrac{\\sqrt{17}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -450034,7 +451903,9 @@ "id": "017388", "content": "已知函数$f(x), g(x)$的定义域均为$\\mathbf{R}$, 且$f(x)+g(2-x)=5$, $g(x)-f(x-4)=7$. 若$y=g(x)$的图像关于直线$x=2$对称, $g(2)=4$, 则$\\displaystyle\\sum_{k=1}^{22} f(k)=$\\bracket{20}.\n\\fourch{$-21$}{$-22$}{$-23$}{$-24$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -450054,7 +451925,9 @@ "id": "017389", "content": "从甲、乙等$5$名同学中随机选$3$名参加社区服务工作, 则甲、乙都入选的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -450074,7 +451947,9 @@ "id": "017390", "content": "过四点$(0,0)$, $(4,0)$, $(-1,1)$, $(4,2)$中的三点的一个圆的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -450094,7 +451969,9 @@ "id": "017391", "content": "记函数$f(x)=\\cos (\\omega x+\\varphi)$($\\omega>0$, $0<\\varphi<\\pi$)的最小正周期为$T$, 若$f(T)=\\dfrac{\\sqrt{3}}{2}$, $x=\\dfrac{\\pi}{9}$为$f(x)$的零点, 则$\\omega$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -450114,7 +451991,9 @@ "id": "017392", "content": "已知$x=x_1$和$x=x_2$分别是函数$f(x)=2 a^x-\\mathrm{e} x^2$($a>0$且$a \\neq 1$)的极小值点和极大值点, 若$x_1=latex,scale = 2]\n\\draw (0,0,0) node [below] {$E$} coordinate (E);\n\\draw ({sqrt(3)},0,0) node [right] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (0,0,-1) node [below] {$C$} coordinate (C);\n\\draw ($(D)!0.7!(B)$) node [above] {$F$} coordinate (F);\n\\draw (A)--(B)--(D)--cycle(A)--(F);\n\\draw [dashed] (A)--(D)(D)--(C)--(B)(F)--(C)--(A)(D)--(E)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$BED \\perp$平面$ACD$;\\\\\n(2) 设$AB=BD=2$, $\\angle ACB=60^{\\circ}$, 点$F$在$BD$上, 当$\\triangle AFC$的面积最小时, 求$CF$与平面$ABD$所成角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450174,7 +452057,9 @@ "id": "017395", "content": "某地经过多年的环境治理, 已将荒山改造成了绿水青山, 为估计一林区某种树木的总材积量, 随机选取了$10$棵这种树木, 测量每棵树的根部横截面积 (单位: $m^2$) 和材积量 (单位: $m^3$), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n样本号$i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 总和 \\\\\\hline\n根部横截面积$x_i$ & 0.04 & 0.06 & 0.04 & 0.08 & 0.08 & 0.05 & 0.05 & 0.07 & 0.07 & 0.06 & 0.6 \\\\\\hline\n材积量$y_i$& 0.25 & 0.40 & 0.22 & 0.54 & 0.51 & 0.34 & 0.36 & 0.46 & 0.42 & 0.40 & 3.9 \\\\\\hline\n\\end{tabular}\n\\end{center}\n并计算得$\\displaystyle\\sum_{i=1}^{10} x_i^2=0.038$, $\\displaystyle\\sum_{i=1}^{10} y_i^2=1.6158$, $\\displaystyle\\sum_{i=1}^{10} x_i y_i=0.2474$.\\\\\n(1) 估计该林区这种树木平均一棵的根部横截面积与平均一棵的材积量;\\\\\n(2) 求该林区这种树木的根部横截面积与材积量的样本相关系数 (精确到 0.01);\\\\\n(3) 现测量了该林区所有这种树木的根部横截面积, 并得到所有这种树木的根部横截面积总和为$186 \\mathrm{m}^2$. 已知树木的材积量与其根部横截面积近似成正比. 利用以上数据给出该林区这种树木的总材积量的估计值.\\\\\n附: 相关系数$r=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})(y_i-\\overline {y})}{\\sqrt{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})^2 \\sum_{i=1}^n(y_i-\\overline {y})^2}}$, $\\sqrt{1.896} \\approx 1.377$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450194,7 +452079,10 @@ "id": "017396", "content": "已知椭圆$E$的中心为坐标原点, 对称轴为$x$轴, $y$轴, 且过$A(0,-2)$, $B(\\dfrac{3}{2},-1)$两点.\\\\\n(1) 求$E$的方程;\\\\\n(2) 设过点$P(1,-2)$的直线交$E$于$M, N$两点, 过$M$且平行于$x$的直线与线段$AB$交于点$T$, 点$H$满足$\\overrightarrow{MT}=\\overrightarrow{TH}$, 证明: 直线$HN$过定点.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450214,7 +452102,9 @@ "id": "017397", "content": "已知函数$f(x)=\\ln (1+x)+a x e^{-x}$.\\\\\n(1) 当$a=1$时, 求曲线$f(x)$在点$(0, f(0))$处的切线方程;\\\\\n(2) 若$f(x)$在区间$(-1,0)$, $(0,+\\infty)$各恰有一个零点, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450234,7 +452124,9 @@ "id": "017398", "content": "在直角坐标系$xOy$中, 曲线$C$的方程为$\\begin{cases}x=\\sqrt{3} \\cos 2 t \\\\ y=2 \\sin t\\end{cases}$($t$为参数). 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 已知直线$l$的极坐标方程为$\\rho \\sin (\\theta+\\dfrac{\\pi}{3})+m=0$.\\\\\n(1) 写出$l$的直角坐标方程;\\\\\n(2) 若$l$与$C$有公共点, 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450254,7 +452146,9 @@ "id": "017399", "content": "已知$a, b, c$为正数, 且$a^{\\frac{3}{2}}+b^{\\frac{3}{2}}+c^{\\frac{3}{2}}=1$, 证明:\\\\\n(1) $a b c \\leq \\dfrac{1}{9}$;\\\\\n(2) $\\dfrac{a}{b+c}+\\dfrac{b}{a+c}+\\dfrac{c}{a+b} \\leq \\dfrac{1}{2 \\sqrt{abc}}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450274,7 +452168,9 @@ "id": "017400", "content": "集合$M=\\{2,4,6,8,10\\}$, $N=\\{x |-1=latex, node distance = 10pt]\n\\node [draw, rounded corners] (start) {开始};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of start] (step1) {输入$a=1$, $b=1$, $n=1$};\n\\node [draw, below = of step1] (step2) {$b=b+2a$};\n\\node [draw, below = of step2] (step3) {$a=b-a$, $n=n+1$};\n\\node [draw, diamond, aspect = 2, below = of step3] (switch) {$|\\dfrac{b^2}{a^2}-2|<0.01$};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of switch] (step4) {输出$n$};\n\\node [draw, rounded corners, below = of step4] (end) {结束};\n\\coordinate [left = 15pt of switch] (stepx);\n\\foreach \\i/\\j in {start/step1,step1/step2,step2/step3,step3/switch,step4/end}\n{\\draw [->] (\\i)--(\\j);};\n\\draw [->] (switch) -- node [right] {是} (step4);\n\\draw (switch) -- (stepx) node[below, midway] {否};\n\\draw [->] (stepx) -- (stepx|-step1) -> (step1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$3$}{$4$}{$5$}{$6$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -450414,7 +452322,10 @@ "id": "017407", "content": "右图是下列四个函数中的某个函数在区间$[-3,3]$的大致图像, 则函数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (-3,0.2) -- (-3,0) node [below] {$-3$};\n\\draw [domain = -3:3, samples = 100] plot (\\x,{(-\\x*\\x*\\x+3*\\x)/(\\x*\\x+1)});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$y=\\dfrac{-x^3+3 x}{x^2+1}$}{$y=\\dfrac{x^3-x}{x^2+1}$}{$y=\\dfrac{2 x \\cos x}{x^2+1}$}{$y=\\dfrac{2 \\sin x}{x^2+1}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -450434,7 +452345,9 @@ "id": "017408", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E, F$分别为$AB, BC$的中点, 则\\bracket{20}.\n\\twoch{平面$B_1EF \\perp BDD_1$}{平面$B_1EF \\perp A_1BD$}{平面$B_1EF\\parallel A_1AC$}{平面$B_1EF\\parallel A_1C_1D$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -450454,7 +452367,9 @@ "id": "017409", "content": "已知等比数列$\\{a_n\\}$的前$3$项和为$168$, $a_2-a_5=42$, 则$a_6=$\\bracket{20}.\n\\fourch{$14$}{$12$}{$6$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -450474,7 +452389,10 @@ "id": "017410", "content": "函数$f(x)=\\cos x+(x+1) \\sin x+1$在区间$[0,2 \\pi]$的最小值、最大值分别为\\bracket{20}.\n\\fourch{$-\\dfrac{\\pi}{2}$, $\\dfrac{\\pi}{2}$}{$-\\dfrac{3 \\pi}{2}$, $\\dfrac{\\pi}{2}$}{$-\\dfrac{\\pi}{2}$, $\\dfrac{\\pi}{2}+2$}{$-\\dfrac{3 \\pi}{2}$, $\\dfrac{\\pi}{2}+2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -450494,7 +452412,9 @@ "id": "017411", "content": "已知球$O$的半径为$1$, 四棱锥的顶点为$O$, 底面的四个顶点均在球$O$的的球面上, 当该四棱锥的体积最大时, 其高为\\bracket{20}.\n\\fourch{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{3}$}{$\\dfrac{\\sqrt{2}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -450514,7 +452434,9 @@ "id": "017412", "content": "记$S_n$为等差数列$\\{a_n\\}$的前$n$项和. 若$2S_3=3S_2+6$, 则公差$d=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -450534,7 +452456,9 @@ "id": "017413", "content": "从甲、乙等$5$名同学中随机选$3$名参加社区服务工作, 则甲、乙都入选的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -450554,7 +452478,9 @@ "id": "017414", "content": "过四点$(0,0)$, $(4,0)$, $(-1,1)$, $(4,2)$中的三点的一个圆的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -450574,7 +452500,9 @@ "id": "017415", "content": "若$f(x)=\\ln |a+\\dfrac{1}{1-x}|+b$是奇函数, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -450594,7 +452522,9 @@ "id": "017416", "content": "记$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$, 已知$\\sin C \\sin (A-B)=\\sin B \\sin (C-A)$.\\\\\n(1) 若$A=2B$, 求$C$;\\\\\n(2) 证明: $2 a^2=b^2+c^2$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450614,7 +452544,9 @@ "id": "017417", "content": "如图, 四面体$ABCD$中, $AD \\perp CD$, $AD=CD$, $\\angle ADB=\\angle BDC$, $E$为$AC$中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [below] {$E$} coordinate (E);\n\\draw ({sqrt(3)},0,0) node [right] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (0,0,-1) node [below] {$C$} coordinate (C);\n\\draw ($(D)!0.7!(B)$) node [above] {$F$} coordinate (F);\n\\draw (A)--(B)--(D)--cycle(A)--(F);\n\\draw [dashed] (A)--(D)(D)--(C)--(B)(F)--(C)--(A)(D)--(E)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$BED \\perp$平面$ACD$;\\\\\n(2) 设$AB=BD=2$, $\\angle ACB=60^{\\circ}$, 点$F$在$BD$上, 当$\\triangle AFC$面积最小时, 求三棱锥$F-ABC$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450634,7 +452566,9 @@ "id": "017418", "content": "某地经过多年的环境治理, 已将荒山改造成了绿水青山, 为估计一林区某种树木的总材积量, 随机选取了$10$棵这种树木, 测量每棵树的根部横截面积 (单位: $m^2$) 和材积量 (单位: $m^3$), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n样本号$i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 总和 \\\\ \\hline\n根部横截面积$x_i$ & 0.04 & 0.06 & 0.04 & 0.08 & 0.08 & 0.05 & 0.05 & 0.07 & 0.07 & 0.06 & 0.6 \\\\ \\hline\n材积量$y_i$& 0.25 & 0.40 & 0.22 & 0.54 & 0.51 & 0.34 & 0.36 & 0.46 & 0.42 & 0.40 & 3.9 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n并计算得$\\displaystyle\\sum_{i=1}^{10} x_i^2=0.038$, $\\displaystyle\\sum_{i=1}^{10} y_i^2=1.6158$, $\\displaystyle\\sum_{i=1}^{10} x_i y_i=0.2474$.\\\\\n(1) 估计该林区这种树木平均一棵的根部横截面积与平均一棵的材积量;\\\\\n(2) 求该林区这种树木的根部横截面积与材积量的样本相关系数 (精确到 0.01);\\\\\n(3) 现测量了该林区所有这种树木的根部横截面积, 并得到所有这种树木的根部横截面积总和为$186 \\mathrm{m}^2$. 已知树木的材积量与其根部横截面积近似成正比. 利用以上数据给出该林区这种树木的总材积量的估计值.\\\\\n附: 相关系数$r=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})(y_i-\\overline {y})}{\\sqrt{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})^2 \\sum_{i=1}^n(y_i-\\overline {y})^2}}$, $\\sqrt{1.896} \\approx 1.377$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450654,7 +452588,9 @@ "id": "017419", "content": "已知函数$f(x)=a x-\\dfrac{1}{x}-(a+1) \\ln x$.\\\\\n(1) 当$a=0$时, 求$f(x)$的最大值;\\\\\n(2) 若$f(x)$恰有一个零点, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450674,7 +452610,10 @@ "id": "017420", "content": "已知椭圆$E$的中心为坐标原点, 对称轴为$x$轴, $y$轴, 且过$A(0,-2), B(\\dfrac{3}{2},-1)$两点.\\\\\n(1) 求$E$的方程;\\\\\n(2) 设过点$P(1,-2)$的直线交$E$于$M, N$两点, 过$M$且平行于$x$轴的直线与线段$AB$交于点$T$, 点$H$满足$\\overrightarrow{MT}=\\overrightarrow{TH}$, 证明: 直线$HN$过定点.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450694,7 +452633,9 @@ "id": "017421", "content": "在直角坐标系$x O y$中, 曲线$C$的方程为$\\begin{cases}x=\\sqrt{3} \\cos 2 t, \\\\ y=2 \\sin t\\end{cases}$($t$为参数). 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 已知直线$l$的极坐标方程为$\\rho \\sin (\\theta+\\dfrac{\\pi}{3})+m=0$.\\\\\n(1) 写出$l$的直角坐标方程;\\\\\n(2) 若$l$与$C$有公共点, 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450714,7 +452655,9 @@ "id": "017422", "content": "已知$a, b, c$为正数, 且$a^{\\frac{3}{2}}+b^{\\frac{3}{2}}+c^{\\frac{3}{2}}=1$, 证明:\\\\\n(1) $a b c \\leq \\dfrac{1}{9}$;\\\\\n(2) $\\dfrac{a}{b+c}+\\dfrac{b}{a+c}+\\dfrac{c}{a+b} \\leq \\dfrac{1}{2 \\sqrt{abc}}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -450734,7 +452677,9 @@ "id": "017423", "content": "已知$\\sin \\alpha=\\dfrac{4}{5}$, 则$\\cos (\\alpha+\\dfrac{\\pi}{2})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac 45$", "solution": "", @@ -450754,7 +452699,9 @@ "id": "017424", "content": "复数$(a-1)+(2 a-1) \\mathrm{i}(a \\in \\mathbf{R})$在复平面的第二象限内, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(\\dfrac 12,1)$", "solution": "", @@ -450774,7 +452721,9 @@ "id": "017425", "content": "$(2 x+\\dfrac{1}{x})^6$二项展开式中, 常数项为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$160$", "solution": "", @@ -450794,7 +452743,9 @@ "id": "017426", "content": "点$P(2,16)$、$Q(\\log _23, t)$都在同一个指数函数的图像上, 则$t=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$9$", "solution": "", @@ -450814,7 +452765,9 @@ "id": "017427", "content": "同一平面内的两个不平行的单位向量$\\overrightarrow {a}, \\overrightarrow {b}$, $\\overrightarrow {a}$在$\\overrightarrow {b}$上的投影向量为$\\overrightarrow{a_0}$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}-\\overrightarrow{a_0} \\cdot \\overrightarrow {b}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$0$", "solution": "", @@ -450834,7 +452787,9 @@ "id": "017428", "content": "一个正方体和一个球的表面积相同, 则正方体的体积$V_1$和球的体积$V_2$的比值$\\dfrac{V_1}{V_2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{6\\pi}}{6}$", "solution": "", @@ -450854,7 +452809,9 @@ "id": "017429", "content": "$P(x_0, y_0)$为抛物线$x^2=4 y$上一点, 其中$y_0<4$, $F$为抛物线焦点, 直线$l$方程为$y=4$, $PH \\perp l$, $H$为垂足, 则$|PF|+|PH|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -450874,7 +452831,9 @@ "id": "017430", "content": "函数$y=x^3$在区间$[0,2]$的平均变化率与在$x=x_0$($0 \\leq x_0 \\leq 2$)处的瞬时变化率相同, 则正数$x_0=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{3}}{3}$", "solution": "", @@ -450894,7 +452853,9 @@ "id": "017431", "content": "若数列$\\{a_n\\}$满足: 对于任意正整数$n$都有$\\displaystyle\\sum_{i=1}^n a_i=2^{n+1}-n-2$成立, 则$\\displaystyle\\sum_{i=1}^{+\\infty}(\\dfrac{a_i}{4^i})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$\\dfrac{2}{3}$", "solution": "", @@ -450914,7 +452875,9 @@ "id": "017432", "content": "正方体$ABCD-A_1B_1C_1D_1$的棱长为$4$, $P$在平面$BCC_1B_1$上, $A, P$之间的距离为$5$, 则$C_1$、$P$之间的最短距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$4\\sqrt{2}-3$", "solution": "", @@ -450934,7 +452897,9 @@ "id": "017433", "content": "如图: 已知$\\triangle ABC$中, $\\angle A=30^{\\circ}$, 边长为$1$的正方形$DEFG$为$\\triangle ABC$的内接正方形, 则$AB+AC$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw ({sqrt(3)},0) node [below] {$D$} coordinate (D);\n\\draw (D) ++ (1,0) node [below] {$E$} coordinate (E);\n\\draw (D) ++ (0,1) node [above left] {$G$} coordinate (G);\n\\draw (E) ++ (0,1) node [above right] {$F$} coordinate (F);\n\\draw ($(A)!1.4!(G)$) node [above] {$C$} coordinate (C);\n\\draw ($(C)!{1.4/0.4}!(F)$) node [below] {$B$} coordinate (B);\n\\draw (A)--(B)--(C)--cycle(D)--(G)--(F)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$3+\\dfrac{5\\sqrt{3}}{3}$", "solution": "", @@ -450954,7 +452919,9 @@ "id": "017434", "content": "设$f(x)=x^2$($x \\geq 1$), $g(x)=(x-2)^2+b$($x \\geq 3$), $A, D$为曲线$y=f(x)$上两点, $B, C$为曲线$y=g(x)$上两点, 且四边形$ABCD$为矩形, 则实数$b$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-1,0)$", "solution": "", @@ -450974,7 +452941,9 @@ "id": "017435", "content": "``$x>1$''是``$x \\geq 1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -450994,7 +452963,9 @@ "id": "017436", "content": "如图, 直角坐标系中有$4$条圆锥曲线$C_i$($i=1,2,3,4$), 其离心率分别为$e_i$. 则$4$条圆锥曲线的离心率的大小关系是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse (1.5 and 0.4);\n\\draw (0,0) ellipse (1.5 and 0.9);\n\\draw [domain = -1.5:1.5] plot ({sqrt(\\x*\\x+1)},\\x);\n\\draw [domain = -1.5:1.5] plot ({-sqrt(\\x*\\x+1)},\\x);\n\\draw [domain = -1.5:1.5] plot ({sqrt(\\x*\\x/4+1)},\\x);\n\\draw [domain = -1.5:1.5] plot ({-sqrt(\\x*\\x/4+1)},\\x);\n\\draw (75:1.5 and 0.4) node [below] {$C_1$};\n\\draw (75:1.5 and 0.9) node [below] {$C_2$};\n\\draw ({sqrt(2)},1) node [right] {$C_3$};\n\\draw ({sqrt(1.44/4+1)},1.2) node [left] {$C_4$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$e_20$)图像是类似椭圆的封闭曲线, $T$上动点$P$($P$在第一象限) 到直线$y=-x$距离的最大值为$M(a)$. 当实数$a$变化时, 求$M(a)$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{3 \\sqrt{2}}{2}$}{$2 \\sqrt{2}$}{$\\sqrt{3}$}{$\\sqrt{5}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -451054,7 +453029,9 @@ "id": "017439", "content": "已知扇形$OAB$的半径为$1$, $\\angle AOB=\\dfrac{\\pi}{3}$, $P$是圆弧上一点(不与$A, B$重合), 过$P$作$PM \\perp OA$, $PN \\perp OB$, $M, N$为垂足.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (2,0) node [below right] {$A$} coordinate (A);\n\\draw (60:2) node [above] {$B$} coordinate (B);\n\\draw (20:2) node [above right] {$P$} coordinate (P);\n\\draw ($(O)!(P)!(A)$) node [below] {$M$} coordinate (M);\n\\draw ($(O)!(P)!(B)$) node [left] {$N$} coordinate (N);\n\\draw (O)--(A) arc (0:60:2) --cycle (O)--(P)(P)--(M)(P)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$|PM|=\\dfrac{1}{2}$, 求$PN$的长;\\\\\n(2) 设$\\angle AOP=x, PM, PN$的线段之和为$y$, 求$y$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) $(\\dfrac{\\sqrt{3}}{2},1]$", "solution": "", @@ -451074,7 +453051,9 @@ "id": "017440", "content": "已知三棱锥$P-ABC$, $PA \\perp$平面$ABC$, $PA=6$, $AC=4$, $AB \\perp BC$, $M, N$分别在线段$PB, PC$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (3,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (0,6,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!(A)!(C)$) node [above right] {$N$} coordinate (N);\n\\draw ($(B)!0.25!(P)$) node [right] {$M$} coordinate (M);\n\\draw (A)--(M)--(N)(A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (C)--(A)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$PB$与平面$ABC$所成角大小为$\\dfrac{\\pi}{3}$, 求三棱锥$P-ABC$的体积$V$;\\\\\n(2) 若$PC \\perp$平面$AMN$, 求证: $AM \\perp$平面$PBC$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $4\\sqrt{3}$; (2) 证明略", "solution": "", @@ -451094,7 +453073,10 @@ "id": "017441", "content": "某数学学习小组的$5$位学生在一次考试后调整了学习方法, 一段时间后又参加了第二次考试. 两次考试的成绩如下表所示(满分$100$分):\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline & 学生 1 & 学生 2 & 学生 3 & 学生 4 & 学生 5 \\\\\n\\hline 第一次 & 82 & 89 & 78 & 92 & 81 \\\\\n\\hline 第二次 & 83 & 90 & 75 & 95 & 76 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 在$5$位学生中依次抽取$3$位学生. 在前$2$位学生中至少有$1$位学生第一次成绩高于第二次成绩的条件下, 求第三位学生第二次考试成绩高于第一次考试成绩的概率;\\\\\n(2) 设$x_i$($i=1,2, \\cdots, 5$)表示第$i$位学生第二次考试成绩减去第一次考试成绩的值. 从数学学习小组$5$位学生中随机选取$2$位, 得到数据$x_i, x_j$($1 \\leq i, j \\leq 5$, $i \\neq j$), 定义随机变量$X$如下: $X=\\begin{cases}0, & 0 \\leq|x_i-x_j|<3, \\\\1, & 3 \\leq|x_i-x_j|<6, \\\\2, & |x_i-x_j| \\geq 6, \\end{cases}$ 求$X$的分布列和数学期望$E[X]$和方差.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 57$; (2) $E[X]=1$, $D[X]=0.8$", "solution": "", @@ -451114,7 +453096,9 @@ "id": "017442", "content": "已知双曲线$T: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$), 离心率为$e$, 圆$O: x^2+y^2=R^2$($R>0$).\\\\\n(1) 若$e=2$, 双曲线$T$的右焦点为$F(2,0)$, 求双曲线方程;\\\\\n(2) 若圆$O$过双曲线$T$的右焦点$F$, 圆$O$与双曲线$T$的四个交点恰好四等分圆周, 求$\\dfrac{b^2}{a^2}$的值;\\\\\n(3) 若$R=1$, 不垂直于$x$轴的直线$l: y=k x+m$与圆$O$相切, 且$l$与双曲线$T$交于点$A, B$时总有$\\angle AOB=\\dfrac{\\pi}{2}$, 求离心率$e$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $x^2-\\dfrac{y^2}{3}=1$; (2) $1+\\sqrt{2}$; (3) $(\\sqrt{2},+\\infty)$", "solution": "", @@ -451134,7 +453118,9 @@ "id": "017443", "content": "定义: 若曲线$C_1$和曲线$C_2$有公共点$P$, 且在$P$处的切线相同, 则称$C_1$与$C_2$在点$P$处相切.\\\\\n(1) 设$f(x)=1-x^2$, $g(x)=x^2-8 x+m$. 若曲线$y=f(x)$与曲线$y=g(x)$在点$P$处相切, 求$m$的值;\\\\\n(2) 设$h(x)=x^3$. 若圆$M: x^2+(y-b)^2=R^2$($R>0$)与曲线$y=h(x)$在点$Q$($Q$在第一象限)处相切, 求$b$的最小值;\\\\\n(3) 若函数$y=f(x)$是定义在$\\mathbf{R}$上的连续可导函数, 导函数为$y=f'(x)$, 且满足$|f'(x)| \\geq|f(x)|$和$|f(x)|<\\sqrt{2}$都恒成立. 是否存在点$P$, 使得曲线$y=f(x) \\sin x$和曲线$y=1$在点$P$处相切? 证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $m=9$; (2) $\\dfrac{4\\sqrt{3}}{9}$; (3) 不存在, 证明略", "solution": "", @@ -451154,7 +453140,9 @@ "id": "017444", "content": "已知集合$A=\\{x, x^2+1,-1\\}$中的最大元素为$2$, 则实数$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -451174,7 +453162,9 @@ "id": "017445", "content": "函数$y=2 \\cos x$的严格减区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[2k\\pi,2k\\pi+\\pi ]$, $k\\in \\mathbf{Z}$", "solution": "", @@ -451194,7 +453184,9 @@ "id": "017446", "content": "若函数$y=f(x)$为偶函数, 且当$x<0$时, $f(x)=2^x-1$, 则$f(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$-\\dfrac{1}{2}$", "solution": "", @@ -451214,7 +453206,9 @@ "id": "017447", "content": "若某圆锥高为$3$, 其侧面积与底面积之比为$2: 1$, 则该圆锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$3\\pi$", "solution": "", @@ -451234,7 +453228,9 @@ "id": "017448", "content": "已知样本数据$2$、$4$、$8$、$m$的极差为$10$, 其中$m>0$, 则该组数据的方差为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$\\dfrac{59}{4}$", "solution": "", @@ -451254,7 +453250,9 @@ "id": "017449", "content": "在财务审计中, 我们可以用``本$\\cdot$福特定律''来检验数据是否造假. 本$\\cdot$福特定律指出, 在一组没有人为编造的自然生成的数据 (均为正实数) 中, 首位非零的数字是$1 \\sim 9$这九个事件不是等可能的. 具体来说, 随机变量$X$是一组没有人为编造的首位非零数字, \n则$P(X=k)=\\lg \\dfrac{k+1}{k}$, $k=1,2, \\cdots, 9$. 则根据本$\\cdot$福特定律, 首位非零数字是$1$与首位非零数字是$8$的概率之比约为\\blank{50}(保留至整数).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$6$", "solution": "", @@ -451274,7 +453272,9 @@ "id": "017450", "content": "若$(1-2 x)^{2023}=a_0+a_1 x+\\cdots+a_{2023} x^{2023}$, 则$\\dfrac{a_1}{2}+\\dfrac{a_2}{2^2}+\\cdots+\\dfrac{a_{2023}}{2^{2023}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$-1$", "solution": "", @@ -451294,7 +453294,9 @@ "id": "017451", "content": "若向量$\\overrightarrow {a}$与$\\overrightarrow {b}$不共线也不垂直, 且$\\overrightarrow {c}=\\overrightarrow {a}-(\\dfrac{\\overrightarrow {a} \\cdot \\overrightarrow {a}}{\\overrightarrow {a} \\cdot \\overrightarrow {b}}) \\overrightarrow {b}$, 则向量夹角$\\langle\\overrightarrow {a}, \\overrightarrow {c}\\rangle=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{2}$", "solution": "", @@ -451314,7 +453316,9 @@ "id": "017452", "content": "已知复数$z$在复平面内对应的点是$A$, 其共轭复数$\\overline {z}$在复平面内对应的点是$B$, $O$是坐标原点, 若$A$在第一象限, 且$OA \\perp OB$, 则$\\dfrac{z+\\overline {z}}{z-\\overline {z}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-\\mathrm{i}$", "solution": "", @@ -451334,7 +453338,9 @@ "id": "017453", "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点分别为$F_1$、$F_2$, $\\Gamma$的渐近线与圆$x^2+y^2=a^2$在第一象限的交点为$M$, 线段$MF_2$与$\\Gamma$交于点$N$, $O$为坐标原点. 若$MF_1\\parallel ON$, 则$\\Gamma$的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$\\sqrt{2}$", "solution": "", @@ -451354,7 +453360,9 @@ "id": "017454", "content": "若项数为$10$的数列$\\{a_n\\}$, 满足$1 \\leq|a_{i+1}-a_i| \\leq 2$($i=1,2, \\cdots, 9$), 且$a_1=a_{10} \\in[-1,0]$, 则数列$\\{a_n\\}$中最大项的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$8$", "solution": "", @@ -451374,7 +453382,9 @@ "id": "017455", "content": "若实数$a$使得存在两两不同的实数$x$、$y$、$z$, 有$\\dfrac{x^3+a}{y+z}=\\dfrac{y^3+a}{z+x}=\\dfrac{z^3+a}{x+y}=-3$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$(-2,0)\\cup (0,2)$", "solution": "", @@ -451394,7 +453404,9 @@ "id": "017456", "content": "我国古代数学著作《九章算术》中有如下问题: ``今有善走男, 日增等里, 首日行走一百里, 九日共行一千二百六十里, 问日增几何?\", 该问题中, ``善走男''第$5$日所走的路程里数为\\bracket{20}.\n\\fourch{$110$}{$120$}{$130$}{$140$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -451414,7 +453426,9 @@ "id": "017457", "content": "``$(\\log _a 2) x^2+(\\log _b 2) y^2=1$表示焦点在$y$轴上的椭圆''的一个充分非必要条件是\\bracket{20}.\n\\fourch{$0=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (A)--(B_1)--(C);\n\\draw [dashed] (A)--(C)(B)--(D)(A)--(D_1)(A_1)--(D)(A_1)--(C)(A)--(C_1)(B_1)--(D);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$AB_1$、$AC$、$AD_1$的长度}{$AC$、$B_1D$、$A_1C$的长度}{$B_1C$、$A_1D$、$B_1D$的长度}{$AC_1$、$BD$、$CC_1$的长度}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -451454,7 +453470,10 @@ "id": "017459", "content": "设关于$x$、$y$的表达式$F(x, y)=\\cos ^2 x+\\cos ^2 y-\\cos (x y)$, 当$x$、$y$取遍所有实数时, $F(x, y)$\\bracket{20}.\n\\twoch{既有最大值, 也有最小值}{有最大值, 无最小值}{无最大值, 有最小值}{既无最大值, 也无最小值}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -451474,7 +453493,9 @@ "id": "017460", "content": "在平面直角坐标系$x O y$中, $A(\\dfrac{\\sqrt{2}}{2}, \\dfrac{\\sqrt{2}}{2})$在以原点$O$为圆心半径等$1$的圆上, 将射线$OA$绕\n原点$O$逆时针方向旋转$\\alpha$后交该圆于点$B$, 设点$B$的横坐标为$f(\\alpha)$, 纵坐标$g(\\alpha)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,0) circle (1);\n\\draw (0,0) -- (45:1) node [above right] {$A$} coordinate (A);\n\\draw (0,0) -- (130:1) node [above left] {$B$} coordinate (B);\n\\draw (0,0) pic [draw, scale = 0.5, \"$\\alpha$\", angle eccentricity = 2] {angle = A--O--B};\n\\end{tikzpicture}\n\\end{center}\n(1) 如果$\\sin \\alpha=m, 0=latex]\n\\draw (0,0,0) node [below] {$M$} coordinate (M);\n\\draw (2.5,0,0) node [below] {$B$} coordinate (B);\n\\draw (0,0,-2) node [left] {$N$} coordinate (N);\n\\draw (M) ++ (0,1.5,0) node [left] {$A$} coordinate (A);\n\\draw (N) ++ (0,1.5,0) node [above] {$D$} coordinate (D);\n\\draw (N) ++ (1.5,0,0) node [right] {$C$} coordinate (C);\n\\draw (A)--(M)--(B)--cycle(A)--(C)--(B)(A)--(D)--(C);\n\\draw [dashed] (M)--(N)--(C)(M)--(C)(D)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$AMB\\parallel$平面$DNC$;\\\\\n(2) 若$MC \\perp CB$, 求证: $BC \\perp AC$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略", "solution": "", @@ -451514,7 +453537,9 @@ "id": "017462", "content": "某科技公司为确定下一年度投入某种产品的研发费, 需了解年研发费$x$(单位: 万元) 对年销售量$y$(单位: 百件) 和年利润 (单位: 万元) 的影响, 现对近$6$年的年研发费$x_i$和年销售量$y_i$($i=1,2, \\cdots, 6$)数据作了初步处理, 得到下面的散点图及一些统计量的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {5/105,8/182,11/233,14/258,17/280,20/295}\n{\\filldraw ({\\i/5},{\\j/100}) circle (0.03);}; \n\\end{tikzpicture}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline $\\displaystyle\\sum_{i=1}^6(x_i-\\overline {x})^2$&$\\displaystyle\\sum_{i=1}^6(y_i-\\overline {y})^2$&$\\displaystyle\\sum_{i=1}^6(\\mu_i-\\overline {\\mu})^2$&$\\displaystyle\\sum_{i=1}^6(x_i-\\overline {x})(y_i-\\overline {y})$&$\\displaystyle\\sum_{i=1}^6(\\mu_i-\\overline {\\mu})(y_i-\\overline {y})$\\\\\n\\hline 157.5 & 16800 & 4.5 & 1254 & 270 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n表中$\\mu_i=\\ln x_i$, $\\overline {\\mu}=\\dfrac{1}{6} \\displaystyle\\sum_{i=1}^6 \\mu_i$.\\\\\n(1) 根据散点图判断$\\hat{y}=\\hat{a}+\\hat{b} x$与$\\hat{y}=\\hat{c}+\\hat{d} \\ln x$哪一个更适宜作为年研发费$x$的回归方程类型; (给出判断即可, 不必说明理由)\\\\\n(2) 根据 (1) 的判断结果及表中数据, 建立$y$关于$x$的回归方程;\\\\\n(3) 已知这种产品的年利润$z=0.5 y-x$, 根据 (2) 的结果, 当年研发费为多少时, 年利润$z$的预报值最大?\\\\\n附: 对于一组数据$(w_1, v_1),(w_2, v_2), \\cdots,(w_n, v_n)$, 其回归直线$\\hat{v}=\\hat{\\alpha}+\\widehat{\\beta} w$的斜率和截距的最小二乘估计分别为$\\widehat{\\beta}=\\dfrac{\\displaystyle\\sum_{i=1}^n(w_i-\\overline {w})(v_i-\\overline {v})}{\\displaystyle\\sum_{i=1}^n(w_i-\\overline {w})^2}$, $\\widehat{\\alpha}=\\overline {v}-\\widehat{\\beta} \\overline {w}$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) $\\hat{y}=\\hat{c}+\\hat{d}\\ln x$更好; (2) $\\hat{y}=60\\ln x+12$; (3) $30$万元", "solution": "", @@ -451534,7 +453559,9 @@ "id": "017463", "content": "贝塞尔曲线是计算机图形学和相关领域中重要的参数曲线. 法国数学象卡斯特利奥对贝塞尔曲线进行了图形化应用的测试, 提出了 De Casteljau 算法: 已知三个定点, 根据对应的比例, 使用递推画法, 可以画出抛物线. 反之, 已知抛物线上三点的切线, 也有相应成比例的结论. 如图所示, 抛物线$\\Gamma: x^2=2 p y$, 其中$p>0$为一给定的实数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.5:1.5, samples = 100] plot (\\x,{\\x*\\x});\n\\foreach \\i/\\j/\\k in {-1.2/above/A,-0.3/above/B,0.7/right/C}\n{\\filldraw (\\i,{\\i*\\i}) node [\\j] {$\\k$} coordinate (\\k) circle (0.03);};\n\\draw (A) ++ (1,-2.4) coordinate (A_1);\n\\draw (B) ++ (1,-0.6) coordinate (B_1);\n\\draw (C) ++ (1,1.4) coordinate (C_1);\n\\path [name path = la, draw] ($(A)!-0.2!(A_1)$) -- ($(A)!1.3!(A_1)$);\n\\path [name path = lb, draw] ($(B)!-0.7!(B_1)$) -- ($(B)!1.3!(B_1)$);\n\\path [name path = lc, draw] ($(C)!-1.5!(C_1)$) -- ($(C)!0.5!(C_1)$);\n\\path [name intersections = {of = la and lb, by = D}];\n\\path [name intersections = {of = la and lc, by = E}];\n\\path [name intersections = {of = lb and lc, by = F}];\n\\foreach \\i/\\j in {D/below left,E/left,F/below}\n{\\filldraw (\\i) node [\\j] {$\\i$} circle (0.03);};\n\\end{tikzpicture}\n\\end{center}\n(1) 写出抛物线$\\Gamma$的焦点坐标及准线方程;\\\\\n(2) 若直线$l: y=k x-2 p k+2 p$与抛物线只有一个公共点, 求实数$k$的值;\\\\\n(3) 如图, $A, B, C$是$H$上不同的三点, 过三点的三条切线分别两两交于点$D, E, F$, 证明: $\\dfrac{|AD|}{|DE|}=\\dfrac{|EF|}{|FC|}=\\dfrac{|DB|}{|BF|}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) 焦点为$(0,\\dfrac{p}{2})$, 准线为$y=-\\dfrac{p}{2}$; (2) $k=2$; (3) 证明略", "solution": "", @@ -451554,7 +453581,10 @@ "id": "017464", "content": "设$y=f(x)$是定义域为$\\mathbf{R}$的函数, 如果对任意的$x_1$、$x_2 \\in \\mathbf{R}$($x_1 \\neq x_2$), $|f(x_1)-f(x_2)|<|x_1-x_2|$均成立, 则称$y=f(x)$是``平缓函数''.\\\\\n(1) 若$f_1(x)=\\dfrac{1}{x^2+1}$, $f_2(x)=\\sin x$, 试判断$y=f_1(x)$和$y=f_2(x)$是否为``平缓函数''? 并说明理由; (参考公式: $x>0$时, $\\sin x1$使得函数$y=A \\cdot g(x)$为``平缓函数\". 现定义数列$\\{x_n\\}$满足: $x_1=0$, $x_n=g(x_{n-1})$($n=2,3,4, \\cdots$), 试证明: 对任意的正整数$n$, $g(x_n) \\leq \\dfrac{A|g(0)|}{A-1}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "解答题", "ans": "(1) $y=f_1(x)$与$y=f_2(x)$都是$\\mathbf{R}$上的``平缓函数''; (2) 证明略; (3) 证明略", "solution": "", @@ -451574,7 +453604,9 @@ "id": "017465", "content": "已知集合$A=\\{x \\| x | \\leq 1\\}, B=\\{-1,1,3,5\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$\\{-1,1\\}$", "solution": "", @@ -451594,7 +453626,9 @@ "id": "017466", "content": "复数$z=\\dfrac{1-2 \\mathrm{i}}{3+\\mathrm{i}}$的模为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{2}}{2}$", "solution": "", @@ -451614,7 +453648,9 @@ "id": "017467", "content": "不等式$\\dfrac{x+3}{x-1} \\geq 0$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$(-\\infty,-3]\\cup (1,+\\infty)$", "solution": "", @@ -451634,7 +453670,9 @@ "id": "017468", "content": "已知幂函数$y=f(x)$的图像过点$(\\dfrac{1}{2}, 8)$, 则$f(-2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$-\\dfrac{1}{8}$", "solution": "", @@ -451654,7 +453692,9 @@ "id": "017469", "content": "已知函数$f(x)=\\sin 2 x+2 \\sqrt{3} \\cos ^2 x$, 则函数$f(x)$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi$", "solution": "", @@ -451674,7 +453714,9 @@ "id": "017470", "content": "由函数的观点, 不等式$2^x+\\log _4 x=17$的解是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$x=4$", "solution": "", @@ -451694,7 +453736,9 @@ "id": "017471", "content": "$(\\dfrac{1}{x}-\\sqrt{x})^8$的展开式中含$x$项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$28$", "solution": "", @@ -451714,7 +453758,9 @@ "id": "017472", "content": "某单位为了解该单位党员开展学习党史知识活动情况, 随机抽取了部分党员, 对他们一周的党史学习时间进行了统计, 统计数据如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 党史学习时间(小时) & 7 & 8 & 9 & 10 & 11 \\\\\n\\hline 党员人数 & 6 & 10 & 9 & 8 & 7 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n则该单位党员一周学习党史时间的第$40$百分位数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$8.5$", "solution": "", @@ -451734,7 +453780,9 @@ "id": "017473", "content": "若存在实数$a$, 使得$x=1$是方程$(x+a)^2=3 x+b$的解, 但不是方程$x+a=\\sqrt{3 x+b}$的解, 则实数$b$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-3,+\\infty)$", "solution": "", @@ -451754,7 +453802,9 @@ "id": "017474", "content": "若随机变量$X \\sim N(105,19^2)$, $Y \\sim N(100,9^2)$, 若$P(X \\leq A)=P(Y \\leq A)$, 那么实数$A$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$95.5$", "solution": "", @@ -451774,7 +453824,9 @@ "id": "017475", "content": "已知曲线$C_1: |y|=x+2$与曲线$C_2: (x-a)^2+y^2=4$恰有两个公共点, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$(-4,0)cup \\{2\\sqrt{2}-2\\}$", "solution": "", @@ -451794,7 +453846,9 @@ "id": "017476", "content": "函数$y=f(x)$是最小正周期为$4$的偶函数, 且在$x \\in[-2,0]$时, $f(x)=2 x+1$, 若存在$x_1$、$x_2$、$\\cdots$、$x_n$满足$0 \\leq x_1=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{pow(\\x-0.2,3)-\\x+0.2});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{-(\\x+1)*(\\x-1.3)*(\\x+0.5)*(\\x-0.2)/1.5});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{(\\x+1)*(\\x-1.3)*(\\x+0.5)*(\\x-0.2)/1.5});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{-2.5*(0.05 + 0.192*\\x - 0.44* \\x*\\x - 0.2*pow(\\x,3)+ pow(\\x,4)/4)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.4,0) -- (1.4,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,0.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.4] plot (\\x,{2.5*(0.15 + 0.192*\\x - 0.44* \\x*\\x - 0.2*pow(\\x,3)+ pow(\\x,4)/4)});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -451854,7 +453912,9 @@ "id": "017479", "content": "已知函数$f(x)=a x^2+|x+a+1|$为偶函数, 则不等式$f(x)>0$的解集为\\bracket{20}.\n\\fourch{$\\varnothing$}{$(-1,0) \\cup(0,1)$}{$(-1,1)$}{$(-\\infty,-1) \\cup(1,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -451874,7 +453934,9 @@ "id": "017480", "content": "已知$n \\in \\mathbf{N}$, $n\\ge 1$, 集合$A=\\{\\sin (\\dfrac{k \\pi}{n}) | k \\in \\mathbf{N},\\ 0 \\leq k \\leq n\\}$, 若集合$A$恰有$8$个子集, 则$n$的可能值有\\bracket{20}个.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -451894,7 +453956,9 @@ "id": "017481", "content": "已知$\\{a_n\\}$为等差数列, $\\{b_n\\}$为等比数列, $a_1=b_1=1$, $a_5=5(a_4-a_3)$, $b_5=4(b_4-b_3)$.\\\\\n(1) 求$\\{a_n\\}$和$\\{b_n\\}$的通项公式;\\\\\n(2) 记$\\{a_n\\}$的前$n$项和为$S_n$, 求证: $S_n S_{n+2}=latex]\n\\draw (0,0,0) node [above right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(B)(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$AB$与$PC$所成角的大小;\\\\\n(2) 求二面角$B-PC-D$的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{\\pi}{4}$; (2) $\\dfrac{\\sqrt{3}}{3}$", "solution": "", @@ -451934,7 +454000,9 @@ "id": "017483", "content": "流行性感冒简称流感, 是流感病毒引起的急性呼吸道感染, 也是一种传染性强、传播速度快的疾病, 了解引起流感的某些细菌、病毒的生存条件、繁殖习性等对于预防流感的传播有极其重要的意义, 某科研团队在培养基中放入一定是某种细菌进行研究. 经过$2$分钟菌落的覆盖面积为$48 \\text{mm}^2$, 经过$3$分钟覆盖面积为$64 \\text{mm}^2$, 后期其蔓延速度越来越快; 菌落的覆盖面积$y$(单位: $\\text{mm}^2$) 与经过时间$x$(单位: $\\text{min}$) 的关系现有三个函数模型: \\textcircled{1} $y=k a^x$($k>0$, $a>1$); \\textcircled{2} $y=\\log _b x$($b>1$); \\textcircled{3} $y=p \\sqrt{x}+q$($p>0$)可供选择.\\\\\n(1) 选出你认为符合实际的函数模型, 说明理由, 并求出该模型的解析式;\\\\\n(2) 在理想状态下, 至少经过多少分钟培养基中菌落的覆盖面积能超过$300 \\text{mm}^2$?(结果保留到整数)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $y=27\\times (\\dfrac{4}{3})^x$, $x\\ge 0$; (2) $9$分钟", "solution": "", @@ -451954,7 +454022,9 @@ "id": "017484", "content": "在平面直角坐标系$x O y$中, 已知椭圆$E: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$的左、右焦点分别为$F_1$、$F_2$, 点$A$在椭圆$E$上且在第一象限内, $AF_2 \\perp F_1F_2$, 直线$AF_1$与椭圆$E$相交于另一点$B$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\path [name path = elli,draw] (0,0) ellipse (2 and {sqrt(3)});\n\\draw (-1,0) node [above left] {$F_1$} coordinate (F_1);\n\\draw (1,0) node [below] {$F_2$} coordinate (F_2);\n\\path [name path = AF2] (F_2) --++ (0,2);\n\\path [name intersections = {of = AF2 and elli, by = A}];\n\\draw (A) node [above] {$A$} --(F_2); \n\\path [name path = AB] (A) -- ($(F_1)!-0.5!(A)$);\n\\path [name intersections = {of = AB and elli, by = B}];\n\\draw (A)--(B) node [left] {$B$} -- (O) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\triangle AF_1F_2$的周长;\\\\\n(2) 在$x$轴上任取一点$P$, 直线$AP$与椭圆$E$的右准线相交于点$Q$, 求$\\overrightarrow{OP} \\cdot \\overrightarrow{QP}$的最小值;\\\\\n(3)设点$M$在椭圆$E$上, 记$\\triangle OAB$与$\\triangle MAB$的面积分别为$S_1$、$S_2$, 若$S_2=3S_1$, 求点$M$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $6$; (2) $-4$; (3) $(2,0)$或$(-\\dfrac{2}{7},-\\dfrac{12}{7})$", "solution": "", @@ -451974,7 +454044,9 @@ "id": "017485", "content": "记$y=f'(x)$、$y=g'(x)$分别为函数$y=f(x)$、$y=g(x)$的导函数, 若存在$x_0 \\in \\mathbf{R}$, 满足$f(x_0)=g(x_0)$且$f'(x_0)=g'(x_0)$, 则称$x_0$为函数$f(x)$与$g(x)$的一个``$S$点''.\\\\\n(1) 证明: 函数$y=x$与$y=x^2+2 x-2$不存在``$S$点'';\\\\\n(2) 若函数$y=a x^2-1$与$y=\\ln x$存在``$S$点'', 求实数$a$的值;\\\\\n(3) 已知$f(x)=-x^2+a$, $g(x)=\\dfrac{b \\mathrm{e}^x}{x}$, 若存在实数$a>0$, 使函数$y=f(x)$与$y=g(x)$在区间$(0,+\\infty)$内存在``$S$点'', 求实数$b$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\mathrm{e}}{2}$; (3) $(-\\dfrac{27}{\\mathrm{e}^3},0)\\cup (0,+\\infty)$", "solution": "", @@ -451994,7 +454066,9 @@ "id": "017486", "content": "某班有男生$26$人, 女生$9$人, 已知男生中 A 档的有$8$人, 女生中 A 档的有$3$人. 现从班级中随机挑选一人, 再选出同性别的另一人, 求在第一次选出的是 A 档生的情况下, 第二次选出的是非 A 档生的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452014,7 +454088,9 @@ "id": "017487", "content": "由频率分布直方图绘制概率密度曲线, 小明认为形状接近钟形曲线, 可以近似为正态分布$N(\\mu, \\sigma^2)$, 则该正态分布是$N(70.9,14^2)$, 若该抽象基本合理, 并用此正态分布估计本校学生的本次测试得分情况, 则随机抽取本校一名学生, 求该生得分高于$84.9$分的概率.\n(已知$\\Phi(1) \\approx 0.8413$, $\\Phi(2) \\approx 0.9772$, $\\Phi(3) \\approx 0.9987$.$\\Phi(x)$表示标准正态分布的密度函数从$-\\infty$到$x$的累计面积)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452034,7 +454110,9 @@ "id": "017488", "content": "某盒子甲中有$5$个红球, $10$个黑球\n(1) 求第一次摸出红球的情况下, 第二次摸出黑球的概率\n(2) 求第二次摸球, 摸出黑球的概率\n(3) 摸出$2$个球, 用随机变量$X$表示该两个球中红球的个数, 求$X$的分布.\n(4) 另一个盒子乙中有$7$个红球, $8$个墨球, 从两个盒子中随机挑选出一个, 并从该盒子中先后随机抽取两个球, 求在第一次取出的是红球的情况下, 第二次取出的是黑球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452054,7 +454132,9 @@ "id": "017489", "content": "在抛掷一枚均匀硬币两次的试验中, 恰一次朝上的概率为\\blank{50}; 至少有一枚正面朝上的概率为\\blank{50}; 在至少有一枚反面朝上的条件下, 第一次反面朝上的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452074,7 +454154,9 @@ "id": "017490", "content": "从$1,2,3,4,5$中随机取出两个不同的数, 则其和为奇数的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452094,7 +454176,9 @@ "id": "017491", "content": "甲乙两人射击同一个标靶, 每人一发, 其中甲命中概率为$\\dfrac{2}{3}$, 乙命中概率为$\\dfrac{1}{2}$, 若甲乙的射击相互独立, 则标靬被击中的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452114,7 +454198,9 @@ "id": "017492", "content": "某种疾病的患病率为$0.50$, 患该种疾病且血检呈阳性的概率为$0.49$, 则已知在患该种疾病的条件下血检呈阳性的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452134,7 +454220,9 @@ "id": "017493", "content": "已知不透明的袋子中有$3$个相同的白球和$3$个相同的黑球, 分别求下列概率: \\\\\n(1) 有放回地依次摸$3$个球, 每次摸一个, 则恰摸到$2$个白球的概率为\\blank{50}, 若$3$次摸到的白球数记为$X$, 则$E[X]=$\\blank{50}, $D[X]=$\\blank{50};\\\\\n(2)不放回地依次摸$3$个球, 每次摸一个, 则第三次摸到白球的概率为恰摸到$2$个白球的概率为若$3$次摸到的白球总数记为$Y$, 则$E[Y]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452154,7 +454242,9 @@ "id": "017494", "content": "已知$X\\sim N(1,2^2)$, 若$P(X>3)=\\alpha$, 则$P(-1 \\leq X \\leq 3)=$(用$a$表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452174,7 +454264,9 @@ "id": "017495", "content": "高三(1)班甲、乙两同学报名参加$A, B, C$三所高校的自主招生考试, 因为三所高校考试时间相同, 所以甲、乙只能随机报考其中一所高校.\\\\\n(1) 写出合适的样本空间;\\\\\n(2) 求甲、乙两人报考不同高校的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452194,7 +454286,9 @@ "id": "017496", "content": "有$5$条线段, 长度分别为$1,3,5,7,9$. 从这五条线段中任取三条, 求所取线段长能构成三角形三边长的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452214,7 +454308,9 @@ "id": "017497", "content": "任取$m \\in\\{2,5,8,9\\}$, $n \\in\\{1,3,5,7\\}$, 求方程$\\dfrac{x^2}{m}+\\dfrac{y^2}{n}=1$表示的焦点在$x$轴上的椭圆的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452234,7 +454330,9 @@ "id": "017498", "content": "两个篮球运动员甲和乙罚球时命中的概率分别是$0.7$和$0.6$, 两人各设一次, 假设事件``甲命中''与``乙命中'', 是独立的. 求至少一人命中的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452254,7 +454352,9 @@ "id": "017499", "content": "甲乙二人争夺一场围棋比赛的冠军, 若比赛为``三局两胜''制 (无平局), 甲在每局比赛中获胜的概率均为$\\dfrac{2}{3}$, 且各局比赛结果相互独立, 则在甲获得冠军的条件下, 比赛进行了三局的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452274,7 +454374,9 @@ "id": "017500", "content": "设盒中装有$5$只球, 其中$3$只是黑球, $2$只是白球, 现从盒中随机地摸出两只, 并换进$2$只黑球之后, 再从盒中摸出$2$只, 第二次摸出的$2$只全是黑球概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452294,7 +454396,9 @@ "id": "017501", "content": "某品牌冰柜由甲、乙、丙三个工厂生产, 其中甲厂占$25 \\%$, 乙厂占$35 \\%$, 丙厂占$40 \\%$, 且各厂的次品率分别为$5 \\%, 4 \\%, 2 \\%$. 如果某人已经买到一台次品冰柜, 则该冰柜由甲厂生产的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452314,7 +454418,9 @@ "id": "017502", "content": "某中学数学竞赛培训共开设有初等代数、初等几何、初等数论和微积分初步共四门课程, 要求初等代数、初等几何都要合格, 且初等数论和微积分初步至少有一门合格, 则能取得参加数学竞赛赛的资格, 现有甲、乙、丙三位同学报名参加数学竞赛培训, 每一位同学对这四门课程考试是否合格相互独立, 其合格的概率均相同, (见下表), 且每一门课程是否合格相互独立.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 课程 & 初等代数 & 初等几何 & 初等数论 & 微积分初步 \\\\\n\\hline 合格的概率 &$\\dfrac{3}{4}$&$\\dfrac{2}{3}$&$\\dfrac{2}{3}$&$\\dfrac{1}{2}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 求甲同学取得参加数学竞赛复赛的资格的概率;\\\\\n(2) 记$\\xi$表示三位同学中取得参加数学竞赛复赛的资格的人数, 求$\\xi$的分布及期望$E[\\xi]$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452334,7 +454440,9 @@ "id": "017503", "content": "生产方提供$50$箱的一批产品, 其中有$2$箱不合格产品. 采购方接收该批产品的准则是: 从该批产品中任取$5$箱产品进行检测, 若至多有$1$箱不合格产品, 便接收该批产品. 问: 该批产品被接收的概率是多少?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452354,7 +454462,9 @@ "id": "017504", "content": "当前, 以``立德树人''为目标的课程改革正在有序推进. 高中联招对初三毕业学生进行体育测试, 是激发学生、家长和学校积极开展体育活动, 保证学生健康成长的有效措施. ``某地区 2019 年初中毕业生升学体育考试规定, 考生必须参加立定跳远、投实心球、 1 分钟跳绳三项测试, 三项考试满分为$50$分, 其中立定跳远$15$分, 投实心球$15$分, 1 分钟跳绳$20$分. 某学校在初三上学期开始时要掌握全年级学生每分钟跳绳的情况, 随机抽取了$100$名学生进行测试, 得到如下频率分布直方图, 且规定计分规则如表:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.1, yscale = 100]\n\\draw [->] (150,0) -- (152,0) -- (153,0.002) -- (155,-0.002) -- (156,0) -- (225,0) node [below right] {每分钟跳绳个数};\n\\draw [->] (150,0) -- (150,0.065) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {165/0.005,175/0.009,185/0.05,195/0.03,205/0.006}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {165/0.005/below left,175/0.009/left,185/0.05/left,195/0.03/left,205/0.006/left}\n{\\draw [dashed] (\\i,\\j) -- (150,\\j) node [\\k] {$\\j$};};\n\\draw (215,0) node [below] {$215$};\n\\end{tikzpicture}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 每分钟咇绳令数 & {$[165,175)$} & {$[175,185)$} & {$[185,195)$} & {$[195,205)$} & {$[205,215)$} \\\\\n\\hline 得分 & 16 & 17 & 18 & 19 & 20 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 现从样本的$100$名学生中, 任意选取$2$人, 求两人得分之和不大于$33$分的概率;\\\\\n(2) 若该校初三年级所有学生的跳绳个数$X$服从正态分布$N(\\mu , \\sigma^2)$, 用样本数据的平均值和方差估计总体的期望和方差 (结果四舍五入到整数), 已知样本方差$S^2 \\approx 77.8$(各组数据用中点值代替). 根据往年经验, 该校初三年级学生经过一年的训练, 正式测试时每人每分钟跳绳个数都有明显进步, 假设明年正式测试时每人每分钟跳绳个数比初三上学期开始时个数增加$10$个, 利用现所得正态分布模型:\\\\ (I) 预估全年级恰好有$1000$名学生, 正式测试时每分钟跳$193$个以上的人数. (结果四舍五入到整数);\\\\\n(II) 若在该地区$2020$年所有初三毕业生中任意选取$3$人, 记正式测试时每分钟跳$202$个以上的人数为$\\xi$, 求随机变量$\\xi$的分布和期望.\\\\\n附: 若随机变量$X$服从正态分布$N(\\mu, \\sigma^2)$, $\\sigma=\\sqrt{77.8} \\approx 9$, 则$P(\\mu-\\sigma=latex]\n\\draw (0,0) node [draw] (A) {元件A};\n\\draw (0,1) node [draw] (A1) {元件A};\n\\draw (0,-1) node [draw] (A2) {元件A};\n\\draw (2,0) node [draw] (B) {元件B};\n\\draw (A)--(B);\n\\draw (A) --++ (-1.5,0) (B)--++ (1,0);\n\\draw (A1) --++ (-1,0) --++ (0,-2) -- (A2); \n\\draw (A1) --++ (1,0) --++ (0,-2) -- (A2); \n\\end{tikzpicture}\n\\end{center}\n(1) 求该装置正常工作超过$10000$小时的概率;\\\\\n(2) 某城市$5G$基站建设需购进$1200$台该装置, 估计该批装置能正常工作超过$10000$小时的件数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452654,7 +454792,10 @@ "id": "017519", "content": "2023 年春节期间, 我国高速公路继续执行``节假日高速免费政策'', 某路桥公司为掌握春节期间车辆出行的高皘情况, 在某高速收费点处记录了大年初三上午$9: 20 \\sim 10: 40$这一时间段内通过的车辆数, 统计发现这一时间段内共有$600$辆车通过该收费点, 它们通过该收费点的时刻的频率分布直方图如图所示, 其中时间段$9: 20 \\sim 9: 40$记作区间$[20,40)$, $9: 40 \\sim 10: 00$记作$[40,60)$, $10: 00 \\sim 10: 20$记作$[60, 80)$, $10: 20 \\sim 10: 40$记作$[80, 100]$, 比方: $10$点$04$分, 记作时刻$64$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.06, yscale = 100]\n\\draw [->] (0,0) -- (120,0) node [below] {时间};\n\\draw [->] (0,0) -- (0,0.03) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {20/0.005,40/0.015,60/0.020,80/0.010}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (20,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {20/0.005,40/0.015,60/0.020,80/0.010}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}\n(1) 估计这$600$辆车在$9: 20 \\sim 10: 40$时间段内通过该收费点的时刻的平均值(同一组中的数据用该组区间的中点值代表);\\\\\n(2) 为了对数据进行分析, 现采用分层抽样的方法从这$600$辆车中抽取$10$辆, 再从这$10$辆车中随机抽取$4$辆, 记$X$为$9: 20\\sim 10: 00$之间通过的车辆数, 求$X$的分布列与数学期望;\\\\\n(3) 由大数据分析可知, 车辆在春节期间每天通过该收费点的时刻$T$服从正态分布$N(\\mu, \\sigma^2)$, 其中$\\mu$可用这$600$辆车在$9: 20\\sim 10: 40$之间通过该收点的时刻的平均值近似代替, $\\sigma^2$可用样本的方差近似代替 (同一组中的数据用该组区间的中点值代表), 已知大年初五全天共有$1000$辆车通过该收费点, 估计在 $9: 46\\sim 10: 40$之间通过的车辆数 (结果保留到整数).\\\\\n参考数据: 若$T \\sim N(\\mu, a^2)$; 则$P(\\mu-\\sigmap_2>p_1>0$, 记该棋手连胜两盘的概率为$p$, 则\\bracket{20}.\n\\twoch{$p$与该棋手和甲、乙、丙的比赛次序无关}{该棋手在第二盘与甲比赛, $p$最大}{该棋手在第二盘与乙比赛, $p$最大}{该棋手在第二盘与丙比赛, $p$最大}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -452754,7 +454903,9 @@ "id": "017524", "content": "一种微生物群体可以经过自身繁殖不断生存下来, 设一个这种微生物为第$0$代, 经过一次繁蝒后为第$1$代, 再经过一次繁殖后为第$2$代, $\\cdots$, 该微生物每代繁殖的个数是相互独立的且有相同的分布列, 设$X$㕈示$1$个微生物个体繁殖下一代的个数, $P(X=i)=p_t$($i=0,1,2,3$).\\\\\n(1) 已知$p_0=0.4$, $p_1=0.3$, $p_2=0.2$, $p_3=0.1$, 求$E[X]$;\\\\\n(2) 设$p$表示该种微生物经过多代繁殖后临近灭绝的概率, $p$是关于$x$的方程: $p_0+p_1 x+p_2 x^2+p_3 x^3=x$的最小正实根, 求证: 当$E[X] \\leq 1$时, $p=1$, 当$E[X]>1$时, $p<1$;\\\\\n(3) 根据你的理解说明(2)问结论的实际含义.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452774,7 +454925,9 @@ "id": "017525", "content": "从某企业生产的某种产品中随机抽取$100$件, 测量这些产品的一项质量指标值, 由测量表得如下频数分布表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 质量指标值 & {$[70,80)$} & {$[80,90)$} & {$[90,100)$} & {$[100,110)$} &$[110,120]$\\\\\n\\hline 频数 & 14 & 20 & 36 & 18 & 12 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n估计这种产品质量指标值的平均数为(同一组中的数据用该组区间的中点值作代表) \\bracket{20}.\n\\fourch{$100$}{$98.8$}{$96.6$}{$94.4$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -452794,7 +454947,9 @@ "id": "017526", "content": "某公司生产三种型号的轿车, 产量分别为$120$辆、 $600$辆和$200$辆.为检验该公司的产品质量, 先用分层抽样的方法抽取$23$辆进行检验, 这三种型号的轿车应分别抽取\\blank{50}, \\blank{50}和\\blank{50}辆.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452814,7 +454969,9 @@ "id": "017527", "content": "为了解某校高三学生的视力情况, 随机抽查了该校$100$名高三学生的视力情况, 得到频率分布直方图如图所示, 由于不慎将部分数据丢失, 仅知道后$5$组的频数和为$62$. 设视力在$4.6$到$4.8$之间的学生数为$m$, 最大频率为$0.32$, 则$m$的值为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 7, yscale = 1]\n\\draw [->] (4.2,0) -- (4.25,0) -- (4.26,0.2) -- (4.28,-0.2) -- (4.29,0)-- (5.3,0) node [below] {视力};\n\\draw [->] (4.2,0) -- (4.2,3.6) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (4.2,0) node [below left] {$O$};\n\\foreach \\i/\\j in {4.4/0.5,4.5/1.1,4.6/2.2,4.7/3.2,4.8/1.6,4.9/1.1,5.0/0.5,5.1/0.2}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (0.1,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {4.4/0.5,4.5/1.1}\n{\\draw [dashed] (\\i,\\j) -- (4.2,\\j) node [left] {$\\k$};};\n\\draw (5.2,0) node [below] {$5.2$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$27$}{$48$}{$54$}{$64$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -452834,7 +454991,9 @@ "id": "017528", "content": "如图为甲、乙两位同学在$5$次数学测试中得分的茎叶图, 则平均成绩较小的那位同学的成绩的方差为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{ccc|c|ccc}\n\\multicolumn{3}{r|}{甲} & & \\multicolumn{3}{l}{乙} \\\\\n& 9 & 8 & 8 & 7 & 9\\\\\n2 & 1 & 0 & 9 & 0 & 1 & 8\n\\end{tabular}\n\\end{center}\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -452854,7 +455013,9 @@ "id": "017529", "content": "下列数据的第$10$百分位数是\\blank{50}; 第$35$百分位数是\\blank{50}.\n\\begin{center}\n\\begin{tabular}{ccccc}\n10 & 10 & 10 & 12 & 14 \\\\\n15 & 15 & 23 & 35 & 60\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -452874,7 +455035,9 @@ "id": "017530", "content": "变量$x, y$之间的一组相关数据如表所示: 若$x, y$之间的线性回归方程为$y=\\hat{a} x+12.28$, 则$\\hat{a}$的值为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$x$& 4 & 5 & 6 & 7 \\\\\n\\hline$y$& 8.2 & 7.8 & 6.6 & 5.4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\fourch{$-0.92$}{$-0.94$}{$-0.96$}{$-0.98$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -452894,7 +455057,9 @@ "id": "017531", "content": "某校抽取了$50$名高三年级学生, 测量他们的身高数据. 但由于某种原因这些原始样本数据不可查得, 但已知按照分层随机抽样原则抽取了样本, 其中男生$21$名, 身高样本平均数为$174.8 \\text{cm}$, 方差为$20.8$; 女生$29$名, 身高样本平均数为$163.1 \\text{cm}$, 方差为$18.4$. 试用这些已知的数据求该$50$名高三年级学生身高的样本平均数和方差, 并估计高三年级学生身高的总体方差. (结果精确到$0.01$)", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452914,7 +455079,9 @@ "id": "017532", "content": "某校举行了一次数学竞赛, 为了了解本次竞赛学生的成绩情况, 从中抽取了部分学生的分数 (得分取正整数, 满分为$100$分) 作为样本 (样本容量为$n$) 进行统计, 按照$[50,60)$, $[60,70)$, $[70,80)$, $[80,90)$, $[90,100]$的分组作出频率分布直方图, 已知得分在$[50,60)$, $[90,100]$的频数分别为$16$, $4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.1, yscale = 60]\n\\draw [->] (35,0) -- (110,0) node [below] {成绩(分)};\n\\draw [->] (35,0) -- (35,0.055) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (35,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.016,60/0.030,70/0.040,80/0.010,90/0.004}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {50/0.016,60/0.030/a,70/0.040,80/0.010,90/0.004/b}\n{\\draw [dashed] (\\i,\\j) -- (35,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求样本容量$n$和频率分布直方图中的$a, b$的值;\\\\\n(2) 估计本次竞赛学生成绩的平均数 (同一组中的数据用该组区间的中点值代表);\\\\\n(3) 在选取的样本中, 若男生和女生人数相同, 我们规定成绩在$70$分或以上称为``优秀'', $70$分以下称为``不优秀'', 其中男女生中成绩优秀的分别有$24$人和$30$人, 请完成列联表, 并判断是否有$90 \\%$的把控认为``学生的成绩优秀与否与性别有关''?\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 男生 & 女生 & 总计 \\\\\n\\hline 优秀 & & & \\\\\n\\hline 不优秀 & & & \\\\\n\\hline 总计 & & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$P(\\chi^2 \\geq k)$& 0.10 & 0.05 & 0.010 & 0.005 & 0.001 \\\\\n\\hline$k$& 2706 & 3.841 & 6.635 & 7.879 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n附: $\\chi^2=\\dfrac{\\dot{n}(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中$n=a+b+c+d$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452934,7 +455101,10 @@ "id": "017533", "content": "由于疫情, 学生在家经过了几个月的线上学习, 某高中学校为了了解学生在家学习情况, 复学后进行了复学摸底考试, 并对学生进行了问卷调查, 如表 (单位: 人) 是对高二年级数学成绩及``认为自己在家学习态度是否端正''的问卷调查的统计结果, 其中成绩不低于$120$分为优秀, 成绩不低于$90$分且小于$120$分的为及格, 成绩小于$90$分的为不及格.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 优秀 & 及格 & 不及格 \\\\\n\\hline 学习态度端正 & 91 & 300 &$a$\\\\\n\\hline 学习态度不端正 & 9 & 200 & 322 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n按成绩用分层抽样的方法在高二年级中抽取$50$人, 其中优秀的人数为$5$.\\\\\n(1) 求$a$的值;\\\\\n(2) 用分层抽样的方法在及格的学生中抽取一个容量为$5$的样本. 将该样本看成一个总体, 从中任取$2$人, 求至少有$1$人学习不端正的概率;\\\\\n(3) 在及格的学生中随机抽取了$10$人, 他们的分数如茎叶图所示, 已知这$10$名学生的平均分为$104.5$, 求$a>b$的概率.\n\\begin{center}\n\\begin{tabular}{c|cccc}\n9 & 2 & $a$ & $b$ \\\\\n10 & 0 & 5 & 6 & 8 \\\\\n11 & 3 & 6 & 7\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452954,7 +455124,9 @@ "id": "017534", "content": "我国是世界上严重缺水的宝家之一, 城市缺水问题较为突出, 某市政府为了节约生活用水, 计划在本市试行居民生活用水定额管理, 即确定一个居民月用水墨标准$x$, 用水量不超过$x$的部分按平价收费, 超出$x$的部分按议价收费. 下面是居民去均用水量的抽样频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 1.5, yscale = 7]\n\\draw [->] (0,0) -- (5.5,0) node [below] {月均用水量(吨)};\n\\draw [->] (0,0) -- (0,0.63) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {0/0.08,0.5/0.16,1/0.3,1.5/0.4,2/0.52,2.5/0.3,3/0.12,3.5/0.08,4/0.04}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (0.5,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0.5/0.16,1.5/0.4,2/0.52,2.5/0.3/a,3/0.12,3.5/0.08,4/0.04}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (4.5,0) node [below] {$4.5$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求直方图中$a$的值;\\\\\n(2) 试估计该市居民月均用水量的众数, 平均数;\\\\\n(3) 设该市有$30$万居民, 估计全市居民中月均用水量不低于$3$吨的人数, 并说明理由;\\\\\n(4) 如果希望$85 \\%$的居民月均用水量不超过标准$x$, 那么标准$x$定为多少比较合理?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452974,7 +455146,9 @@ "id": "017535", "content": "垃圾是人类日常生活和生产中产生的废弃物, 由于排出量大, 成分复杂多样, 且具有污染性, 所以需要无害化、减量化处理. 某市为调查产生的垃圾数量, 采用简单随机抽样的方法抽取$20$个县城进行了分析, 得到样本数据$(x_i, y_i)$($i=1,2, \\cdots, 20)$, 其中$x_i$和$y_i$分别表示第$i$个县城的人口 (单位: 万人) 和该县年垃圾产生总量 (单位: 吨), 并计算得$\\displaystyle\\sum_{i=1}^{20} x_i=80$, $\\displaystyle\\sum_{i=1}^{20} y_i=4000$, $\\displaystyle\\sum_{i=1}^{20}(x_i-\\overline {x})^2=80$, $\\displaystyle\\sum_{i=1}^{20}(y_i-\\overline {y})^2=8000$, $\\displaystyle\\sum_{i=1}^{20}(x_i-\\overline {x})(y_i-\\overline {y})=7000$.\\\\\n(1) 请用相关系数说明该组数据中$y$与$x$间的关系可用线性回归模型进行拟合;\\\\\n(2) 求$y$关于$x$的线性回归方程;\\\\\n(3) 某科研机构研发了两款垃圾处理机器, 如表是以往两款垃圾处理机器的使用年限 (整年) 统计表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline \\backslashbox{款式}{台数}{使用年限} & 1 年 & 2 年 & 3 年 & 4 年 & 5 年 \\\\\n\\hline 甲款 & 5 & 20 & 15 & 10 & 50 \\\\\n\\hline 乙款 & 15 & 20 & 10 & 5 & 50 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n某环保机构若考虑购买其中一款垃圾处理器, 以使用年限的频率估计概率. 根据以往经验估计, 该机构选择购买哪一款垃圾处理机器, 才能使用更长久?\n参考公式: 相关系数$r=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})(y_i-\\overline {y})}{\\displaystyle\\sqrt{\\sum_{i=1}^n(x_i-\\overline {x})^2} \\sqrt{\\displaystyle\\sum_{i=1}^n(y_i-\\overline {y})^2}}$. 对于一组具有线性相关关系的数据$(x_i, y_i)$($i=1,2, \\cdots, n$), 其回归直线$y=\\hat{a} x+\\hat{b}$的斜率和截距\n的最小二乘估计分别为: $\\hat{a}=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})(y_i-\\overline {y})}{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})^2}$, $\\hat{b}=\\overline {y}-\\hat{a} \\overline {x}$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -452994,7 +455168,9 @@ "id": "017536", "content": "已知回归方程$y=5 x+1$, 则该方程在样本$(1, 4)$处的离差为\\bracket{20}.\n\\fourch{$-2$}{$1$}{$2$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453014,7 +455190,9 @@ "id": "017537", "content": "某歌唱兴趣小组由$15$个编号为$01,02, \\cdots, 15$的学生个体组成, 现要从中选取$3$名学生参加合唱团, 选取方法是从随机数表的第$1$行的第$18$列开始由左往右依次选取两个数字, 则选出来的第$3$名同学的编号为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{cccccccccccccccccc}\n49 & 54 & 43 & 54 & 82 & 17 & 37 & 93 & 23 & 78 & 30 & 35 & 20 & 96 &23 & 84 & 26 & 34 \\\\\n91 & 64 & 50 & 25 & 83 & 92 & 12 & 06 & 76 & 57 & 23 & 55 & 06 & 88 & 77 & 04 & 74 & 47 \\\\\n67 & 21 & 76 & 33 & 50 & 25 & 83 & 92 & 12 & 06 & 76 & 49 & 54 & 43 & 54 & 82 & 74 & 47\n\\end{tabular}\n\\end{center}\n\\fourch{$02$}{$09$}{$12$}{$03$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453034,7 +455212,9 @@ "id": "017538", "content": "某校从高一年级学生中随机抽取部分学生, 将他们的模块测试成绩分为$6$组:\n$[40,50)$, $[50,60)$, $[60,70)$, $[70,80)$, $[80,90)$, $[90,100]$加以统计; 得到如图所示的频率分布直方图, 已知高一年级共有学生$600$名, 据此估计, 该模块测试成绩不少于$60$分的学生人数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.08, yscale = 100]\n\\draw [->] (30,0) -- (32,0) -- (33,0.002) -- (35,-0.002) -- (36,0) -- (115,0) node [below] {分数};\n\\draw [->] (30,0) -- (30,0.04) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (30,0) node [below left] {$O$};\n\\foreach \\i/\\j in {40/0.005,50/0.015,60/0.030,70/0.025,80/0.015,90/0.010}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {40/0.005,60/0.030,70/0.025,80/0.015,90/0.010}\n{\\draw [dashed] (\\i,\\j) -- (30,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$588$}{$480$}{$450$}{$120$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453054,7 +455234,9 @@ "id": "017539", "content": "某校对甲、乙两个数学兴趣小组的同学进行了知识测试, 现从两兴趣小组的成员中各随机选取$15$人的测试成绩(单位: 分) 用茎叶图表示, 如图, 根据茎叶图, 对甲, 乙两兴趣小组的测试成绩作比较, 下列统计结论正确的有\\blank{50}.\n\\begin{center}\n\\begin{tabular}{ccccc|c|ccc}\n\\multicolumn{5}{r|}{甲} & & \\multicolumn{3}{l}{乙}\\\\\n&&&3&6&9&2&1\\\\\n&&2&5&8&8&4&6\\\\\n2&5&7&9&9&7&2&5&7\\\\\n&1&3&5&8&6&1&4&4\\\\\n&&&&9&5&2&3&6\\\\\n&&&&&4&6&9\n\\end{tabular}\n\\end{center}\n\\textcircled{1} 甲兴趣小组测试成绩的平均分高于乙兴趣小组测试成绩的平均;\\\\\\textcircled{2} 甲兴趣小组测试成绩较乙兴趣小组测试成绩更分散;\\\\ \\textcircled{3} 甲兴趣小组测试成绩的中位数大于乙兴趣小组测试成绩的中位数;\\\\ \\textcircled{4} 甲兴趣小组测试成绩的众数小于乙兴趣小组测试成绩的众数.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453074,7 +455256,9 @@ "id": "017540", "content": "已知$100$个数据的$75$百分位数是$9.3$, 则下列说法正确的是\\bracket{20}.\n\\onech{这$100$个数据中恰有$75$个数小于或等于$9.3$}{把这$100$个数据从小到大排列后, $9.3$是第$75$个数据}{把这$100$个数据从小到大排列后, $9.3$是第$75$个数据和第$76$个数据的平均数}{把这$100$个数据从小到大排列后, $9.3$是第$75$个数据和第$74$个数据的平均数}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453094,7 +455278,9 @@ "id": "017541", "content": "蟋蛒鸣叫可以说是大自然优美、和谐的音乐, 殊不知蟋蜶鸣叫的频率$x$(每分钟鸣叫的次数) 与气温$y$(单位: $^{\\circ} \\mathrm{C}$) 存在较强的线性相关关系. 某地观测人员根据如表的观测数据, 建立了$y$关于$x$的线性回归方程$y=0.25 x+k$, 则当蟋蟀每分钟鸣叫$56$次时, 该地当时的气温预报值为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$x$(次数/分钟) & 20 & 30 & 40 & 50 & 60 \\\\\n\\hline$y$($^\\circ \\mathrm{C}$)& 25 & 27.5 & 29 & 32.5 & 36 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\fourch{$33^{\\circ} \\mathrm{C}$}{$34^{\\circ} \\mathrm{C}$}{$35^{\\circ} \\mathrm{C}$}{$35.5^{\\circ} \\mathrm{C}$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453114,7 +455300,9 @@ "id": "017542", "content": "张老师将某位高三学生$10$次选填题专测的成绩进行统计, 得到的统计结果如图所示, 但学习委员在将成绩登记在册的时候将$62$与$68$均登记成了$65$, 则两个成绩相比, 不变的数字特征是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{c|cccc}\n5 & 3 & 5 & 7\\\\\n6 & 2 & 3 & 5 & 8\\\\\n7 & 5 & 5 & 7\n\\end{tabular}\n\\end{center}\n\\fourch{众数}{中位数}{平均数}{方差}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453134,7 +455322,9 @@ "id": "017543", "content": "为了考察某校各班参加课外书法小组的人数, 在全校随机抽取$5$个班级, 把每个班级参加该小组的人数作为样本数据. 已知样本平均数为$7$, 样本方差为$4$, 且样本数据互不相同, 求样本数据中的最大值.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453154,7 +455344,9 @@ "id": "017544", "content": "由正整数组成的一组数据$x_1, x_2, x_3, x_4$, 其平均数和中位数都是$2$, 且标准差等于$1$, 则这组数据为\\blank{50}(从小到大排列).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453174,7 +455366,9 @@ "id": "017545", "content": "在发生某公共卫生事件期间, 有专业机构认为该事件在一段时间没有发生在规模群体感染的标志为``连续$10$天, 每天新增颢似病例不超过$7$人''. 根据过去$10$天甲、乙、丙、丁四地新增疑似病例数据, 一定符合该标志的是\\bracket{20}.\n\\twoch{甲地: 总体均值为$3$, 中位数为$4$}{乙地: 总体均值为$1$, 总体方差大于$0$}{主地: 中位数为$2$, 众数为$3$}{丁地: 总体均值为$2$, 总体方差为$3$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453194,7 +455388,9 @@ "id": "017546", "content": "近年来, 随着互眹网技术的快速发展, 共享经济覆盖的范围迅速扩张, 继共享单车、共享汽车之后, 共享房屋以``民宿''、``农家乐''等形式开始在很多平台上线. 某创业者计划在某景区附近租赁一套农房发展成特色``农家乐'', 为了确定未来发展方向, 此创业者对该景区附近六家``农家乐''跟踪调查了$100$天. 得到的统计数据如下表, $x$为收费标准 (单位: 元/日), $t$为入住天数 (单位: 天), 以频率作为各自的``入住率'', 收费标准$x$与``入住率''$y$的散点图如图\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline $x$& 50 & 100 & 150 & 200 & 300 & 400 \\\\\n\\hline $t$& 90 & 65 & 45 & 30 & 20 & 20 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (6,0) node [below] {收费标准};\n\\draw [->] (0,0) -- (0,3.3) node [left] {入住率};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {100,200,300,400,500}\n{\\draw ({\\i/100},0.1) -- ({\\i/100},0) node [below] {$\\i$};};\n\\foreach \\i in {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1}\n{\\draw [gray!50] (5,{\\i*3}) --++ (-5,0) node [left, black] {$\\i$};};\n\\foreach \\i/\\j in {50/0.9,100/0.65,150/0.45,200/0.3,300/0.2,400/0.2}\n{\\filldraw ({\\i/100},{\\j*3}) circle (0.03);};\n\\end{tikzpicture}\n\\end{center}\n(1) 若从以上六家``农家乐''中随机抽取两家深入调查, 记$\\xi$为``入住率''超过$0.6$的农家乐的个数, 求$\\xi$的概率分布列;\\\\\n(2) 令$z=\\ln x$, 由散点图判断$y=\\hat{a} x+\\hat{b}$与$y=\\hat{a} z+\\hat{b}$哪个更合适于此模型(给出判断即可, 不必说明理由)?, 并根据你的判断结果求回归方程; ($\\hat{a}$结果保留一位小数)\\\\\n(3) 若一年按$365$天计算, 试估计收费标准为多少时, 年销售额$L$最太? (年销售额$L=365\\times$入住率$\\times$收费标准$x$)", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453214,7 +455410,10 @@ "id": "017547", "content": "某种疾病可分为I、II两种类型, 为了解该疾病类型与性别的关系, 在某地区随机抽取了患该疾病的病人进行调查, 其中男性人数为$z$, 女性人数为$2 z$, 男性患I型病的人数占男性病人的$\\dfrac{5}{6}$, 女性患I型病的人数占性病人的$\\dfrac{1}{3}$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & I 型病 & II型病 & 合计 \\\\\n\\hline 男 & & & \\\\\n\\hline 女 & & & \\\\\n\\hline 合计 & & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 完成$2 \\times 2$联表; 若在犯错误的概率不超过$0.005$的前提下认为``所患疾病类型''与``性别''有关, 求男性患者至少有多少人?\\\\\n(2) 某药品研发公司欲安排甲乙两个研发团队来研发此疾病的治疗药物, 两个团队各至多安排$2$个接种周期进行试验. 每人每次接种花费$m$($m>0$)元. 甲团队研发的药物每次接种后产生抗体的概率为$p$; 根据以往试验统计, 甲团队平均花费为$-2 m p^2+6 m$; 乙团队研发的药物每次接种后产生抗体的概率为$q$, 每个周期必须完成$3$次接种, 若一个周期内至少出现$2$次抗体, 则该周期结束后终止试验, 否则进入第三个接种周期. 假设两个研发团队每次接种后产生抗体与否均相互独立. 若$p=2 q$, 从两个团队试验的平均花费考虑, 该公司应选择哪个团队进行药品研发?\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline $P(\\chi^2 \\geq k_0)$& 0.10 & 0.05 & 0.01 & 0.005 & 0.001 \\\\\n\\hline $k_0$& 2.706 & 3.841 & 6.635 & 7.879 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453234,7 +455433,9 @@ "id": "017548", "content": "为了解某地农村经济情况, 对该地农户家庭年收入进行抽样调查, 将农户家庭年收入的调查数据整理得到如下频率分布直方图. 根据此频率分布直方图, 下面结论中不正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.8, yscale = 14]\n\\draw [->] (1.5,0) -- (16,0) node [below] {收入/万};\n\\draw [->] (1.5,0) -- (1.5,0.26) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (1.5,0) node [below left] {$O$};\n\\foreach \\i/\\j in {2.5/0.02,3.5/0.04,4.5/0.10,5.5/0.14,6.5/0.20,7.5/0.20,8.5/0.10,9.5/0.10,10.5/0.04,11.5/0.02,12.5/0.02,13.5/0.02}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {5.5/0.14,6.5/0.20,8.5/0.10,10.5/0.04,11.5/0.02}\n{\\draw [dashed] (\\i,\\j) -- (1.5,\\j) node [left] {$\\k$};};\n\\draw (14.5,0) node [below] {$14.5$};\n\\end{tikzpicture}\n\\end{center}\n\\onech{该地农户家庭年收入低手 4.5 万元的农户比率估计为$6 \\%$}{该地农户家庭年收入不低于 10.5 万元的农户比率估计为$10 \\%$}{估计该地农户家庭年收入的平均值不超过 6.5 万元}{估计该地有一半以上的农户, 其家庭年收入介于 4.5 万元至 8.5 万元之间\\blank{50}}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453254,7 +455455,9 @@ "id": "017549", "content": "为了解甲、乙两种离子在小鼠体内的残留程度, 进行如下试验: 将$200$只小鼠随机分成 A、B 两组, 每组$100$只, 其中 A 组小鼠给服甲离子溶液, B 组小鼠给服乙离子溶液, 每组小鼠给服的溶液体积相同、摩尔浓度相同. 经过一段时间后用某种科学方法测算出残留在小鼠体内离子的百分比. 根据试验数据分别得到如下直方图:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {6.5/0.05/0.05,5.5/0.1/0.1,1.5/0.15/0.15,4.5/0.2/0.2,3.5/0.3/0.3}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {1.5/0.15/0.15,2.5/0.2/0.2,3.5/0.3/0.3,4.5/0.2/0.2,5.5/0.1/0.1,6.5/0.05/0.05}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$7.5$};\n\\draw (4.5,-0.1) node {甲离子残留百分比直方图};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/b,7.5/0.15/0.15,6.5/0.2/0.2,5.5/0.35/a}\n{\\draw [dashed] ({\\i-1},\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/0.1,4.5/0.15/0.15,5.5/0.35/0.35,6.5/0.2/0.2,7.5/0.15/0.15}\n{\\draw ({\\i-1},0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$8.5$};\n\\draw (4.5,-0.1) node {乙离子残留百分比直方图};\n\\end{tikzpicture}\n\\end{center}\n记$C$为事件: ``乙离子残留在体内的百分比不低于$5.5$'', 根据直方图得到$P(C)$的估计值为$0.70$.\\\\\n(1) 求乙离子残留百分比直方图中$a, b$的值;\\\\\n(2) 分别估计甲、乙离子残留百分比的平均值(同一组中的数据用该组区间的中点值为代表).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453274,7 +455477,10 @@ "id": "017550", "content": "在校运动会上, 只有甲、乙、丙三名同学参加铅球比赛, 比赛成绩达到$9.50 \\text{m}$上 (含$9.50 \\text{m}$) 的同学将获得优秀奖. 为预测获得优秀奖的人数及冠军得主, 收集了甲、乙、丙以往的比赛成绩, 并整理得到如下数据 (单位: $\\text{m}$):\\\\\n甲: $9.80,9.70,9.55,9.54,9.48,9.42,9.40,9.35,9.30,9.25$;\\\\\n乙: $9.78,9.56,9.51,9.36,9.32,9.23$\\\\\n两: $9.85,9.65,9.20,9.16$.\\\\\n假设用频率估计概率, 且甲、乙、丙的比赛成绩相互独立.\\\\\n(1) 估计甲在校运动会铅球比赛中获得优秀奖的概率;\\\\\n(2) 设$X$是甲、乙、丙在校运动会铅球比赛中获得优秀奖的总人数, 估计$X$的数学期望$E[X]$;\\\\\n(3) 在校运动会铅球比赛中, 甲、乙、丙谁获得冠军的概率估计值最大? (结论不要求证明)", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453294,7 +455500,9 @@ "id": "017551", "content": "已知集合$A=\\{x | x<-1$或$2 \\leq x<3\\}$, $B=\\{x |-2 \\leq x<4\\}$, 则$A \\cup B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453314,7 +455522,9 @@ "id": "017552", "content": "若集合$A=\\{x | x \\leq 2\\}$与集合$B=\\{x | x \\geq a\\}$满足$A \\cap B=\\varnothing$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453334,7 +455544,9 @@ "id": "017553", "content": "集合$A=\\{0,2, a\\}$, $B=\\{1, a^2\\}$, 若$A \\cup B=\\{0,1,2,4,16\\}$, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453354,7 +455566,9 @@ "id": "017554", "content": "满足条件$M \\cup\\{1\\}=\\{1,2,3\\}$的集合$M$的个数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453374,7 +455588,9 @@ "id": "017555", "content": "若全集$U=\\mathbf{R}$, $f(x)$、$g(x)$均为$x$的二次函数, $P=\\{x | f(x)<0\\}$, $Q=\\{x|g(x)| \\geq 0\\}$, 则不等式组$\\begin{cases}f(x)<0, \\\\ g(x)<0\\end{cases}$的解集可用$P$、$Q$表示为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453394,7 +455610,9 @@ "id": "017556", "content": "对于集合$A$、$B$, 定义两种运算``$-$''和``$\\oplus$'': $A-B=\\{x | x \\in A$且$x \\notin B\\}$, $A \\oplus B=(A-B) \\cup(B-A)$, 设$A=\\{y | y=x^2-3 x,\\ x \\in \\mathbf{R}\\}$, $B=\\{y | y=-2^x,\\ x \\in \\mathbf{R}\\}$, 则$A \\oplus B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453414,7 +455632,9 @@ "id": "017557", "content": "设曲线$C_1$和$C_2$的方程分别为$F_1(x, y)=0$和$F_2(x, y)=0$, 则点$P(a, b) \\notin C_1 \\cap C_2$的一个充分条件为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453434,7 +455654,9 @@ "id": "017558", "content": "已知函数$y=f(x)$, $x \\in[a, b]$, 那么集合$\\{(x, y) | y=f(x), x \\in[a, b]\\} \\cap\\{(x, y) | x=2\\}$中元素的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453454,7 +455676,10 @@ "id": "017559", "content": "若集合$A=\\{x | y=\\sqrt{2 x-1}\\}$, $B=\\{x|| x \\leq 1\\}$, 则$A \\cap B$是\\bracket{20}.\n\\fourch{$\\{x | \\dfrac{1}{2} \\leq x \\leq 1\\}$}{$\\{x | x \\leq-1\\}$}{$\\{x |-1 \\leq x \\leq \\dfrac{1}{2}\\}$}{$\\{x | x \\geq 1\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453474,7 +455699,10 @@ "id": "017560", "content": "函数$f(x)=x^2+m x+1$的图像关于直线$x=1$对称的充要条件是\\bracket{20}.\n\\fourch{$m=-2$}{$m=2$}{$m=-1$}{$m=1$}", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453494,7 +455722,9 @@ "id": "017561", "content": "定义集合的一种运算``$\\ast$'': $A \\ast B=\\{z | z=x y,\\ x \\in A,\\ y \\in B\\}$. 设$A=\\{1,2\\}$, $B=\\{0,2\\}$, 则集合$A \\ast B$的所有元素之和为\\bracket{20}.\n\\fourch{$0$}{$2$}{$3$}{$6$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453514,7 +455744,10 @@ "id": "017562", "content": "设集合$M=\\{(x, y) | y=x^2+2 b x+1\\}$, $N=\\{(x, y)|y=2 a(x+b)|(a, b \\in \\mathbf{R}\\}$, 若$M \\cap N=\\varnothing$, 则所有满足条件的点$(a, b)$在坐标平面上对应图形的面积为\\bracket{20}.\n\\fourch{$1$}{$4$}{$\\pi$}{$4 \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -453534,7 +455767,9 @@ "id": "017563", "content": "已知全集$U=\\mathbf{R}$, 集合$A=\\{x | \\log _{0.5}(3-x) \\geq-2\\}, B=\\{x | \\dfrac{5}{x+2} \\geq 1\\}$, 求$\\overline {A} \\cap B$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453554,7 +455789,9 @@ "id": "017564", "content": "设集合$A=\\{x|| x-a |<2\\}, B=\\{x | \\dfrac{2 x-1}{x+2}<1\\}$, 若$A \\subseteq B$, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453574,7 +455811,9 @@ "id": "017565", "content": "已知$p: x^2-4 x-32 \\leq 0$, $q: [x-(1-m)][x-(1+m)] \\leq 0$($m>0$), 若$p$是$q$的充分非必要条件, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453594,7 +455833,9 @@ "id": "017566", "content": "设集合$A=\\{x | x=m^2-n^2,\\ m \\in \\mathbf{Z},\\ n \\in \\mathbf{Z}\\}$.\\\\\n(1) 求证: $11 \\in A$, $12 \\in A$, $2 k+1 \\in A$($k \\in \\mathbf{Z}$);\\\\\n(2) 用反证法证明: $10 \\notin \\mathbf{A}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453614,7 +455855,9 @@ "id": "017567", "content": "设非空数集$S$满足: 若$a$、$b \\in S$, 则$a+b \\in S$, $a-b \\in S$, 则称集合$S$为闭集合. 如整数集$\\mathbf{Z}$, 有理数集$\\mathbf{Q}$等都是闭集合.\\\\\n(1) 写出一个闭集合$S$, 要求满足$S \\subset \\mathbf{R}$, 且$S \\neq \\mathbf{Z}$, $S \\neq \\mathbf{Q}$. 请加以证明;\\\\\n(2) 求证: 对于任意两个满足$S_1 \\subset \\mathbf{R}$, $S_2 \\subset \\mathbf{R}$的闭集合$S_1$、$S_2$, 存在$c \\in \\mathbf{R}$, 但$c \\notin S_1 \\cup S_2$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453634,7 +455877,9 @@ "id": "017568", "content": "不等式$x^2-2 x-3 \\leq 0$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453654,7 +455899,9 @@ "id": "017569", "content": "关于$x$的不等式$|x-2|>m$的解集为$\\mathbf{R}$的充要条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453674,7 +455921,9 @@ "id": "017570", "content": "不等式$\\dfrac{x-1}{x+2}>1$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453694,7 +455943,9 @@ "id": "017571", "content": "已知集合$A=\\{x | \\dfrac{x-7}{3-x}>0\\}$, 函数$y=\\lg (-x^2+6 x-8)$的定义域为集合$B$, 那么集合$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453714,7 +455965,9 @@ "id": "017572", "content": "已知$a, b \\in \\mathbf{R}$, $a>b$且$a b=1$, 则$\\dfrac{a^2+b^2}{a-b}$的最小值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453734,7 +455987,9 @@ "id": "017573", "content": "若$a>0$, 则不等式$-a<\\dfrac{1}{x}<3$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453754,7 +456009,9 @@ "id": "017574", "content": "若对于任意的$x>0$, 不等式$\\dfrac{x}{x^2+3 x+1} \\leq a$恒成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453774,7 +456031,9 @@ "id": "017575", "content": "已知实数$x>0$, $y>0$, 且$x+y=2$, 则$\\dfrac{2}{x}+\\dfrac{x}{2 y}$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453794,7 +456053,9 @@ "id": "017576", "content": "设$a>b>0$, $m>0$, 记$x=\\dfrac{b}{a}$, $y=\\dfrac{b+m}{a+m}$, 则\\bracket{20}.\n\\fourch{$x>y$}{$x \\geq y$}{$x1$($a>0$).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453854,7 +456119,9 @@ "id": "017579", "content": "如图所示, 某公园要在一块绿地的中央修建两个相同的矩形的池塘, 每个面积为$10000$平方米, 池塘前方要留$4$米宽的走道, 其余各方为$2$米宽的走道, 问每个池塘的长宽各为多少米时, 池塘的占地总面积最少?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0) rectangle (2,2.5) (3,2.5) rectangle (5,0);\n\\draw (-1,-2) rectangle (6,3.5);\n\\draw (1,3) node {走道$2$米} (4,3) node {走道$2$米};\n\\draw (1,-1) node {走道$4$米} (4,-1) node {走道$4$米};\n\\draw (1,1.25) node {池塘} (4,1.25) node {池塘};\n\\draw (-0.5,1.25) node [rotate = 90] {走道$2$米} (2.5,1.25) node [rotate = 90] {走道$2$米} (5.5,1.25) node [rotate = 90] {走道$2$米};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453874,7 +456141,9 @@ "id": "017580", "content": "若$a>0, b>0$, 且$\\dfrac{1}{a}+\\dfrac{1}{b}=\\sqrt{a b}$.\\\\\n(1) 求$a^3+b^3$的最小值;\\\\\n(2) 是否存在$a, b$, 使得$2 a+3 b=6$成立, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453894,7 +456163,9 @@ "id": "017581", "content": "已知关于$x$的不等式$|x^2-4 x+a|+|x-3| \\leq 5$的解集为$M$, 且$M \\subseteq(-\\infty, 3]$. 求整数$a$的值, 并解此不等式.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453914,7 +456185,9 @@ "id": "017582", "content": "已知函数$y=\\dfrac{x^2}{a x+b}(a$、$b$为常数), 且关于$x$的方程$y-x+12=0$有两个实根$x_1=3$、$x_2=4$.\\\\\n(1) 求函数$y$的表达式;\\\\\n(2) 设$k>1$, 解关于$x$的不等式$y<\\dfrac{(k+1) x-k}{2-x}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -453934,7 +456207,9 @@ "id": "017583", "content": "若$f(x)=\\log _{(\\sqrt{2}-1)} x$, 则$f(3+2 \\sqrt{2})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453954,7 +456229,9 @@ "id": "017584", "content": "如果$\\lg 108=a$, $\\lg 72=b$, 则$\\lg 48=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453974,7 +456251,9 @@ "id": "017585", "content": "函数$y=4 \\times 2^x$的图像可以由函数$y=2^x$的图像向\\blank{50}平移\\blank{50}个单位得到.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -453994,7 +456273,9 @@ "id": "017586", "content": "函数$y=(a^2-1)^x$是单调减函数, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454014,7 +456295,9 @@ "id": "017587", "content": "函数$f(x)=a^x$($a>0$, $a \\neq 1$)在$[1,2]$中的最大值比最小值大$\\dfrac{a}{2}$, 则$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454034,7 +456317,9 @@ "id": "017588", "content": "已知$x^2+y^2-4 x-2 y+5=0$, 则$\\log _x(y^x)$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454054,7 +456339,9 @@ "id": "017589", "content": "关于$x$的方程$5^x=\\dfrac{a+3}{5-a}$有负根, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454074,7 +456361,9 @@ "id": "017590", "content": "当$a>0$且$a \\neq 1$时, 不论$a$为何值, 函数$y=a^{x-1}+1$的图像都通过定点\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454094,7 +456383,9 @@ "id": "017591", "content": "已知函数$f(x)=\\begin{cases}3^x, & x \\leq 0, \\\\ \\log _2 x,& x>0,\\end{cases}$那么$f(f(\\dfrac{1}{4}))$的值为\\bracket{20}.\n\\fourch{$9$}{$\\dfrac{1}{9}$}{$-9$}{$-\\dfrac{1}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454114,7 +456405,9 @@ "id": "017592", "content": "方程$\\log _2(x+4)=2^x$的根的情况是\\bracket{20}.\n\\fourch{仅有一根}{有两个正根}{有一正根和一个负根}{有两个负根}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454134,7 +456427,9 @@ "id": "017593", "content": "解方程$\\lg (2 x^2-2 x)=\\lg (x^2-5 x+4)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -454154,7 +456449,9 @@ "id": "017594", "content": "设函数$f(x)=2^{|x+1|-|x-1|}$, 求使$f(x) \\geq 2 \\sqrt{2}$的$x$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -454174,7 +456471,9 @@ "id": "017595", "content": "设$A$、$B$是函数$y=\\log _2 x$图像上两点, 其横坐标分别为$a$和$a+4$, 直线$l: x=a+2$与函数$y=\\log _2 x$图像交于点$C$, 与直线$AB$交于点$D$.\\\\\n(1) 求点$D$的坐标;\\\\\n(2) 当$\\triangle ABC$的面积大于$1$时, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -454194,7 +456493,9 @@ "id": "017596", "content": "设$a>0$且$a \\neq 1$, $f(x)=\\log _a(x+\\sqrt{x^2-1})$($x \\geq 1$).\\\\\n(1) 求函数$f(x)$的反函数$f^{-1}(x)$及其定义域;\\\\\n(2) 若$f^{-1}(n)<\\dfrac{3^n+3^{-n}}{2}$($n \\in \\mathbf{N}$, $n \\geq 1$), 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -454214,7 +456515,9 @@ "id": "017597", "content": "在\\textcircled{1} $y=x$与$y=\\sqrt{x^2}$; \\textcircled{2} $y=(\\sqrt{x})^2$与$y=\\sqrt{x^2}$; \\textcircled{3} $y=|x|$与$y=\\dfrac{x^2}{x}$; \\textcircled{4} $y=|x|$与$y=\\sqrt{x^2}$; \\textcircled{5} $y=x^0$与$y=1$五组函数中, 表示相同函数的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454234,7 +456537,9 @@ "id": "017598", "content": "函数$y=\\lg (-6 x^2+13 x-6)$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454254,7 +456559,9 @@ "id": "017599", "content": "已知偶函数$f(x)$, 当$x>0$时, 解析式为$f(x)=x^2-x$, 则当$x<0$时, $f(x)$的解析式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454274,7 +456581,9 @@ "id": "017600", "content": "函数$y=\\lg (x^2+x-a)$的定义域为$\\mathbf{R}$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454294,7 +456603,9 @@ "id": "017601", "content": "已知函数$f(x)=x^2-(a+2) x+2-a$, 若集合$A=\\{x | f(x)<0, x \\in \\mathbf{Z}\\}$有且只有一个元素, 则正实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454314,7 +456625,9 @@ "id": "017602", "content": "函数$f(x)=\\lg \\dfrac{1-x}{1+x}$是\\bracket{20}.\n\\twoch{奇函数}{偶函数}{既是奇函数又是偶函数}{非奇非偶函数}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454334,7 +456647,9 @@ "id": "017603", "content": "若函数$y=f(x)$在$[a, b]$上是单调函数, 则使得$y=f(x+3)$必为单调函数的区间是\\bracket{20}.\n\\fourch{$[a, b+3]$}{$[a+3, b+3]$}{$[a-3, b-3]$}{$[a+3, b]$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454354,7 +456669,9 @@ "id": "017604", "content": "若定义在闭区间$[-a, a]$($a>0$)上的函数$y=f(x)$为奇函数, 则下列各图中可以成为它的图像的是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0,0.5) -- (1,1.5) (0,-0.5) -- (-1,-1.5);\n\\filldraw (0,-0.5) circle (0.05);\n\\filldraw [fill = white] (0,0.5) circle (0.05);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0,0.5) -- (1,1.5) (0,-0.5) -- (-1,-1.5);\n\\filldraw (0,0) circle (0.05);\n\\filldraw [fill = white] (0,0.5) circle (0.05);\n\\filldraw [fill = white] (0,-0.5) circle (0.05);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0,0.5) -- (1,1.5) (0,-0.5) -- (-1,-1.5);\n\\filldraw (0,0.5) circle (0.05);\n\\filldraw [fill = white] (0,-0.5) circle (0.05);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0,0.5) -- (1,1.5) (0,-0.5) -- (-1,-1.5);\n\\filldraw [fill = white] (0,0.5) circle (0.05);\n\\filldraw [fill = white] (0,-0.5) circle (0.05);\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454374,7 +456691,9 @@ "id": "017605", "content": "函数$y=\\sqrt{x+1}-\\sqrt{x-1}$的值域为\\bracket{20}.\n\\fourch{$(-\\infty, \\sqrt{2}]$}{$(0, \\sqrt{2}]$}{$[\\sqrt{2},+\\infty)$}{$[0,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454394,7 +456713,9 @@ "id": "017606", "content": "函数$y=f(x)$对任意实数$x$都有$f(x)0)$, 方程$f(x)-x=0$的两个根$x_1, x_2$满足: $00$恒成立, 求$m$的取值范围;\\\\\n(3) 讨论函数$y=f(x)$的零点个数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -454494,7 +456823,9 @@ "id": "017611", "content": "若$\\cos (\\theta+\\dfrac{\\pi}{3})=1$, 则$\\cos \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454514,7 +456845,9 @@ "id": "017612", "content": "若$\\sin \\theta=k \\cos \\theta$, 则$\\sin \\theta \\cdot \\cos \\theta$的值等于 (用$k$表示).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -454534,7 +456867,9 @@ "id": "017613", "content": "已知$\\alpha \\in(0, \\pi)$, 且有$1-2 \\sin 2 \\alpha=\\cos 2 \\alpha$, 则$\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454554,7 +456889,9 @@ "id": "017614", "content": "已知角$\\alpha$的顶点是坐标原点, 始边与$x$轴的正半轴集合, 它的终边过点$P(-\\dfrac{3}{5}, \\dfrac{4}{5})$, 则$\\cos 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454574,7 +456911,9 @@ "id": "017615", "content": "在$\\triangle ABC$中, 若$AB=4$, $AC=7$, $BC$边上的中线长为$\\dfrac{7}{2}$, 那么边$BC$的长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454594,7 +456933,9 @@ "id": "017616", "content": "若将$\\sqrt{3} \\cos x-\\sin x$化为$A \\cos (\\omega x+\\varphi)$的形式, 其中$A>0$, $\\omega>0$, $\\varphi \\in(0,2 \\pi)$, 则$\\varphi=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454614,7 +456955,9 @@ "id": "017617", "content": "若一个扇形的半径变为原来的一半, 其弧长变为原来的$\\dfrac{3}{2}$, 则该弧所对的圆心角与原圆心角的比值为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$2$}{$\\dfrac{1}{3}$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454634,7 +456977,9 @@ "id": "017618", "content": "若$\\tan \\theta$和$\\tan (\\dfrac{\\pi}{4}-\\theta)$是方程$x^2+p x+q=0$的两个根, 则$p$与$q$之间的关系是\\bracket{20}.\n\\fourch{$p+q+1=0$}{$p-q+1=0$}{$p+q-1=0$}{$p-1-1=0$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454654,7 +456999,9 @@ "id": "017619", "content": "若$\\triangle ABC$的三个内角的余弦值分别等于$\\triangle A' B' C'$的三个内角的正弦值, 则可确定\\bracket{20}.\n\\onech{$\\triangle ABC$和$\\triangle A' B' C'$都是锐角三角形}{$\\triangle ABC$和$\\triangle A' B' C'$都是钝角三角形}{$\\triangle ABC$是锐角三角形, $\\triangle A' B' C'$为钝角三角形}{$\\triangle ABC$是钝角三角形, $\\triangle A' B' C'$为锐角三角形}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454674,7 +457021,9 @@ "id": "017620", "content": "设函数$f(x)=\\alpha \\sin x+b \\cos x$, 其中$a>0$, $b>0$, 若$f(x) \\leq f(\\dfrac{\\pi}{4})$对任意的$x \\in \\mathbf{R}$恒成立, 则下列说法正确的是\\bracket{20}.\n\\onech{$f(\\dfrac{\\pi}{2})>f(\\dfrac{\\pi}{6})$}{$f(x)$的图像关于直线$x=\\dfrac{3 \\pi}{4}$对称}{$f(x)$在$[\\dfrac{\\pi}{4}, \\dfrac{5 \\pi}{4}]$上单调增}{过点$(a, b)$的直线与函数$f(x)$的图像必有公共点}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -454694,7 +457043,9 @@ "id": "017621", "content": "已知$\\sin (\\dfrac{\\pi}{4}-x)=\\dfrac{5}{13}$, $0=latex, scale = 0.2]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (16,0) node [below] {$B$} coordinate (B);\n\\path [name path = arca] (A) ++ (15,0) arc (0:35:15);\n\\path [name path = arcb] (B) ++ (-9,0) arc (180:110:9);\n\\path [name intersections = {of = arca and arcb, by = P}] (P) node [above] {$P$};\n\\draw (A)--(B)--(P)--cycle;\n\\draw ($(A)!0.5!(B)$) node [below] {居民生活区};\n\\draw [->] (B)++(0,5) --++ (0,3) node [midway, right] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) 当$AP=15 \\text{km}$时, 求$\\angle APB$的值;\\\\\n(2) 发电厂尽量远离居民区, 要求$\\triangle PAB$的面积最大, 问此时发电厂与两个垃圾中转站的距离各为多少?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -454774,7 +457131,10 @@ "id": "017625", "content": "函数$y=\\lg (\\sin x-\\cos x)$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454794,7 +457154,9 @@ "id": "017626", "content": "函数$f(x)=\\cos ^2 x+\\dfrac{\\sqrt{3}}{2} \\sin 2 x$, $x \\in \\mathbf{R}$的单调增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454814,7 +457176,9 @@ "id": "017627", "content": "已知函数$y=\\sin (a x+2)$($x \\in \\mathbf{R}$)的最小正周期是$\\pi$, 则实数$a$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454834,7 +457198,9 @@ "id": "017628", "content": "已知函数$f(x)=\\sin x+a \\cos ^2 \\dfrac{x}{2}$($a$为常数, $a \\in \\mathbf{R}$), 且$x=\\dfrac{\\pi}{2}$是方程$f(x)=0$的解. 当$x \\in[0, \\pi]$时, 函数$f(x)$值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454854,7 +457220,9 @@ "id": "017629", "content": "已知函数$y=\\tan (\\omega x+\\dfrac{\\pi}{6})$的图像关于点$(\\dfrac{\\pi}{3}, 0)$对称, 且$|\\omega| \\leq 1$, 则实数$\\omega$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -454874,7 +457242,9 @@ "id": "017630", "content": "函数$f(x)=\\sin x$, 对于$x_10$)在区间$[0, \\pi]$上至少有两个不同的解, 求$\\omega$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -454994,7 +457374,9 @@ "id": "017636", "content": "如图, 某公园有一块直角三角形$ABC$的空地, 其中$\\angle ACB=\\dfrac{\\pi}{2}$, $\\angle ABC=\\dfrac{\\pi}{6}$, $AC$长$\\alpha$千米, 现要在空地上围出一块正角形区域$DEF$建文化景观区, 其中$D$、$E$、$F$分别在$BC$、$AC$、$AB$上, 设$\\angle DEC=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$E$} coordinate (E);\n\\draw (E) ++ (70:1) node [right] {$F$} coordinate (F);\n\\draw (E) ++ (130:1) node [left] {$D$} coordinate (D);\n\\draw (D)--(E)--(F)--cycle;\n\\path [name path = BC] (D) ++ (0,1.9) --++ (0,-2.8);\n\\path [name path = AC] (E) ++ (1,0) --++ (-1.7,0);\n\\path [name path = AB] (F) ++ (120:2) --++ (120:-3.1);\n\\path [name intersections = {of = BC and AC, by = C}] (C) node [below left] {$C$};\n\\path [name intersections = {of = AB and AC, by = A}] (A) node [below right] {$A$};\n\\path [name intersections = {of = AB and BC, by = B}] (B) node [above] {$B$};\n\\draw (A)--(B)--(C)--cycle;\n\\draw (E) pic [draw, scale = 0.4] {angle = D--E--C};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\theta=\\dfrac{\\pi}{3}$, 求$\\triangle DEF$的边长;\\\\\n(2) 当$\\theta$多大时, $\\triangle DEF$的边长最小? 并求出最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455014,7 +457396,9 @@ "id": "017637", "content": "已知函数$f(x)=2 \\sqrt{3} \\sin x \\cos x+2 \\cos ^2 x-1$($x \\in \\mathbf{R}$).\\\\\n(1) 求函数$f(x)$的最小正周期及在区间$[0, \\dfrac{\\pi}{2}]$上的最大值和最小值;\\\\\n(2) 若$f(x_0)=\\dfrac{6}{5}$, $x_0 \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$, 求$\\cos 2 x_0$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455034,7 +457418,9 @@ "id": "017638", "content": "对于三个数$a$、$b$、$c$能构成三角形的三边, 则称这三个数为``三角形数''. 对于``三角形数''$a$、$b$、$c$, 如果函数$y=f(x)$使得三个数$f(a)$、$f(b)$、$f(c)$仍为``三角形数'', 则称$y=f(x)$为这三个数的``保三角形函数''.\\\\\n(1) 对于``三角形数''$\\alpha$、$2 \\alpha$、$\\dfrac{\\pi}{4}+\\alpha$, 其中$\\dfrac{\\pi}{8}<\\alpha<\\dfrac{\\pi}{4}$且$\\tan \\alpha=p$, 判断函数$f(x)=\\tan x$是否是这三个数的``保三角形函数'', 并说明理由;\\\\\n(2) 对于``三角形数''$\\alpha$、$\\alpha+\\dfrac{\\pi}{6}$、$\\alpha+\\dfrac{\\pi}{3}$, 其中$\\dfrac{\\pi}{6}<\\alpha<\\dfrac{7 \\pi}{12}$, 判断函数$f(x)=\\sin x$是否是这三个数的``保三角形函数'', 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455054,7 +457440,9 @@ "id": "017639", "content": "已知$\\overrightarrow {a}=3 \\overrightarrow {i}+\\overrightarrow {j}$, $\\overrightarrow {b}=x \\overrightarrow {i}-6 \\overrightarrow {j}$, 且$\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则实数$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455074,7 +457462,9 @@ "id": "017640", "content": "已知向量$|\\overrightarrow {a}|=3$, $\\overrightarrow {b}=(1,2)$, 且$\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则$\\overrightarrow {a}$的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455094,7 +457484,9 @@ "id": "017641", "content": "已知$\\triangle ABC$是边长为$1$的等边三角形, 点$D$、$E$分别是边$AB$、$BC$的中点, 联接$DE$并延长到点$F$, 使得$DE=2EF$, 则$\\overrightarrow{AF} \\cdot \\overrightarrow{BC}$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455114,7 +457506,9 @@ "id": "017642", "content": "已知平行四边形$ABCD$的三个顶点$A(0,0)$, $B(3,1)$, $C(4,1)$, 则$D$点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455134,7 +457528,10 @@ "id": "017643", "content": "已知向量$\\overrightarrow {a}=(\\cos \\theta, \\sin \\theta)$, 向量$\\overrightarrow {b}=(\\sqrt{3},-1)$, 则$|\\overrightarrow {a}-\\overrightarrow {b}|$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455154,7 +457551,9 @@ "id": "017644", "content": "已知点$A(-1,1)$和坐标原点$O$, 若点$M(x, y)$为平面区域$\\begin{cases}x+y \\geq 2, \\\\ x \\leq 1, \\\\ y \\leq 2\\end{cases}$上的一个动点, 则$\\overrightarrow{OA} \\cdot \\overrightarrow{OM}$的取值范围是\\bracket{20}.\n\\fourch{$[-1,0]$}{$[0,1]$}{$[0,2]$}{$[-1,2]$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455174,7 +457573,10 @@ "id": "017645", "content": "将函数$y=\\log _3(2 x+1)-4$的图像按向量$\\overrightarrow {a}$平移后得到的是函数$y=\\log _3^{2 x}$的图像, 则$\\overrightarrow {a}$的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455194,7 +457596,9 @@ "id": "017646", "content": "在等腰梯形$ABCD$中, 已知$AB\\parallel DC$, $AB=2$, $BC=1$, $\\angle ABC=60^{\\circ}$, 动点$E$和$F$分别在线段$BC$和$DC$上, 且$\\overrightarrow{BE}=\\lambda \\overrightarrow{BC}$, $\\overrightarrow{DF}=\\dfrac{1}{9 \\lambda} \\overrightarrow{DC}$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{AF}$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2.5]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (60:1) node [above] {$D$} coordinate (D);\n\\draw (B) ++ (120:1) node [above] {$C$} coordinate (C);\n\\draw ($(B)!0.8!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(D)!{0.8/9}!(C)$) node [above right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\draw [->] (A)--(E);\n\\draw [->] (A)--(F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455214,7 +457618,9 @@ "id": "017647", "content": "设$\\overrightarrow {a}$与$\\overrightarrow {b}$为非零向量, 给出下列命题:\\\\\n\\textcircled{1} 若$\\overrightarrow {a}$与$\\overrightarrow {b}$平行, 则$\\overrightarrow {a}$与$\\overrightarrow {b}$向量的方向相同或相反;\\\\\n\\textcircled{2} 若$\\overline{AB}=\\overrightarrow {a}, \\overline{CD}=\\overrightarrow {b}, \\overrightarrow {a}$与$\\overrightarrow {b}$共线, 则$A$、$B$、$C$、$D$四点必在一条直线上;\\\\\n\\textcircled{3} 若$\\overrightarrow {a}$与$\\overrightarrow {b}$共线, 则$|\\overrightarrow {a}|+|\\overrightarrow {b}|=|\\overrightarrow {a}+\\overrightarrow {b}|$;\\\\\n\\textcircled{4}若$\\overrightarrow {a}$与$\\overrightarrow {b}$反向, 则$\\overrightarrow {a}=-\\dfrac{|\\overrightarrow {a}|}{|\\overrightarrow {b}|} \\cdot \\overrightarrow {b}$.\\\\\n其中正确命题的个数有\\bracket{20}.\n\\fourch{1 个}{2 个}{3 个}{4 个}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455234,7 +457640,9 @@ "id": "017648", "content": "设$\\overrightarrow{x_0}, \\overrightarrow{x_1}, \\overrightarrow{x_2}$为平面上的三个向量, 且满足$\\overrightarrow{x_0}\\cdot \\overrightarrow{x_1}=\\dfrac{1}{k}$, $\\overrightarrow{x_1}\\cdot\\overrightarrow{x_k}=\\dfrac{1}{k+1}$, $\\overrightarrow{x_2} \\cdot\\overrightarrow{x_k}=\\dfrac{1}{k+2}$($k=1,2$), 则能使$a \\overrightarrow{x_1}+b \\overrightarrow{x_2}=\\overrightarrow{x_0}$成立的常数$a$、$b$的值是\\bracket{20}.\n\\fourch{$a=6$, $b=6$}{$a=-6$, $b=6$}{$a=6$, $b=-6$}{$a=-6$, $b=-6$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455254,7 +457662,9 @@ "id": "017649", "content": "已知非零向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$, 则``$\\overrightarrow {a} \\cdot \\overrightarrow {c}=\\overrightarrow {b} \\cdot \\overrightarrow {c}$''是``$\\overrightarrow {a}=\\overrightarrow {b}$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455274,7 +457684,10 @@ "id": "017650", "content": "定义: 如果一个向量列从第二项起, 每一项与它的前一项的差都等于同一个常向量, 那么这个向量列叫做等差向量列, 这个常向量叫做等差向量列的公差.\n已知向量列$\\{\\overrightarrow {a}_n\\}$是以$\\overrightarrow{a_1}=(1,3)$为首项, 公差$\\overrightarrow {d}=(1,0)$的等差向量列. 若向量$\\overrightarrow{a_n}$与非零向量$\\overrightarrow{b_n}=(x_n, x_{n+1})$($n \\in \\mathbf{N}$, $n \\geq 1$)垂直, 则$\\dfrac{x_{10}}{x_1}=$\\bracket{20}.\n\\fourch{$\\dfrac{44800}{729}$}{$\\dfrac{4480}{243}$}{$-\\dfrac{44800}{729}$}{$-\\dfrac{4480}{243}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元", + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455294,7 +457707,9 @@ "id": "017651", "content": "已知$\\overrightarrow {p}=(2,-3)$, $\\overrightarrow {q}=(1,2), \\overrightarrow {a}=(9,4)$, 且$\\overrightarrow {a}=m \\cdot \\overrightarrow {p}+n \\cdot \\overrightarrow {q}$, 求实数$m$、$n$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455314,7 +457729,10 @@ "id": "017652", "content": "平面内有向量$\\overrightarrow{OA}=(1,7)$, $\\overrightarrow{OB}=(5,1)$, $\\overrightarrow{OP}=(2,1)$, 点$X$为直线$OP$上的一个动点.\\\\\n(1) 当$\\overrightarrow{XA} \\cdot \\overrightarrow{XB}$取最小值时, 求$\\overrightarrow{OX}$的坐标;\\\\\n(2) 当点$X$满足 (1) 的条件和结论时, 求$\\cos \\angle AXB$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455334,7 +457752,10 @@ "id": "017653", "content": "已知向量$\\overrightarrow {a}=(\\cos x, \\sin x)$, $\\overrightarrow {b}=(3,-\\sqrt{3})$, $x \\in[0, \\pi]$.\\\\\n(1) 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 求$x$的值;\\\\\n(2) 记$f(x)=\\overrightarrow {a} \\cdot \\overrightarrow {b}$, 求函数$y=f(x)$的最大值和最小值及对应的$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455354,7 +457775,10 @@ "id": "017654", "content": "在$\\triangle ABC$中, 内角$A, B, C$的对边分别为$a, b, c$, 且$a>c$, 已知$\\overrightarrow{BA} \\cdot \\overrightarrow{BC}=2$, $\\cos B=\\dfrac{1}{3}$, $b=3$.\\\\\n(1) 求$a$和$c$的值;\\\\\n(2) 求$\\cos (B-C)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455374,7 +457798,9 @@ "id": "017655", "content": "已知$x, y \\in \\mathbf{R}$, $\\overrightarrow {i}$, $\\overrightarrow {j}$分别为直角坐标系中$x, y$轴正方向上的单位向量. 若向量$\\overrightarrow {a}=x \\overrightarrow {i}+(y+\\sqrt{3}) \\overrightarrow {j}$, $\\overrightarrow {b}=x \\overrightarrow {i}+(y-\\sqrt{3}) \\overrightarrow {j}$, 且$|\\overrightarrow {a}|+|\\overrightarrow {b}|=4$.\\\\\n(1) 求点$M(x, y)$的轨迹$C$的方程;\\\\\n(2) 过点$Q(-2,0)$作直线$l$与曲线$C$交于$A, B$两点, 设$P$是过点$(-\\dfrac{5}{17}, 0)$且以$\\overrightarrow {j}$为方向向量的直线上一动点, 满足$\\overrightarrow{OP}=\\overrightarrow{OA}+\\overrightarrow{OB}$($O$为坐标原点), 问是否存在这样的直线$l$, 使得四边形$OAPB$为矩形? 若存在, 求出直线$l$的方程; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455394,7 +457820,9 @@ "id": "017656", "content": "设$z_1=2-\\mathrm{i}$, $z_2=1+3 \\mathrm{i}$, 则复数$z=\\dfrac{z_1}{\\mathrm{i}}+\\dfrac{\\overline {z}_2}{5}$的虚部为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455414,7 +457842,9 @@ "id": "017657", "content": "已知$\\dfrac{a+b \\mathrm{i}}{2-\\mathrm{i}}=3+\\mathrm{i}$($a, b \\in \\mathbf{R}$, $\\mathrm{i}$为虚数单位), 则$a+b$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455434,7 +457864,9 @@ "id": "017658", "content": "复数$\\dfrac{(1-\\mathrm{i})(1+\\mathrm{i})}{\\mathrm{i}}$在复平面中所对应的点到原点的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455454,7 +457886,9 @@ "id": "017659", "content": "已知复数$z_1=x+2 \\mathrm{i}$, $z_2=-2+\\mathrm{i}$且$|z_1|<|z_2|$, 则实数$x$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455474,7 +457908,9 @@ "id": "017660", "content": "在复数范围内分解因式: $2 x^2+3 x+2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455494,7 +457930,9 @@ "id": "017661", "content": "已知$OACB$是复平面上的平行四边形, $O$是原点, 点$A$对应的复数是$3+\\mathrm{i}$, 向量$\\overrightarrow{OB}$对应的复数是$2+4 \\mathrm{i}$, 则点$C$对应的复数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455514,7 +457952,9 @@ "id": "017662", "content": "已知复数$z$满足$|z|=1$, 则$|z+\\mathrm{i}+1|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455534,7 +457974,9 @@ "id": "017663", "content": "已知$z_1$、$z_2$是复数, 定义复数的一种运算``$\\otimes$''为: $z_1 \\otimes z_2=\\begin{cases}z_1 z_2, & |z_1|>|z_2|, \\\\ z_1+z_2 & |z_1| \\leq|z_2|,\\end{cases}$ 若$z_1=2+\\mathrm{i}$且$z_1 \\otimes z_2=3+4 \\mathrm{i}$, 则复数$z_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455554,7 +457996,9 @@ "id": "017664", "content": "为求解方程$x^5-1=0$的虚根, 可以把原方程变形为$(x-1)(x^4+x^3+x^2+x+1)=0$, 再变形为$(x-1)(x^2+a x+1)(x^2+b x+1)=0$, 由此可得原方程的一个虚数根为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455574,7 +458018,9 @@ "id": "017665", "content": "设$C$是由全体复数组成的集合, $A$是由全体实数组成的集合, $B$是由全体纯虚数组成的集合, 全集$U=C$, 则下列结论正确的是\\bracket{20}.\n\\fourch{$A \\cup B=C$}{$\\overline {A} \\supset B$}{$A \\cap \\overline {B}=\\varnothing$}{$B \\cap \\overline {B}=C$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455594,7 +458040,9 @@ "id": "017666", "content": "已知复数$z$满足$|z|=2$, 且$(z-a)^2=a$, 则实数$a$不可能取值\\bracket{20}.\n\\fourch{$\\dfrac{1+\\sqrt{3}}{2}$}{$\\dfrac{1-\\sqrt{17}}{2}$}{$1$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455614,7 +458062,9 @@ "id": "017667", "content": "下列类比推理命题(其中$\\mathbf{Q}$为有理数集, $\\mathbf{R}$为实数集, $\\mathbf{C}$为复数集):\\\\\n\\textcircled{1} ``若$a, b \\in \\mathbf{R}$, 则$a-b=0 \\Rightarrow a=b$''类比推出``若$a, b \\in \\mathbf{C}$, 则$a-b=0 \\Rightarrow a=b$'';\\\\\n\\textcircled{2} ``若$a, b, c, d \\in \\mathbf{R}$, 则复数$a+b \\mathrm{i}=c+d \\mathrm{i} \\Rightarrow a=c, b=d$''类比推出``若$a, b, c, d \\in \\mathbf{Q}$, 则$a+b \\sqrt{2}=c+d \\sqrt{2} \\Rightarrow a=c, b=d$'';\\\\\n\\textcircled{3} ``若$a, b \\in \\mathbf{R}$, 则$a-b>0 \\Rightarrow a>b$''类比推出``若$a, b \\in \\mathbf{C}$, 则$a-b>0 \\Rightarrow a>b$''.\\\\\n其中类比结论正确的个数是\\bracket{20}.\n\\fourch{0}{1}{2}{3}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455634,7 +458084,9 @@ "id": "017668", "content": "已知复数$z=(m^2+5 m+6)+(m^2-2 m-15) \\mathrm{i}$, 当实数$m$为何值时:\\\\\n(1) $z$为实数;\\\\\n(2) $z$为虚数;\\\\\n(3) $z$为纯虚数;\\\\\n(4) 复数$z$对应的点$Z$在第四象限.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455654,7 +458106,9 @@ "id": "017669", "content": "设复数$z$的共轭复数为$\\overline {z}$, 已知$(1+2 \\mathrm{i}) \\overline {z}=4+3 \\mathrm{i}$.\\\\\n(1) 求复数$z$及$\\dfrac{z}{\\overline {z}}$;\\\\\n(2) 求满足$|z_1-1|=|z|$的复数$z_1$对应的点的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455674,7 +458128,9 @@ "id": "017670", "content": "已知关于$x$的实系数一元二次方程$a x^2+b x+c=0$有两个虚数根$x_1$、$x_2$, 且$(1-3 a \\mathrm{i}) \\mathrm{i}=c-\\dfrac{a}{\\mathrm{i}}$($\\mathrm{i}$为虚数单位), $|x_1-x_2|=1$, 求实数$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455694,7 +458150,9 @@ "id": "017671", "content": "对于任意的复数$z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$), 定义运算$P(z)=x^2[\\cos (y \\pi)+\\mathrm{i}\\sin(y \\pi)]$.\\\\\n(1) 集合$A=\\{\\omega|\\omega=P(z), \\ | z | \\leq 1, \\ \\mathrm{Re} z$、$\\mathrm{Im} z \\in \\mathbf{Z}\\}$, 试用列举法写出集合$A$;\\\\\n(2) 若$z=2+y \\mathrm{i}$($y \\in \\mathbf{R}$), $P(z)$为纯虚数, 求$|z|$的最小值;\\\\\n(3) 直线$l: y=x-9$上是否存在格点$(x, y)$(坐标$x, y$均为整数的点), 使复数$z=x+y \\mathrm{i}$经运算$P$后, $P(z)$对应的点也在直线$l$上? 若存在, 求出所有的点; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455714,7 +458172,9 @@ "id": "017672", "content": "已知$\\alpha$、$\\beta$是不同的两个平面, 直线$a \\subset \\alpha$, 直线$b \\subset \\beta$, 命题$p: a$与$b$没有公共点; 命题$q: \\alpha\\parallel \\beta$, 则$p$是$q$的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455734,7 +458194,9 @@ "id": "017673", "content": "设直线$m$与平面$\\alpha$相交但不垂直, 给出以下说法:\\\\\n\\textcircled{1} 在平面$\\alpha$内有且只有一条直线与直线$m$垂直;\\\\\n\\textcircled{2} 过直线$m$有且只有一个平面与平面$\\alpha$垂直;\\\\\n\\textcircled{3} 与直线$m$垂直的直线不可能与平面$\\alpha$平行;\\\\\n\\textcircled{4} 与直线$m$平行的平面不可能与平面$\\alpha$垂直.\\\\\n其中错误的是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455754,7 +458216,9 @@ "id": "017674", "content": "在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, 点$M$和$N$分别是矩形$ABCD$和$BB_1C_1C$的中心, 则过点$A$、$M$、$N$的平面截正方体所得截面的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455774,7 +458238,9 @@ "id": "017675", "content": "在$30^{\\circ}$的二面角的一个面内有一点$P$, 点$P$到另一个面的距离是$10$, 则点$P$到二面角棱的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455794,7 +458260,9 @@ "id": "017676", "content": "若将一个$45^{\\circ}$的直角三角板的一直角边放在一桌面上, 另一直角边与桌面所成角为$45^{\\circ}$, 则此时该三角板的斜边与桌面所成角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455814,7 +458282,9 @@ "id": "017677", "content": "如图所示, 在四棱锥$P-ABCD$中, $PA \\perp$底面$ABCD$, 且底面各边都相等, $M$是$PC$上的一动点, 当点$M$满足\\blank{50}时, 平面$MBD$$\\perp$平面$PCD$. (只要填写一个你认为是正确的条件即可)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,-2) node [right] {$C$} coordinate (C);\n\\draw (0,0,-2) node [left] {$D$} coordinate (D);\n\\draw (0,2.5,0) node [above] {$P$} coordinate (P);\n\\draw ($(C)!0.45!(P)$) node [above] {$M$} coordinate (M);\n\\draw (P)--(A)--(B)--(C)--cycle(B)--(M)(B)--(P);\n\\draw [dashed] (A)--(D)--(M)(P)--(D)--(B)(D)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455834,7 +458304,9 @@ "id": "017678", "content": "设$a$、$b$是两条不同的直线, $\\alpha$、$\\beta$是两个不同的平面, 则下面四个命题中错误的是\\bracket{20}.\n\\twoch{若$a \\perp b$, $a \\perp \\alpha$, $b \\not \\subset \\alpha$, 则$b\\parallel \\alpha$}{若$a \\perp b$, $a \\perp \\alpha, b \\perp \\beta$, 则$\\alpha \\perp \\beta$}{若$a \\perp \\beta$, $\\alpha \\perp \\beta$, 则$a\\parallel \\alpha$或$a \\subset \\alpha$}{若$a \\perp \\alpha$, $\\alpha \\perp \\beta$, 则$a \\perp \\beta$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455854,7 +458326,9 @@ "id": "017679", "content": "如图, $\\alpha \\perp \\beta$, $\\alpha \\cap \\beta=l$, $A \\in \\alpha$, $B \\in \\beta$, $A, B$到$l$的距离分别是$a$和$b$, $AB$与$\\alpha, \\beta$所成的角分别是$\\theta$和$\\varphi$, $AB$在$\\alpha, \\beta$内的射影长分别是$m$和$n$. 若$a>b$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (S) (0,0,-4) coordinate (T);\n\\draw ($(S)!0.4!(T)$) coordinate (P) ($(S)!0.75!(T)$) coordinate (Q);\n\\draw (S) --++ (1.5,0) node [above] {$\\beta$} --++ (0,0,-4) -- (T) (S) --++ (0,1.5) node [midway, right] {$l$} node [right] {$\\alpha$} --++ (0,0,-4) -- (T) -- (S);\n\\draw (P) --++ (0,1.2) node [midway, right] {$a$} node [above] {$A$} coordinate (A);\n\\draw (Q) --++ (0.8,0) node [midway, below] {$b$} node [right] {$B$} coordinate (B) -- (A);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\theta>\\varphi$, $m>n$}{$\\theta>\\varphi$, $mn$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -455874,7 +458348,9 @@ "id": "017680", "content": "如图, $P$为$\\triangle ABC$所在平面外一点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(C)$) ++ (0,{2*sqrt(6)/3},0) node [above] {$P$} coordinate (P);\n\\draw ($(C)!0.5!(P)$) node [above right] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(P)--cycle(B)--(D)(B)--(P);\n\\draw [dashed] (A)--(D)(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$AP=AC$, $BP=BC$, $D$为$PC$的中点, 证明: $PC \\perp$平面$ABD$;\\\\\n(2) 若$AP=BP$, $AC=BC$, 证明: $PC \\perp AB$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455894,7 +458370,9 @@ "id": "017681", "content": "正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$分别为$A_1B_1$、$A_1D_1$的中点, $E$、$F$分别是$B_1C_1$、$C_1D_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [below left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A_1)!0.5!(B_1)$) node [below] {$M$} coordinate (M);\n\\draw ($(A_1)!0.5!(D_1)$) node [left] {$N$} coordinate (N);\n\\draw ($(B_1)!0.5!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$F$} coordinate (F);\n\\draw (A)--(M)--(N)(B)--(E)--(F);\n\\draw [dashed] (B)--(D)--(F)(A)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $E$、$F$、$B$、$D$共面;\\\\\n(2) 求证: 平面$AMN\\parallel$平面$EFDB$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455914,7 +458392,9 @@ "id": "017682", "content": "如图所示, $PA \\perp$平面$ABCD$, 矩形$ABCD$的边长$AB=1$, $BC=2$, $E$为$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (2,0,1) node [right] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(E)--(D);\n\\draw [dashed] (B)--(A)--(D)(P)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $PE \\perp DE$;\\\\\n(2) 如果$PA=2$, 求异面直线$AE$与$PD$所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455934,7 +458414,9 @@ "id": "017683", "content": "如图, 已知正方形$ABCD$和矩形$ACEF$所在的平面互相垂直, 且$AB=1$, $AF=1$, 点$M$是线段$EF$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(235:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$D$} coordinate (D);\n\\draw (D) ++ (\\l,0,0) node [below right] {$A$} coordinate (A);\n\\draw (D) ++ (\\l,0,-\\l) node [right] {$B$} coordinate (B);\n\\draw (D) ++ (0,0,-\\l) node [left] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\l,0) node [above] {$F$} coordinate (F);\n\\draw (C) ++ (0,\\l,0) node [above] {$E$} coordinate (E);\n\\draw ($(E)!0.5!(F)$) node [above] {$M$} coordinate (M);\n\\draw (D)--(A)--(B)--(F)--(E)--cycle(D)--(F)(A)--(F);\n\\draw [dashed] (A)--(C)(B)--(D)(D)--(C)--(B)(E)--(C)(A)--(M)(B)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AM\\parallel$平面$BDE$;\\\\\n(2) 求二面角$A-DF-B$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455954,7 +458436,9 @@ "id": "017684", "content": "如图, 已知$ABCD$为正方形, $PD \\perp$平面$ABCD$, $PD=AD=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (D) ++ (0,\\l,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)(P)--(B)(P)--(C);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C)(A)--(C)(B)--(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$PC$与平面$PBD$所成角的大小;\\\\\n(2) 求异面直线$PC$与$BD$所成角的大小;\\\\\n(3) 在线段$PB$上是否存在一点$E$, 使得$PC \\perp$平面$ADE$. 若存在, 确定点$E$的位置; 不存在, 则说明理由?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -455974,7 +458458,9 @@ "id": "017685", "content": "一个圆锥的底面圆的半径和高分别为$2$和$6$, 则该圆锥的侧面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -455994,7 +458480,9 @@ "id": "017686", "content": "正三棱锥底面边长为$a$, 侧棱与底面所成的角的大小为$45^{\\circ}$, 则它的斜高等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456014,7 +458502,9 @@ "id": "017687", "content": "某地球仪上一点$A$位于北纬$30^{\\circ}$的纬线上, 纬线的长度为$12 \\pi \\text{cm}$, 则该地球仪的表面积是\\blank{50}$\\text{cm}^2$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456034,7 +458524,9 @@ "id": "017688", "content": "如图所示, 在正三棱柱$ABC-A_1B_1C_1$中, $AA_1=6$, 异面直线$BC_1$与$AA_1$所成角的大小为$\\dfrac{\\pi}{6}$, 该三棱柱的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\def\\l{1}\n\\def\\h{{sqrt(3)}}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\draw (B)--(C_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456054,7 +458546,9 @@ "id": "017689", "content": "如图所示, 边长为$2$的正方形$ABCD$中, $E$为$AB$的中点, 将它沿$EC$、$ED$折起, 使$EA, EB$重合, 组成一个四面体, 这个四面体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [above] {$D$} coordinate (D);\n\\draw (2,2) node [above] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw (A) rectangle (C) (D)--(E)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456074,7 +458568,9 @@ "id": "017690", "content": "在三棱柱$ABC-A_1B_1C_1$中, $AB=BC=2$, $\\angle ABC=120^{\\circ}$, 侧面$A_1ACC_1 \\perp$底面$ABC$, 侧棱$AA_1$与底面$ABC$成$60^{\\circ}$角, $D$为$AC$的中点, $\\angle A_1D\\mathrm{C}_1=90^{\\circ}$, 则三棱柱$ABC-A_1B_1C_1$的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456094,7 +458590,9 @@ "id": "017691", "content": "一个圆锥的侧面积是其底面积的$2$倍, 则该圆锥的母线与底面所成的角为\\bracket{20}.\n\\fourch{$30^{\\circ}$}{$45^{\\circ}$}{$60^{\\circ}$}{$75^{\\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456114,7 +458612,9 @@ "id": "017692", "content": "如图所示, 三棱锥的四个顶点$P$、$A$、$B$、$C$在同一个球面上, 顶点$P$在平面$ABC$上的射影是$H$, 若球心在直线$PH$上, 则点$H$一定是$\\triangle ABC$的\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (1,1.5,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(C)$) node [above right] {$H$} coordinate (H);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(C)(P)--(H);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{重心}{垂心}{内心}{外心}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456134,7 +458634,9 @@ "id": "017693", "content": "三棱锥$P-ABC$的四个顶点都在球$O$的球面上, 已知$PA$、$PB$、$PC$两两垂直, $PA=1$, $PB+PC=4$, 当三棱锥的体积最大时, 球$O$的体积为\\bracket{20}.\n\\fourch{$36 \\pi$}{$9 \\pi$}{$\\dfrac{9}{2} \\pi$}{$\\dfrac{9}{4} \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456154,7 +458656,9 @@ "id": "017694", "content": "如图所示, 在正方体$ABCD-A_1B_1C_1D_1$中, 给出下列四个命题:\\\\\n\\textcircled{1} 点$P$在直线$BC_1$上运动时, 三棱锥$A-D_1PC$的体积不变;\\\\\n\\textcircled{2} 点$P$在直线$BC_1$上运动时, 直线$AP$与平面$ACD_1$所成角的大小不变;\\\\\n\\textcircled{3} 点$P$在直线$BC_1$上运动时, 二面角$P-AD_1-C$的大小不变;\\\\\n\\textcircled{4} 点$P$是平面$ABCD$上到点$D$和$C_1$距离相等的动点, 则$P$的轨迹是过点$B$的直线.\\\\\n其中的真命题是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (B)--(C_1);\n\\draw [dashed] (A)--(C)--(D_1)--cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456174,7 +458678,9 @@ "id": "017695", "content": "如图所示, 在正三棱柱$ABC-A_1B_1C_1$中, $AA_1=A_1B_1=4$, 点$D$、$E$分别为棱$AA_1$、$A_1B_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-125:0.5cm)}]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ({\\l/2},0,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (C) ++ (0,\\h) node [below right] {$C_1$} coordinate (C_1);\n\\draw (B) ++ (0,\\h) node [right] {$B_1$} coordinate (B_1);\n\\draw (A) -- (C) -- (B) (A) -- (A_1) (C) -- (C_1) (B) -- (B_1) (A_1) -- (C_1) -- (B_1) (A_1) -- (B_1);\n\\draw [dashed] (A) -- (B);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$D$} coordinate (D);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$E$} coordinate (E);\n\\draw (E)--(C_1)--(B)(C_1)--(D);\n\\draw [dashed] (E)--(D)--(B)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$BC_1$与平面$ABB_1A_1$所成的角的大小 (结果用反三角函数值表示);\\\\\n(2) 求四面体$BDEC_1$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456194,7 +458700,9 @@ "id": "017696", "content": "如图所示, 已知圆柱$OO_1$的轴截面$ABCD$为正方形, $E$为上底面圆周上一点, 且$2CE=\\sqrt{3} CD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\filldraw (0,0) node [below] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2) node [below] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O) ++ (-1,0) node [left] {$A$} coordinate (A);\n\\draw ($(O)!-1!(A)$) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,2) node [left] {$D$} coordinate (D);\n\\draw (B) ++ (0,2) node [right] {$C$} coordinate (C);\n\\draw (O_1) ++ (120:1 and 0.25) node [above] {$E$} coordinate (E);\n\\draw (A) arc (180:360:1 and 0.25);\n\\draw [dashed] (A) arc (180:0:1 and 0.25) (A)--(C)(A)--(E)(A)--(B);\n\\draw (C) arc (0:360:1 and 0.25) -- (C) -- (E) (A)--(D)(B)--(C)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AE \\perp CE$;\\\\\n(2) 求平面$ACE$与圆$O$所在平面所成的锐二面角的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456214,7 +458722,9 @@ "id": "017697", "content": "如图$1$所示, 四边形$PBCD$是直角梯形, $\\angle PBC=90^{\\circ}$, $CD=1$, $PB=BC=2$, 点$A$是$PB$的中点, $E$是$BC$的中点, 现沿$AD$将平面$PAD$折起, 设$\\angle PAB=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (0,1) node [left] {$A$} coordinate (A);\n\\draw (0,2) node [left] {$P$} coordinate (P);\n\\draw (2,1) node [right] {$D$} coordinate (D);\n\\filldraw (1,0) node [below] {$E$} coordinate (E) circle (0.03);\n\\draw (P)--(B)--(C)--(D)--cycle(A)--(D);\n\\draw (1,-0.5) node [below] {图1};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\filldraw (1,0) node [below] {$E$} coordinate (E) circle (0.03);\n\\draw (B) ++ (0,0,-1) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (A) ++ (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (B)--(C)--(D)--(P)--(A)--cycle(A)--(E)(P)--(C);\n\\draw [dashed] (A)--(D);\n\\draw (1,-0.5) node [below] {图2};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (B) ++ (0,0,-1) node [above right] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (A) ++ (0,{sqrt(2)/2},{-sqrt(2)/2}) node [above] {$P$} coordinate (P);\n\\draw (B)--(C)--(D)--(P)--cycle(B)--(D);\n\\draw [dashed] (A)--(D)(B)--(A)--(P);\n\\draw (1,-0.5) node [below] {图3};\n\\end{tikzpicture}\n\\end{center}\n(1) 如图 2, 当$\\theta=90^{\\circ}$时, 求异面直线$PC$与$AE$所成的角的大小;\\\\\n(2) 如图 3, 当$\\theta$为多大时, 三棱锥$P-ABD$的体积为$\\dfrac{\\sqrt{2}}{6}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456234,7 +458744,9 @@ "id": "017698", "content": "从自动打包机包装的食盐中, 随机抽取$20$袋, 测得各袋的质量分别为 (单位: 克): $494$, $496$, $494$, $495$, $498$, $497$, $501$,$502$, $504$, $496$, $497$, $503$, $506$, $508$, $507$, $492$, $496$, $500$, $501$, $499$. 则该自动包装机包装的袋装食盐质量在 $497.5$到$501.5$之间的概率约是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456254,7 +458766,9 @@ "id": "017699", "content": "从一副混合后的扑克牌($52$张)中, 随机抽取$1$张. 事件$A$为``抽得红桃K'', 事件$B$为``抽得黑桃'', 则概率$P(A \\cup B)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456274,7 +458788,9 @@ "id": "017700", "content": "连续$4$次抛掷一枚硬币, 至少出现$1$次正面的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456294,7 +458810,9 @@ "id": "017701", "content": "甲、乙两人进行三局比赛, 并规定如果有一人胜满两局, 则比赛结束. 若每局比赛甲获胜的概率为$\\dfrac{2}{3}$, 则比赛以甲$2$胜$1$负而结束的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456314,7 +458832,9 @@ "id": "017702", "content": "某水产试验基地实行某种鱼的人工孵化, $10000$个鱼卵能孵出$8513$尾鱼苗. 要孵出$50000$尾鱼苗, 大概需要准备\\blank{50}个鱼卵.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456334,7 +458854,9 @@ "id": "017703", "content": "已知事件$A$、$B$是独立的, 如果$P(A)=0.3$, $P(B)=0.8$, 那么$P(A \\cap B)=$\\blank{50}, $P(A \\cap \\overline {B})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456354,7 +458876,9 @@ "id": "017704", "content": "甲、乙两人进行$5$局$3$胜制的比赛, 每局两人获胜的可能性相等. 若已知第一局甲胜, 则甲最终获胜的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456374,7 +458898,9 @@ "id": "017705", "content": "某保险公司把被保险人分为$3$类: ``谨慎的''、``一般的''、``冒失的''. 统计资料表明, 这$3$类被保险人在一年内发生事故的概率依次为$0.05$、$0.15$和$0.30$. 若``谨慎的''被保险人占$20 \\%$, , ``一般的''被保险人占$50 \\%$, ``冒失的''被保险人占$30 \\%$, 随机抽取该保险公司的一位被保险人, 此人在一年内出事故的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456394,7 +458920,9 @@ "id": "017706", "content": "在进行抛掷一枚均匀硬币的试验时, 以下说法正确的是\\bracket{20}.\n\\onech{若前一次抛掷得正面朝上, 则后一次抛掷一定得反面朝上}\n{在$1000$次抛掷中, 正面朝上应有$5000$次}{随着抛掷次数的逐渐增加, 正面朝上出现的频率的近似值为$\\dfrac{1}{2}$}{随着抛掷次数的逐渐增加, 正面朝上出现的频率越来越趋于稳定}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456414,7 +458942,9 @@ "id": "017707", "content": "如果事件$A, B$互斥, 那么\\bracket{20}.\n\\fourch{$A \\cup B$是必然事件}{$\\overline {A} \\cup \\overline {B}$是必然事件}{$\\overline {A}$与$\\overline {B}$互斥}{$\\overline {A}$与$\\overline {B}$一定不互斥}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456434,7 +458964,9 @@ "id": "017708", "content": "若独立地重复一个伯努利试验$n$次, 记成功的频率为$f(n)$, 则随着$n$的逐渐增大, 有$\\cdots$\\bracket{20}.\n\\twoch{$f(n)$与某个常数越来越接近}{$f(n)$与某个常数差的绝对值逐渐减少}{$f(n)$无限趋近于某个常数}{$f(n)$的值趋于稳定}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456454,7 +458986,9 @@ "id": "017709", "content": "$6$张奖券中有$2$张是有奖的, 先后由甲、乙两人各抽一张. 对于以下两种情况, 分别研究``甲中奖''和``乙中奖''这两个事件是否独立? \\textcircled{1} 抽后返回奖券; \\textcircled{2} 抽后不返回奖券. \\bracket{20}.\n\\fourch{\\textcircled{1} 独立; \\textcircled{2} 不独立}{\\textcircled{1} 不独立;$\\textcircled{2}$独立}{\\textcircled{1} 不独立; \\textcircled{2} 不独立}{\\textcircled{1} 独立; \\textcircled{2}独立}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456474,7 +459008,9 @@ "id": "017710", "content": "掷两颗骰子, 求:\\\\\n(1) 它们的点数都是偶数的概率;\\\\\n(2) 它们的点数之和是偶数的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456494,7 +459030,9 @@ "id": "017711", "content": "甲、乙两名实习生加工零件为一等品的概率分别是$0.75$和$0.8$. 甲、乙每人独立地加工一个零件, 求:\\\\\n(1) 只有甲生产的零件为一等品的概率;\\\\\n(2) 至少有一个零件为一等品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456514,7 +459052,9 @@ "id": "017712", "content": "已知关于$x$的一元二次方程$x^2-b x+c=0$. 其中$b$、$c$是分别掷两颗骰子得到的点数. 求下列事件的概率:\\\\\n(1) 方程有两个不相等的实根;\\\\\n(2) $x=2$是方程的根.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456534,7 +459074,9 @@ "id": "017713", "content": "罐子中有$b$个黑球, $r$个红球. 从中随机摸出一个球, 观察其颜色后放回, 并在罐子中加人同色球$c$个. 再从罐子中第二次摸出一个球, 求第二次摸出的是黑球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456554,7 +459096,9 @@ "id": "017714", "content": "抛掷一枚均匀的硬币两次.\\\\\n(1) 已知第一次出现正面, 求第二次也出现正面的概率;\\\\\n(2) 已知其中有一次出现正面, 求另一次也出现正面的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456574,7 +459118,9 @@ "id": "017715", "content": "把$1$、$2$、$3$、$4$、$5$、$6$、$7$、$8$、$9$、$10$分别写在$10$个形状大小相同的卡片上, 从中抽取一张卡片, 设事件$A$表示``卡片上是偶数'', 事件$B$表示``卡片上是$5$的倍数'', 事件$C$表示``卡片上是合数''.\\\\\n(1) 事件$A$与事件$B$是否独立?\\\\\n(2) 事件$A$与事件$C$是否独立?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456594,7 +459140,9 @@ "id": "017716", "content": "在五个数字$1,2,3,4,5$中, 若随机取出三个数字, 则剩下两个数字都是奇数的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456614,7 +459162,9 @@ "id": "017717", "content": "今年我市约有$50000$名高一学生参加地理高中学业水平考, 为了了解这$50000$名学生的地理成绩, 准备从中随机抽取$2500$名学生的地理成绩进行统计分析, 那么某位高一学生地理成绩被抽中的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456634,7 +459184,9 @@ "id": "017718", "content": "$10$件产品中有$3$件次品, 从中随机取出$5$件, 则恰含$1$件次品的概率是\\blank{50}(结果用数值表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456654,7 +459206,9 @@ "id": "017719", "content": "某林场有树苗$30000$棵, 其中松树苗$4000$棵. 为调查树苗的生长情况, 采用分层抽样的方法抽取一个容量为$150$的样本, 则样本中松树苗的数量为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456674,7 +459228,9 @@ "id": "017720", "content": "已知一组数据: $125,121,123,125,127,129,125,128,130,129,126,124,125,127,126$. 则这组数据的第$25$百分位数和第$80$百分位数分别是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456694,7 +459250,9 @@ "id": "017721", "content": "为了了解某地区高三学生的身体发育情况, 抽查了该地区$100$名年龄为$17.5$岁至$18$岁的男生体重$(\\text{kg})$, 得到频率分布直方图如下:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.45, yscale = 50]\n\\draw [->] (53.5,0) -- (53.6,0) -- (53.7,0.003) -- (53.9,-0.003) -- (54,0) -- (80.5,0) node [below] {体重(kg)};\n\\draw [->] (53.5,0) -- (53.5,0.09) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {54.5/0.01,56.5/0.03,58.5/0.05,60.5/0.05,62.5/0.07,64.5/0.08,66.5/0.07,68.5/0.06,70.5/0.04,72.5/0.03,74.5/0.01}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (2,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {56.5/0.03,58.5/0.05,62.5/0.07}\n{\\draw [dashed] (\\i,\\j) -- (53.5,\\j) node [left] {$\\k$};};\n\\draw (76.5,0) node [below] {$76.5$};\n\\end{tikzpicture}\n\\end{center}\n根据上图可得这$100$名学生中体重在$[56.5,64.5)$的学生人数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456714,7 +459272,9 @@ "id": "017722", "content": "从$4$名男同学和$6$名女同学中随机选取$3$人参加某社团活动, 选出的$3$人中男女同学都有的概率为\\blank{50}(结果用数值表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456734,7 +459294,9 @@ "id": "017723", "content": "同时抛掷两枚质地均匀的骰子(一种各面上分别标有$1,2,3,4,5,6$个点的正方体玩具), 观察向上的点数, 则两个点数之积不小于$4$的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456754,7 +459316,9 @@ "id": "017724", "content": "随机抽取某果园一批橙子中的$10$个, 经称重分别为$245$、$260$、$235$、$240$、$245$、$255$、$250$、$225 、$$240$、$255$克, 试用这些橙子去估计该批次橙子, 则该批次橙子的重量的标准差是\\blank{50}克(精确到小数点后两位).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456774,7 +459338,9 @@ "id": "017725", "content": "总体由编号为$01$, $02$, $\\cdots$, $29$, $30$的$30$个个体组成. 利用所给的随机数表选取$6$个个体, 选取的方法是从随机数表第$1$行的第$3$列和第$4$列数字开始, 由左到右一次选取两个数字, 则选出来的第$5$个个体的编号为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{cccccccc}\n1712 & 1340 & 3320 & 3826 & 1389 & 5103 & 7417 & 7637 \\\\ \n1304 & 0774 & 2119 & 3056 & 6218 & 3735 & 9683 & 5087\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456794,7 +459360,9 @@ "id": "017726", "content": "从个位数与十位数之和为奇数的两位数中任取一个, 其个位数为$0$的概率是\\bracket{20}.\n\\fourch{$\\dfrac{1}{9}$}{$\\dfrac{2}{9}$}{$\\dfrac{1}{3}$}{$\\dfrac{4}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456814,7 +459382,9 @@ "id": "017727", "content": "某校为了了解学生的课外阅读情况, 随机调查了$50$名学生, 得到他们在某一天各自课外阅读所用时间的数据, 结果用条形图表示. 根据条形图要得这$50$名学生这一天平均每人的课外阅读时间为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 1.5, yscale = 0.1]\n\\draw [->] (0,0) -- (3.5,0) node [below] {时间(小时)};\n\\draw [->] (0,0) -- (0,25) node [left] {人数(人)};\n\\draw (0,0) rectangle (0.3,5) (0,0) ++ (0.15,0) node [below] {$0$};\n\\draw (0.5,0) rectangle (0.8,20) (0.5,0) ++ (0.15,0) node [below] {$0.5$};\n\\draw (1,0) rectangle (1.3,10) (1,0) ++ (0.15,0) node [below] {$1.0$};\n\\draw (1.5,0) rectangle (1.8,10) (1.5,0) ++ (0.15,0) node [below] {$1.5$};\n\\draw (2,0) rectangle (2.3,5) (2,0) ++ (0.15,0) node [below] {$2.0$};\n\\foreach \\i/\\j in {2/5,1.5/10,0.5/20}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\j$};};\n\\draw (0,15) node [left] {$15$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$0.6$小时}{$0.9$小时}{$1.0$小时}{$1.5$小时}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456834,7 +459404,9 @@ "id": "017728", "content": "一个容量为$20$的样本数据, 分组后, 组距与频数如下:\n$[10,20), 2$; $[20,30), 3$; $[30,40), 4$; $[40,50), 5$; $[50,60), 4$; $[60,70), 2$, 则样本在$[0,50)$上的频率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{20}$}{$\\dfrac{1}{4}$}{$\\dfrac{1}{2}$}{$\\dfrac{7}{10}$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456854,7 +459426,9 @@ "id": "017729", "content": "某社区安置了$15$个体温检测点, 每个检测点每天检测的人数都是随机的, 不受位置等因素影响, 如图是由某天检测人数绘制的茎叶图, 则某个检测点某天检测人数达$145$及以上的概率是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{c|ccccccc}\n13 & 0 & 2 & 4 & 6 \\\\\n14 & 0 & 5 & 5 & 5 & 6 & 8 & 8\\\\\n15 & 2 & 3 & 3 & 4 \n\\end{tabular}\n\\end{center}\n\\fourch{$\\dfrac{7}{15}$}{$\\dfrac{8}{15}$}{$\\dfrac{1}{3}$}{$\\dfrac{2}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -456874,7 +459448,9 @@ "id": "017730", "content": "某居民小区所有$263$户家庭人口数分组列表如下, 求:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline 家庭人口数 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\\n\\hline 家庭数 & 20 & 29 & 59 & 50 & 46 & 36 & 19 & 4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 总体平均数、众数、中位数;\\\\\n(2) 求总体标准差.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456894,7 +459470,9 @@ "id": "017731", "content": "某中学从甲班选出$6$名、乙班选出$7$名学生参加数学竞赛, 将他们的成绩 (满分$100$分) 进行统计分析, 绘制成如图所示的茎叶图, 已知甲班学生成绩的众数是$83$, 乙班学生成绩的平均数是$86$.\n\\begin{center}\n\\begin{tabular}{cc|c|ccc}\n\\multicolumn{2}{r|}{甲} & & \\multicolumn{3}{l}{乙}\\\\\n8 &) 9 & 7 & 6\\\\\n$x$ & 3 & 8 & 1 & 2 & $y$\\\\\n6 & 2 & 9 & 1 & 1 & 6\n\\end{tabular}\n\\end{center}\n(1) 求$x$、$y$的值;\\\\\n(2) 设成绩在$85$分以上 (含$85$分) 的学生为优秀学生, 从甲、乙两班的优秀学生中各取$1$人, 记甲班选取的学生成绩不低于乙班选取的学生成绩为事件$A$, 求事件$A$发生的概率$P(A)$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456914,7 +459492,9 @@ "id": "017732", "content": "有$10$张卡片, 其号码分别为$1,2,3, \\cdots, 10$. 从中任取$3$张.\\\\\n(1) 求恰有一张号码为$3$的倍数的概率;\\\\\n(2) 求至少有一张号码为$3$的倍数的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456934,7 +459514,9 @@ "id": "017733", "content": "我国是世界上严重缺水的国家之一, 城市缺水问题较为突出. 某市为了节约生活用水, 计划在本市试行居民生活用水定额管理(即确定一个居民月均用水量标准, 用水量不超过$a$的部分按照平价收费, 超过$a$的部分按照议价收费). 为了较为合理地确定出这个标准, 通过抽样获得了$100$位居民某年的月均用水量(单位: $\\text{t}$), 制作了频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 1, yscale = 4]\n\\draw [->] (0,0) -- (5.2,0) node [below] {月均用水量/t};\n\\draw [->] (0,0) -- (0,0.75) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i in {0.1,0.2,0.3,0.4,0.5,0.6}\n{\\draw [dashed] (3.5,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i/\\j in {0/0.1,0.5/0.2,1/0.3,1.5/0,2/0.6,2.5/0.3,3/0.1\n}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (0.5,0) --++ (0,-\\j);};\n\\draw (3.5,0) node [below] {$3.5$};\n\\end{tikzpicture}\n\\end{center}\n(1) 由于某种原因频率分布直方图部分数据丢失, 请在图中将其补充完整;\\\\\n(2) 用样本估计总体, 如果希望$80 \\%$的居民每月的用水量不超过标准, 则月均用水量的最低标准定为多少吨? 并说明理由;\\\\\n(3) 从频率分布直方图中估计该$100$位居民月均用水量的平均数. (同一组中的数据用该区间的中点值代表)", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -456954,7 +459536,9 @@ "id": "017734", "content": "直线$l_1: x+m y+1=0$与直线$l_2: y=2 x-1$垂直, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456974,7 +459558,9 @@ "id": "017735", "content": "已知过点$(0,1)$的直线$l: x \\tan \\alpha-y-3 \\tan \\beta=0$的一个法向量为$(2,-1)$, 则$\\tan (\\alpha+\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -456994,7 +459580,9 @@ "id": "017736", "content": "点$P(1,1)$到直线$x \\cos \\theta+y \\sin \\theta=2$的距离为$d$, 则$d$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457014,7 +459602,9 @@ "id": "017737", "content": "已知点$M(2,4)$, $N(5,-4)$, 点$P$在$y$轴上, 且$\\angle MPN=90^{\\circ}$, 则点$P$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457034,7 +459624,9 @@ "id": "017738", "content": "已知直线$l: y=k x-\\sqrt{3}$与直线$2 x+3 y-6=0$的交点位于第一象限, 则直线$l$的倾斜角的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457054,7 +459646,9 @@ "id": "017739", "content": "若直线$l$过点$P(0,1)$, 且被两平行直线$l_1: 2 x+y-6=0$和$l_2: 4 x+2 y-5=0$截得长为$\\dfrac{7}{2}$的线段, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457074,7 +459668,9 @@ "id": "017740", "content": "设$m$、$n \\in \\mathbf{R}$, 若直线$(m+1) x+(n+1) y-2=0$与圆$(x-1)^2+(y-1)^2=1$相切, 则$m+n$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457094,7 +459690,9 @@ "id": "017741", "content": "设$m \\in \\mathbf{R}$, 过定点$A$的动直线$x+m y=0$与过定点$B$的动直线$m x-y-2 m+4=0$交于点$P(x, y)$, 则$|PA| \\cdot|PB|$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457114,7 +459712,9 @@ "id": "017742", "content": "若两直线$l_1: (a-1) x-3 y-2=0$与$l_2: x-(a+1) y+2=0$平行, 则$a$的值为\\bracket{20}.\n\\fourch{$0$}{$2$}{$-2$}{$\\pm 2$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -457134,7 +459734,9 @@ "id": "017743", "content": "已知点$M, N$分别在直线$l_1: x+y=0$与直线$l_2: x+y-3=0$, 且$MN \\perp l_1$, 点$P(-1,-3)$, $Q(\\dfrac{7}{2}, \\dfrac{1}{2})$, 则$|PM|+|QN|$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{15}}{2}$}{$\\sqrt{15}$}{$\\sqrt{13}$}{$3 \\sqrt{3}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -457154,7 +459756,9 @@ "id": "017744", "content": "如图, 在平面直角坐标系$x O y$中, 点$A(0,3)$, 直线$l: y=2 x-4$, 设圆$C$的半径为$1$, 圆心在$l$上. 若圆心$C$也在直线$y=x-1$上, 过点$A$作圆$C$的切线, 求切线的方程.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.25]\n\\draw [->] (-2,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-6) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (0,3) node [right] {$A$} coordinate (A) circle (0.12);\n\\draw (-1,-6) -- (5,6) node [right] {$l$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457174,7 +459778,9 @@ "id": "017745", "content": "直线$l$被两条直线$l_1: 4 x+y+3=0$和$l_2: 3 x-5 y-5=0$截得的线段$AB$的中点为$P(-1,2)$, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457194,7 +459800,9 @@ "id": "017746", "content": "直线$l$与两坐标轴构成等腰直角三角形, 且点$P(4,3)$到直线$l$的距离为$3 \\sqrt{2}$, 求此直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457214,7 +459822,9 @@ "id": "017747", "content": "在平面直角坐标系$x O y$中, $O$为坐标原点. 定义$P(x_1, y_1)$、$Q(x_2, y_2)$两点之间的``直角距离''为$d(P, Q)=|x_1-x_2|+|y_1-y_2|$. 已知$B(1,0)$, 点$M$为直线$x-y+2=0$上的动点, 求$d(B, M)$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457234,7 +459844,9 @@ "id": "017748", "content": "设集合$L=\\{l|$直线$l$与直线$y=3 x$相交, 且以交点的横坐标为斜率$\\}$.\\\\\n(1) 是否存在直线$l_0$使$l_0 \\in L$且$l_0$过点$(1,5)$, 若存在, 请写出$l_0$的方程, 若不存在, 请说明理由;\\\\\n(2) 点$P(-3,5)$与集合$L$中的哪一条直线的距离最小?\\\\\n(3) 设$a \\in(0,+\\infty)$, 点$P(-3, a)$与集合$L$中的直线的距离最小值记为$f(a)$, 求$f(a)$的解析式.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457254,7 +459866,9 @@ "id": "017749", "content": "已知直线$l$过点$P(-\\dfrac{5}{2}, \\dfrac{3}{2})$, 且直线$l$的一个方向向量为$\\overrightarrow {m}=(3,3)$. 一组直线$l_1$, $l_2$, $\\cdots$, $l_n$, $\\cdots$, $l_{2 n}$($n \\in \\mathbf{N}$, $n \\geq 1$)都与直线$l$平行且与椭圆$C: \\dfrac{x^2}{10}+\\dfrac{y^2}{6}=1$均有交点, 它们到直线$l$的距离依次为$d, 2 d, \\cdots, n d, \\cdots, 2 n d$($d>0$), 直线$l_n$恰好过椭圆$C$的中心, 试用$n$表示$d$的关系式, 并求出直线$l_i$($i=1,2, \\cdots, 2 n$)的方程 (用$n$、$i$表示).", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457274,7 +459888,9 @@ "id": "017750", "content": "已知圆$O_1: x^2+y^2=1$, 圆$O_2: x^2+y^2+2 x-4 y+3=0$, 则两圆的圆心距是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457294,7 +459910,9 @@ "id": "017751", "content": "已知方程$2 x^2+m y^2=1$表示焦点在$y$轴上的椭圆, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457314,7 +459932,9 @@ "id": "017752", "content": "已知双曲线$\\dfrac{x^2}{m}-\\dfrac{y^2}{4}=1$的一条渐近线方程为$y=x$, 则实数$m$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457334,7 +459954,9 @@ "id": "017753", "content": "若直线$a x-y+1=0$经过抛物线$y^2=4 x$的焦点, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457354,7 +459976,9 @@ "id": "017754", "content": "以原点为中心的椭圆的两焦点在$x$轴上, 长轴是短轴的$2$倍, 且过点$(2,-1)$, 则该椭圆的标准方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457374,7 +459998,9 @@ "id": "017755", "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{(a+3)^2}=1$($a>0$)的一条渐近线方程为$y=2 x$, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457394,7 +460020,9 @@ "id": "017756", "content": "设$P$是曲线$y^2=4 x$上的一个动点, 则点$P$到点$(0,1)$的距离与点$P$到直线$x=-1$的距离之和的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457414,7 +460042,9 @@ "id": "017757", "content": "设双曲线$\\dfrac{x^2}{12}-\\dfrac{y^2}{4}=1$的右焦点为$F(c, 0)$, 点$P$到$F(c, 0)$的距离与到直线$x+4=0$的距离相等, 则点$P$的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457434,7 +460064,9 @@ "id": "017758", "content": "已知椭圆$\\dfrac{x^2}{a}+y^2=1$($a>1$)和双曲线$\\dfrac{x^2}{m}-y^2=1$($m>0$)有相同焦点, 则\\bracket{20}.\n\\fourch{$a=m+2$}{$m=a+2$}{$a^2=m^2+2$}{$m^2=a^2+2$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -457454,7 +460086,9 @@ "id": "017759", "content": "无论$m$为任何实数, 直线$l: y=x+m$与双曲线$C: \\dfrac{x^2}{2}-\\dfrac{y^2}{b^2}=1$($b>0$)恒有公共点, 则双曲线$C$的焦距的取值范围是\\bracket{20}.\n\\fourch{$(2 \\sqrt{2},+\\infty)$}{$(4,+\\infty)$}{$(2 \\sqrt{6},+\\infty)$}{$(4 \\sqrt{2},+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -457474,7 +460108,9 @@ "id": "017760", "content": "抛物线$C$的顶点为坐标原点$O$, 焦点在$x$轴上, 直线$l: x=1$交$C$于$P, Q$两点, 且$OP \\perp OQ$. 已知点$M(2,0)$, 且$\\odot M$与$l$相切. 求$C$, $\\odot M$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457494,7 +460130,9 @@ "id": "017761", "content": "已知$\\odot M: x^2+(y-2)^2=1$, $Q$是$x$轴上的动点, $QA$、$QB$分别切$\\odot M$于$A$、$B$两点.\\\\ \n(1) 如果$|AB|=\\dfrac{4 \\sqrt{2}}{3}$, 求直线$MQ$的方程;\\\\\n(2) 求动弦$AB$的中点$P$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457514,7 +460152,9 @@ "id": "017762", "content": "已知抛物线$y^2=2 p x$($p>0$), 其准线方程为$x+1=0$, 直线$l$过点$T(t, 0)$($t>0$)且与抛物线交于$A$、$B$两点, $O$为坐标原点.\\\\\n(1) 求抛物线方程, 并证明: $\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值与直线$l$倾斜角的大小无关;\\\\\n(2) 若$P$为抛物线上的动点, 记$|PT|$的最小值为函数$d(t)$, 求$d(t)$的解析式.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457534,7 +460174,9 @@ "id": "017763", "content": "已知离心率为$\\dfrac{1}{2}$的椭圆$E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左顶点及右焦点分别为点$A$、$F$, 且$|AF|=3$.\\\\\n(1) 求$E$的方程;\\\\\n(2) 过点$F$的直线$l$与$E$交于$M, N$两点, $P$是直线$l$上异于$F$的点, 且$|MF| \\cdot|PN|=|NF| \\cdot|PM|$, 证明: 点$P$在定直线上.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457554,7 +460196,9 @@ "id": "017764", "content": "已知$F_1(-2,0)$, $F_2(2,0)$, 点$P$满足$|PF_1|-|PF_2|=2$, 记点$P$的轨迹为$E$.\\\\\n(1) 求轨迹$E$的方程;\\\\\n(2) 若直线$l$过点$F_2$且法向量为$\\overrightarrow {n}=(a, 1)$, 直线与轨迹$E$交于$P$、$Q$两点.\\\\\n(I) 过$P$、$Q$作$y$轴的垂线$PA$、$QB$, 垂足分别为$A$、$B$, 记$|PQ|=\\lambda|AB|$, 试确定$\\lambda$的取值范围;\\\\\n(II) 在$x$轴上是否存在定点$M$, 无论直线$l$绕点$F_2$怎样转动, 使$\\overrightarrow{MP} \\cdot \\overrightarrow{MQ}=0$恒成立? 如果存在, 求出定点$M$; 如果不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457574,7 +460218,9 @@ "id": "017765", "content": "若向量$\\overrightarrow {b}$与向量$\\overrightarrow {a}=(2,-1,2)$共线, 且满足$\\overrightarrow {a} \\cdot \\overrightarrow {b}+18=0$, 则向量$\\overrightarrow {b}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457594,7 +460240,9 @@ "id": "017766", "content": "若向量$\\overrightarrow {a}=(1, \\lambda, 2)$与$\\overrightarrow {b}=(2,-1,2)$的夹角的余弦值为$\\dfrac{8}{9}$, 则$\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457614,7 +460262,9 @@ "id": "017767", "content": "如图, 已知矩形$ABCD$中, $AB=1, BC=a, PA \\perp$平面$ABCD$, 若在$BC$上只有一点$Q$满足$PQ \\perp DQ$, 则$a$的值等于\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (2,0,1) node [right] {$C$} coordinate (C);\n\\draw ($(B)!0.6!(C)$) node [below] {$Q$} coordinate (Q);\n\\draw (A) ++ (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(Q)--(D)(A)--(B)--(C)--(D)--cycle(A)--(P);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457634,7 +460284,9 @@ "id": "017768", "content": "已知空间四边形$ABCD$的每条边和对角线的长都等于$1$, 点$E$、$F$分别是$AB$、$AD$的中点, 则$\\overrightarrow{EF} \\cdot \\overrightarrow{DC}$等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457654,7 +460306,9 @@ "id": "017769", "content": "在正三棱柱$ABC-A_1B_1C_1$中, $AB=\\sqrt{2} BB_1$, 则$AB_1$与$C_1B$所成的角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457674,7 +460328,9 @@ "id": "017770", "content": "在棱长为$a$的正方体$OABC-O' A' B' C'$中, $E$、$F$分别为棱$AB$、$BC$上的动点, 且$AE=BF$, 则$A' F$与$C' E$的位置关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457694,7 +460350,9 @@ "id": "017771", "content": "如图, 在平行六面体$ABCD-A_1B_1C_1D_1$中, 若$AB=2$, $BC=2$, $AA_1=1$, $\\angle A_1AD=\\angle A_1AB=\\angle ABD=60^{\\circ}$, 则$BD_1$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (1,0,{-sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw ($(B)+(D)-(A)$) node [right] {$C$} coordinate (C);\n\\draw ($1/6*(B)+1/6*(D)$) ++ (0,{sqrt(6)/3},0) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)-(A)+(B)$) node [above] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)-(A)+(C)$) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)-(A)+(D)$) node [above] {$D_1$} coordinate (D_1);\n\\draw (A)--(B)--(C)--(C_1)--(D_1)--(A_1)--cycle(B)--(B_1)--(C_1)(A_1)--(B_1);\n\\draw [dashed] (A)--(D)--(C)(D)--(D_1)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457714,7 +460372,9 @@ "id": "017772", "content": "如图, 四面体$ABCD$中, $AB, BC, BD$两两垂直, 且$AB=BC=2$, $E$是$AC$中点, 异面直线$AD$与$BE$所成的角为$\\arccos \\dfrac{\\sqrt{10}}{10}$, 则四面体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (0,2.5,0) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.5 !(C)$)node [below] {$E$} coordinate (E);\n\\draw (A)--(C)--(D)--cycle;\n\\draw [dashed] (A)--(B)--(C)(D)--(B)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457734,7 +460394,9 @@ "id": "017773", "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $AB=AC=1$, $AA_1=2$, $\\angle B_1A_1C_1=90^{\\circ}$, $D$为$BB_1$的中点, 则异面直线$C_1D$与$A_1C$所成角的余弦值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(B_1)$) node [left] {$D$} coordinate (D);\n\\draw (B)--(C)--(C_1)--(A_1)--(B_1)--cycle(B_1)--(C_1)(C_1)--(D);\n\\draw [dashed] (A_1)--(A)--(C)(A)--(B)(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457754,7 +460416,9 @@ "id": "017774", "content": "已知点$P$在正方体$ABCD-A_1B_1C_1D_1$的对角线$BD_1$上, $\\angle PDA=60^{\\circ}$, 则$DP$与$CC_1$所成角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457774,7 +460438,10 @@ "id": "017775", "content": "已知点$O, A, B, C$为空间不共面的四点, 且向量$\\overrightarrow {a}=\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}$, 向量$\\overrightarrow {b}=\\overrightarrow{OA}+\\overrightarrow{OB}-\\overrightarrow{OC}$, 则与向量$\\overrightarrow {a}$、$\\overrightarrow {b}$不能共同构成空间基底的向量是\\bracket{20}.\n\\fourch{$\\overrightarrow{OA}$}{$\\overrightarrow{OB}$}{$\\overrightarrow{OC}$}{$\\overrightarrow{OA}$或$\\overrightarrow{OB}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -457794,7 +460461,10 @@ "id": "017776", "content": "已知$\\overrightarrow{AB}=(1,5,-2)$, $\\overrightarrow{BC}=(3,1, z)$, 若$\\overrightarrow{AB} \\perp \\overrightarrow{BC}, \\overrightarrow{BP}=(x-1, y,-3)$, 且$BP \\perp$平面$ABC$, 则实数$x, y, z$分别为\\bracket{20}.\n\\fourch{$\\dfrac{33}{7},-\\dfrac{15}{7}, 4$}{$\\dfrac{40}{7},-\\dfrac{15}{7}, 4$}{$\\dfrac{40}{7},-2,4$}{$4, \\dfrac{40}{7},-15$}", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -457814,7 +460484,10 @@ "id": "017777", "content": "空间四边形$ABCD$的各边和对角线均相等, $E$是$BC$的中点, 那么\\bracket{20}.\n\\twoch{$\\overrightarrow{AE} \\cdot \\overrightarrow{BC}<\\overrightarrow{AE} \\cdot \\overrightarrow{CD}$}{$\\overrightarrow{AE} \\cdot \\overrightarrow{BC}=\\overrightarrow{AE} \\cdot \\overrightarrow{CD}$}{$\\overrightarrow{AE} \\cdot \\overrightarrow{BC}>\\overrightarrow{AE} \\cdot \\overrightarrow{CD}$}{$\\overrightarrow{AE} \\cdot \\overrightarrow{BC}$与$\\overrightarrow{AE} \\cdot \\overrightarrow{CD}$的大小不能比较}", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -457834,7 +460507,9 @@ "id": "017778", "content": "设动点$P$在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$的对角线$BD_1$上, 记$\\dfrac{D_1P}{D_1B}=\\lambda$. 当$\\angle APC$为钝角时, 则$\\lambda$的取值范围是\\bracket{20}.\n\\fourch{$(0, \\dfrac{1}{3})$}{$(0, \\dfrac{1}{2})$}{$(\\dfrac{1}{2}, 1)$}{$(\\dfrac{1}{3}, 1)$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -457854,7 +460529,9 @@ "id": "017779", "content": "如图, 在棱长为$a$的正方体$OABC-O_1A_1B_1C_1$中, $E, F$分别是棱$AB, BC$上的动点, 且$AE=BF=x$, 其中$0 \\leq x \\leq a$, 以$O$为原点建立空间直角坐标系$O x y z$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [above right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$O$} coordinate (O);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (O) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (O) ++ (0,\\l,0) node [above left] {$O_1$} coordinate (O_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (O_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (O) -- (O_1);\n\\draw ($(A)!0.75!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(B)!0.75!(C)$) node [below right] {$F$} coordinate (F);\n\\draw [->] (A) -- ($(O)!1.3!(A)$) node [right] {$x$};\n\\draw [->] (C) -- ($(O)!1.3!(C)$) node [below] {$y$};\n\\draw [->] (O_1) -- ($(O)!1.3!(O_1)$) node [left] {$z$};\n\\draw [dashed] (A_1)--(F)(C_1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1F \\perp C_1E$;\\\\\n(2) 若$A_1, E, F, C_1$四点共面, 求证: $\\overrightarrow{A_1F}=\\dfrac{1}{2} \\overrightarrow{A_1C_1}+\\overrightarrow{A_1E}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457874,7 +460551,9 @@ "id": "017780", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$、$G$分别为$AB$、$B_1C_1$、$AA_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$D$} coordinate (D);\n\\draw (D) ++ (\\l,0,0) node [below right] {$A$} coordinate (A);\n\\draw (D) ++ (\\l,0,-\\l) node [right] {$B$} coordinate (B);\n\\draw (D) ++ (0,0,-\\l) node [left] {$C$} coordinate (C);\n\\draw (D) -- (A) -- (B);\n\\draw [dashed] (D) -- (C) -- (B);\n\\draw (D) ++ (0,\\l,0) node [left] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,\\l,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [above right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above left] {$C_1$} coordinate (C_1);\n\\draw (D_1) -- (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw (D) -- (D_1) (A) -- (A_1) (B) -- (B_1);\n\\draw [dashed] (C) -- (C_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(B_1)!0.5!(C_1)$) node [above] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$G$} coordinate (G);\n\\draw (D)--(G)--(B)(A)--(D_1);\n\\draw [dashed] (D)--(B)(F)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证$EF \\perp$平面$GBD$;\\\\\n(2) 求异面直线$AD_1$与$EF$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457894,7 +460573,9 @@ "id": "017781", "content": "如图, 在四棱锥$P-ABCD$中, $PA \\perp$底面$ABCD$, 底面$ABCD$是正方形, $PA=AB$, $E$、$F$、$G$分别是线段$PA, PD, CD$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,0,2) node [below] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [left] {$E$} coordinate (E);\n\\draw ($(P)!0.5!(D)$) node [right] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(D)$) node [below right] {$G$} coordinate (G);\n\\draw ($(P)!0.25!(B)$) node [left] {$H$} coordinate (H);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C)(F)--(G);\n\\draw [dashed] (B)--(A)--(D)(E)--(F)(E)--(G)(E)--(H)(P)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证$PB\\parallel$平面$EFG$;\\\\\n(2) 若点$H$在侧棱$PB$上, 且$PB=4PH$, 求证: $HE \\perp$平面$EFG$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457914,7 +460595,9 @@ "id": "017782", "content": "在直三棱柱$ABC-A_1B_1C_1$中, 底面是以$\\angle ABC$为直角的等腰三角形, $AC=2 a$, $BB_1=3 a$, $D, E$分别为线段$A_1C_1, B_1C$的中点.\\\\\n(1) 求直线$BE$与$A_1C$所成的角;\\\\\n(2) 在线段$AA_1$上是否存在点$F$, 使$CF \\perp$平面$B_1DF$; 若存在, 求出$|\\overrightarrow{AF}|$; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -457934,7 +460617,10 @@ "id": "017783", "content": "已知空间四边形$OABC$, 点$M$、$N$分别为$OA$、$BC$的中点, 且$\\overrightarrow{OA}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, $\\overrightarrow{OC}=\\overrightarrow {c}$, 用$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$表示$\\overrightarrow{MN}$, 则$\\overrightarrow{MN}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457954,7 +460640,9 @@ "id": "017784", "content": "在空间直角坐标系$O-x y z$中, 平面$OAB$的一个法向量$\\overrightarrow {n}=(2,-2,1)$, 已知$P(-1,3,2)$, 则点$P$到平面$OAB$的距离$d$等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457974,7 +460662,10 @@ "id": "017785", "content": "在四面体$OABC$中, 空间的一点$M$满足$\\overrightarrow{OM}=\\dfrac{1}{2} \\overrightarrow{OA}+\\dfrac{1}{6} \\overrightarrow{OB}+\\lambda \\cdot \\overrightarrow{OC}$, 若$\\overrightarrow{MA}$、$\\overrightarrow{MB}$、$\\overrightarrow{MC}$共面, 则$\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -457994,7 +460685,9 @@ "id": "017786", "content": "如图所示, 在三棱柱$ABC-A_1B_1C_1$中, $AA_1 \\perp$底面$ABC$, $AB=BC=AA_1$, $\\angle ABC=90^{\\circ}$, 点$E$、$F$分别是棱$AB$、$BB_1$的中点, 则直线$EF$和$BC_1$所成的角是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\draw ($(A)!0.5!(B)$) node [below left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$F$} coordinate (F);\n\\draw (E)--(F)(B)--(C_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458014,7 +460707,9 @@ "id": "017787", "content": "正四棱锥$S-ABCD$中, $O$为顶点在底面上的射影, $P$为侧棱$SD$的中点, 且$SO=OD$, 则直线$BC$与平面$PAC$的夹角等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458034,7 +460729,9 @@ "id": "017788", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 平面$A_1BD$与平面$C_1BD$的夹角的余弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458054,7 +460751,9 @@ "id": "017789", "content": "正方体$ABCD-A_1B_1C_1D_1$中, 二面角$A-BD_1-B_1$的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458074,7 +460773,9 @@ "id": "017790", "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$的棱长为$1$, $O$是底面$A_1B_1C_1D_1$的中心, 则点$O$到平面$ABC_1D_1$的距离为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ($(A_1)!0.5!(C_1)$) node [below] {$O$} coordinate (O) circle (0.03);\n\\draw (A_1) -- (C_1)--(B);\n\\draw [dashed] (A)--(D_1); \n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458094,7 +460795,10 @@ "id": "017791", "content": "在平行六面体$ABCD-A_1B_1C_1D_1$中, 向量$\\overrightarrow{AB}$、$\\overrightarrow{AD}$、$\\overrightarrow{AA_1}$两两的夹角均为$60^{\\circ}$, 且$|\\overrightarrow{AB}|=1$, $|\\overrightarrow{AD}|=2$, $|\\overrightarrow{AA_1}|=3$, 则$|\\overrightarrow{AC_1}|$等于\\bracket{20}.\n\\fourch{$5$}{$6$}{$4$}{$8$}", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -458114,7 +460818,9 @@ "id": "017792", "content": "正$\\triangle ABC$与正$\\triangle BCD$所在平面垂直, 则二面角$A-BD-C$的正弦值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{5}}{5}$}{$\\dfrac{\\sqrt{3}}{3}$}{$\\dfrac{2 \\sqrt{5}}{5}$}{$\\dfrac{\\sqrt{6}}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -458134,7 +460840,9 @@ "id": "017793", "content": "如图, 正方体的棱长为$1, C$、$D$、$M$分别为三条棱的中点, $A$、$B$是顶点, 则点$M$到截面$ABCD$的距离为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$A$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ($(B)!0.5!(C)$) node [right] {$M$} coordinate (M) circle (0.03);\n\\draw ($(B_1)!0.5!(C_1)$) node [right] {$C$} coordinate (S);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$D$} coordinate (T);\n\\draw (B)--(S)--(T);\n\\draw [dashed] (B)--(D)--(T);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{1}{3}$}{$\\dfrac{\\sqrt{2}}{4}$}{$\\dfrac{\\sqrt{3}}{4}$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -458154,7 +460862,10 @@ "id": "017794", "content": "沿着正四面体$O-ABC$的三条棱$\\overrightarrow{OA}$、$\\overrightarrow{OB}$、$\\overrightarrow{OC}$的方向有大小分别等于$1$, $2$和$3$的三个力$\\overrightarrow{f_1}, \\overrightarrow{f_2}, \\overrightarrow{f_3}$, 则它们的合力的大小为\\bracket{20}.\n\\fourch{$6$}{$5$}{$4$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -458174,7 +460885,9 @@ "id": "017795", "content": "如图所示, 正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是正方形$ADD_1A_1$和$ABCD$的中心, $G$是$CC_1$的中点. 设$GF$、$C_1E$与$AB$所成的角分别为$\\alpha, \\beta$. 求$\\alpha+\\beta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$A$} coordinate (A);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$C$} coordinate (C);\n\\draw (B) -- (A) -- (D);\n\\draw [dashed] (B) -- (C) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (A) ++ (0,\\l,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (C) ++ (0,\\l,0) node [above left] {$C_1$} coordinate (C_1);\n\\draw (B_1) -- (A_1) -- (D_1) -- (C_1) -- cycle;\n\\draw (B) -- (B_1) (A) -- (A_1) (D) -- (D_1);\n\\draw [dashed] (C) -- (C_1);\n\\draw ($(C)!0.5!(C_1)$) node [left] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw ($(A_1)!0.5!(D)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (G)--(F)(C_1)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458194,7 +460907,9 @@ "id": "017796", "content": "如图, 在正四棱柱$ABCD-A_1B_1C_1D_1$中, 已知$AB=1$, $BB_1=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\m) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\m) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\n,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above right] {$D_1$} coordinate (D1);\n\\draw (A) ++ (0,\\n,0) node [above left] {$A_1$} coordinate (A1);\n\\draw (B1) -- (C1) -- (D1) -- (A1) -- cycle;\n\\draw (D) -- (D1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (A) -- (A1);\n\\draw (B1)--(D1);\n\\draw [dashed] (A1)--(C)(B1)--(A)--(D1);\n\\end{tikzpicture}\n\\end{center} \n(1) 求异面直线$A_1C$与直线$AD_1$所成的角的大小;\\\\\n(2) 求点$C$到平面$AB_1D_1$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458214,7 +460929,9 @@ "id": "017797", "content": "如图, 在六面体$ABCD-A_1B_1C_1D_1$中, 四边形$ABCD$是边长为$2$的正方形, 四边形$A_1B_1C_1D_1$是边长为$1$的正方形, $DD_1 \\perp$平面$A_1B_1C_1D_1$, $DD_1 \\perp$平面$ABCD$, $DD_1=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (D_1) ++ (0,0,{\\l/2}) node [left] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ ({\\l/2},0,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (D_1) ++ ({\\l/2},0,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (A)--(C)(B)--(D);\n\\draw (A_1)--(C_1)(B_1)--(D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1C_1$与$AC$共面, $B_1D_1$与$BD$共面;\\\\\n(2) 求二面角$A-BB_1-C$的大小 (结果用反三角函数值表示).", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458234,7 +460951,9 @@ "id": "017798", "content": "如图, 四棱锥$P-ABCD$中, $PA \\perp$底面$ABCD$. 四边形$ABCD$中, $AB \\perp AD$, $AB+AD=4$, $CD=\\sqrt{2}$, $\\angle CDA=45^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (2,0,1) node [below] {$C$} coordinate (C);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(P)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$PAB \\perp$平面$PAD$;\\\\\n(2) 设$AB=AP$, 在线段$AD$上是否存在一个点$G$, 使得点$G$到点$P$、$B$、$C$、$D$的距离都相等? 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458254,7 +460973,9 @@ "id": "017799", "content": "已知$\\{a_n\\}$为等差数列, 若$a_1=6$, $a_3+a_5=0$, 则数列$\\{a_n\\}$的通项公式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458274,7 +460995,9 @@ "id": "017800", "content": "若等比数列$\\{a_n\\}$满足$a_2+a_4=20$, $a_3+a_5=40$, 则公比$q=$\\blank{50}; 前$n$项和$S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458294,7 +461017,9 @@ "id": "017801", "content": "数列$\\{a_n\\}$满足$a_1=3+\\dfrac{\\sqrt{6}}{2}$, $a_{n+1}=[a_n]+\\dfrac{1}{\\{a_n\\}}$, 其中$[a_n]$、$\\{a_n\\}$分别表示$a_n$的整数部分和小数部分, 则$a_{2020}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458314,7 +461039,9 @@ "id": "017802", "content": "用数学归纳法证明``$(n+1)(n+2) \\cdots (n+n-1)(n+n)=2^n \\cdot 1 \\cdot 3 \\cdot 5 \\cdot (2 n-1)$($n \\in \\mathbf{N}$, $n \\geq 1$)''时, 从``$n=k$''到``$n=k+1$'', 左边需要添加的代数式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458334,7 +461061,9 @@ "id": "017803", "content": "设$S_n$是公比为$q$的等比数列$\\{a_n\\}$的前$n$项的和, 若对任意的正整数$k$, 都有$\\displaystyle\\sum_{i=k+2}^{+\\infty} a_i=a_k$成立, 则$q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458354,7 +461083,9 @@ "id": "017804", "content": "设等差数列$\\{a_n\\}$的各项都是正数, 其前$n$项和为$S_n$, 公差为$d$. 若数列$\\{\\sqrt{S_n}\\}$也是公差为$d$的等差数列, 则$\\{a_n\\}$的通项公式为$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458374,7 +461105,9 @@ "id": "017805", "content": "已知数列$\\{a_n\\}$满足$a_1=m$($m \\in \\mathbf{N}$, $m \\geq 1$), $a_{n+1}=\\begin{cases}\\dfrac{a_n}{2}, & a_n \\text {为偶数}, \\\\ 3 a_n+1, & a_n \\text {为奇数}.\\end{cases}$ 若$a_6=1$, 则$m$的所有可能的取值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458394,7 +461127,9 @@ "id": "017806", "content": "已知数列$\\{a_n\\}$是以$q$为公比的等比数列. 若$b_n=-2 a_n$, 则数列$\\{b_n\\}$是\\bracket{20}.\n\\twoch{以$q$为公比的等比数列}{以$-q$为公比的等比数列}{以$2 q$为公比的等比数列}{以$-2 q$为公比的等比数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -458414,7 +461149,9 @@ "id": "017807", "content": "设等差数列$\\{a_n\\}$的公差为$d$, $d \\neq 0$. 若$\\{a_n\\}$的前$10$项之和大于其前$21$项之和, 则\\bracket{20}.\n\\fourch{$d<0$}{$d>0$}{$a_{16}<0$}{$a_{16}>0$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -458434,7 +461171,9 @@ "id": "017808", "content": "已知数列$\\{a_n\\}$满足$a_n+a_{n+4}=a_{n+1}+a_{n+3}$($n \\in \\mathbf{N}$, $n \\geq 1$), 那么\\bracket{20}.\n\\fourch{$\\{a_n\\}$是等差数列}{$\\{a_{2 n-1}\\}$是等差数列}{$\\{a_{2 n}\\}$是等差数列}{$\\{a_{3 n}\\}$是等差数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -458454,7 +461193,9 @@ "id": "017809", "content": "已知等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 公比是正数的等比数列$\\{b_n\\}$的前$n$项和为$T_n$. 已知$a_1=1$, $b_1=3$, $a_3+b_3=17$, $T_3-S_3=12$, 求$\\{a_n\\}$, $\\{b_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458474,7 +461215,9 @@ "id": "017810", "content": "我们把所有真约数 (除其本身之外的正约数) 的和等于它本身的正整数叫做``完全数''. 如$6$、$28$、$496$都是``完全数'': $6=1+2+3$; $28=1+2+4+7+14$; $496=1+2+4+8+16+31+62+124+248$. 试判断命题``若$2^n-1$是质数, 则$2^{n-1}(2^n-1)$($n \\in \\mathbf{N}$, $n \\geq 2)$是``完全数'''的真假, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458494,7 +461237,9 @@ "id": "017811", "content": "已知数列$\\{b_n\\}$是公差不为$0$的等差数列, $b_1=\\dfrac{3}{2}$, 数列$\\{a_n\\}$是等比数列, 且$a_1=b_1$, $a_2=-b_3$, $a_3=b_4$, 数列$\\{a_n\\}$的前$n$项和为$S_n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 若$\\lambda \\leq S_n-\\dfrac{1}{S_n} \\leq \\mu$对任意$n \\in \\mathbf{N}$($n \\geq 1$)都成立, 求$\\mu-\\lambda$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458514,7 +461259,9 @@ "id": "017812", "content": "已知数列$\\{a_n\\}$, $\\{b_n\\}$与函数$f(x)$, $\\{a_n\\}$是首项$a_1=15$, 公差$d \\neq 0$的等差数列, $\\{b_n\\}$满足: $b_n=f(a_n)$.\\\\\n(1) 若$a_4, a_7, a_8$成等比数列, 求$d$的值;\\\\\n(2) 若$d=2$, $f(x)=|x-21|$, 求$\\{b_n\\}$的前$n$项和$S_n$;\\\\\n(3) 若$d=-1$, $f(x)=\\mathrm{e}^x$, $T_n=b_1 \\cdot b_2 \\cdot b_3 \\cdots b_n$, 问$n$为何值时, $T_n$的值最大?", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458534,7 +461281,9 @@ "id": "017813", "content": "已知函数$f(x)=\\mathrm{e}^x$, 若$f'(x_0)=1$, 则$x_0=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458554,7 +461303,9 @@ "id": "017814", "content": "已知函数$f(x)=x+\\tan x$, 则$f'(\\dfrac{\\pi}{3})$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458574,7 +461325,9 @@ "id": "017815", "content": "已知定义在$\\mathbf{R}$上的函数$f(x)$, 其导函数为$f'(x)$, 满足$f'(x)>2$, 且$f(2)=1$, 则不等式$f(x)>2 x-3$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458594,7 +461347,9 @@ "id": "017816", "content": "已知函数$f(x)=(x+1) \\ln x$, 则曲线$y=f(x)$在$(1, f(1))$处的切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458614,7 +461369,9 @@ "id": "017817", "content": "已知曲线$f(x)=x \\mathrm{e}^x-\\dfrac{1}{\\mathrm{e}} \\ln x+1$在点$(x_0, f(x_0))$处的切线的斜率为$\\dfrac{1}{\\mathrm{e}}$, $x_0+\\ln x_0=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458634,7 +461391,9 @@ "id": "017818", "content": "设曲线$y=\\dfrac{1}{2} x^2$在点$A(1, \\dfrac{1}{2})$处的切线与曲线$y=x \\ln x$在点$P$处的切线互相平行, 则点$P$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458654,7 +461413,9 @@ "id": "017819", "content": "如图为函数$f(x)$的导函数的图像, 则下列判断正确的是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-4,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-3,-2,-1,1,2,3,4}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\draw [domain = -3:1] plot (\\x,{sin(45*\\x+45)*2});\n\\draw [domain = 1:5] plot (\\x,{sin(90*\\x)*2});\n\\draw [dashed] (1,0) -- (1,2) (3,0) -- (3,-2);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $f(x)$在$(-3,1)$上单调增;\n\\textcircled{2} $x=-1$是$f(x)$的极小值点;\n\\textcircled{3} $f(x)$在$(2,4)$上单调减, 在$(-1,2)$上单调增;\n\\textcircled{4} $x=2$是$f(x)$的极小值点.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458674,7 +461435,9 @@ "id": "017820", "content": "已知函数$f(x)=\\dfrac{1+\\ln x}{x}$在区间$(a, a+\\dfrac{2}{3})$($a>0$)上存在极值, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458694,7 +461457,9 @@ "id": "017821", "content": "如图, 已知圆柱和半径为$\\sqrt{3}$的半球$O$, 圆柱的下底面在半球$O$底面所在平面上, 圆柱的上底面内接于球$O$, 则该圆柱体积的最大值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [right] {$O$} coordinate (O) circle (0.03);\n\\draw [dashed] (O) ellipse (1.6 and 0.4);\n\\draw (2,0) arc (0:180:2);\n\\draw (2,0) arc (360:180:2 and 0.5);\n\\draw [dashed] (2,0) arc (0:180:2 and 0.5);\n\\draw [dashed] (1.6,1.2) arc (0:180:1.6 and 0.4);\n\\draw (1.6,1.2) arc (360:180:1.6 and 0.4);\n\\draw [dashed] (1.6,0) -- (1.6,1.2) (-1.6,0) -- (-1.6,1.2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458714,7 +461479,9 @@ "id": "017822", "content": "已知函数$f(x)$图像的对称轴为$x=1$, 当$x>1$时, $f(x)=\\dfrac{x}{\\ln x}$, 若$f^2(x)-2 m f(x)+4 m=0$有$8$个不同的实数解, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458734,7 +461501,9 @@ "id": "017823", "content": "设函数$f(x)=\\ln x+1$, 则$\\displaystyle\\lim _{\\Delta x \\to 0} \\dfrac{f(1+5 \\Delta x)-f(1)}{\\Delta x}=$\\bracket{20}.\n\\fourch{$1$}{$5$}{$\\dfrac{1}{5}$}{$0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -458754,7 +461523,9 @@ "id": "017824", "content": "有一机器人的运动方程为$s(t)=t^2+3 t$, ($t$是时间, $s$是位移), 则该机器人在时刻$t=2$时的瞬时速度为\\bracket{20}.\n\\fourch{$5$}{$7$}{$10$}{$13$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -458774,7 +461545,9 @@ "id": "017825", "content": "函数$y=f(x)$的图像如图所示, $f'(x)$是$f(x)$的导函数, 则下列数值排序正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (0,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 1.2:3.5] plot (\\x,{\\x*\\x*\\x/12+0.2});\n\\draw [dashed] (2,0) node [below] {$2$} --++ (0,{8/12+0.2}) --++ (-2,0);\n\\draw [dashed] (3,0) node [below] {$3$} --++ (0,{27/12+0.2}) --++ (-3,0);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$f'(2)=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,2) node [above] {$C$} coordinate (C);\n\\draw (0,2) node [above] {$D$} coordinate (D);\n\\draw [dashed] (A) rectangle (C);\n\\draw (A) --++ (2,0.5) node [right] {$E$} coordinate (E);\n\\draw (A) --++ (0.5,2) node [above] {$F$};\n\\draw (E) arc ({atan(1/4)}:{atan(4)}:{sqrt(4+0.25)});\n\\draw (1,1) node {I};\n\\draw (2,0) node [above left] {II};\n\\draw (0,2) node [below right] {II};\n\\draw (2,2) node [below left] {III};\n\\end{tikzpicture}\n\\end{center}\n(1) 要使观赏区的年收人不低于$5$万元, 求$\\theta$的最大值;\\\\\n(2) 试问: 当$\\theta$为多少时, 年总收人最大?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458854,7 +461633,9 @@ "id": "017829", "content": "已知函数$f(x)=\\ln x$.\\\\\n(1) 当$x \\in[1,+\\infty)$时, 证明: 函数$f(x)$的图像恒在函数$g(x)=\\dfrac{2}{3} x^3-\\dfrac{1}{2} x^2$的图像的下方;\\\\\n(2) 讨论方程$f(x)+k x=0$的根的个数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -458874,7 +461655,9 @@ "id": "017830", "content": "个位数是$0$, 且各位上的数字互不相同的三位数共有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458894,7 +461677,9 @@ "id": "017831", "content": "$20$部5G手机中, 有$15$部为一等品, $5$部为二等品, 现从中取$4$部手机, 求取到的手机中至多有$1$部为二等品的概率\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458914,7 +461699,9 @@ "id": "017832", "content": "观察下列各式: $\\mathrm{C}_1^0=4^0$; $\\mathrm{C}_3^0+\\mathrm{C}_3^1=4^1$; $\\mathrm{C}_5^0+\\mathrm{C}_5^1+\\mathrm{C}_5^2=4^2$; $\\mathrm{C}_7^0+\\mathrm{C}_7^1+\\mathrm{C}_7^2+\\mathrm{C}_7^3=4^3$; $\\cdots$; 照此规律, 当$n \\in \\mathbf{N}$($n \\geq 1$)时, $C_{2 n-1}^0+C_{2 n-1}^2+\\cdots+C_{2 n-1}^{n-1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458934,7 +461721,9 @@ "id": "017833", "content": "满足$a, b \\in\\{-1,0,1,2\\}$, 且关于$x$的方程$a x^2+2 x+b=0$有实数解, 则有序数对$(a, b)$的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458954,7 +461743,9 @@ "id": "017834", "content": "设虚数$a=2+\\mathrm{i}$, 化简$1-\\mathrm{C}_{12}^1 a+\\mathrm{C}_{12}^2 a^2-\\mathrm{C}_{12}^3 a^3+\\cdots+\\mathrm{C}_{12}^{12} a^{12}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458974,7 +461765,9 @@ "id": "017835", "content": "从单词``equation''中选取$5$个不同的字母排成一排, 含有``qu''(其中``qu''相连且顺序不变) 的不同排列共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -458994,7 +461787,9 @@ "id": "017836", "content": "如图, 在由二项式系数所构成的杨辉三角形中, 第\\blank{50}行中从左至右第$14$与第$15$个数的比为$2: 3$.\n\\begin{center}\n\\begin{tabular}{cc}\n第0行 & 1 \\\\\n第1行 & 1 \\quad 1 \\\\\n第2行 & 1 \\quad 2 \\quad 1 \\\\\n第3行 & 1 \\quad 3 \\quad 3 \\quad 1 \\\\\n第4行 & 1 \\quad 4 \\quad 6 \\quad 4 \\quad 1 \\\\\n第5行 & 1 \\quad 5 \\quad 10 \\quad 10 \\quad 5 \\quad 1 \\\\\n$\\cdots$ & $\\cdots$ \\quad $\\cdots$ \\quad $\\cdots$ \\quad $\\cdots$\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459014,7 +461809,9 @@ "id": "017837", "content": "设常数$a \\in \\mathbf{R}$, 若$(x^2+\\dfrac{a}{x})^5$的二项展开式中含$x^7$项的系数为$-10$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459034,7 +461831,9 @@ "id": "017838", "content": "一次国际会议, 从某大学外语系选出$11$名翻译, 其中$5$人只会英语, $4$人只会日语, 两人既会英语, 也会日语, 现从这$11$人中选出$4$名当英语翻译, $4$名当日语翻译, 不同的选法有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459054,7 +461853,9 @@ "id": "017839", "content": "设集合$A=\\{(x_1, x_2, x_3, x_4, x_5) | x_i \\in\\{-1,0,1\\},\\ i=1,2,3,4,5\\}$, 那么集合$A$中满足条件``$1 \\leq|x_1|+|x_2|+|x_3|+|x_4|+|x_5| \\leq 3$''的元素个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459074,7 +461875,9 @@ "id": "017840", "content": "某班级要从$4$名男生、 $2$名女生中选派$4$人参加某项社区服务, 如果要求至少有$1$名女生, 那么不同的选派方案种数为\\bracket{20}.\n\\fourch{$14$}{$24$}{$28$}{$48$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459094,7 +461897,9 @@ "id": "017841", "content": "$(1+\\dfrac{1}{x^2})(1+x)^6$的展开式中含$x^2$的项的系数为\\bracket{20}.\n\\fourch{$15$}{$20$}{$30$}{$35$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459114,7 +461919,9 @@ "id": "017842", "content": "使$(3 x+\\dfrac{1}{x \\sqrt{x}})^n$($n \\in \\mathbf{N}$, $n \\geq 1$)的展开式中含有常数项的最小的$n$为\\bracket{20}.\n\\fourch{$4$}{$5$}{$6$}{$7$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459134,7 +461941,9 @@ "id": "017843", "content": "设$(\\dfrac{\\sqrt{2}}{2}+x)^{2 n}=a_0+a_1 x+a_2 x^2+\\cdots+a_{2 n-1} x^{2 n-1}+a_{2 n} x^{2 n}$, 则$\\displaystyle\\lim_{n\\to\\infty}[(a_0+a_2+a_4+\\cdots+a_{2 n})^2-(a_1+a_3+a_5+\\cdots+a_{2 n-1})^2]=$\\bracket{20}.\n\\fourch{$-1$}{$0$}{$1$}{$\\dfrac{\\sqrt{2}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459154,7 +461963,9 @@ "id": "017844", "content": "从$5$名男生、 $3$名女生中选$5$名担任$5$门不同学科的课代表, 求符合下列条件的不同选取方法数.\\\\\n(1) $5$门课代表中必须有女生;\\\\\n(2) 英语课代表由女生担任.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459174,7 +461985,9 @@ "id": "017845", "content": "若$(x+2)^n=x^n+\\cdots+a x^3+b x^2+c x+d$($n \\in \\mathbf{N}$, $n \\geq 3$), 且$c: d=4: 1$, 求$a: b$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459194,7 +462007,9 @@ "id": "017846", "content": "已知函数$f(x)=\\sqrt{2} \\sin \\dfrac{\\pi}{3} x$, $A=\\{1,2,3,4,5, \\cdots, 10\\}$, 现从$A$中随机取两个不同数$a$、$b$, 求满足$f(a) \\cdot f(b)=0$的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459214,7 +462029,9 @@ "id": "017847", "content": "若$m, n \\in\\{x | x=a_2 \\times 10^2+a_1 \\times 10+a_0, \\ a_i \\in\\{1,2,3,4,5,6,7\\}(i=0,1,2)\\}$, 且$m+n=636$, 求实数对$(m, n)$表示平面上不同点的个数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459234,7 +462051,9 @@ "id": "017848", "content": "将一枚质地均匀的硬币抛掷$2$次, 设事件$A$为``第一次出现正面'', 事件$B$为``第二次出现正面''. 求$P(A | B)$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459254,7 +462073,9 @@ "id": "017849", "content": "一个袋子里装有大小与质地相同的$8$个红球、$6$个白球, 甲、乙两人依次随机不放回地摸出$1$个球, 在甲摸到红球的条件下, 乙摸到白球的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459274,7 +462095,9 @@ "id": "017850", "content": "郑一颗骰子并观察出现的点数. 已知出现的点数不超过$3$, 求出现的点数是偶数的概率\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459294,7 +462117,9 @@ "id": "017851", "content": "某超市仓库中的家用空调的$60 \\%$来自 A 公司, $40 \\%$来自 B 公司, A、B 两个公司的一等品率分别是$80 \\%$和$90 \\%$. 现从仓库中任提取一台空调, 则它是一等品的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459314,7 +462139,9 @@ "id": "017852", "content": "一个不透明的袋中装有分别写上数字$1$、$2$、$3$、$4$、$5$、$6$的$6$张卡片, 现随机取出$2$张卡片, 用$X$表示卡片上数字较大的数, 则$X$的分布为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459334,7 +462161,9 @@ "id": "017853", "content": "某手机芯片生产厂, 第一车间的次品率为$0.15$, 第二车间的次品率为$0.12$, 两个车间生产的芯片都混合放在一个仓库里, 假设第一、二车间生产的芯片数量比例为$2: 3$, 今有一客户从手机芯片仓库中随机买一个芯片, 则该芯片是合格品的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459354,7 +462183,9 @@ "id": "017854", "content": "已知随机变量$X \\sim B(n, p)$, $E[X]=120$, $D[X]=48$, 则$n=$\\blank{50}, $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459374,7 +462205,9 @@ "id": "017855", "content": "将大小质地完全相同的球分别装人三个盒子, 每盒$10$个. 其中, 第一个盒子中有$7$个球标有字母$A$, $3$个球标有字母$B$; 第二个盒子中有红球和白球各$5$个; 第三个盒子中有红球$8$个, 白球$2$个. 试验按如下规则进行: 先在第一个盒子中任取一个球, 若取得标有字母$A$的球, 则在第二个盒子中任取一个球; 若在第一个盒子中取得标有字母$B$的球, 则在第三个盒子中任取一个球. 如果第二次取出的是红球, 那么试验成功, 则试验成功的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459394,7 +462227,9 @@ "id": "017856", "content": "已知随机事件$A$、$B$满足$P(B | A)=\\dfrac{1}{2}$, $P(A \\cap B)=\\dfrac{3}{8}$, 则$P(A)=$\\bracket{20}.\n\\fourch{$\\dfrac{3}{16}$}{$\\dfrac{13}{16}$}{$\\dfrac{3}{4}$}{$\\dfrac{1}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459414,7 +462249,9 @@ "id": "017857", "content": "小东步行上学途中要经过两个交通路口, 第一个路口遇见红灯的概率为$0.5$, 两个路口连续遇到红灯的概率为$0.35$, 则甲在第一个路口遇到红灯的条件下, 第二个路口遇到红灯的概率为\\bracket{20}.\n\\fourch{$0.6$}{$0.7$}{$0.8$}{$0.9$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459434,7 +462271,9 @@ "id": "017858", "content": "一个不透明的袋中装有除颜色外完全相同的$5$个红球和$2$个白球, 若采用不放回地依次取$2$个小球, 则在第$1$次取到红球的条件下, 第$2$次取到红球的概率是\\bracket{20}.\n\\fourch{$\\dfrac{3}{5}$}{$\\dfrac{3}{10}$}{$\\dfrac{2}{3}$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459454,7 +462293,9 @@ "id": "017859", "content": "某射击选手每次射击击中目标的概率是$0.8$, 这名选手在$10$次独立的射击中, 恰有$8$次击中目标的概率为\\bracket{20}.\n\\fourch{$\\mathrm{C}_{10}^8 \\times 0.8^8 \\times 0.2^2$}{$0.8^8 \\times 0.2^2$}{$\\mathrm{C}_{10}^2 \\times 0.2^8 \\times 0.8^2$}{$0.2^8 \\times 0.8^2$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459474,7 +462315,9 @@ "id": "017860", "content": "已知随机变量$X \\sim B(6, \\dfrac{1}{3})$, 则$P(X=2)=$\\bracket{20}.\n\\fourch{$\\dfrac{3}{16}$}{$\\dfrac{4}{243}$}{$\\dfrac{13}{243}$}{$\\dfrac{80}{243}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459494,7 +462337,9 @@ "id": "017861", "content": "将编号为$1$、$2$、$3$、$4$的球随机放入编号为$1$、$2$、$3$、$4$的四个盒子里, 当球的编号与盒子的编号相同时称为一个配对. 用$X$表示配对数, 求$E[X]$、$D[X]$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459514,7 +462359,9 @@ "id": "017862", "content": "已知甲袋子中装有$7$张分别写上数字$1$、$2$、$3$、$4$、$5$、$6$、$7$的卡片, 乙袋子中装有$6$张分别写上数字$8$、$9$、$10$、$11$、$12$、$14$的卡片. 现从甲袋中任取$2$张卡片放人乙袋中, 再从乙袋中任取$2$张卡片, 求从乙袋中取出的$2$张卡片上的数字都是奇数的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459534,7 +462381,9 @@ "id": "017863", "content": "一个不透明的袋中装有$6$个黄色、 $4$个白色的乒乓球(只有颜色不同, 大小质地完全相同), 不放回抽取, 每次任取一球, 取两次, 求:\\\\\n(1) 第二次才取到黄球的概率;\\\\\n(2) 取出的两个球的其中之一是黄球时, 另一个也是黄球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459554,7 +462403,9 @@ "id": "017864", "content": "某蜂蜜生产公司的瓶装蜂蜜标识质量是$500 \\text{g}$, 已知每瓶蜂蜜的实际质量服从$\\mu=500$、$\\sigma^2=1.5^2$的正态分布. 该公司董事长承诺: 有$99 \\%$的把握保证顾客随意买一瓶蜂蜜其质量误差不超过$4.5 \\text{g}$, 试问董事长的话是否有道理?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459574,7 +462425,9 @@ "id": "017865", "content": "函数$y=\\lg (x-3)+\\dfrac{(x-2)^0}{x+1}$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459594,7 +462447,9 @@ "id": "017866", "content": "已知集合$A=\\{1,2,3\\}$, $B=\\{1, m\\}$, 若$3-m \\in A$, 则非零实数$m$的数值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459614,7 +462469,9 @@ "id": "017867", "content": "设函数$y=\\dfrac{(x+1)(x+a)}{x}$为奇函数, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459634,7 +462491,9 @@ "id": "017868", "content": "用反证法证明命题``已知$x, y \\in (0,+\\infty)$, 且$x+y>2$, 求证: $\\dfrac{1+x}{y}$与$\\dfrac{1+y}{x}$中至少有一个小于$2$''时, 应首先假设\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459654,7 +462513,9 @@ "id": "017869", "content": "已知集合$A=\\{x | x^2-16 \\leq 0,\\ x \\in \\mathbf{R}\\}$, $B=\\{x|| x-3 | \\leq a,\\ x \\in \\mathbf{R}\\}$, 若$B \\subseteq A$, 则正实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459674,7 +462535,9 @@ "id": "017870", "content": "已知幂函数$f(x)=x^{(m^2+m)^{-1}}$($m \\in \\mathbf{N}$, $m \\geq 1$), 经过点$(2, \\sqrt{2})$, 则满足条件$f(2-a)>f(a-1)$的实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459694,7 +462557,9 @@ "id": "017871", "content": "已知$y=f(x)$在$\\mathbf{R}$上是严格减函数, 则满足$f(|\\dfrac{1}{x}|)0$, 且$a \\neq 1$. 若$P \\cap Q$只有一个子集, 则$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459774,7 +462645,9 @@ "id": "017875", "content": "设$\\max \\{a, b\\}$表示实数$a, b$中的较大者, 则函数$f(x)=\\max \\{|x+1|,|x-2|\\}$($x \\in \\mathbf{R}$)的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459794,7 +462667,9 @@ "id": "017876", "content": "已知函数$y=f(x)$的定义域为$\\mathbf{R}$, $f(1)=3$, 对任意两个不等的实数$a$、$b$都有$\\dfrac{f(a)-f(b)}{a-b}>1$, 则不等式$f(2^x-1)<2^x+1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459814,7 +462689,9 @@ "id": "017877", "content": "已知$a \\in \\mathbf{R}$, 若存在定义域为$\\mathbf{R}$的函数$f(x)$同时满足下列两个条件, \\textcircled{1} 对任意$x_0 \\in \\mathbf{R}$, $f(x_0)$的值为$x_0$或$x_0^2$; \\textcircled{2} 关于$x$的方程$f(x)=a$无实数解; 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459834,7 +462711,9 @@ "id": "017878", "content": "已知函数$y=f(x)$是定义域为$\\mathbf{R}$的偶函数, 且当$x \\geq 0$时, $f(x)=\\begin{cases}(\\dfrac{1}{2})^x, & 0 \\leq x<2, \\\\ \\log _{16} x, & x \\geq 2,\\end{cases}$若关于$x$的方程$[f(x)]^2+a \\cdot f(x)+b=0$($a$、$b \\in \\mathbf{R}$)有且仅有$7$个不同实数根, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -459854,7 +462733,9 @@ "id": "017879", "content": "已知函数$D(x)=\\begin{cases} 1,& x \\in \\mathbf{Q}, \\\\ 0, & x \\notin \\mathbf{Q}, \\end{cases}$则下列结论中错误的是\\bracket{20}.\n\\twoch{函数$D(x)$的值域为$\\{0,1\\}$}{函数$D(x)$是偶函数}{函数$D(x)$不是周期函数}{函数$D(x)$不是单调函数}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459874,7 +462755,9 @@ "id": "017880", "content": "已知集合$M$、$P$都是非空集合, 若命题``$M$中的元素都是$P$中的元素''是假命题, 则下列说法必定为真命题的是\\bracket{20}.\n\\twoch{$M \\cap P=\\varnothing$}{$M$中至多有一个元素不属于$P$}{$P$中有不属于$M$的元素}{$M$中有不属于$P$的元素}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459894,7 +462777,9 @@ "id": "017881", "content": "已知$f(x)$是定义在$[a, b]$上的函数, 如果存在常数$M>0$, 对区间$[a, b]$的任意划分: $a=x_00$恒成立; 命题$q_2: f(x)$单调增, 存在$x_0<0$使得$f(x_0)=0$; 则下列说法正确的是\\bracket{20}.\n\\twoch{$q_1$、$q_2$都是$p$的充分条件}{只有$q_1$是$p$的充分条件}{只有$q_2$是$p$的充分条件}{$q_1$、$q_2$都不是$p$的充分条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -459934,7 +462821,9 @@ "id": "017883", "content": "已知二次函数$f(x)=a x^2+b x$, 对任意$x \\in \\mathbf{R}$均有$f(x-4)=f(2-x)$成立, 且函数$f(x)$的图像过点$A(1, \\dfrac{3}{2})$.\\\\\n(1) 求函数$y=f(x)$的表达式;\\\\\n(2) 若不等式$f(x-t) \\leq x$的解集为$[4, m]$, 求实数$t$、$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459954,7 +462843,9 @@ "id": "017884", "content": "已知函数$f(x), g(x)$满足关系$g(x)=f(x) \\cdot f(x+\\alpha)$, 其中$\\alpha$是常数.\\\\\n(1) 若$f(x)=\\cos x+\\sin x$, 且$\\alpha=\\dfrac{\\pi}{2}$, 求$g(x)$的解析式, 并写出$g(x)$的增区间;\\\\\n(2) 设$f(x)=2^x+\\dfrac{1}{2^x}$, 若$g(x)$的最小值为$6$, 求常数$\\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459974,7 +462865,9 @@ "id": "017885", "content": "已知函数$y=f(x)$, 其中$f(x)=\\dfrac{a x^2+b x+c}{x+d}$($a$、$b$、$c$、$d \\in \\mathbf{R}$, $x \\neq-d$).\\\\\n(1) 若$a=0$, 函数$f(x)$的图像关于点$(-1,3)$成中心对称, 求$b$、$d$的值;\\\\\n(2) 若$f(x)$满足条件 (1), 且对任意$x_0 \\in[3,10]$, 总有$f(x_0) \\in[3,10]$, 求$c$的取值范围;\\\\\n(3) 若$b=0$, 函数$f(x)$是奇函数, $f(1)=0$, $f(-2)=-\\dfrac{3}{2}$, 且对任意, $x \\in[1,+\\infty)$时, 不等式$f(m x)+m f(x)<0$恒成立, 求负实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -459994,7 +462887,9 @@ "id": "017886", "content": "已知函数$y=f(x)$, 其中$f(x)=m(x-2 m) \\cdot(x+m+3)$, $g(x)=2^x-2$, 若同时满足:\\\\\n(1) 对任意$x \\in \\mathbf{R}$, 恒有$f(x)<0$或$g(x)<0$成立;\\\\\n(2) 存在$x \\in(-\\infty,-4)$, 使得$f(x) \\cdot g(x)<0$, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460014,7 +462909,9 @@ "id": "017887", "content": "定义: 若对定义域为$\\mathbf{R}$的函数$y=f(x)$, 总存在数对$(k, m)$($k$、$m$是常数, 且$m \\neq 0$)对$x \\in \\mathbf{R}$满足$f(x+k)=m \\cdot f(k-x)$, 则说函数$y=f(x)$存在理想数对$(k, m)$.\\\\\n(1) 若$f(x)=2^x$, 试判断函数$f(x)$是否存在理想数对, 并说明理由;\\\\\n(2) 若函数$f(x)=a x+b$($a \\neq 0$), 证明: 函数$f(x)$总存在理想数对;\\\\\n(3) 若函数$f(x)=x^2+a x+b$($a \\neq 0$)存在理想数对$(1, m)$, 且对$x \\in[2,+\\infty)$, $f(x)+1 \\geq f(b)$恒成立, 求实数$b$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460034,7 +462931,9 @@ "id": "017888", "content": "若半径为$1$的圆上一段圆弧所对的弦长为$1$, 则该弦所对应的弧长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460054,7 +462953,9 @@ "id": "017889", "content": "正三角形$ABC$的边长为$1, G$是其重心, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AG}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460074,7 +462975,9 @@ "id": "017890", "content": "若$\\tan \\dfrac{\\alpha}{2}=2$, 则$\\sin (\\alpha+\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460094,7 +462997,10 @@ "id": "017891", "content": "已知函数$f(x)=a \\sin x+b \\cos x$($x \\in[a^2-2, a]$)是奇函数, 则$a+b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460114,7 +463020,9 @@ "id": "017892", "content": "在$\\triangle ABC$中, $AC=3$, $3 \\sin A=2 \\sin B$, 且$\\cos C=\\dfrac{1}{4}$, 则$AB=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460134,7 +463042,9 @@ "id": "017893", "content": "函数$y=\\dfrac{1}{2} \\sin (2 x-\\dfrac{\\pi}{3})$的减区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460154,7 +463064,10 @@ "id": "017894", "content": "已知向量$\\overrightarrow {a}=(\\sin x, \\cos x)$, $\\overrightarrow {b}=(\\sin x, \\sin x)$, 则函数$f(x)=\\overrightarrow {a} \\cdot \\overrightarrow {b}$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460174,7 +463087,9 @@ "id": "017895", "content": "已知钝角$\\alpha$的终边经过点$P(\\sin 2 \\theta, \\sin 4 \\theta)$, 且$\\cos \\theta=0.5$, 则$\\alpha$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460194,7 +463109,9 @@ "id": "017896", "content": "在直角坐标系$x o y$中, 已知三点$A(a, 1), B(2, b), C(3,4)$, 若向量$\\overrightarrow{OA}, \\overrightarrow{OB}$在向量$\\overrightarrow{OC}$方向上的投影相同, 则$3 a-4 b$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460214,7 +463131,9 @@ "id": "017897", "content": "方程$|\\sin \\dfrac{\\pi x}{2}|=\\sqrt{x}-1$的实数解的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460234,7 +463153,9 @@ "id": "017898", "content": "在椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$上任意一点$P, Q$与$P$关于$x$轴对称, 且$F_1$、$F_2$是其左右焦点, 若有$\\overrightarrow{F_1P} \\cdot \\overrightarrow{F_2P} \\leq 1$, 则$\\overrightarrow{F_1P}$与$\\overrightarrow{F_2Q}$的夹角范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460254,7 +463175,9 @@ "id": "017899", "content": "已知$O$是$\\triangle ABC$的外心, 且外接圆半径为$2$, $AB=2$, $AC=3$, 若$\\overrightarrow{AO}=\\dfrac{1}{2} \\overrightarrow{AB}+\\dfrac{1}{2} \\overrightarrow{AC}$, 则$\\cos \\angle BAC=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460274,7 +463197,9 @@ "id": "017900", "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$是平面内两个互相垂直的单位向量, 若向量$\\overrightarrow {c}$满足$(\\overrightarrow {c}-\\overrightarrow {a}) \\cdot(\\overrightarrow {c}-\\overrightarrow {b})=0$, 则$|\\overrightarrow {c}|$的最大值是\\bracket{20}.\n\\fourch{$1$}{$2$}{$\\sqrt{2}$}{$\\dfrac{\\sqrt{2}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -460294,7 +463219,9 @@ "id": "017901", "content": "在$\\triangle ABC$中, ``$\\cos A=2 \\sin B \\sin C$''是``$\\triangle ABC$为等腰三角形''的\\bracket{20}.\n\\twoch{必要非充分条件}{充分非必要条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -460314,7 +463241,9 @@ "id": "017902", "content": "设函数$f(x)=\\sin ^2 x+b \\sin x+c$, 则$f(x)$的最小正周期\\bracket{20}.\n\\twoch{与$b$有关, 且与$c$有关}{与$b$无关, 但与$c$有关}{与$b$无关, 且与$c$无关}{与$b$有关, 但与$c$无关}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -460334,7 +463263,9 @@ "id": "017903", "content": "已知在$\\triangle ABC$中, $P_0$是边$AB$上的一个定点, 满足$\\overrightarrow{P_0B}=\\dfrac{1}{4} \\overrightarrow{AB}$, 且对于边$AB$上任意一点$P$, 恒有$\\overrightarrow{PB} \\cdot \\overrightarrow{PC} \\geq \\overrightarrow{P_0B} \\cdot \\overrightarrow{P_0C}$, 则\\bracket{20}.\n\\fourch{$B=\\dfrac{\\pi}{2}$}{$A=\\dfrac{\\pi}{2}$}{$AB=AC$}{$AC=BC$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -460354,7 +463285,10 @@ "id": "017904", "content": "在锐角$\\triangle ABC$中, $\\sin A=\\sin ^2B+\\sin (\\dfrac{\\pi}{4}+B) \\sin (\\dfrac{\\pi}{4}-B)$.\\\\\n(1) 求角$A$的值;\\\\\n(2) 若$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=12$, 求$\\triangle ABC$的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460374,7 +463308,9 @@ "id": "017905", "content": "已知$\\triangle ABC$的外接圆半径为$1$, 圆心为$O$, 且$3 \\overrightarrow{OA}+4 \\overrightarrow{OB}+5 \\overrightarrow{OC}=\\overrightarrow{0}$, 求$\\triangle ABC$的面积.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460394,7 +463330,9 @@ "id": "017906", "content": "已知函数$f(x)=\\sin 2 x$, $g(x)=\\cos (2 x+\\dfrac{\\pi}{6})$, 直线$x=t$($t \\in \\mathbf{R}$)与函数$f(x)$、$g(x)$的图像分别交于$M$、$N$两点.\\, \\\n(1) 当$t=\\dfrac{\\pi}{4}$时, 求$|MN|$的值;\\\\\n(2) 求$| MN|$在$t \\in[0, \\dfrac{\\pi}{2}]$时的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460414,7 +463352,10 @@ "id": "017907", "content": "设$O$为坐标原点, 动点$M$在椭圆$C: \\dfrac{x^2}{2}+y^2=1$上, 过$M$作$x$轴的垂线, 垂足为$N$, 点$P$满足$\\overrightarrow{NP}=\\sqrt{2} \\cdot \\overrightarrow{NM}$.\\\\\n(1) 求动点$P$的轨迹方程;\\\\\n(2) 设点$Q$在直线$x=-3$上, 且$\\overrightarrow{OP} \\cdot \\overrightarrow{PQ}=1$. 证明: 过点$P$且垂直于$OQ$的直线$l$过曲线$C$的左焦点$F$.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460434,7 +463375,10 @@ "id": "017908", "content": "已知虚数$z_1=\\cos \\alpha+\\mathrm{i} \\sin \\alpha$, $z_2=\\cos \\beta+\\mathrm{i} \\sin \\beta$.\\\\\n(1) 若$|z_1-z_2|=\\dfrac{2}{5} \\sqrt{5}$, 求$\\cos (\\alpha-\\beta)$的值;\\\\\n(2) 若$z_1, z_2$是方程$3 x^2-2 x+c=0$的两个根, 求实数$c$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460454,7 +463398,9 @@ "id": "017909", "content": "如图, 一个半圆和长方形组成的铁皮, 长方形的边$AD$为半圆的直径, $O$为半圆的圆心, $AB=1$, $BC=2$, 现要将此铁皮剪出一个等腰三角形$PMN$, 其底边$MN \\perp BC$, 点$P$在边$AB$上, 设$\\angle MOD=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (O) --++ (-1,0) node [left] {$A$} coordinate (A);\n\\draw (O) --++ (1,0) node [right] {$D$} coordinate (D);\n\\draw (A) --++ (0,-1) node [below] {$B$} coordinate (B) --++ (2,0) node [below] {$C$} coordinate (C) -- (D) arc (0:180:1);\n\\draw (50:1) node [above] {$M$} coordinate (M);\n\\draw ($(B)!(M)!(C)$) node [below] {$N$} coordinate (N);\n\\draw ($(A)!($(M)!0.5!(N)$)!(B)$) node [below left] {$P$} coordinate (P);\n\\draw (P)--(M)--(N)--cycle(O)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\theta=30^{\\circ}$, 求三角形铁皮$PMN$的面积;\\\\\n(2) 求剪下的三角形铁皮$PMN$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460474,7 +463420,9 @@ "id": "017910", "content": "数列$\\{a_n\\}$的前$n$项和$S_n=2 n^2-n+2$, 则$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460494,7 +463442,9 @@ "id": "017911", "content": "在等比数列$\\{a_n\\}$中, $a_2=8$, $a_5=64$, 则公比$q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460514,7 +463464,9 @@ "id": "017912", "content": "若等差数列$\\{a_n\\}$的前三项和$S_3=9$, 且$a_1=1$, 则$a_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460534,7 +463486,9 @@ "id": "017913", "content": "若等差数列$\\{a_n\\}$中, $S_{10}-S_5=40$, 则$a_8=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460554,7 +463508,9 @@ "id": "017914", "content": "已知$\\{a_n\\}$是等差数列, $a_1+a_2=4$, $a_7+a_8=28$, 则该数列前$10$项和$S_{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460574,7 +463530,9 @@ "id": "017915", "content": "设$S_n$表示等比数列$\\{a_n\\}$($n \\in \\mathbf{N}$, $n \\geq 1$)的前$n$项和, 已知$\\dfrac{S_{10}}{S_5}=3$, 则$\\dfrac{S_{15}}{S_5}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460594,7 +463552,9 @@ "id": "017916", "content": "数列$\\{a_n\\}$是严格增数列, 且对任意$n \\in \\mathbf{N}$, $n \\geq 1$, 都有$a_n=n^2-\\lambda n$, 则实数$\\lambda$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460614,7 +463574,9 @@ "id": "017917", "content": "设$a_n=\\begin{cases}2^{n-1}, & 1 \\leq n \\leq 2, \\ n \\in \\mathbf{N}, \\\\ \\dfrac{1}{3^n}, & n \\geq 3, \\ n \\in \\mathbf{N}.\\end{cases}$数列$\\{a_n\\}$的前$n$项和为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty} S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460634,7 +463596,9 @@ "id": "017918", "content": "数列$\\{a_n\\}$满足$\\sum_{i=1}^n 2^{i-1} a_i=5-3 n$, 则$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460654,7 +463618,9 @@ "id": "017919", "content": "观察下列等式:\n\\begin{align*}\n1& =1 \\\\\n1-4&=-3=-(1+2) \\\\\n1-4+9&=6=(1+2+3) \\\\\n1-4+9-16&=-10=-(1+2+3+4)\n\\end{align*}\n写出一个更一般的等式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460674,7 +463640,9 @@ "id": "017920", "content": "在数列$\\{a_n\\}$中, $a_n=4 n-\\dfrac{5}{2}$, $\\displaystyle\\sum_{i=1}^n a_i=a n^2+b n$, $n$为正整数, 其中$a, b$为常数, 则$a b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460694,7 +463662,9 @@ "id": "017921", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=(-1)^n \\cdot n+2^n$,$ n \\in \\mathbf{N}$, $n \\geq 1$, 则这个数列的前$2 n$项和$S_{2 n}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460714,7 +463684,9 @@ "id": "017922", "content": "下面四个判断中, 正确的是\\bracket{20}.\n\\onech{式子$1+k+k^2+\\cdots+k^n$($n \\in \\mathbf{N}$, $n \\geq 1$), 当$n=1$时恒为$1$}{式子$1+k+k^2+\\cdots+k^{n-1}$($n \\in \\mathbf{N}$, $n \\geq 1$), 当$n=1$时恒为$1-k$}{式子$\\dfrac{1}{1}+\\dfrac{1}{2}+\\dfrac{1}{3}+\\cdots+\\dfrac{1}{2 n+1}$($n \\in \\mathbf{N}$, $n \\geq 1$), 当$n=1$时恒为$1+\\dfrac{1}{2}+\\dfrac{1}{3}$}{设$f(n)=\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\dfrac{1}{3 n+1}$($n \\in \\mathbf{N}$, $n \\geq 1$), 则$f(k+1)=f(k)+\\dfrac{1}{3 k+2}+\\dfrac{1}{3 k+3}+\\dfrac{1}{3 k+4}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -460734,7 +463706,9 @@ "id": "017923", "content": "在数列$\\{a_n\\}$中, $a_1=2$, $a_{n+1}=a_n+\\ln (1+\\dfrac{1}{n})$, 则$a_n=$\\bracket{20}.\n\\fourch{$2+\\ln n$}{$2+(n-1) \\ln n$}{$2+n \\ln n$}{$1+n+\\ln n$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -460754,7 +463728,9 @@ "id": "017924", "content": "农民收人由工资性收人和其它收人两部分构成$2003$年某地区农民人均收人为$3150$元 (其中工资性收人为$1800$元, 其它收人为$1350$元), 预计该地区自$2004$年起的$5$年内, 农民的工资性收人将以每年$6 \\%$的年增长率增长, 其它收人每年增加$160$元. 根据以上数据, 2008 年该地区农民人均收人介于\\bracket{20}.\n\\fourch{$4200$元至$4400$元}{$4400$元至$4600$元}{$4600$元至$4800$元}{$4800$元至$5000$元}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -460774,7 +463750,9 @@ "id": "017925", "content": "已知数列$\\{a_n\\}$是等比数列, 给出下列六个数列: \\textcircled{1} $\\{k a_n\\}(k \\neq 0)$; \\textcircled{2} $\\{a_{2 n-1}\\}$; \\textcircled{3} $\\{a_{n+1}-a_n\\}$; \\textcircled{4} $\\{a_n a_{n+1}\\}$; \\textcircled{5} $\\{n a_n\\}$; \\textcircled{6} $\\{a_n^3\\}$, 其中仍然构成等比数列的个数为\\bracket{20}.\n\\fourch{$4$}{$5$}{$6$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -460794,7 +463772,9 @@ "id": "017926", "content": "已知数列$\\{x_n\\}$的首项$x_1=3$, 通项公式$x_n=2^n p+n q(n \\in \\mathbf{N}, n \\geq 1, p, q$为常数), 且$x_1, x_4, x_5$成等差数列, 求:\\\\\n(1) $p, q$的值;\\\\\n(2) 数列$\\{x_n\\}$的前$n$项的和$S_n$的公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460814,7 +463794,9 @@ "id": "017927", "content": "等比数列$\\{a_n\\}$首项为$1$, 公比为$q$, 前$n$项和是$T_n$, 此数列的各项倒数组成一个新数列$\\{\\dfrac{1}{a_n}\\}$, 求新数列的前$n$项和$S_n$, 且用$T_n$表示$S_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460834,7 +463816,9 @@ "id": "017928", "content": "设$x$轴上有一点列: $P_0(x_0, 0)$, $P_1(x_1, 0)$, $P_2(x_2, 0)$ $\\cdots$, 且$\\overrightarrow{P_n P_{n+2}}=\\lambda \\overrightarrow{P_{n+2} P_{n+1}}$, 其中$n \\in \\mathbf{N}$, $\\lambda>0$, $x_0=0$, $x_1=1$.\\\\\n(1) 设$a_{n+1}=x_{n+1}-x_n$, 求证: 数列$\\{a_{n+1}\\}$是等比数列;\\\\\n(2) 求$\\displaystyle\\lim_{n\\to\\infty} x_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460854,7 +463838,9 @@ "id": "017929", "content": "等比数列$\\{a_n\\}$, $a_1=1000$, $q=\\dfrac{1}{10}$, 数列$\\{b_n\\}$满足$b_k=\\dfrac{1}{k}(\\lg a_1+\\lg a_2+\\cdots+\\lg a_k)$($k \\in \\mathbf{N}$, $k \\geq 1$).\\\\\n(1) 求数列$\\{b_n\\}$的前$n$项和的最大值;\\\\\n(2) 求数列$\\{|b_n|\\}$的前$n$项和$S'_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460874,7 +463860,9 @@ "id": "017930", "content": "已知有限数列$\\{a_n\\}$, 若满足$|a_1-a_2| \\leq|a_1-a_3| \\leq \\cdots \\leq|a_1-a_m|$, $m$是项数, 则称$\\{a_n\\}$满足性质$P$.\\\\\n(1) 判断数列$3$、$2$、$5$、$1$和$4$、$3$、$2$、$5$、$1$是否具有性质$P$, 请说明理由;\\\\\n(2) 若$a_1=1$, 公比为$q$的等比数列, 项数为$10$, 具有性质$P$, 求$q$的取值范围;\\\\\n(3) 若$\\{a_n\\}$是$1,2, \\cdots, m$的一个排列($m \\geq 4$), $\\{b_n\\}$符合$b_k=a_{k+1}$($k=1,2, \\cdots, m-1$), $\\{a_n\\}$, $\\{b_n\\}$都具有性质$P$, 求所有满足条件的$\\{a_n\\}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460894,7 +463882,9 @@ "id": "017931", "content": "已知数列$\\{a_n\\}$中, $a_1=a$($a \\in \\mathbf{R}$, $a \\neq-\\dfrac{1}{2}$), $a_n=2 a_{n-1}+\\dfrac{1}{n}+\\dfrac{1}{n(n+1)}$($n \\geq 2$, $n \\in \\mathbf{N})$. 又数列$\\{b_n\\}$满足: $b_n=a_n+\\dfrac{1}{n+1}$($n \\in \\mathbf{N}$, $n \\geq 1$).\\\\\n(1) 求证: 数列$\\{b_n\\}$是等比数列;\\\\\n(2) 若数列$\\{a_n\\}$是单调增数列, 求实数$a$的取值范围;\\\\\n(3) 若数列$\\{b_n\\}$的各项皆为正数, $c_n=\\log _{\\frac{1}{2}} b_n$, 设$T_n$是数列$\\{c_n\\}$的前$n$和, 问: 是否存在整数$a$, 使得数列$\\{T_n\\}$是单调减数列? 若存在, 求出整数$a$; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -460914,7 +463904,9 @@ "id": "017932", "content": "过直线$3 x+2 y-7=0$与直线$4 x-y-2=0$的交点, 并且与直线$2 x-y+3=0$平行的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460934,7 +463926,9 @@ "id": "017933", "content": "已知两直线$l_1: x+y \\sin \\theta-1=0$和$l_2: 2 x \\sin \\theta+y+1=0$, 当$l_1 \\perp l_2$时, $\\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460954,7 +463948,9 @@ "id": "017934", "content": "若过点$A(1,0)$, 且与$y$轴的夹角为$45^{\\circ}$的直线与圆$x^2+y^2=4$交于两点$P$、$Q$, 则$|PQ|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460974,7 +463970,9 @@ "id": "017935", "content": "已知圆$C: x^2+y^2=4$, 过点$P(2,5)$作直线与圆相切, 则切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -460994,7 +463992,9 @@ "id": "017936", "content": "直线$l: 2 x-y-4=0$绕它与$x$轴的交点逆时针旋转$\\dfrac{\\pi}{4}$后, 所得的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461014,7 +464014,9 @@ "id": "017937", "content": "已知$F_1$、$F_2$是椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$的两个焦点, $P$是椭圆上任意一点, 则$|PF_1| \\cdot|PF_2|$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461034,7 +464036,9 @@ "id": "017938", "content": "设$\\theta \\in[0,2 \\pi)$, 若圆$(x-\\cos \\theta)^2+(y-\\sin \\theta)^2=r^2$($r>0$)与直线$2 x-y-10=0$有交点, 则$r$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461054,7 +464058,9 @@ "id": "017939", "content": "直线$l$的倾角为$45^{\\circ}$, 它与已知圆$C: x^2+y^2=16$相交于$A$、$B$两点, 若弦$AB$的长为$\\sqrt{46}$, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461074,7 +464080,9 @@ "id": "017940", "content": "动点$P$在直线$x+y=0$上运动, 过点$P$作圆$x^2+y^2+4 x+4 y+7=0$的切线, 则切线长的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461094,7 +464102,9 @@ "id": "017941", "content": "如果实数$x$、$y$满足等式$(x-2)^2+y^2=3$, 那么$\\dfrac{y}{x}$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461114,7 +464124,9 @@ "id": "017942", "content": "若$x$轴上一点$P$到$A(2,2), B(-1,1)$的距离之差$|PA|-|PB|$最大, 则点$P$坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461134,7 +464146,9 @@ "id": "017943", "content": "有以下$4$个命题: \\textcircled{1} 斜率相等的两直线一定平行; \\textcircled{2} 两直线平行, 则两直线的斜率一定相等; \\textcircled{3} 两直线的斜率之积为$-1$, 则两直线一定互相垂直; \\textcircled{4} 两直线互相垂直, 则两直线的斜率之积等$-1$. 其中假命题的个数是\\bracket{20}.\n\\fourch{1 个}{2 个}{3 个}{4 个}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -461154,7 +464168,9 @@ "id": "017944", "content": "若直线$y=a x+2$和直线$y=3 x+b$关于直线$y=x$对称, 那么\\bracket{20}.\n\\fourch{$a=\\dfrac{1}{3}$, $b=6$}{$a=\\dfrac{1}{3}$, $b=-6$}{$a=3$, $b=-2$}{$a=3$, $b=6$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -461174,7 +464190,9 @@ "id": "017945", "content": "设点$A(2,-3), B(-3,-2)$, 直线$l$过点$P(1,1)$且与线段$AB$相交, 则$l$的斜率$k$的取值范围是\\bracket{20}.\n\\fourch{$k \\geq \\dfrac{3}{4}$或$k \\leq-4$}{$k \\geq \\dfrac{3}{4}$或$k \\leq-\\dfrac{1}{4}$}{$-4 \\leq k \\leq \\dfrac{3}{4}$}{$-\\dfrac{3}{4} \\leq k \\leq 4$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -461194,7 +464212,9 @@ "id": "017946", "content": "圆$x^2+2 x+y^2+4 y-3=0$上到直线$x+y+1=0$的距离为$\\sqrt{2}$的点的个数只有\\bracket{20}.\n\\fourch{1 个}{2 个}{3 个}{4 个}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -461214,7 +464234,9 @@ "id": "017947", "content": "已知直线$l_1$、$l_2$的斜率是方程$6 x^2+x-1=0$的两个根, 求直线$l_1$与直线$l_2$的夹角的大小.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461234,7 +464256,9 @@ "id": "017948", "content": "已知直线$l_1$经过直线$2 x+y-5=0$与直线$3 x-2 y-4=0$的交点, 且和直线$l: x+y+3=0$垂直.\\\\\n(1) 求直线$l_1$的方程;\\\\\n(2) 求直线$l_1$关于点$(1,-1)$的对称的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461254,7 +464278,9 @@ "id": "017949", "content": "设点$P$是圆$x^2+y^2-10 x-10 y+40$上的任意一点, $O$为坐标原点, 求线段$OP$的中点$M$的轨迹方程, 并指出曲线的形状.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461274,7 +464300,9 @@ "id": "017950", "content": "已知圆$C: x^2+y^2+x-6 y+m=0$与直线$l: x+2 y-3=0$相交于两点$M$、$N$, 且$OM \\perp ON$($O$为坐标原点), 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461294,7 +464322,9 @@ "id": "017951", "content": "已知圆$(x+4)^2+y^2=25$的圆心为$M_1$, 圆$(x-4)^2+y^2=1$的圆心角为$M_2$, 一动圆$P$与这两个圆都外切, 求动圆圆心$P$的轨迹.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461314,7 +464344,9 @@ "id": "017952", "content": "已知圆$C: x^2+y^2-2 x-2 y+1=0$, 点$A(2 a, 0)$, $B(0,2 b)$, 其中$a>1$, $b>1$, $O$为坐标原点, 圆$C$与直线$AB$相切.\\\\\n(1) 求线段$AB$的中点$P$的轨迹方程;\\\\\n(2) 求直线$AB$的方程, 使$\\triangle AOB$的面积最小.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461334,7 +464366,9 @@ "id": "017953", "content": "准线方程为$y=1$的抛物线标准方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461354,7 +464388,9 @@ "id": "017954", "content": "若双曲线的渐近线方程$y= \\pm 3 x$, 它的一个焦点是$(\\sqrt{10}, 0)$, 则双曲线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461374,7 +464410,9 @@ "id": "017955", "content": "OBSOLETE椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{2}=1$的焦点为$F_1, F_2$, 点$P$在椭圆上, 若$|PF_1|=4$, 则此椭圆的长轴长的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461395,7 +464433,9 @@ "id": "017956", "content": "若以椭圆上一点和两个焦点为顶点的三角形的最大面积为$1$, 则此椭圆的长轴长的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461415,7 +464455,9 @@ "id": "017957", "content": "平面内到点$(1,2)$的距离和到点$(3,4)$的距离之和等于$2 \\sqrt{2}$的点的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461435,7 +464477,9 @@ "id": "017958", "content": "设双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的右顶点为$A$, 右焦点为$F$, 过点$F$平行于双曲线的一条渐近线的直线与双曲线交于点$B$, 则$\\triangle AFB$的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461455,7 +464499,9 @@ "id": "017959", "content": "与椭圆$\\dfrac{x^2}{40}+\\dfrac{y^2}{15}=1$有公共焦点, 且离心率为$\\dfrac{5}{3}$的双曲线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461475,7 +464521,9 @@ "id": "017960", "content": "已知$A$、$B$是抛物线$x^2=4 y$上的两点, 线段$AB$的中点为$M(2,2)$, 则$AB=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461495,7 +464543,9 @@ "id": "017961", "content": "从双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左焦点$F$引圆$x^2+y^2=a^2$的切线, 切点为$T$, 延长$FT$交双曲线右去于点$P$, 若$M$是线段$FP$的中点, $O$为坐标原点, 则$|MO|-|MT|$的值\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461515,7 +464565,9 @@ "id": "017962", "content": "关于曲线$C: x^4-y^3=1$, 给出下列四个结论:\\\\\n\\textcircled{1} 曲线$C$是双曲线;\\\\\n\\textcircled{2} 关于$y$轴对称;\\\\\n\\textcircled{3} 关于坐标原点中心对称;\\\\\n\\textcircled{4} 与$x$轴所围成封闭图形面积小于$2$.\\\\\n则其中正确结论的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461535,7 +464587,9 @@ "id": "017963", "content": "``$m>n>0$''是``方程$m x^2+n y^2=1$表示焦点在$y$轴上的椭圆''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -461555,7 +464609,9 @@ "id": "017964", "content": "设$P(x, y)$是曲线$C: \\sqrt{\\dfrac{x^2}{25}}+\\sqrt{\\dfrac{y^2}{9}}=1$上的点, $F_1(-4,0)$、$F_2(4,0)$, 则$|PF_1|+|PF_2|$\\bracket{20}.\n\\fourch{小于$10$}{大于$10$}{不大于$10$}{不小于$10$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -461575,7 +464631,9 @@ "id": "017965", "content": "已知点$P(4, m)$是直线$l: \\begin{cases}x=1+3 t, \\\\ y=-5+t,\\end{cases}$($t \\in \\mathbf{R}$, $t$是参数)和圆$C: \\begin{cases}x=1+5 \\cos \\theta, \\\\ y=5 \\sin \\theta,\\end{cases}$($\\theta \\in \\mathbf{R}$, $\\theta$是参数)的公共点, 过点$P$作圆$C$的切线, 则切线方程是\\bracket{20}.\n\\fourch{$3 x-4 y-28=0$}{$3 x+4 y-28=0$}{$3 x-y-13=0$}{$x-3 y-16=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -461595,7 +464653,9 @@ "id": "017966", "content": "已知抛物线的顶点在原点, 焦点在$y$轴上, 抛物线上一点$M(m,-3)$到焦点的距离为$5$, 求$m$的值以及抛物线方程和准线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461615,7 +464675,9 @@ "id": "017967", "content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), $c$是半焦距长, 若$\\dfrac{c}{a}=\\dfrac{\\sqrt{3}}{2}$, 椭圆上任意三个顶点构成的三角形的面积为$\\dfrac{1}{2}$.\\\\\n(1) 求椭圆的方程;\\\\\n(2) 若过$P(\\lambda, 0)$的直线$l$与椭圆交于不同的两点$A$、$B$, 且$\\overrightarrow{AP}=2 \\overrightarrow{PB}$, 求实数$\\lambda$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461635,7 +464697,9 @@ "id": "017968", "content": "已知双曲线$C: \\dfrac{x^2}{4}-y^2=1$, $P$是$C$上的任意一点.\\\\\n(1) 求证: 点$P$到双曲线$C$的两条渐近线的距离的乘积是一个常数;\\\\\n(2) 设点$A$的坐标为$(3,0)$, 求$|PA|$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461655,7 +464719,9 @@ "id": "017969", "content": "设圆$C$与两圆$(x+\\sqrt{5})^2+y^2=4,(x-\\sqrt{5})^2+y^2=4$中的一个内切, 另一个外切.\\\\\n(1) 求圆$C$的圆心轨迹$L$的方程;\\\\\n(2) 已知点$M(\\dfrac{3 \\sqrt{5}}{5}, \\dfrac{4 \\sqrt{5}}{5})$, $F(\\sqrt{5}, 0)$, 且$P$为$L$上动点, 求 $||MP|-| FP||$的最大值及此时点$P$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461675,7 +464741,9 @@ "id": "017970", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右焦点的坐标为$(2,0)$, 且长轴长为短轴长的$\\sqrt{2}$倍. 椭圆$\\Gamma$的上、下顶点分别为$A$、$B$, 经过点$P(0,4)$的直线$l$与椭圆$\\Gamma$相交于$M$、$N$两点 (不同于$A$、$B$两点).\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 若直线$BM \\perp l$, 求点$M$的坐标;\\\\\n(3) 设直线$AN$、$BM$相交于点$Q(m, n)$, 求证: $n$是定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461695,7 +464763,9 @@ "id": "017971", "content": "椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右顶点为$A(a, 0)$, 焦距为$2 c$($c>0$), 左、右焦点分别为$F_1, F_2$, $P(x_0, y_0)$为椭圆$C$上的任一点.\\\\\n(1) 写出向量$\\overrightarrow{PF_1}$、$\\overrightarrow{PF_2}$的坐标(用含$x_0$、$y_0$、$c$的字母表示);\\\\\n(2) 若$\\overrightarrow{PF_1} \\cdot \\overrightarrow{PF_2}$的最大值为$3$, 最小值为$2$, 求$a$、$b$的值;\\\\\n(3) 在满足 (2) 的条件下, 若直线$l: y=k x+m$与椭圆$C$交于$M$、$N$两点$(M$、$N$与椭圆的左右顶点不重合), 且以$MN$为直径的圆经过点$A$, 求证: 直线$l$必经过定点, 并求出定点的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -461715,7 +464785,9 @@ "id": "017972", "content": "若一个球的体积为$4 \\sqrt{3} \\pi$, 则它的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461735,7 +464807,9 @@ "id": "017973", "content": "如图, 棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$分别是棱$AB$、$CC_1$的中点, 则异面直线$D_1M$与$BN$所成角的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$N$} coordinate (N);\n\\draw [dashed] (D_1) -- (M);\n\\draw (B)--(N);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461755,7 +464829,9 @@ "id": "017974", "content": "如果一条直线与两个平行平面中的一个平行, 那么这条直线与另一个平面的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461775,7 +464851,9 @@ "id": "017975", "content": "如图, 在半径为$3$的球面上有$A$、$B$、$C$三点, $\\angle ABC=90^{\\circ}$, $BA=BC$, 球心$O$到平面$ABC$的距离是$\\dfrac{3 \\sqrt{2}}{2}$, 则$B$、$C$两点的球面距离是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\draw (O) circle (3);\n\\draw (O) ++ (3,0) arc (0:-180:3 and 0.75);\n\\draw [dashed] (O) ++ (3,0) arc (0:180:3 and 0.75);\n\\draw (O) ++ (0,{-3/sqrt(2)}) coordinate (O_1);\n\\draw (O_1) ++ ({3/sqrt(2)},0) coordinate (S) arc (0:-180:{3/sqrt(2)} and {3/4/sqrt(2)});\n\\draw [dashed] (O_1) ++ ({3/sqrt(2)},0) coordinate (S) arc (0:180:{3/sqrt(2)} and {3/4/sqrt(2)});\n\\filldraw (O_1) ++ (-60:{3/sqrt(2)} and {3/4/sqrt(2)}) node [above] {$C$} coordinate (C) circle (0.03);\n\\filldraw (O_1) ++ (120:{3/sqrt(2)} and {3/4/sqrt(2)}) node [above] {$A$} coordinate (A) circle (0.03);\n\\filldraw (O_1) ++ (-150:{3/sqrt(2)} and {3/4/sqrt(2)}) node [left] {$B$} coordinate (B) circle (0.03);\n\\draw [dashed] (A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461795,7 +464873,9 @@ "id": "017976", "content": "已知正三棱锥的侧棱长是底面边长的$2$倍, 则侧棱与底面所成角的余弦值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461815,7 +464895,9 @@ "id": "017977", "content": "将函数$y=-\\sqrt{1-x^2}$的图像绕着$y$轴旋转一周所得的几何容器的容积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461835,7 +464917,9 @@ "id": "017978", "content": "设圆锥底面圆周上两点$A$、$B$间的距离为$2$, 圆锥顶点到直线$AB$的距离为$\\sqrt{3}$, $AB$和圆锥的轴的距离为$1$, 则该圆锥的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [right] {$O$} coordinate (O);\n\\draw (O) ++ (0,{sqrt(2)}) node [above] {$S$} coordinate (S);\n\\draw (O) ++ ({-sqrt(2)},0) node [left] {$A$} coordinate (A);\n\\draw (O) ++ (-80:{sqrt(2)} and {sqrt(2)/4}) node [below] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [below left] {$C$} coordinate (C);\n\\draw (A) arc (180:360:{sqrt(2)} and {sqrt(2)/4});\n\\draw [dashed] (A) arc (180:0:{sqrt(2)} and {sqrt(2)/4});\n\\draw (A)--(S)--($(A)!2!(O)$);\n\\draw [dashed] (S)--(O)(A)--(O)--(B)--cycle(O)--(C)(S)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461855,7 +464939,9 @@ "id": "017979", "content": "正三棱锥的一个侧面的面积与底面积之比为$2: 3$, 则这个三棱锥的侧面和底面所成二面角的度数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461875,7 +464961,9 @@ "id": "017980", "content": "若地球表面上北纬$60^{\\circ}$圈上有$A$、$B$两点, 它们的纬度圈上的弧长为$\\dfrac{\\pi}{4} R$, 则$A$、$B$的球面距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461895,7 +464983,9 @@ "id": "017981", "content": "在 Rt$\\triangle ABC$中, $\\angle C=90^{\\circ}$, $\\angle A=60^{\\circ}$, $AC=4$, $PC \\perp$平面$ABC$, $PC=6$, $Q$是$AB$边上动点, 则$PQ$与平面$ABC$所成角的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461915,7 +465005,9 @@ "id": "017982", "content": "如图所示, 一个盛满溶液的玻璃杯, 其形状为一个倒置的圆锥, 现放一个球状物体完全浸没于杯中, 球面与圆锥侧面相切, 且与玻璃杯口所在平面相切, 则溢出溶液的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0) coordinate (O);\n\\draw (O) ++ (60:4) coordinate (B);\n\\draw (O) ++ (120:4) coordinate (A);\n\\draw (O) ++ (0,{4*sqrt(3)/3}) coordinate (T);\n\\draw [dashed] (T) circle ({2*sqrt(3)/3});\n\\draw (O)--(A)(O)--(B)(B) ($(A)!0.5!(B)$) ellipse (2 and 0.5);\n\\draw (O) --++ (0,-1) (O) ++ (-2,-1) --++ (4,0);\n\\draw (O) ++ (-30:0.1) --++ (-30:0.8) (B) ++ (-30:0.1) --++ (-30:0.8);\n\\draw [<->] (O) ++ (-30:0.5) --++ (60:4) node [midway, fill=white, sloped] {$4$};\n\\draw (A) ++ (90:0.3) --++ (90:1) (B) ++ (90:0.3) --++ (90:1);\n\\draw [<->] (A) ++ (90:1) --++ (4,0) node [midway, fill=white, sloped] {$4$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461935,7 +465027,9 @@ "id": "017983", "content": "如图所示, 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$分别是$A_1B_1$和$BB_1$的中点, 那么直线$AM$和$CN$所成角的余弦值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(B_1)$) node [left] {$N$} coordinate (N);\n\\draw (A)--(M)(C)--(N);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461955,7 +465049,9 @@ "id": "017984", "content": "已知$A$、$B$、$C$三点不共线, $O$为平面$ABC$外一点, 若由向量$\\overrightarrow{OP}=\\dfrac{1}{5} \\overrightarrow{OA}+\\dfrac{2}{3} \\overrightarrow{OB}+\\lambda \\overrightarrow{OC}$确定的点$P$与$A$、$B$、$C$共面, 那么$\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461975,7 +465071,9 @@ "id": "017985", "content": "正四棱锥的侧棱长为$2 \\sqrt{3}$, 侧棱与底面所成的角为$60^{\\circ}$, 则该棱锥的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw (0,3,0) node [above] {$P$} coordinate (P);\n\\draw (O) ++ ({-sqrt(3)},0,{sqrt(3)}) node [below] {$A$} coordinate (A);\n\\draw (A) ++ ({2*sqrt(3)},0,0) node [below] {$B$} coordinate (B);\n\\draw ($(A)!2!(O)$) node [right] {$C$} coordinate (C);\n\\draw ($(B)!2!(O)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(P);\n\\draw [dashed] (A)--(C)(B)--(D)(P)--(O);\n\\draw (C) pic [draw, scale = 0.5, angle eccentricity = 2, \"$60^\\circ$\"] {angle = P--C--A};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -461995,7 +465093,9 @@ "id": "017986", "content": "在棱长为$10$的正方体$ABCD-A_1B_1C_1D_1$中, $P$为左侧面$ADD_1A_1$上一点, 已知点$P$到$A_1D_1$的距离为$3, P$到$AA_1$的距离为$2$, 则过点$P$且与$A_1C$平行的直线相交的面是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.2]\n\\def\\l{10}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (C)--(A_1);\n\\draw [dashed] ($(A)!0.7!(A_1)$) --++ (0,0,-2) node [right] {$P$} --++ (0,3,0);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$ABCD$}{$BB_1C_1C$}{$CC_1D_1D$}{$AA_1B_1B$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462015,7 +465115,9 @@ "id": "017987", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $E, F, G, H$分别为$AA_1$, $AB_1$, $BB_1$, $B_1C_1$的中点, 则异面直线$EF$与$GH$所成的角等于\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [below left ] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(B)$) node [below] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(B_1)$) node [left] {$G$} coordinate (G);\n\\draw ($(B_1)!0.5!(C_1)$) node [above] {$H$} coordinate (H);\n\\draw (E)--(F)(G)--(H);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$45^{\\circ}$}{$60^{\\circ}$}{$90^{\\circ}$}{$120^{\\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462035,7 +465137,9 @@ "id": "017988", "content": "如图, 空间四边形$ABCD$的四条边及对角线长都是$a$, 点$E$、$F$、$G$分别是$AB$、$AD$、$CD$的中点, 则$a^2$等于\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ($1/3*(D)+1/3*(B)+1/3*(C)$) ++ (0,{2*sqrt(6)/3},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)(A)--(C)(A)--(D)(B)--(C)--(D);\n\\draw [dashed] (B)--(D);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(D)$) node [right] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(D)$) node [below right] {$G$} coordinate (G);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\draw [dashed] (B)--(D)(E)--(F)--(G)--cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$2 \\overrightarrow{BA} \\cdot \\overrightarrow{AC}$}{$2 \\overrightarrow{AD} \\cdot \\overrightarrow{BD}$}{$2 \\overrightarrow{FG} \\cdot \\overrightarrow{CA}$}{$2 \\overrightarrow{EF} \\cdot \\overrightarrow{CB}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462055,7 +465159,9 @@ "id": "017989", "content": "设向量$\\overrightarrow {u}=(a, b, 0)$、$\\overrightarrow {v}=(c, d, 1)$, 其中$a^2+b^2=c^2+d^2=1$, 则下列判断错误的是\\bracket{20}.\n\\onech{向量$\\overrightarrow {v}$与$z$轴正方向的夹角为定值(与$c$、$d$之值无关)}{$\\overrightarrow {u} \\cdot \\overrightarrow {v}$的最大值为$\\sqrt{2}$}{$\\overrightarrow {u}$与$\\overrightarrow {v}$夹角的最大值为$\\dfrac{3 \\pi}{4}$}{$a d-b c$的最大值为$1$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462075,7 +465181,9 @@ "id": "017990", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 给出以下向量表达式: \n\\textcircled{1} $(\\overrightarrow{A_1D_1}-\\overrightarrow{A_1A})-\\overrightarrow{AB}$; \n\\textcircled{2} $(\\overrightarrow{BC}+\\overrightarrow{BB_1})-\\overrightarrow{D_1C_1}$;\n\\textcircled{3} $(\\overrightarrow{AD}-\\overrightarrow{AB})-2 \\overrightarrow{DD_1}$\n\\textcircled{4} $(\\overrightarrow{B_1D_1}+\\overrightarrow{A_1A})+\\overrightarrow{DD_1}$. \n其中能够化简为向量$\\overrightarrow{BD_1}$的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462095,7 +465203,9 @@ "id": "017991", "content": "已知正方体$ABCD-A_1B_1C_1D_1$中, 点$E$为上底面$A_1C_1$的中心, 若$\\overrightarrow{AE}=\\overrightarrow{AA_1}+x \\overrightarrow{AB}+y \\overrightarrow{AD}$, 则$x$、$y$的值分别为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A_1)!0.5!(C_1)$) node [above] {$E$} coordinate (E);\n\\draw [dashed, ->] (A)--(E);\n\\draw (A_1)--(C_1)(B_1)--(D_1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$x=1$, $y=1$}{$x=1$, $y=\\dfrac{1}{2}$}{$x=\\dfrac{1}{2}$, $y=\\dfrac{1}{2}$}{$x=\\dfrac{1}{2}$, $y=1$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462115,7 +465225,9 @@ "id": "017992", "content": "如图, 已知$M$、$N$分别为四面体$ABCD$的面$BCD$与面$ACD$的重心, $G$为$AM$上一点, 且$GM: GA=1: 3$. 设$\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{AC}=\\overrightarrow {b}$, $\\overrightarrow{AD}=\\overrightarrow {c}$, 试用$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$表示$\\overrightarrow{BG}$, $\\overrightarrow{BN}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ($1/3*(D)+1/3*(B)+1/3*(C)$) ++ (0,{2*sqrt(6)/3},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)(A)--(C)(A)--(D)(B)--(C)--(D);\n\\draw [dashed] (B)--(D);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\draw [dashed] (B)--(D);\n\\filldraw ($1/3*(D)+1/3*(B)+1/3*(C)$) node [below] {$M$} coordinate (M) circle (0.03);\n\\filldraw ($1/3*(D)+1/3*(A)+1/3*(C)$) node [below] {$N$} coordinate (N) circle (0.03);\n\\filldraw ($(A)!0.75!(M)$) node [left] {$G$} coordinate (G) circle (0.03);\n\\draw [dashed] (A)--(M);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462135,7 +465247,9 @@ "id": "017993", "content": "已知$\\overrightarrow {a}=(3,5,-4)$, $\\overrightarrow {b}=(2,1,8)$. 求:\\\\\n(1) $\\overrightarrow {a} \\cdot \\overrightarrow {b}$;\\\\\n(2) $\\overrightarrow {a}$与$\\overrightarrow {b}$夹角$\\theta$的余弦值;\\\\\n(3) 确定$\\lambda, \\mu$的值使得$\\lambda \\overrightarrow {a}+\\mu \\overrightarrow {b}$与$z$轴垂直, 且$(\\lambda \\overrightarrow {a}+\\mu \\overrightarrow {b}) \\cdot(\\overrightarrow {a}+\\overrightarrow {b})=53$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462155,7 +465269,9 @@ "id": "017994", "content": "已知边长为$1$的正方形$ABCD$, 正方形$ABCD$绕$BC$旋转$360^{\\circ}$形成一个圆柱.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,-1,0) node [left] {$D$} coordinate (D);\n\\draw (1,-1,0) node [right] {$C$} coordinate (C);\n\\draw (B) ++ (-110:1 and 0.25) node [below left] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (0,-1,0) node [below] {$D_1$} coordinate (D_1);\n\\draw (B) ellipse (1 and 0.25);\n\\draw (D) arc (180:360:1 and 0.25);\n\\draw [dashed] (D) arc (180:0:1 and 0.25);\n\\draw [dashed] (B)--(C)--(D)(C)--(D_1);\n\\draw (D_1)--(A_1)--(B)--(A)--(D);\n\\draw (2,0,0) --++ (0,-1,0);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆柱的表面积;\\\\\n(2) 正方形$ABCD$绕$BC$逆时针旋转$\\dfrac{\\pi}{2}$到$A_1BCD_1$, 求$AD_1$与平面$ABCD$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462175,7 +465291,9 @@ "id": "017995", "content": "如图所示, 正方形$ABCD$、$ABEF$的边长都是$1$, 且$AD \\perp$平面$ABEF$, 点$M$在$AC$上移动, 点$N$在$BF$上移动, 若$CM=BN=a(0=latex]\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$F$} coordinate (F);\n\\draw (2,0,-2) node [right] {$E$} coordinate (E);\n\\draw (0,0,-2) node [left] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$D$} coordinate (D);\n\\draw (D) ++ (0,0,-2) node [above] {$C$} coordinate (C);\n\\draw (A)--(F)--(E)--(B)--(C)--(D)--cycle(A)--(B);\n\\draw ($(A)!0.7!(C)$) node [left] {$M$} coordinate (M);\n\\draw ($(F)!0.7!(B)$) node [below] {$N$} coordinate (N);\n\\draw (A)--(C)(B)--(F)(M)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$MN$的长;\\\\\n(2) 当$a$为何值时, $MN$取得最小值? 并求最小值;\\\\\n(3) 当$2$为何值时, $MN$与平面$ABEF$所成角为$30^{\\circ}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462195,7 +465313,9 @@ "id": "017996", "content": "如图所示, 在四棱锥$P-ABCD$中, $PA \\perp$平面$ABCD$, 正方形$ABCD$的边长为$2, PA=4$, 设$E$为侧棱$PC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,0,2) node [below] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,4,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (P)--(A)--(B)(A)--(D);\n\\draw ($(P)!0.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw (B)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求正四棱锥$E-ABCD$的体积$V$;\\\\\n(2) 求直线$BE$与平面$PCD$所成角$\\theta$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462215,7 +465335,9 @@ "id": "017997", "content": "已知正四棱锥$P-ABCD$的全面积为$2$, 记正四棱锥的高为$h$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0,1) node [left] {$B$} coordinate (B);\n\\draw (1,0,1) node [right] {$C$} coordinate (C);\n\\draw (1,0,-1) node [right] {$D$} coordinate (D);\n\\draw (-1,0,-1) node [left] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(C)(P)--(B)(P)--(D)(B)--(C)--(D);\n\\draw [dashed] (B)--(A)--(D)(A)--(P);\n\\filldraw ($(A)!0.5!(C)$) coordinate (O) circle (0.03);\n\\draw [dashed] (P)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 用$h$表示底面边长, 并求正四棱锥体积$V$的最大值;\\\\\n(2) 当$V$取最大值时, 求异面直线$AB$和$PD$所成角的大小. (结果用反三角函数值表示)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462235,7 +465357,9 @@ "id": "017998", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=2$, 过$A_1$、$C_1$、$B$三点的平面截去长方体的一个角后, 得到如图所示的几何体$ABCD-A_1C_1D_1$, 且这个几何体的体积为$10$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (C) -- (C1) (A1)--(B)--(C1);\n\\draw [dashed] (D) -- (D1); \n\\end{tikzpicture}\n\\end{center}\n(1) 求棱$A_1A$的长;\\\\\n(2) 求点$D$到平面$A_1BC_1$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462255,7 +465379,9 @@ "id": "017999", "content": "如图所示, 点$P$在圆柱$OO_1$的底面圆$O$上, $\\angle AOP=120^{\\circ}$, 圆$O$的直径$AB=4$, 圆柱的高$OO_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\filldraw (0,0) node [above] {$O$} coordinate (O) circle (0.03);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,3) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,3) node [right] {$B_1$} coordinate (B_1);\n\\filldraw (O) ++ (0,3) node [above] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O) ++ (-50:2 and 0.5) node [below] {$P$} coordinate (P);\n\\draw (A)--(A_1)--(B_1)--(B)arc (0:-180:2 and 0.5);\n\\draw (O_1) ellipse (2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw [dashed] (A)--(B)--(P)--cycle(A_1)--(P)(A_1)--(B)(O)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆柱的表面积和三棱锥$A_1-APB$的体积;\\\\\n(2) 求点$A$到平面$A_1PO$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462275,7 +465401,9 @@ "id": "018000", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AD=AA_1=1$, $AB=2$, 点$E$在棱$AD$上移动.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, scale = 1.5]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (D1)--(E)(A1)--(D);\n\\draw [dashed] (A)--(C)--(D1)--cycle(D)--(E)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $D_1E \\perp A_1D$;\\\\\n(2) 当$E$为$AB$的中点时, 求点$E$到平面$ACD_1$的距离;\\\\\n(3) $AE$等于何值时, 二面角$D_1-EC-D$的大小为$\\dfrac{\\pi}{4}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462295,7 +465423,9 @@ "id": "018001", "content": "如图, 四棱锥$P-ABCD$中, 底面$ABCD$是平行四边形, $PG \\perp$平面$ABCD$, 垂足为$G$, $G$在$AD$上, 且$PG=4$, $AG=\\dfrac{1}{3} GD$, $BG \\perp GC$, $GB=GC=2$, $E$是$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0,0) node [above right] {$G$} coordinate (G);\n\\draw (G) ++ (0,4,0) node [above] {$P$} coordinate (P);\n\\draw (G) ++ ({-sqrt(2)/2},0) node [left] {$A$} coordinate (A);\n\\draw (G) ++ ({3*sqrt(2)/2},0) node [right] {$D$} coordinate (D);\n\\draw (G) ++ (0,0,{sqrt(2)}) node [below] {$E$} coordinate (E);\n\\draw (E) ++ ({-sqrt(2)},0) node [left] {$B$} coordinate (B);\n\\draw (E) ++ ({sqrt(2)},0) node [right] {$C$} coordinate (C);\n\\draw ($(P)!0.75!(C)$) node [right] {$F$} coordinate (F);\n\\draw (B)--(C)--(D)--(P)--cycle (P)--(C)(D)--(F);\n\\draw [dashed] (B)--(A)--(D)(B)--(G)--(C)(E)--(G)--(P)(A)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$GE$与$PC$所成的角的余弦值;\\\\\n(2) 求点$D$到平面$PBG$的距离;\\\\\n(3) 若$F$点是棱$PC$上一点, 且$DF \\perp GC$, 求$\\dfrac{PF}{FC}$的值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462315,7 +465445,9 @@ "id": "018002", "content": "若二项式$(3 x^2-\\dfrac{2}{\\sqrt[3]{x}})^n$($n \\in \\mathbf{N}$, $n \\geq 1$)展开式中含有常数项, 则$n$的最小取值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462335,7 +465467,9 @@ "id": "018003", "content": "$2^{6 n-3}+3^{2 n-1}$($n \\in \\mathbf{N}$, $n \\geq 1$)除以$11$所得的余数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462355,7 +465489,9 @@ "id": "018004", "content": "若$(1-2 x)^5$展开式中的第$2$项小于第$1$项, 且不小于第$3$项, 则实数$x$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462375,7 +465511,9 @@ "id": "018005", "content": "复数$z=a+b \\mathrm{i}$($a$、$b \\in \\mathbf{Z}$), 且$z^3=2+11 \\mathrm{i}$, 则$a+b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462395,7 +465533,9 @@ "id": "018006", "content": "从$\\{1,2,3,4,5\\}$中随机选取一个数$a$, 从$\\{1,2,3\\}$中随机选取一个数$b$, 则关于$x$的方程$x^2+2 a x+b^2=0$有两个虚数根的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462415,7 +465555,9 @@ "id": "018007", "content": "若$(2 x+\\sqrt{3})^4=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4$, 则$(a_0+a_2+a_4)^2-(a_1+a_3)^2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462435,7 +465577,9 @@ "id": "018008", "content": "计算: $\\mathrm{C}_7^3+\\mathrm{C}_7^4+\\mathrm{C}_8^5+\\mathrm{C}_9^6=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462455,7 +465599,9 @@ "id": "018009", "content": "方程: $\\mathrm{C}_{13}^{x+1}=\\mathrm{C}_{13}^{2 x-3}$的解为$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462475,7 +465621,9 @@ "id": "018010", "content": "方程: $\\mathrm{C}_{x+2}^{x-2}+\\mathrm{C}_{x+2}^{x-3}=\\dfrac{1}{10} \\mathrm{P}_{x+3}^3$的解为$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462495,7 +465643,9 @@ "id": "018011", "content": "投掷两颗骰子, 得到其向上的点数分别为$m$和$n$, 则复数$z=(m+n \\mathrm{i})(n-4 m \\mathrm{i})$($\\mathrm{i}$是虚数单位) 为实数的概率为\\blank{50}. (结果用最简分数表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462515,7 +465665,9 @@ "id": "018012", "content": "用红、黄、蓝三种颜色分别去涂图中标号为$1,2,3, \\cdots, 9$的$9$个小正方形, 需满足任意相邻(有公共边)的小正方形所涂颜色都不相同, 且标号为``$1$、$5$、$9$''的小正方形涂相同的颜色. 则符合条件的所有涂法中, 恰好满足``$1$、$3$、$5$、$7$、$9$''为同一颜色, ``$2$、$4$、$6$、$8$''为同一颜色的概率为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline 1 & 2 & 3 \\\\\n\\hline 4 & 5 & 6 \\\\\n\\hline 7 & 8 & 9 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462535,7 +465687,9 @@ "id": "018013", "content": "停车场可把$12$辆车停放在一排上, 现有$8$辆车停放, 而恰有$4$个空位连在一起, 这样的事件发生的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462555,7 +465709,9 @@ "id": "018014", "content": "组合数$\\mathrm{C}_n^r$($n>r \\geq 1$, $n$、$r \\in \\mathbf{Z})$恒等于\\bracket{20}.\n\\fourch{$\\dfrac{r+1}{n+1} \\mathrm{C}_{n-1}^{r-1}$}{$(n+1)(r+1) \\mathrm{C}_{n-1}^{r-1}$}{$n r \\mathrm{C}_{n-1}^{r-1}$}{$\\dfrac{n}{4} \\mathrm{C}_{n-1}^{r-1}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462575,7 +465731,9 @@ "id": "018015", "content": "12 名同学合影, 站成前排$4$人后排$8$人, 现摄影师要从后排$8$人中抽$2$人调整到前排, 若其他人的相对顺序不变, 则不同调整方法的总数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_8^2 \\mathrm{P}_3^2$}{$\\mathrm{C}_8^2 \\mathrm{P}_6^6$}{$\\mathrm{C}_8^2\\mathrm{P}_6^2$}{$\\mathrm{C}_8^2\\mathrm{P}_5^2$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462595,7 +465753,9 @@ "id": "018016", "content": "二项式$(\\sqrt{3} \\mathrm{i}-x)^{10}$的展开式中的第八项是\\bracket{20}.\n\\fourch{$-135 x^3$}{$3645 x^2$}{$360 \\sqrt{3} \\mathrm{i} x^7$}{$3240 \\sqrt{3} \\mathrm{i} x^3$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462615,7 +465775,9 @@ "id": "018017", "content": "已知$10$件产品中有$5$件一等品, $3$件二等品和$2$件三等品, 从中任取$3$件. 试求下列事件的概率:\\\\\n(1) $2$件为一等品, $1$件是二等品;\\\\\n(2) 至少有一件是一等品;\\\\\n(3) 取到了三等品.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462635,7 +465797,9 @@ "id": "018018", "content": "已知关于$x$的方程$x^2+k x+k^2-3 k=0$($k \\in \\mathbf{R}$)有一个模为$1$的根, 求实数$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462655,7 +465819,9 @@ "id": "018019", "content": "在二项式定理这节教材中有这样一个性质: $\\mathrm{C}_n^0+\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\mathrm{C}_n^3+\\cdots+\\mathrm{C}_n^n=2^n$($n \\in \\mathbf{N}$, $n \\geq 1$).\\\\\n(1) 计算$1 \\cdot \\mathrm{C}_3^0+2 \\cdot \\mathrm{C}_3^1+3 \\cdot \\mathrm{C}_3^2+4 \\cdot \\mathrm{C}_3^3$的值方法如下:\n设$S=1 \\cdot \\mathrm{C}_3^0+2 \\cdot \\mathrm{C}_3^1+3 \\cdot \\mathrm{C}_3^2+4 \\cdot \\mathrm{C}_3^3$, \n又$S=4 \\cdot \\mathrm{C}_3^3+3 \\cdot \\mathrm{C}_3^2+2 \\cdot \\mathrm{C}_3^1+1 \\cdot \\mathrm{C}_3^0$, \n相加得$2S=5 \\cdot \\mathrm{C}_3^0+5 \\cdot \\mathrm{C}_3^1+5 \\cdot \\mathrm{C}_3^2+5 \\cdot \\mathrm{C}_3^3$, 即$2S=5 \\cdot 2^3$, \n所以$S=5 \\cdot 2^2=20$.\n利用类似方法求值: $1 \\cdot \\mathrm{C}_2^0+2 \\cdot \\mathrm{C}_2^1+3 \\cdot \\mathrm{C}_2^2, 1 \\cdot \\mathrm{C}_4^0+2 \\cdot \\mathrm{C}_4^1+3 \\cdot \\mathrm{C}_4^2+4 \\cdot \\mathrm{C}_4^3+5 \\cdot \\mathrm{C}_4^4$;\\\\\n(2) 将 (1) 的情况推广到一般结论, 并给予证明;\\\\\n(3) 设$S_n$是首项为$a$, 公比为$q$的等比数列$\\{a_n\\}$的前$n$项和, 求:\n$S_1 \\mathrm{C}_n^0+S_2 \\mathrm{C}_n^1+S_3 \\mathrm{C}_n^2+S_4 \\mathrm{C}_n^3 S_1 \\mathrm{C}_n^0+\\cdots+S_{n+1} \\mathrm{C}_n^n$($n \\in \\mathbf{N}$, $n \\geq 1$).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -462675,7 +465841,9 @@ "id": "018020", "content": "如果一个函数的瞬时变化率处处为$0$, 则这个函数的图像是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462695,7 +465863,9 @@ "id": "018021", "content": "已知$f(x)=\\dfrac{2}{x}$, 且$f'(m)=-\\dfrac{1}{2}$, 则$m$的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462715,7 +465885,9 @@ "id": "018022", "content": "曲线$y=x^2+\\ln x$在点$(1,1)$处的切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462735,7 +465907,9 @@ "id": "018023", "content": "若函数$f(x)=a x^4+b x^2+c$满足$f'(1)=2$, 则$f'(-1)$等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462755,7 +465929,9 @@ "id": "018024", "content": "函数$f(x)=\\ln x-x$的单调增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462775,7 +465951,9 @@ "id": "018025", "content": "函数$y=\\dfrac{x^2}{x+3}$的导数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462795,7 +465973,9 @@ "id": "018026", "content": "若函数$f(x)=x(x-a)^2$在$x=2$处取得极小值, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462815,7 +465995,9 @@ "id": "018027", "content": "已知函数$f(x)=\\dfrac{1}{2} x-\\sin x$, $x \\in[0, \\pi]$, 则$f(x)$的最小值为\\blank{50}, 最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462835,7 +466017,9 @@ "id": "018028", "content": "曲线$y=e^{-2 x}+1$在点$(0,2)$处的切线与直线$y=0$和$y=x$围成的三角形的面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462855,7 +466039,9 @@ "id": "018029", "content": "如图是$y=f(x)$的导函数图像, 现有四种说法:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-4,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-3,-2,-1,1,2,3,4}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\draw [domain = -3.2:4.2, samples = 100] plot (\\x,{(\\x - 4)*(\\x - 2)*(\\x + 1)*(\\x + 3)*(\\x + 4)/100}); \n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $f(x)$在$(-1.5,0.5)$上是严格增函数;\\\\\n\\textcircled{2} $x=-1$是$f(x)$的极小值点;\\\\\n\\textcircled{3} $f(x)$在$(-1,2)$上是严格增函数;\\\\\n\\textcircled{4} $x=2$是$f(x)$的极小值点;\\\\\n以上说法正确的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462875,7 +466061,9 @@ "id": "018030", "content": "若函数$y=e^x-2 m x$有小于零的极值点, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462895,7 +466083,9 @@ "id": "018031", "content": "若函数$f(x)=\\dfrac{1}{3} x^3+x^2-1$在区间$(m, m+3)$上存在最小值, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -462915,7 +466105,9 @@ "id": "018032", "content": "已知可导函数$f(x)$的导函数为$f'(x)$, 则``$f'(x_0)=0$''是``$x=x_0$是函数$f(x)$的一个极值点''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -462935,7 +466127,9 @@ "id": "018033", "content": "函数$f(x)$的图像如图所示, $f'(x)$为函数$f(x)$的导函数, 下列数值排序正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (0,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.2:3.2] plot (\\x,{sqrt(13-(\\x-3.5)^2)});\n\\draw [dashed] (2,{sqrt(13-(2-3.5)^2)}) node [above] {$A$} -- (2,0) node [below] {$2$};\n\\draw [dashed] (3,{sqrt(13-(3-3.5)^2)}) node [above] {$B$} -- (3,0) node [below] {$3$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$01$), $C_2: \\dfrac{x^2}{4}+y^2=1$的离心率分别为$e_1, e_2$. 若$e_2=\\sqrt{3} e_1$, 则$a=$\\bracket{20}.\n\\fourch{$\\dfrac{2 \\sqrt{3}}{3}$}{$\\sqrt{2}$}{$\\sqrt{3}$}{$\\sqrt{6}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463155,7 +466369,9 @@ "id": "018044", "content": "过点$(0,-2)$与圆$x^2+y^2-4 x-1=0$相切的两条直线的夹角为$\\alpha$, 则$\\sin \\alpha=$\\bracket{20}.\n\\fourch{$1$}{$\\dfrac{\\sqrt{15}}{4}$}{$\\dfrac{\\sqrt{10}}{4}$}{$\\dfrac{\\sqrt{6}}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463175,7 +466391,9 @@ "id": "018045", "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 设甲: $\\{a_n\\}$为等差数列: 乙: $\\{\\dfrac{S_n}{n}\\}$为等差数列, 则\\bracket{20}.\n\\twoch{甲是乙的充分条件但不是必要条件}{甲是乙的必要条件但不是充分条件}{甲是乙的充要条件}{甲既不是乙的充分条件也不是乙的必要条件}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463195,7 +466413,9 @@ "id": "018046", "content": "已知$\\sin (\\alpha-\\beta)=\\dfrac{1}{3}$, $\\cos \\alpha \\sin \\beta=\\dfrac{1}{6}$, 则$\\cos (2 \\alpha+2 \\beta)=$\\bracket{20}.\n\\fourch{$\\dfrac{7}{9}$}{$\\dfrac{1}{9}$}{$-\\dfrac{1}{9}$}{$-\\dfrac{7}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463215,7 +466435,9 @@ "id": "018047", "content": "有一组样本数据$x_1, x_2, \\cdots, x_6$, 其中$x_1$是最小值, $x_6$是最大值, 则\\blank{50}.\\\\\n\\textcircled{1} $x_2, x_3, x_4, x_5$的平均数等于$x_1, x_2, \\cdots, x_6$的平均数;\\\\\n\\textcircled{2} $x_2, x_3, x_4, x_5$的中位数等于$x_1, x_2, \\cdots, x_6$的中位数;\\\\\n\\textcircled{3} $x_2, x_3, x_4, x_5$的标准差不小于$x_1, x_2, \\cdots, x_6$的标准差;\\\\\n\\textcircled{4} $x_2, x_3, x_4, x_5$的极差不大于$x_1, x_2, \\cdots, x_6$的极差.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463235,7 +466457,9 @@ "id": "018048", "content": "噪声污染问题越来越受到重视. 用声压级来度量声音的强弱, 定义声压级$L_p=20 \\times \\lg \\dfrac{p}{p_0}$, 其中常数$p_0$($p_0>0$) 是听觉下限阈值, $p$是实际声压. 下表为不同声源的声压级:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline 声源 & 与声源的距离/m & 声压级/dB \\\\\n\\hline 燃油轮 & 10 &$60 \\sim 90$\\\\\n\\hline 混合动力汽车 & 10 &$50 \\sim 60$\\\\\n\\hline 电动汽车 & 10 & 40 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n已知在距离燃油汽车、混合动力汽车、电动汽车$10 \\text{m}$处测得实际声压分别为$p_1$, $p_2$, $p_3$, 则\\blank{50}.\\\\\n\\textcircled{1} $p_1 \\geq p_2$; \\textcircled{2} $p_2>10 p_1$; \\textcircled{3} $p_3=100 p_0$; \\textcircled{4} $p_1 \\leq 100 p_2$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463255,7 +466479,9 @@ "id": "018049", "content": "已知函数$f(x)$的定义域为$\\mathbf{R}$, $f(x y)=y^2 f(x)+x^2 f(y)$, 则\\blank{50}.\\\\\n\\textcircled{1} $f(0)=0$; \\textcircled{2} $f(1)=0$; \\textcircled{3} $f(x)$是偶函数; \\textcircled{4} $x=0$为$f(x)$的极小值点.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463275,7 +466501,9 @@ "id": "018050", "content": "下列物体中, 能够被整体放入棱长为$1$ (单位: m ) 的正方体容器 (容器壁厚度忽略不计) 内的有\\blank{50}.\\\\\n\\textcircled{1} 直径为$0.99 \\mathrm{m}$的球体;\\\\\n\\textcircled{2} 所有棱长均为$1.4 \\mathrm{m}$的四面体;\\\\\n\\textcircled{3} 底面直径为$0.01 \\mathrm{m}$, 高为$1.8 \\mathrm{m}$的圆柱体;\\\\\n\\textcircled{4} 底面直径为$1.2 \\mathrm{m}$, 高为$0.01 \\mathrm{m}$的圆柱体.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463295,7 +466523,9 @@ "id": "018051", "content": "某学校开设了$4$门体育类选修课和$4$门艺术类选修课, 学生需从这$8$门课中选修$2$门或$3$门课, 并且每类选修课至少选修$1$门, 则不同的选课方案共有\\blank{50}种(用数字作答).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463315,7 +466545,9 @@ "id": "018052", "content": "在正四棱台$ABCD-A_1B_1C_1D_1$中, $AB=2$, $A_1B_1=1$, $AA_1=\\sqrt{2}$, 则该棱台的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463335,7 +466567,9 @@ "id": "018053", "content": "已知函数$f(x)=\\cos \\omega x-1$($\\omega>0$) 在区间$[0,2 \\pi]$有且仅有$3$个零点, 则$\\omega$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463355,7 +466589,10 @@ "id": "018054", "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的左、右焦点分别为$F_1, F_2$. 点$A$在$C$上, 点$B$在$y$轴上, $\\overrightarrow{F_1A} \\perp \\overrightarrow{F_1B}$, $\\overrightarrow{F_2A}=-\\dfrac{2}{3}\\overrightarrow{F_2B}$, 则$C$的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463375,7 +466612,9 @@ "id": "018055", "content": "已知在$\\triangle ABC$中, $A+B=3C$, $2 \\sin (A-C)=\\sin B$. \\\\\n(1) 求$\\sin A$;\\\\\n(2) 设$AB=5$, 求$AB$边上的高.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463395,7 +466634,9 @@ "id": "018056", "content": "如图, 在正四楼柱$ABCD-A_1B_1C_1D_1$中, $AB=2$, $AA_1=4$. 点$A_2, B_2, C_2, D_2$分别在棱$AA_1, BB_1, CC_1, DD_1$上, $AA_2=1$, $BB_2=DD_2=2$, $CC_2=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$D$} coordinate (D);\n\\draw (D) ++ (\\l,0,0) node [below right] {$A$} coordinate (A);\n\\draw (D) ++ (\\l,0,-\\l) node [right] {$B$} coordinate (B);\n\\draw (D) ++ (0,0,-\\l) node [left] {$C$} coordinate (C);\n\\draw (D) -- (A) -- (B);\n\\draw [dashed] (D) -- (C) -- (B);\n\\draw (D) ++ (0,{2*\\l},0) node [left] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,{2*\\l},0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,{2*\\l},0) node [above right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,{2*\\l},0) node [above left] {$C_1$} coordinate (C_1);\n\\draw (D_1) -- (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw (D) -- (D_1) (A) -- (A_1) (B) -- (B_1);\n\\draw [dashed] (C) -- (C_1);\n\\draw ($(A)!0.25!(A_1)$) node [right] {$A_2$} coordinate (A_2);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$B_2$} coordinate (B_2);\n\\draw ($(C)!0.75!(C_1)$) node [left] {$C_2$} coordinate (C_2);\n\\draw ($(D)!0.5!(D_1)$) node [left] {$D_2$} coordinate (D_2);\n\\draw (D_2)--(A_2);\n\\draw [dashed] (B_2)--(C_2);\n\\draw ($(B)!0.75!(B_1)$) node [right] {$P$} coordinate (P);\n\\draw (P)--(A_2);\n\\draw [dashed] (P)--(C_2)(C_2)--(D_2)(C_2)--(A_2);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $B_2C_2\\parallel A_2D_2$;\\\\\n(2) 点$P$在棱$BB_1$上, 当二面角$P-A_2C_2-D_2$为$150^{\\circ}$时, 求$B_2P$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463415,7 +466656,9 @@ "id": "018057", "content": "已知函数$f(x)=a(\\mathrm{e}^x+a)-x$.\\\\\n(1) 讨论$f(x)$的单调性;\\\\\n(2) 证明: 当$a>0$时, $f(x)>2 \\ln a+\\dfrac{3}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463435,7 +466678,9 @@ "id": "018058", "content": "设等差数列$\\{a_n\\}$的公差为$d$, 且$d>1$. 令$b_n=\\dfrac{n^2+n}{a_n}$, 记$S_n, T_n$分别为数列$\\{a_n\\},\\{b_n\\}$的前$n$项和.\\\\\n(1) 若$3 a_2=3 a_1+a_3$, $S_3+T_3=21$, 求$\\{a_n\\}$的通项公式;\\\\\n(2) 若$\\{b_n\\}$为等差数列, 且$S_{99}-T_{99}=99$, 求$d$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463455,7 +466700,9 @@ "id": "018059", "content": "甲乙两人投篮, 每次由其中一人投篮, 规则如下: 若命中则此人继续投篮, 若未命中则换为对方投篮. 无论之前投篮情况如何, 甲每次投篮的命中率均为$0.6$, 乙每次投篮的命中率均为$0.8$, 由抽签确定第$1$次投篮的人选, 第一次投篮的人是甲, 乙的概率各为$0.5$.\\\\\n(1) 求第$2$次投篮的人是乙的概率;\\\\\n(2) 求第$i$次投篮的人是甲的概率;\\\\\n(3) 已知: 若随机变量$X_i$服从两点分布, 且$P(X_i=1)=1-P(X_i=0)=q_i$, $i=1,2, \\cdots, n$, 则$E[\\displaystyle\\sum_{i=1}^n X_i]=\\displaystyle\\sum_{i=1}^n q_i$. 记前$n$次(即从第$1$次到第$n$次投篮)中甲投篮的次数为$Y$, 求$E[Y]$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463475,7 +466722,9 @@ "id": "018060", "content": "在直角坐标系$x O y$中, 点$P$到$x$轴的距离等于点$P$到点$(0, \\dfrac{1}{2})$的距离, 记动点$P$的轨迹为$W$.\\\\\n(1) 求$W$的方程;\\\\\n(2) 已知矩形$ABCD$有三个顶点在$W$上, 证明: 矩形$ABCD$的周长大于$3 \\sqrt{3}$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463495,7 +466744,9 @@ "id": "018061", "content": "不等式$|x-2|<1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463515,7 +466766,9 @@ "id": "018062", "content": "已知$\\overrightarrow {a}=(-2,3)$, $\\overrightarrow {b}=(1,2)$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463535,7 +466788,9 @@ "id": "018063", "content": "已知$\\{a_n\\}$为等比数列, 且$a_1=3$, $q=2$, 则该数列的前$6$项之和$S_6=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463556,7 +466811,9 @@ "id": "018064", "content": "已知$\\tan \\alpha=3$, 则$\\tan 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463576,7 +466833,9 @@ "id": "018065", "content": "已知$f(x)=\\begin{cases}2^x, & x>0 \\\\ 1, & x \\leq 0\\end{cases}$, 则$f(x)$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463596,7 +466855,9 @@ "id": "018066", "content": "已知当$z=1+\\mathrm{i}$, 则$|1-\\mathrm{i} \\cdot z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463616,7 +466877,9 @@ "id": "018067", "content": "已知$x^2+y^2-4 y-m=0$的面积为$\\pi$, 则实数$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463636,7 +466899,9 @@ "id": "018068", "content": "在$\\triangle ABC$中, $a=4$, $b=5$, $c=6$, 则$\\sin A=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463656,7 +466921,9 @@ "id": "018069", "content": "国内生产总值 (GDP) 是衡量地区经济状况的最佳指标, 根据统计数据显示, 某市在$2020$年间经济高质量增长, GDP 稳步增长, 第一季度和第四季度的 GDP 分别为$231$百亿元和$242$百亿元, 且四个季度 GDP 的中位数与平均数相等, 则$2020$年 GDP 总额为\\blank{50}百亿元.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463676,7 +466943,9 @@ "id": "018070", "content": "已知$(1+2023 x)^{100}+(2023-x)^{100}=a_0+a_1 x+a_2 x^2+\\cdots+a_{100} x^{100}$, 其中$a_0, a_1, a_2 \\cdots a_{100} \\in \\mathbf{R}$, 若$0 \\leq k \\leq 100$, $k \\in \\mathbf{N}$, 且$a_k<0$时, $k$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463696,7 +466965,9 @@ "id": "018071", "content": "公园修建斜坡, 假设斜坡起点在水平面上, 斜坡与水平面的夹角为$\\theta$, 斜坡终点距离水平面的垂直高度为$4$米, 游客每走一米消耗的体能为$(1.025-\\cos \\theta)$, 要使游客从斜坡底走到斜坡顶端所消耗的总体能最少, 则$\\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463716,7 +466987,9 @@ "id": "018072", "content": "空间内存在三点$A$、$B$、$C$, 满足$AB=AC=BC=1$, 在空间内取不同两点(不计顺序), 使得这两点与$A$、$B$、$C$可以构成正四棱锥的五个顶点, 不同的方案数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -463736,7 +467009,9 @@ "id": "018073", "content": "已知$P=\\{1,2\\}$, $Q=\\{2,3\\}$, 若$M=\\{x | x \\in P$且$x \\notin Q\\}$, 则$M=$\\bracket{20}.\n\\fourch{$\\{1\\}$}{$\\{2\\}$}{$\\{1,2\\}$}{$\\{1,2,3\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463756,7 +467031,9 @@ "id": "018074", "content": "根据如下身高和体重散点图, 下列说法正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 0.1, yscale = 0.05]\n\\foreach \\i in {140,150,...,190,200}\n{\\draw [gray] (\\i,40) -- (\\i,120);\n\\draw (\\i,40) node [below] {$\\i$};};\n\\foreach \\j in {40,50,...,120}\n{\\draw [gray] (140,\\j) -- (200,\\j);\n\\draw (140,\\j) node [left] {$\\j$};};\n\\draw (170,25) node {身高/cm};\n\\draw (130,80) node [rotate = 90] {体重/kg};\n\\foreach \\i/\\j in {162.7541947/51.35667837,155.5904334/50.43458716,182.6281063/66.42520322,194.5162186/73.26691491,167.0109575/64.85837148,186.1664161/56.42752272,155.3201017/41.01812489,164.8129716/63.14035077,149.9758607/50.66177296,182.1245727/61.56063672,167.7164262/63.36603393,188.1316785/62.07889087,168.181048/56.74601718,169.0289674/48.2316275,165.9568328/45.69465675,158.6475788/59.43511492,177.9581592/65.94572529,149.8883543/42.89574403,154.1762601/52.35704594,177.5744725/64.40559165,147.8675451/36.73642194,158.874746/50.18512684,169.9543832/56.11235339,178.5879736/72.65914687,164.7971686/64.1538182,148.9397741/40.05063508,169.2046131/56.68262717,161.8658622/50.29256194,183.4761935/65.62962727,172.6295456/56.89824375,145.9413914/42.71276268,165.7868326/48.24900293,144.8721261/43.65404209,154.2605136/48.46520756,164.9177355/56.03416046,152.6628467/51.99903484,162.0341495/49.92958441,198.8067301/88.7414868,191.9745755/65.53331829,176.9590713/61.41886794}\n{\\filldraw (\\i,\\j*1.2) ellipse (0.3 and 0.6);};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{身高越高, 体重越重}{身高越高, 体重越轻}{身高与体重成正相关}{身高与体重成负相关}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463776,7 +467053,9 @@ "id": "018075", "content": "设$a>0$, 函数$y=\\sin x$在区间$[a, 2 a]$上的最小值为$s_a$, 在$[2 a, 3 a]$上的最小值为$t_a$, 当$a$变化时, 以下不可能的情形是\\bracket{20}.\n\\fourch{$s_a>0$且$t_a>0$}{$s_a<0$且$t_a<0$}{$s_a>0$且$t_a<0$}{$s_a<0$且$t_a>0$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463796,7 +467075,9 @@ "id": "018076", "content": "在平面上, 若曲线$\\Gamma$具有如下性质: 存在点$M$, 使得对于任意点$P \\in \\Gamma$, 都有$Q \\in \\Gamma$使得$|PM| \\cdot|QM|=1$.则称这条曲线为``自相关曲线''.\n下列两个命题:\\\\\n\\textcircled{1} 所有椭圆都是``自相关曲线'';\\\\\n\\textcircled{2} 存在是``自相关曲线''的双曲线.\\\\\n的真假情况为\\bracket{20}.\n\\twoch{\\textcircled{1}为假命题, \\textcircled{2}为真命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为真命题}{\\textcircled{1}为假命题, \\textcircled{2}为假命题}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463818,7 +467099,9 @@ "id": "018077", "content": "直四棱柱$ABCD-A_1B_1C_1D_1$中, $AB\\parallel DC$, $AB \\perp AD$, $AB=2$, $AD=3$, $DC=4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (4,0,-3) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-3) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,4,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,4,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,4,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,4,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (A_1)--(B);\n\\draw [dashed] (B)--(D)--(A)(D)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1B \\parallel$平面$DCC_1D$;\\\\\n(2) 若四棱柱体积为$36$, 求二面角$A_1-BD-A$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463839,7 +467122,9 @@ "id": "018078", "content": "函数$f(x)=\\dfrac{x^2+(3 a+1) x+c}{x+a}$($a, c \\in \\mathbf{R}$).\\\\\n(1) 当$a=0$时, 是否存在实数$c$, 使得$f(x)$为奇函数? 说明理由;\\\\\n(2) 函数$f(x)$的图像过点$(1,3)$, 且$f(x)$的图像$x$轴负半轴有两个交点, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463859,7 +467144,9 @@ "id": "018079", "content": "21 世纪汽车博览会于 2023年 6 月 7 日在上海举行, 某汽车模型公司展示了$25$个汽车模型, 其外观和内饰的颜色分布如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline \\backslashbox{内饰}{外观} & 红色外观 & 蓝色外观 \\\\\n\\hline 棕色内饰 & 12 & 8 \\\\\n\\hline 米色内饰 & 2 & 3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 若小明从这些模型中随机拿一个模型, 记事件$A$为小明取到的模型为红色外观, 事件$B$为取到模型有棕色内饰. 求$P(B)$、$P(B|A)$, 并据此判断事件$A$和事件$B$是否独立;\\\\\n(2) 该公司举行了一个抽奖活动, 规定在一次抽奖中, 每人可以一次性从这些模型中拿两个汽车模型, 给出以下假设: \\textcircled{1} 拿到的两个模型会出现三种结果, 即外观和内饰均为同色、 外观内饰都异色、以及仅外观或仅内饰同色; \\textcircled{2} 按结果的可能性大小, 概率越小奖项越高; \\textcircled{3} 奖金额为一等奖$600$元, 二等奖$300$元, 三等奖$150$元, 请你分析奖项对应的结果. 设$X$为一次抽奖获得的奖金额, 写出$X$的分布列并求出$X$的期望.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463879,7 +467166,9 @@ "id": "018080", "content": "曲线$\\Gamma: y^2=4 x$, 第一象限内点$A$在$\\Gamma$上, $A$的纵坐标是$a$.\\\\\n(1) 若$A$到准线的距离为$3$, 求$a$;\\\\\n(2) 若$a=4$, $B$在$x$轴上, $AB$中点在$\\Gamma$上, 求点$B$坐标和坐标原点$O$到$AB$距离;\\\\\n(3) 直线$l: x=-3$, 令$P$是第一象限$\\Gamma$上异于$A$的一点, 直线$PA$交$l$于$Q$, $H$是$P$在$l$上的投影, 若点$A$满足``对于任意$P$都有$|HQ|>4$'', 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463900,7 +467189,10 @@ "id": "018081", "content": "令$f(x)=\\ln x$, 取点$(a_1, f(a_1))$, 过该点作曲线$y=f(x)$的切线交$y$轴于点$(0, a_2)$, 取点$(a_{2},f(a_2))$, 过该点作切线交$y$轴于$(0, a_3)$, 以此类推, 期间若$a_k\\le 0$则停止, 得到数列$\\{a_n\\}$.\\\\\n(1) 若正整数$m \\geq 2$, 证明$a_m=\\ln a_{m-1}-1$;\\\\\n(2) 若正整数$m \\geq 2$, 试比较$a_m$与$a_{m-1}-2$大小;\\\\\n(3) 若正整数$k \\geq 3$, 是否存在$k$使得$a_1, a_2 \\cdots a_k$依次成等差数列? 若存在, 求出$k$的所有取值, 若不存在, 试说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -463921,7 +467213,9 @@ "id": "018082", "content": "在复平面内, $(1+3 \\mathrm{i})(3-\\mathrm{i})$对应的点位于\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463941,7 +467235,9 @@ "id": "018083", "content": "设集合$A=\\{0,-a\\}$, $B=\\{1, a-2,2 a-2\\}$, 若$A \\subseteq B$, 则$a=$\\bracket{20}.\n\\fourch{$2$}{$1$}{$\\dfrac{2}{3}$}{$-1$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463961,7 +467257,9 @@ "id": "018084", "content": "某学校为了解学生参加体育运动的情况, 用比例分配的分层随机抽样法作抽样调查, 拟从初中部和高中部两层共抽取$60$名学生, 已知该校初中部和高中部分别有$400$和$200$名学生, 则不同的抽样结果共有\\bracket{20}种.\n\\fourch{$C_{400}^{45} \\cdot C_{200}^{15}$}{$C_{400}^{20} \\cdot C_{200}^{40}$}{$C_{400}^{30} \\cdot C_{200}^{30}$}{$C_{400}^{40} \\cdot C_{200}^{20}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -463981,7 +467279,9 @@ "id": "018085", "content": "若$f(x)=(x+a) \\ln \\dfrac{2 x-1}{2 x+1}$为偶函数, 则$a=$\\bracket{20}.\n\\fourch{$-1$}{$0$}{$\\dfrac{1}{2}$}{1}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464001,7 +467301,9 @@ "id": "018086", "content": "已知椭圆$\\dfrac{x^2}{3}+y^2=1$的左、右焦点分别为$F_1, F_2$, 直线$y=x+m$与$C$交于$A$、$B$两点, 若$\\triangle F_1AB$的面积是$\\triangle F_2AB$的面积的$2$倍, 则$m=$\\bracket{20}.\n\\fourch{$\\dfrac{2}{3}$}{$\\dfrac{\\sqrt{2}}{3}$}{$-\\dfrac{\\sqrt{2}}{3}$}{$-\\dfrac{2}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464021,7 +467323,9 @@ "id": "018087", "content": "已知函数$f(x)=a \\mathrm{e}^x-\\ln x$在区间$(1,2)$上单调递增, 则$a$的最小值为\\bracket{20}.\n\\fourch{$\\mathrm{e}^2$}{$\\mathrm{e}$}{$\\mathrm{e}^{-1}$}{$\\mathrm{e}^{-2}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464041,7 +467345,9 @@ "id": "018088", "content": "已知$\\alpha$为锐角, $\\cos \\alpha=\\dfrac{1+\\sqrt{5}}{4}$, 则$\\sin \\dfrac{\\alpha}{2}=$\\bracket{20}.\n\\fourch{$\\dfrac{3-\\sqrt{5}}{8}$}{$\\dfrac{-1+\\sqrt{5}}{8}$}{$\\dfrac{3-\\sqrt{5}}{4}$}{$\\dfrac{-1+\\sqrt{5}}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464061,7 +467367,9 @@ "id": "018089", "content": "记$S_n$等比数列$\\{a_n\\}$的前$n$项和, 若$S_4=-5$, $S_6=21S_2$, 则 $S_8=$\\bracket{20}.\n\\fourch{$120$}{$85$}{$-85$}{$-120$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464081,7 +467389,9 @@ "id": "018090", "content": "已知圆锥的顶点为$P$, 底面圆心为$O$, $AB$为底面的直径, $\\angle APB=120^{\\circ}$, $AP=2$, 点$C$在底面圆周上, 且二面角$P-AC-O=45^{\\circ}$, 则\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$P$} coordinate (P);\n\\draw (-30:2) node [right] {$B$} coordinate (B);\n\\draw (-150:2) node [left] {$A$} coordinate (A);\n\\draw (A) arc (180:360:{sqrt(3)} and {sqrt(3)/8});\n\\draw [dashed] (A) arc (180:0:{sqrt(3)} and {sqrt(3)/8});\n\\draw ($(A)!0.5!(B)$) node [right] {$O$} coordinate (O);\n\\draw [dashed] (P)--(O)--(A);\n\\draw (O) ++ (-110:{sqrt(3)} and {sqrt(3)/8}) node [below] {$C$} coordinate (C);\n\\draw [dashed] (A)--(C)--(O);\n\\draw (A)--(P)--(B)(C)--(P);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 该圆锥的体积为$\\pi$;\\\\\n\\textcircled{2} 该圆锥的侧面积为$4 \\sqrt{3} \\pi$;\\\\\n\\textcircled{3} $AC=2 \\sqrt{2}$;\\\\\n\\textcircled{4} $\\triangle PAC$的面积为$\\sqrt{3}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464101,7 +467411,9 @@ "id": "018091", "content": "设$O$为坐标原点, 直线$y=-\\sqrt{3}(x-1)$过抛物线 $C: y^2=2 p x$($p>0$)的焦点, 且与$C$交于$M$、$N$两点, $l$为$C$的准线, 则\\blank{50}.\\\\\n\\textcircled{1} $p=2$; \\textcircled{2} $|MN|=\\dfrac{8}{3}$; \\textcircled{3} 以$MN$为直径的圆与$l$相切; \\textcircled{4} $\\triangle OMN$为等腰三角形.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464121,7 +467433,9 @@ "id": "018092", "content": "若函数$f x=a \\ln x+\\dfrac{b}{x}+\\dfrac{c}{x^2}$($a \\neq 0$)既有极大值又有极小值, 则\\blank{50}.\\\\\n\\textcircled{1} $b c>0$; \\textcircled{2} $a b>0$; \\textcircled{3} $b^2+8 a c>0$; \\textcircled{4} $a c<0$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464141,7 +467455,9 @@ "id": "018093", "content": "在信道内传输$0$, $1$信号, 信号的传输相互独立, 发送$0$时, 收到$1$的概率为$\\alpha$($0<\\alpha<1$), 收到$0$的概率为$1-\\alpha$; 发送$1$时, 收到$0$的概率为$\\beta$($0<\\beta<1$), 收到$1$的概率为$1-\\beta$. 考虑两种传输方案: 单次传输和三次传输. 单次传输是指每个信号只发送$1$次; 三次传输是指每个信号重复发送$3$次. 收到的信号需要译码, 译码规则如下: 单次传输时, 收到的信号即为译码: 三次传输时, 收到的信号中出现次数多的即为译码(例如, 若依次收到$1,0,1$, 则译码为$1$). 以下说法中正确的是\\blank{50}.\\\\\n\\textcircled{1} 采用单次传输方案, 若依次发送$1,0,1$, 则依次收到$1,0,1$的概率为$(1-\\alpha)(1-\\beta)^2$;\\\\\n\\textcircled{2} 采用三次传输方案, 若发送$1$, 则依次收到$1,0,1$的概率为$\\beta(1-\\beta)^2$;\\\\\n\\textcircled{3} 采用三次传输方案, 若发送$1$, 则译码为$1$的概率为$\\beta(1-\\beta)^2+(1-\\beta)^3$;\\\\\n\\textcircled{4} 当$0<\\alpha<0.5$时, 若发送$0$, 则采用三次传输方案译码为$0$的概率大于采用单次传输方案译码为$0$的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464161,7 +467477,9 @@ "id": "018094", "content": "已知向量$\\overrightarrow{a}, \\overrightarrow{b}$满足$|\\overrightarrow{a}-\\overrightarrow{b}|=\\sqrt{3},|\\overrightarrow{a}+\\overrightarrow{b}|=|2 \\overrightarrow{a}-\\overrightarrow{b}|$, 则$|\\overrightarrow{b}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464181,7 +467499,9 @@ "id": "018095", "content": "底面边长为$4$的正四棱锥被平行于其底面的平面所截, 截去一个底面边长为 $2$, 高为$3$的正四棱锥, 所得棱台的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464201,7 +467521,9 @@ "id": "018096", "content": "已知直线$x-m y+1=0$与$\\odot C: (x-1)^2+y^2=4$交于$A, B$两点, 写出满足``$\\triangle ABC$面积为$\\dfrac{8}{5}$''的$m$的一个值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464221,7 +467543,9 @@ "id": "018097", "content": "已知函数$f(x)=\\sin (\\omega x+\\varphi)$, 如图$A, B$是直线$y=\\dfrac{1}{2}$与曲线$y=f(x)$的两个交点, 若$|AB|=\\dfrac{\\pi}{6}$, 则$f(\\pi)=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -0.5:2.5, samples = 100] plot (\\x,{sin(4*\\x/pi*180-120)});\n\\filldraw ({2*pi/3},0) circle (0.03) node [below right] {$\\dfrac{2\\pi}{3}$};\n\\draw (-1,0.5) -- (3,0.5);\n\\filldraw ({5*pi/24},0.5) circle (0.03) node [above left] {$A$};\n\\filldraw ({3*pi/8},0.5) circle (0.03) node [above right] {$B$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464241,7 +467565,9 @@ "id": "018098", "content": "记$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$, 已知三角形$\\triangle ABC$的面积为$\\sqrt{3}$, 点$D$为$BC$的中点, 且$AD=1$.\\\\\n(1) 若$\\angle ADC=\\dfrac{\\pi}{3}$, 求$\\tan B$;\\\\\n(2) 若$b^2+c^2=8$, 求$b$和$c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464261,7 +467587,9 @@ "id": "018099", "content": "$\\{a_n\\}$为等差数列, $b_n=\\begin{cases}a_n-6, & n \\text{为奇数},\\\\ 2 a_n, & n \\text{为偶数}.\\end{cases}$ 记$S_n, T_n$为$\\{a_n\\},\\{b_n\\}$的前$n$项和, $S_4=32$, $T_3=16$.\\\\\n(1) 求$\\{a_n\\}$的通项公式;\\\\\n(2) 证明: 当$n>5$时, $T_n>S_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464281,7 +467609,9 @@ "id": "018100", "content": "某研究小组经过研究发现某种疾病的患病者与未患病者的某项医学指标有明显差异, 经过大量调查, 得到如下的患病者和未患病者该指标的频率分布直方图:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.13, yscale = 130]\n\\draw [->] (90,0) -- (91,0) -- (91.5,0.001) -- (92.5,-0.001) -- (93,0) -- (135,0) node [below] {指标};\n\\draw [->] (90,0) -- (90,0.045) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {95/0.002,100/0.012,105/0.034,110/0.036,115/0.04,120/0.04,125/0.036}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (5,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {95/0.002,100/0.012,105/0.034,115/0.04,125/0.036}\n{\\draw [dashed] (\\i,\\j) -- (90,\\j) node [left] {$\\k$};};\n\\draw (130,0) node [below] {$130$};\n\\draw (112.5,-0.004) node [below] {患病者};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, xscale = 0.13, yscale = 130]\n\\draw [->] (65,0) -- (66,0) -- (66.5,0.001) -- (67.5,-0.001) -- (68,0) -- (110,0) node [below] {指标};\n\\draw [->] (65,0) -- (65,0.045) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {70/0.038,75/0.04,80/0.04,85/0.036,90/0.034,95/0.01,100/0.002}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (5,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {70/0.038,75/0.04,85/0.036,90/0.034,95/0.01,100/0.002}\n{\\draw [dashed] (\\i,\\j) -- (65,\\j) node [left] {$\\k$};};\n\\draw (105,0) node [below] {$130$};\n\\draw (87.5,-0.004) node [below] {未患病者};\n\\end{tikzpicture}\n\\end{center}\n利用该指标制定一个检测标准, 需要确定临界值$c$, 将该指标大于$c$的人判定为阳性, 小于或等于$c$的人判定为阴性, 此检测标准的漏诊率是将患病者判为阴性的概率, 记为$p(c)$; 误诊率是将未患病者判定为阳性的概率, 记为$q(c)$. 假设数据在组内平均分布, 以事件发生的频率作为相应事件发生的概率.\\\\\n(1) 当$p(c)=0.5 \\%$时, 求临界值$c$和误诊率$q(c)$;\\\\\n(2) 设函数$f(c)=p(c)+q(c)$, 当$c \\in[95,105]$时, 求$f(c)$的解析式, 并求$f(c)$在区间$[95,105]$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464301,7 +467631,9 @@ "id": "018101", "content": "在三棱锥$A-BCD$中, $DA=DB=DC$, $BD \\perp CD$, $\\angle ADB=\\angle ADC=60^{\\circ}$, 已知$E$为$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,-2) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [above] {$E$} coordinate (E);\n\\draw (E) ++ (0,{sqrt(2)},0) node [above] {$A$} coordinate (A);\n\\draw ($(A)+(E)-(D)$) node [right] {$F$} coordinate (F);\n\\draw (A)--(D)--(B)--(F)--cycle(A)--(B);\n\\draw [dashed] (D)--(C)--(B)(C)--(A)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $BC \\perp DA$;\\\\\n(2) 点$F$满足$\\overrightarrow{EF}=\\overrightarrow{DA}$, 求二面角$D-AB-F$的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464321,7 +467653,9 @@ "id": "018102", "content": "双曲线$C$中心为坐标原点, 左焦点$F_1(-2 \\sqrt{5}, 0)$, 离心率为$\\sqrt{5}$.\\\\\n(1) 求$C$的方程;\\\\\n(2) 记$C$的左、右顶点分别为$A_1, A_2$, 过点$B(-4,0)$的直线与$C$的左支交于$M, N$两点, $M$在第二象限, 直线$MA_1$与$NA_2$交于点$P$, 证明: $P$在定直线上.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464341,7 +467675,9 @@ "id": "018103", "content": "(1) 证明: 当$0=latex, node distance = 10pt]\n\\node [draw, rounded corners] (start) {开始};\n\\node [draw, below = of start] (init) {$n=1$, $A=1$, $B=2$};\n\\node [draw, diamond, below = of init, aspect = 2] (judge) {$n\\le 3$?};\n\\node [draw, below = of judge] (step1) {$A=A+B$};\n\\node [draw, below = of step1] (step2) {$B=A+B$};\n\\node [draw, below = of step2] (step3) {$n=n+1$};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of step3] (output) {输出$B$};\n\\node [draw, rounded corners, below = of output] (end) {结束};\n\\coordinate [right = 15pt of judge] (stepx);\n\\coordinate [left = 15pt of step3] (stepy);\n\\foreach \\i/\\j in {start/init,init/judge,step1/step2,step2/step3,output/end}\n{\\draw [->] (\\i)--(\\j);};\n\\draw [->] (judge) -- node [right] {是} (step1);\n\\draw (judge) -- (stepx) node [above, midway] {否};\n\\draw [->] (stepx) -- (stepx|-output) -> (output);\n\\draw [->] (step3) -- (stepy) -- (stepy|-judge) -> (judge);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$21$}{$34$}{$55$}{$89$}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464421,7 +467763,9 @@ "id": "018107", "content": "向量$|\\overrightarrow {a}|=|\\overrightarrow {b}|=1$, $|\\overrightarrow {c}|=\\sqrt{2}$, 且$\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$, 则$\\cos \\langle\\overrightarrow {a}-\\overrightarrow {c}, \\overrightarrow {b}-\\overrightarrow {c}\\rangle=$\\bracket{20}.\n\\fourch{$-\\dfrac{1}{5}$}{$-\\dfrac{2}{5}$}{$\\dfrac{2}{5}$}{$\\dfrac{4}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464441,7 +467785,9 @@ "id": "018108", "content": "已知正项等比数列$\\{a_n\\}$中, $a_1=1$, $S_n$为$\\{a_n\\}$前$n$项和, $S_5=5S_3-4$, 则$S_4=$\\bracket{20}.\n\\fourch{$7$}{$9$}{$15$}{$30$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464461,7 +467807,9 @@ "id": "018109", "content": "有$50$加人报名足球倶乐部, $60$人报名乒乓球倶乐部, $70$人报名足球或乒乓球倶乐部, 若已知某人报足球倶乐部, 则其报乒乓球倶乐部的概率为\\bracket{20}.\n\\fourch{$0.8$}{$0.4$}{$0.2$}{$0.1$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464481,7 +467829,9 @@ "id": "018110", "content": "``$\\sin ^2 \\alpha+\\sin ^2 \\beta=1$''是``$\\sin \\alpha+\\cos \\beta=0$''的\\bracket{20}.\n\\twoch{充分条件但不是必要条件}{必要条件但不是充分条件}{充要条件}{既不是充分条件也不是必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464501,7 +467851,9 @@ "id": "018111", "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的离心率为$\\sqrt{5}$, 其中一条渐近线与圆$(x-2)^2+(y-3)^2=1$交于$A, B$两点, 则$|AB|=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{\\sqrt{5}}{5}$}{$\\dfrac{2 \\sqrt{5}}{5}$}{$\\dfrac{4 \\sqrt{5}}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464521,7 +467873,9 @@ "id": "018112", "content": "有五名志愿者参加社区服务, 共服务星期六、星期天两天, 每天从中任选两人参加服务, 则恰有$1$人连续参加两天服务的选择种数为\\bracket{20}.\n\\fourch{$120$}{$60$}{$40$}{$30$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464541,7 +467895,9 @@ "id": "018113", "content": "已知$f(x)$为函数$y=\\cos (2 x+\\dfrac{\\pi}{6})$向左平移$\\dfrac{\\pi}{6}$个单位所得函数, 则$y=f(x)$与$y=\\dfrac{1}{2} x-\\dfrac{1}{2}$的交点个数为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464561,7 +467917,9 @@ "id": "018114", "content": "在四棱锥$P-ABCD$中, 底面$ABCD$为正方形, $AB=4$, $PC=PD=3$, $\\angle PCA=45^{\\circ}$, 则$\\triangle PBC$的面积为\\bracket{20}.\n\\fourch{$2 \\sqrt{2}$}{$3 \\sqrt{2}$}{$4 \\sqrt{2}$}{$5 \\sqrt{2}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464581,7 +467939,9 @@ "id": "018115", "content": "已知椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{6}=1$, $F_1$、$F_2$为两个焦点, $O$为原点, $P$为椭圆上一点, $\\cos \\angle F_1PF_2=\\dfrac{3}{5}$, 则$|PO|=$\\bracket{20}.\n\\fourch{$\\dfrac{2}{5}$}{$\\dfrac{\\sqrt{30}}{2}$}{$\\dfrac{3}{5}$}{$\\dfrac{\\sqrt{35}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464601,7 +467961,9 @@ "id": "018116", "content": "若$y=(x-1)^2+a x+\\sin (x+\\dfrac{\\pi}{2})$为偶函数, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464621,7 +467983,9 @@ "id": "018117", "content": "设$x, y$满足约束条件$\\begin{cases}-2x+3y\\le 3, \\\\ 3x-2y\\le 3,\\\\ x+y\\ge 1,\\end{cases}$ $z=3 x+2 y$, 则$z$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464641,7 +468005,9 @@ "id": "018118", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E, F$分别为棱$CD, A_1B_1$的中点, 则以$EF$为直径的球面与正方体的所有棱的交点总数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464661,7 +468027,9 @@ "id": "018119", "content": "$\\triangle ABC$中, $\\angle BAC=60^{\\circ}$, $AB=2$, $BC=\\sqrt{6}$, $AD$平分$\\angle BAC$与$BC$交于点$D$, 则$AD=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -464681,7 +468049,9 @@ "id": "018120", "content": "已知数列$\\{a_n\\}$中, $a_2=1$, 设$S_n$为$\\{a_n\\}$的前$n$项和, $2S_n=n a_n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求数列$\\{\\dfrac{a_{n+1}}{2^n}\\}$的前$n$项和$T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464701,7 +468071,9 @@ "id": "018121", "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, $AA_1=2$, $A_1C\\perp$底面$ABC$, $\\angle ACB=90^{\\circ}$, 点$A$到平面$BCC_1B_1$的距离为$1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,0,1) node [below] {$A$} coordinate (A);\n\\draw ({sqrt(3)/sqrt(2)},0,{sqrt(3)/sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)},0) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)+(B)-(A)$) node [right] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)+(C)-(A)$) node [above] {$C_1$} coordinate (C_1);\n\\draw (A)--(B)--(B_1)--(A_1)--cycle(A_1)--(C_1)--(B_1);\n\\draw (A)--(B_1);\n\\draw [dashed] (A)--(C)--(B)(C)--(C_1)(C)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1C=AC$;\\\\\n(2) 若$A_1A$到$B_1B$的距离为$2$, 求$AB_1$与平面$BCC_1B_1$所成角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464721,7 +468093,10 @@ "id": "018122", "content": "为研究某药物对小鼠的生长抑制作用, 将$40$只小鼠均分为两组, 分别为对照组 (不加药物) 和实验组 (加药物).\n(1) 从$40$只小鼠中任取两只小鼠, 设对照组小鼠数目为$X$, 求$X$的分布列和期望;\\\\\n(2) 测得$40$只小鼠的体重如下(单位: $g$)(已经按照从小到大排好).\\\\\n对照组:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline 17.3 & 18.4 & 20.1 & 20.4 & 21.5 & 23.2 & 24.6 & 24.8 & 25.0 & 25.4 \\\\\n\\hline 26.1 & 26.3 & 26.4 & 26.5 & 26.8 & 27.0 & 27.4 & 27.5 & 27.6 & 28.3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n实验组:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline 5.4 & 6.6 & 6.8 & 6.9 & 7.8 & 8.2 & 9.4 & 10.0 & 10.4 & 11.2 \\\\\n\\hline 14.4 & 17.3 & 19.2 & 20.2 & 23.6 & 23.8 & 24.5 & 25.1 & 25.2 & 26.0 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(I) 求$40$只小鼠体重的中位数$m$, 并完成下面$2 \\times 2$列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline &$0$)交于$A, B$两点, 且$|AB|=4 \\sqrt{15}$.\\\\\n(1) 求$p$的值;\\\\\n(2) $F$为$y^2=2 p x$的焦点, $M$、$N$为抛物线上两点且$\\overrightarrow{MF} \\cdot \\overrightarrow{NF}=0$, 求$\\triangle MNF$面积的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464761,7 +468138,9 @@ "id": "018124", "content": "已知函数$f(x)=a x-\\dfrac{\\sin x}{\\cos ^3 x}$, $x \\in(0, \\dfrac{\\pi}{2})$.\\\\\n(1) 若$a=8$, 讨论函数$f(x)$的单调性;\\\\\n(2) 若$f(x)<\\sin 2 x$恒成立, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464781,7 +468160,9 @@ "id": "018125", "content": "已知$P(2,1)$, 直线$l: \\begin{cases}x=2+t \\cos \\alpha, \\\\ y=1+t \\sin \\alpha\\end{cases}$($t$为参数), $l$与$x$轴, $y$轴正半轴交于$A, B$两点, $|PA| \\cdot|PB|=4$.\\\\\n(1) 求$\\alpha$的值;\\\\\n(2) 以原点为极点, $x$轴的正半轴为极轴建立极坐标系, 求$l$的极坐标方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -464801,7 +468182,10 @@ "id": "018126", "content": "已知$a>0$, $f(x)=2|x-a|-a$.\\\\\n(1) 求不等式$f(x)=latex, scale = 0.5]\n\\foreach \\i in {0,1,...,6}\n{\\draw [gray] (\\i,0) --++ (0,5);\n\\draw [gray] ({\\i+7},0) --++ (0,5);\n\\draw [gray] (\\i,-6) --++ (0,5);};\n\\foreach \\i in {0,1,...,5}\n{\\draw [gray] (0,\\i) --++ (6,0);\n\\draw [gray] (7,\\i) --++ (6,0);\n\\draw [gray] (0,{\\i-6}) --++ (6,0);};\n\\draw [ultra thick] (2,1) rectangle (4,4) (2,3) -- (4,3);\n\\draw [ultra thick] (9,1) --++ (2,0) --++ (0,2) --++ (-1,0) --++ (0,1) --++ (-1,0) --cycle;\n\\draw [ultra thick] (2,-2) rectangle (4,-4) (2,-3) --++ (2,0);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$24$}{$26$}{$28$}{$30$}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464882,7 +468272,9 @@ "id": "018130", "content": "已知$f(x)=\\dfrac{x \\mathrm{e}^x}{\\mathrm{e}^{a x}-1}$足偶函数, 则$a=$\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$1$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464902,7 +468294,9 @@ "id": "018131", "content": "设$O$为平面坐标系的坐标原点, 在区域$\\{(x, y) | 1 \\leq x^2+y^2 \\leq 4\\}$内随机取一点, 记该点为$A$, 则直线$OA$的倾斜角不大于$\\dfrac{\\pi}{4}$的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{8}$}{$\\dfrac{1}{6}$}{$\\dfrac{1}{4}$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464922,7 +468316,9 @@ "id": "018132", "content": "已知函数$f(x)=\\sin (\\omega x+\\varphi)$在区问$(\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3})$单调递增, 直线$x=\\dfrac{\\pi}{6}$和$x=\\dfrac{2 \\pi}{3}$为函数$y=f(x)$的图像的两条对称轴, 则$f(-\\dfrac{5 \\pi}{12})=$\\bracket{20}.\n\\fourch{$-\\dfrac{\\sqrt{3}}{2}$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464942,7 +468338,9 @@ "id": "018133", "content": "甲乙两位同学从$6$种课外读物中各自选读$2$种, 则这两人选读的课外读物中恰有$1$种相同的选法共有\\bracket{20}.\n\\fourch{$30$ 种}{$60$ 种}{$120$ 种}{$240$ 种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464962,7 +468360,9 @@ "id": "018134", "content": "已知圆锥$PO$的底面半径为$\\sqrt{3}$, $O$为底面圆心, $PA, PB$为圆锥的母线, $\\angle AOB=120^{\\circ}$, 若$\\triangle PAB$的面积等于$\\dfrac{9 \\sqrt{3}}{4}$, 则该圆锥的体积为\\bracket{20}\n\\fourch{$\\pi$}{$\\sqrt{6} \\pi$}{$3 \\pi$}{$3 \\sqrt{6} \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -464982,7 +468382,9 @@ "id": "018135", "content": "已知$\\triangle ABC$为等腰直角三角形, $AB$为斜边, $\\triangle ABD$为等边三角形, 若二面角$C-AB-D$为$150^{\\circ}$, 则直线$CD$与平面$ABC$所成角的正切值为\\bracket{20}.\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{\\sqrt{2}}{5}$}{$\\dfrac{\\sqrt{3}}{5}$}{$\\dfrac{2}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465002,7 +468404,9 @@ "id": "018136", "content": "已知$\\{a_n\\}$是公差为$\\dfrac{2 \\pi}{3}$的等差数列, 若$\\{\\cos a_n | n \\in \\mathbf{N}^*\\}=\\{a, b\\}$, 则$a b=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$-\\dfrac{1}{2}$}{$0$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465022,7 +468426,9 @@ "id": "018137", "content": "已知双曲线$C: x^2-\\dfrac{y^2}{9}=1$, 则以下可能为双曲线$C$的弦的中点的是\\bracket{20}.\n\\fourch{$(1,1)$}{$(-1,2)$}{$(1,3)$}{$(-1,-4)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465042,7 +468448,9 @@ "id": "018138", "content": "已知圆$O$半径为$1, PA$与圆$O$相切, 切点为$A,|OP|=\\sqrt{2}$, 过点$P$的直线与圆$O$交于$B, C$两点, $D$为$BC$中点, 则$\\overrightarrow{PD} \\cdot \\overrightarrow{PA}$的最大值为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}+\\dfrac{\\sqrt{2}}{2}$}{$1+\\dfrac{\\sqrt{2}}{2}$}{$1+\\sqrt{2}$}{$2+\\sqrt{2}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465062,7 +468470,9 @@ "id": "018139", "content": "已知点$A(1, \\sqrt{5})$在抛物线$C: y^2=2 p x$上, 则$A$到$C$的准线的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465082,7 +468492,9 @@ "id": "018140", "content": "若$x, y$满足约束条件$\\begin{cases}x-3 y \\leq-1, \\\\ x+2 y \\leq 9, \\\\ 3 x+y \\geq 7,\\end{cases}$ 则$z=2 x-y$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465102,7 +468514,9 @@ "id": "018141", "content": "已知$\\{a_n\\}$为等比数列, $a_2 a_4 a_5=a_3 a_6$, $a_9 a_{10}=-8$, 则$a_7=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465122,7 +468536,9 @@ "id": "018142", "content": "已知$a \\in(0,1)$, 函数$f(x)=a^x+(1+a)^x$是$(0,+\\infty)$上的增函数, $a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465142,7 +468558,9 @@ "id": "018143", "content": "某厂为比较甲乙两种工艺对橡胶产品伸缩率的处理效应, 进行$10$次配对试验, 每次配对试验选用材质相同的两个橡胶产品, 随机地选其中一个用甲工艺处理, 另一个用乙工艺处理, 测量处理后的橡胶产品的伸缩率, 甲.乙两种工艺处理后的橡胶产品的伸缩率分别记为$x_i$, $y_i$($i=1,2, \\cdots,10$), 试验结果如下表\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 试验序号$i$& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 伸缩率$x_i$& 545 & 533 & 551 & 522 & 575 & 544 & 541 & 568 & 596 & 548 \\\\\n\\hline 伸缩率$y_i$& 536 & 527 & 543 & 530 & 560 & 533 & 522 & 550 & 576 & 536 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n$z_i=x_i-y_i$($i=1,2, \\cdots, 10$), 记$z_1, z_2, \\cdots, z_{10}$的样本平均数为$\\overline {z}$, 样本方差为$s^2$.\\\\\n(1) 求$\\overline {z}$, $s^2$;\\\\\n(2) 判断甲工艺处理后的橡胶产品的伸缩率较乙工艺处理后的橡胶产品的伸缩率是否有显著提高(如果$\\overline {z} \\geq 2 \\sqrt{\\dfrac{s^2}{10}}$, 则认为甲工艺处理后的橡胶产品的伸缩率较乙工艺处理后的橡胶产品的伸缩率有显著提高, 否则不认为有显著提高).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465162,7 +468580,9 @@ "id": "018144", "content": "$\\triangle ABC$中, $\\angle A=120^{\\circ}$, $AB=2$, $AC=1$.\\\\\n(1) 求$\\sin \\angle ABC$;\\\\\n(2) 若$D$为$BC$上的一点, $\\angle BAD=90^{\\circ}$, 求$\\triangle BAD$的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465182,7 +468602,9 @@ "id": "018145", "content": "如图, 三棱锥$P-ABC$中, $\\angle ABC=90^{\\circ}$, $AB=2$, $BC=2 \\sqrt{2}$, $PB=PC=\\sqrt{6}$, $AD=\\sqrt{5} OD$, $BF \\perp AO$, $O, D, E$分别为$BC, PB, AP$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-45:0.5cm)}, scale = 1.5]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw ({sqrt(2)},{sqrt(3)},-1) node [above] {$P$} coordinate (P);\n\\draw (A)--(C)--(P)--(B)--cycle;\n\\draw [dashed] (B)--(C);\n\\draw (A)--(P);\n\\draw ($(B)!0.5!(C)$) node [above left] {$O$} coordinate (O);\n\\draw ($(A)!0.5!(C)$) node [below right] {$F$} coordinate (F);\n\\draw ($(P)!0.5!(B)$) node [left] {$D$} coordinate (D);\n\\draw ($(P)!0.5!(A)$) node [right] {$E$} coordinate (E);\n\\draw (A)--(D)--(E)--(F)(B)--(E);\n\\draw [dashed] (A)--(O)(B)--(F)--(O)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $EF\\parallel$平面$ADO$;\\\\\n(2) 平面$ADO \\perp$平面$BEF$;\\\\\n(3) 求二面角$D-AO-C$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465202,7 +468624,9 @@ "id": "018146", "content": "已知曲线$C$的方程为$\\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1$($a>b>0$), 离心率为$\\dfrac{\\sqrt{5}}{3}$, 曲线过点$A(-2,0)$.\\\\\n(1) 求曲线$C$的方程;\\\\\n(2) 过点$(-2,3)$的直线交曲线$C$于$P, Q$两点, 直线$AP, AQ$与$y$轴交于$M, N$两点, 证明: 线段$MN$的中点是定点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465222,7 +468646,9 @@ "id": "018147", "content": "已知函数$f(x)=(\\dfrac{1}{x}+a) \\ln (x+1)$.\\\\\n(1) 若$a=-1$, 求$f(x)$在$(1, f(1))$处的切线方程;\\\\\n(2) 是否存在$a, b$使$y=f(\\dfrac{1}{x})$图像关于直线$x=b$轴对称?\\\\\n(3) 若$f(x)$在$(0,+\\infty)$上存在极值点, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465242,7 +468668,9 @@ "id": "018148", "content": "在直角坐标系$x O y$中, 以坐标原点$O$为极点, $x$轴正半轴为极轴建立极坐标系, 曲线$C_1$的极坐标方程为$\\rho=2 \\sin \\theta(\\dfrac{\\pi}{4} \\leq \\theta \\leq \\dfrac{\\pi}{2})$, 曲线$C_2: \\begin{cases}x=2 \\cos \\alpha, \\\\ y=2 \\sin \\alpha\\end{cases}$($\\alpha$为参数, $\\dfrac{\\pi}{2}<\\alpha<\\pi$).\\\\\n(1) 写出$C_1$的直角坐标方程;\\\\\n(2) 若直线$y=x+m$既与$C_1$没有公共点, 也与$C_2$没有公共点, 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465262,7 +468690,9 @@ "id": "018149", "content": "已知$f(x)=2|x|+|x-2|$.\\\\\n(1) 求不等式$f(x) \\leq 6-x$的解集;\\\\\n(2) 在直角坐标系$x O y$中, 求不等式组$\\begin{cases}f(x) \\leq y, \\\\ x+y-6 \\leq 0\\end{cases}$所确定的平面区域的面积.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465282,7 +468712,9 @@ "id": "018150", "content": "已知集合$U=\\{1,2,3,4,5\\}$, $A=\\{1,3\\}$, $B=\\{1,2,4\\}$, 则$A \\cup(\\complement_UB)=$\\bracket{20}.\n\\fourch{$\\{1,3,5\\}$}{$\\{1,3\\}$}{$\\{1,2,4\\}$}{$\\{1,2,4,5\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465302,7 +468734,9 @@ "id": "018151", "content": "``$a^2=b^2$''是``$a^2+b^2=2 a b$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465322,7 +468756,9 @@ "id": "018152", "content": "若$a=1.01^{0.5}$, $b=1.01^{0.6}$, $c=0.6^{0.5}$, 则$a, b, c$的大小关系为\\bracket{20}.\n\\fourch{$c>a>b$}{$c>b>a$}{$a>b>c$}{$b>a>c$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465342,7 +468778,9 @@ "id": "018153", "content": "函数$f(x)$的图象如下图所示, 则$f(x)$的解析式可能为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -4:4, samples = 100] plot (\\x,{2*cos(\\x/pi*180)/(\\x*\\x+1)});\n\\draw (2,0) node [above] {$2$} (-2,0) node [above] {$-2$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{2(\\mathrm{e}^x-\\mathrm{e}^{-x})}{x^2+2}$}{$\\dfrac{2 \\sin x}{x^2+1}$}{$\\dfrac{2(\\mathrm{e}^x+\\mathrm{e}^{-x})}{x^2+2}$}{$\\dfrac{2 \\cos x}{x^2+1}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465362,7 +468800,9 @@ "id": "018154", "content": "已知函数$f(x)$的一条对称轴为直线$x=2$, 一个周期为$4$, 则$f(x)$的解析式可能为\\bracket{20}.\n\\fourch{$\\sin (\\dfrac{\\pi}{2} x)$}{$\\cos (\\dfrac{\\pi}{2} x)$}{$\\sin (\\dfrac{\\pi}{4} x)$}{$\\cos (\\dfrac{\\pi}{4} x)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465382,7 +468822,9 @@ "id": "018155", "content": "已知$\\{a_n\\}$为等比数列, $S_n$为数列$\\{a_n\\}$的前$n$项和, $a_{n+1}=2S_n+2$, 则$a_4$的值为\\bracket{20}.\n\\fourch{$3$}{$18$}{$54$}{$152$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465402,7 +468844,9 @@ "id": "018156", "content": "调查某种花萝长度和花瓣长度, 所得数据如图所示, 其中相关系数$r=0.8245$, 下列说法正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\draw [->] (0,0) -- (1.2,0) node [below] {花瓣长度};\n\\draw [->] (0,0) -- (0,1.2) node [left] {花萼长度};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.37/0.23,0.35/0.34,0.54/0.63,0.42/0.44,1.00/0.90,0.77/0.88,0.75/0.75,0.96/0.82,0.65/0.56,0.71/0.61,0.18/0.07,0.42/0.58,0.50/0.80,0.43/0.36,0.50/0.57,0.56/0.50,0.75/0.79,0.55/0.33,0.07/0.21,0.14/0.21,0.43/0.25,0.75/0.52,0.64/0.82,0.72/0.67,0.33/0.18,0.83/0.69,0.56/0.69,0.39/0.21,0.34/0.49,0.93/0.68}\n{\\filldraw (\\i,\\j) circle (0.01);};\n\\draw [dashed, domain = 0.1:1.1] plot (\\x,{0.83*\\x+0.067});\n\\end{tikzpicture}\n\\end{center}\n\\onech{花瓣长度和花萼长度没有相关性}{花瓣长度和花萼长度呈负相关}{花瓣长度和花萼长度呈正相关}{若从样本中抽取一部分, 则这部分的相关系数一定是$0.8245$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465422,7 +468866,9 @@ "id": "018157", "content": "在三棱锥$P-ABC$中, 线段$PC$上的点$M$满足$PM=\\dfrac{1}{3} PC$, 线段$PB$上的点$N$满足$PN=\\dfrac{2}{3} PB$, 则三棱锥$P-AMN$和三棱锥$P-ABC$的体积之比为\\bracket{20}.\n\\fourch{$\\dfrac{1}{9}$}{$\\dfrac{2}{9}$}{$\\dfrac{1}{3}$}{$\\dfrac{4}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465442,7 +468888,9 @@ "id": "018158", "content": "双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左右焦点分别为$F_1$和$F_2$, 过$F_2$作其中一条渐近线的垂线, 垂足为$P$. 已知$PF_2=2$, 直线$PF_1$的斜率为$\\dfrac{\\sqrt{2}}{4}$, 则双曲线的方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{8}-\\dfrac{y^2}{4}=1$}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{8}=1$}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{2}=1$}{$\\dfrac{x^2}{2}-\\dfrac{y^2}{4}=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465462,7 +468910,9 @@ "id": "018159", "content": "已知$\\mathrm{i}$是虚数单位, 化简$\\dfrac{5+14 \\mathrm{i}}{2+3 \\mathrm{i}}$的结果为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465482,7 +468932,9 @@ "id": "018160", "content": "在$(2 x^3-\\dfrac{1}{x})^6$的展开式中, $x^2$的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465502,7 +468954,9 @@ "id": "018161", "content": "过原点的一条直线与圆$C: (x+2)^2+y^2=3$相切, 交曲线$y^2=2 p x$($p>0$)于点$P$, 若$OP=8$, 则$p$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465522,7 +468976,9 @@ "id": "018162", "content": "甲乙丙三个盒子中装有一定量的黑球和白球, 其总数之比为$5: 4: 6$, 这三个盒子中黑球占总数得比例分别为$40 \\%, 25 \\%, 50 \\%$, 现从三个盒子中各取一个球, 取到的三个球都是黑球的概率为\\blank{50}; 将三个盒子混合在一起后任取一个球, 是白球的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465542,7 +468998,9 @@ "id": "018163", "content": "在$\\triangle ABC$中, $\\angle A=60^{\\circ}$, $BC=1$, 点$D$为$BC$的中点, 点$E$为$CD$的中点, 若设$\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{AC}=\\overrightarrow {b}$, 则$\\overrightarrow{AE}$可用$\\overrightarrow {a}, \\overrightarrow {b}$表示为\\blank{50}; 若$\\overrightarrow{BF}=\\dfrac{1}{3} \\overrightarrow{BC}$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{AF}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465562,7 +469020,9 @@ "id": "018164", "content": "若函数$f(x)=a x^2-2 x-|x^2-a x+1|$有且仅有两个零点, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465582,7 +469042,9 @@ "id": "018165", "content": "在$\\triangle ABC$中, 角$A, B, C$所对的边分別是$a, b, c$. 已知$a=\\sqrt{39}$, $b=2$, $\\angle A=120^{\\circ}$.\n(1) 求$\\sin B$的值;\\\\\n(2) 求$c$的值;\\\\\n(3) 求$\\sin (B-C)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465602,7 +469064,9 @@ "id": "018166", "content": "三棱台$ABC-A_1B_1C_1$中, 已知$A_1A \\perp$平面$ABC$, $AB \\perp AC$, $AB=AC=AA_1=2$, $A_1C_1=1$, $N$为线段$AB$的中点, $M$为线段$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (0,0,1) node [left] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (1,0,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw ($(A)!0.5!(B)$) node [left] {$N$} coordinate (N);\n\\draw (B)--(C)--(C_1)--(B_1)--cycle(B_1)--(A_1)--(C_1);\n\\draw [dashed] (B)--(A)--(C)(A)--(A_1);\n\\draw [dashed] (A_1)--(N)(C_1)--(A)--(M);\n\\draw (C_1)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1N\\parallel$平面$C_1MA$;\\\\\n(2) 求平面$C_1MA$与平面$ACC_1A_1$所成角的余弦值;\\\\\n(3) 求点$C$到平面$C_1MA$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465622,7 +469086,9 @@ "id": "018167", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右顶点分别为$A_1, A_2$, 右焦点为$F$, 且$|A_1F|=3$, $|A_2F|=1$.\\\\\n(1) 求椭圆$C$的方程及离心率;\\\\\n(2) 设点$P$是椭圆$C$上一动点 (不与顶点重合), 直线$A_2P$交$y$轴于点$Q$, 若三角形$A_1PQ$的面积是三角形$A_2FP$面积的二倍, 求直线$A_2P$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465642,7 +469108,9 @@ "id": "018168", "content": "已知$\\{a_n\\}$为等差数列, $a_3+a_5=16$, $a_5-a_3=4$.\\\\\n(1) 求$a_n$和$\\displaystyle\\sum_{i=2^{n-1}}^{2^n-1} a_i$;\\\\\n(2) 设$\\{b_n\\}$为等比数列, 当$2^{k-1} \\leq n \\leq 2^k-1$时, $b_k0$时, 证明: $f(x)>1$;\\\\\n(3) 证明: $\\dfrac{5}{6} \\leq \\ln (n !)-(n+\\dfrac{1}{2}) \\ln (n)+n \\leq 1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -465682,7 +469152,9 @@ "id": "018170", "content": "设全集$U=\\{1,2,3,4,5\\}$, 集合$M=\\{1,4\\}$, $N=\\{2,5\\}$, 则$N \\cup \\overline{M}=$\\bracket{20}.\n\\fourch{$\\{2,3,5\\}$}{$\\{1,3,4\\}$}{$\\{1,2,4,5\\}$}{$\\{2,3,4,5\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465702,7 +469174,9 @@ "id": "018171", "content": "$\\dfrac{5(1+\\mathrm{i}^3)}{(2+\\mathrm{i})(2-\\mathrm{i})}=$\\bracket{20}.\n\\fourch{$-1$}{$1$}{$1-\\mathrm{i}$}{$1+\\mathrm{i}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465722,7 +469196,9 @@ "id": "018172", "content": "已知向量$\\overrightarrow{a}=(3,1)$, $\\overrightarrow{b}=(2,2)$, 则$\\cos \\langle\\overrightarrow{a}+\\overrightarrow{b}, \\overrightarrow{a}-\\overrightarrow{b}\\rangle=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{17}$}{$\\dfrac{\\sqrt{17}}{17}$}{$\\dfrac{\\sqrt{15}}{15}$}{$\\dfrac{2 \\sqrt{5}}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465742,7 +469218,9 @@ "id": "018173", "content": "某校文艺部有$4$名学生, 其中高一、高二年级各$2$名. 从这$4$名学生中随机选$2$名组织校文艺汇演, 则这$2$名学生来自不同年级的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{6}$}{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{2}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465762,7 +469240,9 @@ "id": "018174", "content": "已知向量$S_n$为等差数列$\\{a_n\\}$的前$n$项和若$a_2+a_6=10$, $a_4 a_8=45$, 则$S_5=$\\bracket{20}.\n\\fourch{$25$}{$22$}{$20$}{$15$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465782,7 +469262,9 @@ "id": "018175", "content": "执行如图的程序框图, 则输出的$B=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, node distance = 10pt]\n\\node [draw, rounded corners] (start) {开始};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of start] (input) {输入$n=3$, $A=1$, $B=2$, $k=1$};\n\\node [draw, diamond, aspect = 2, below = of input] (switch) {$k\\le n$};\n\\node [draw, below = of switch] (step1) {$A=A+B$};\n\\node [draw, below = of step1] (step2) {$B=A+B$};\n\\node [draw, below = of step2] (step3) {$k=k+1$};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of step3] (output) {输出$B$};\n\\node [draw, rounded corners, below = of output] (end) {结束};\n\\foreach \\i/\\j in {start/input,input/switch,step1/step2,step2/step3,output/end}\n{\\draw [->] (\\i) -- (\\j);};\n\\draw [->] (switch) -- (step1) node [midway, right] {是};\n\\coordinate [right = 15pt of switch] (stepx);\n\\coordinate [left = 15pt of step3] (stepy);\n\\draw [->] (switch) -- (stepx) node [midway, above] {否} -- (stepx|-output) -- (output);\n\\draw [->] (step3) -- (stepy) -- (stepy|-switch) -- (switch);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$21$}{$34$}{$55$}{$89$}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465802,7 +469284,9 @@ "id": "018176", "content": "设$F_1, F_2$为椭圆$C: \\dfrac{x^2}{5}+y^2=1$的两个焦点, 点$P$在$C$上, 若$\\overrightarrow{PF_1} \\cdot \\overrightarrow{PF_2}=0$, 则$|PF_1| \\cdot|PF_2|=$\\bracket{20}.\n\\fourch{$1$}{$2$}{$4$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465822,7 +469306,9 @@ "id": "018177", "content": "曲线$y=\\dfrac{\\mathrm{e}^x}{x+1}$在点$(1, \\dfrac{\\mathrm{e}}{2})$处的切线方程为\\bracket{20}.\n\\fourch{$y=\\dfrac{\\mathrm{e}}{4} x$}{$y=\\dfrac{\\mathrm{e}}{2} x$}{$y=\\dfrac{\\mathrm{e}}{4} x+\\dfrac{\\mathrm{e}}{4}$}{$y=\\dfrac{\\mathrm{e}}{2} x+\\dfrac{3 \\mathrm{e}}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465842,7 +469328,9 @@ "id": "018178", "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的离心率为$\\sqrt{5}$, $C$的一条渐近线与圆$(x-2)^2+(y-3)^2=1$交于$A, B$两点, 则$|AB|=$\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{5}}{5}$}{$\\dfrac{2 \\sqrt{5}}{5}$}{$\\dfrac{3 \\sqrt{5}}{5}$}{$\\dfrac{4 \\sqrt{5}}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465862,7 +469350,9 @@ "id": "018179", "content": "在三棱锥$P-ABC$中, $\\triangle ABC$是边长为$2$的等边三角形, $PA=PB=2$, $PC=\\sqrt{6}$, 则该棱锥的体积为\\bracket{20}.\n\\fourch{$1$}{$\\sqrt{3}$}{$2$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465882,7 +469372,9 @@ "id": "018180", "content": "已知函数$f(x)=\\mathrm{e}^{-(x-1)^2}$. 记$a=f(\\dfrac{\\sqrt{2}}{2})$, $b=f(\\dfrac{\\sqrt{3}}{2})$, $c=f(\\dfrac{\\sqrt{6}}{2})$, 则\\bracket{20}.\n\\fourch{$b>c>a$}{$b>a>c$}{$c>b>a$}{$c>a>b$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465902,7 +469394,9 @@ "id": "018181", "content": "函数$y=f(x)$的图像由$y=\\cos (2 x+\\dfrac{\\pi}{6})$的图像向左平移$\\dfrac{\\pi}{6}$个单位长度得到, 则$y=f(x)$的图像与直线$y=\\dfrac{1}{2} x-\\dfrac{1}{2}$的交点个数为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -465922,7 +469416,9 @@ "id": "018182", "content": "记$S_n$为等比数列$\\{a_n\\}$的前$n$项和. 若$8S_6=7S_3$, 则$\\{a_n\\}$的公比为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465942,7 +469438,9 @@ "id": "018183", "content": "若$f(x)=(x-1)^2+a x+\\sin (x+\\dfrac{\\pi}{2})$为偶函数, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465962,7 +469460,9 @@ "id": "018184", "content": "若$x, y$满足约束条件$\\begin{cases}3 x-2 y \\leq 3, \\\\ -2 x+3 y \\leq 3,\\\\ x+y \\geq 1,\\end{cases}$ 则$z=3 x+2 y$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -465982,7 +469482,9 @@ "id": "018185", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $AB=4$, $O$为$AC_1$的中点, 若该正方体的棱与球$O$的球面有公共点, 则球$O$的半径的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -466002,7 +469504,9 @@ "id": "018186", "content": "记$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$, 已知$\\dfrac{b^2+c^2-a^2}{\\cos A}=2$.\\\\\n(1) 求$b c$;\\\\\n(2) 若$\\dfrac{a \\cos B-b \\cos A}{a \\cos B+b \\cos A}-\\dfrac{b}{c}=1$, 求$\\triangle ABC$的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466022,7 +469526,9 @@ "id": "018187", "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, $A_1C \\perp$平面$ABC$, $\\angle ACB=90^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (0,0,{sqrt(2)}) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)},0) node [above] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)+(C)-(A)$) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)+(B)-(A)$) node [right] {$B_1$} coordinate (B_1);\n\\draw (A)--(B)--(B_1)--(A_1)--cycle(B)--(A_1)--(C_1)--(B_1);\n\\draw [dashed] (A)--(C)--(B)(C)--(A_1)(C)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$ACC_1A_1 \\perp$平面$BB_1C_1C$;\\\\\n(2) 设$AB=A_1B$, $AA_1=2$, 求四棱锥$A_1-BB_1C_1C$的高.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466042,7 +469548,9 @@ "id": "018188", "content": "一项试验旨在研究臭氧效应, 试验方案如下: 选$40$只小白鼠, 随机地将其中$20$只分配到试验组, 另外$20$只分配到对照组, 试验组的小白鼠饲养在高浓度臭氧环境, 对照组的小白鼠饲养在正常环境, 一段时间后统计每只小白鼠体重的增加量(单位: $\\text{g}$). 试验结果如下\\\\\n对照组的小白鼠体重的增加量从小到大排序为:\n\\begin{center}\n\\begin{tabular}{cccccccccc}\n15.2 & 18.8 & 20.2 & 21.3 & 22.5 & 23.2 & 25.8 & 26.5 & 27.5 & 30.1 \\\\\n32.6 & 34.3 & 34.8 & 35.6 & 35.6 & 35.8 & 36.2 & 37.3 & 40.5 & 43.2\n\\end{tabular}\n\\end{center}\n试验组的小白鼠体重的增加量从小到大排序为:\n\\begin{center}\n\\begin{tabular}{cccccccccc}\n7.8 & 9.2 & 11.4 & 12.4 & 13.2 & 15.5 & 16.5 & 18.0 & 18.8 & 19.2 \\\\ 19.8 & 20.2 & 21.6 & 22.8 & 23.6 & 23.9 & 25.1 & 28.2 & 32.3 & 36.5\n\\end{tabular}\n\\end{center}\n(1) 计算试验组的样本平均数;\\\\\n(2) (I) 求$40$只小白鼠体重的增加量的中位数$m$, 再分别统计两样本中小于$m$与不小于$m$的数据的个数, 完成如下列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline &$0$) 交于$A, B$两点, $|AB|=4 \\sqrt{15}$.\\\\\n(1) 求$p$;\\\\\n(2) 设$F$为$C$的焦点, $M, N$为$C$上两点, 且$\\overrightarrow{FM} \\cdot \\overrightarrow{FN}=0$, 求$\\triangle MFN$面积的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466102,7 +469614,9 @@ "id": "018191", "content": "已知$P(2,1)$, 直线$l: \\begin{cases}x=2+t \\cos \\alpha, \\\\ y=1+t \\sin \\alpha\\end{cases}$($t$为参数), $l$与$x$轴, $y$轴正半轴交于$A, B$两点, $|PA| \\cdot|PB|=4$.\\\\\n(1) 求$\\alpha$的值;\\\\\n(2) 以原点为极点, $x$轴的正半轴为极轴建立极坐标系, 求$l$的极坐标方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466122,7 +469636,9 @@ "id": "018192", "content": "已知$a>0$, $f(x)=2|x-a|-a$.\\\\\n(1) 求不等式$f(x)=latex, scale = 0.5]\n\\foreach \\i in {0,1,...,6}\n{\\draw [gray] (\\i,0) --++ (0,5);\n\\draw [gray] ({\\i+7},0) --++ (0,5);\n\\draw [gray] (\\i,-6) --++ (0,5);};\n\\foreach \\i in {0,1,...,5}\n{\\draw [gray] (0,\\i) --++ (6,0);\n\\draw [gray] (7,\\i) --++ (6,0);\n\\draw [gray] (0,{\\i-6}) --++ (6,0);};\n\\draw [ultra thick] (2,1) rectangle (4,4) (2,3) -- (4,3);\n\\draw [ultra thick] (9,1) --++ (2,0) --++ (0,2) --++ (-1,0) --++ (0,1) --++ (-1,0) --cycle;\n\\draw [ultra thick] (2,-2) rectangle (4,-4) (2,-3) --++ (2,0);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$24$}{$26$}{$28$}{$30$}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466203,7 +469725,9 @@ "id": "018196", "content": "在$\\triangle ABC$中, 内角$A, B, C$的对边分别是$a, b, c$, 若$a \\cos B-b \\cos A=c$, 且$C=\\dfrac{\\pi}{5}$, 则$\\angle B=$\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{10}$}{$\\cdot \\dfrac{\\pi}{5}$}{$\\dfrac{3 \\pi}{10}$}{$\\dfrac{2 \\pi}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466223,7 +469747,9 @@ "id": "018197", "content": "已知函数$f(x)=\\dfrac{x \\mathrm{e}^x}{\\mathrm{e}^{a x}-1}$是偶函数, 则实数$a=$\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$1$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466243,7 +469769,9 @@ "id": "018198", "content": "正方形$ABCD$的边长是$2$, $E$是$AB$的中点, 则$\\overrightarrow{EC} \\cdot \\overrightarrow{ED}=$\\bracket{20}.\n\\fourch{$\\sqrt{5}$}{$3$}{$2 \\sqrt{5}$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466263,7 +469791,9 @@ "id": "018199", "content": "已知$O$是平面直角坐标系的原点, 在区域$\\{(x, y) | 1 \\leq x^2+y^2 \\leq 4\\}$内随机取一点$A$, 则直线$OA$的倾斜角不大于$\\dfrac{\\pi}{4}$的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{8}$}{$\\dfrac{1}{6}$}{$\\dfrac{1}{4}$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466283,7 +469813,9 @@ "id": "018200", "content": "函数$f(x)=x^3+a x+2$存在$3$个零点, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty,-2)$}{$(-\\infty,-3)$}{$(-4,-1)$}{$(-3,0)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466303,7 +469835,9 @@ "id": "018201", "content": "某学校举办作文比赛, 共$6$个主题, 每位参赛同学从中随机抽取一个主题准备作文, 则甲、乙两位参赛同学抽到不同主题概率为\\bracket{20}.\n\\fourch{$\\dfrac{5}{6}$}{$\\dfrac{2}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{1}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466323,7 +469857,9 @@ "id": "018202", "content": "函数$f(x)=\\sin (\\omega x+\\varphi)$在区间$(\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3})$上单调递增, 直线$x=\\dfrac{\\pi}{6}$和$x=\\dfrac{2 \\pi}{3}$是函数$y=f(x)$图像的两条对称轴, 则$f(-\\dfrac{5 \\pi}{12})=$\\bracket{20}.\n\\fourch{$-\\dfrac{\\sqrt{3}}{2}$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466343,7 +469879,9 @@ "id": "018203", "content": "已知实数$x, y$满足$x^2+y^2-4 x-2 y-4=0$, 则$x-y$的最大值是\\bracket{20}.\n\\fourch{$1+\\dfrac{3 \\sqrt{2}}{2}$}{4}{$1+3 \\sqrt{2}$}{7}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466363,7 +469901,9 @@ "id": "018204", "content": "设$A, B$为双曲线$x^2-\\dfrac{y^2}{9}=1$上两点, 下列四个点中, 可以为线段$AB$中点的是\\bracket{20}.\n\\fourch{$(1,1)$}{$(-1,2)$}{$(1,3)$}{$(-1,-4)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -466383,7 +469923,9 @@ "id": "018205", "content": "已知点$A(1, \\sqrt{5})$在抛物线$C: y^2=2 p x$上, 则$A$到$C$的准线的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -466403,7 +469945,9 @@ "id": "018206", "content": "若$\\theta \\in(0, \\dfrac{\\pi}{2}), \\tan \\theta=\\dfrac{1}{2}$, 则$\\sin \\theta-\\cos \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -466423,7 +469967,9 @@ "id": "018207", "content": "若$x, y$满足约束条件$\\begin{cases}x-3 y \\leq-1 \\\\x+2 y \\leq 9, \\\\\n3 x+y \\geq 7,\\end{cases}$则$z=2 x-y$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -466443,7 +469989,9 @@ "id": "018208", "content": "已知点$S, A, B, C$均在半径为$2$的球面上, $\\triangle ABC$是边长为$3$的等边三角形, $SA \\perp$平面$ABC$, 则$SA=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -466463,7 +470011,9 @@ "id": "018209", "content": "某厂为比较甲乙两种工艺对橡胶产品伸缩率的处理效应, 进行$10$次配对试验, 每次配对试验选用材质相同的两个橡胶产品, 随机地选其中一个用甲工艺处理, 另一个用乙工艺处理, 测量处理后的橡胶产品的伸缩率. 甲、乙两种工艺处理后的橡胶产品的伸缩率分别记为$x_i$, $y_i$($i=1,2, \\cdots, 10$). 试验结果如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 试验序号$i$& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 伸缩率$x_i$& 545 & 533 & 551 & 522 & 575 & 544 & 541 & 568 & 596 & 548 \\\\\n\\hline 伸缩率$y_i$& 536 & 527 & 543 & 530 & 560 & 533 & 522 & 550 & 576 & 536 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n记$z_i=x_i-y_i$($i=1,2, \\cdots, 10$), 记$z_1, z_2, \\cdots, z_{10}$的样本平均数为$\\overline {z}$, 样本方差为$s^2$.\\\\\n(1) 求$\\overline {z}$, $s^2$;\\\\\n(2) 判断甲工艺处理后的橡胶产品的伸缩率较乙工艺处理后的橡胶产品的伸缩率是否有显著提高(如果$\\overline {z} \\geq 2 \\sqrt{\\dfrac{s^2}{10}}$, 则认为甲工艺处理后的橡胶产品的伸缩率较乙工艺处理后的橡胶产品的伸缩率有显著提高, 否则不认为有显著提高).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466483,7 +470033,9 @@ "id": "018210", "content": "记$S_n$为等差数列$\\{a_n\\}$的前$n$项和, 已知$a_2=11$, $S_{10}=40$.\n(1) 求$\\{a_n\\}$的通项公式;\\\\\n(2) 求数列$\\{|a_n|\\}$的前$n$项和$T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466503,7 +470055,9 @@ "id": "018211", "content": "如图, 在三棱锥$P-ABC$中, $AB \\perp BC$, $AB=2$, $BC=2 \\sqrt{2}$, $PB=PC=\\sqrt{6}$, $BP$、$AP$、$BC$的中点分别为$D$、$E$、$O$, 点$F$在$AC$上, $BF \\perp AO$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-45:0.5cm)}, scale = 1.5]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw ({sqrt(2)},{sqrt(3)},-1) node [above] {$P$} coordinate (P);\n\\draw (A)--(C)--(P)--(B)--cycle;\n\\draw [dashed] (B)--(C);\n\\draw (A)--(P);\n\\draw ($(B)!0.5!(C)$) node [above left] {$O$} coordinate (O);\n\\draw ($(A)!0.5!(C)$) node [below right] {$F$} coordinate (F);\n\\draw ($(P)!0.5!(B)$) node [left] {$D$} coordinate (D);\n\\draw ($(P)!0.5!(A)$) node [right] {$E$} coordinate (E);\n\\draw (A)--(D) (E)--(F)(B)--(E);\n\\draw [dashed] (A)--(O)(B)--(F)--(O)--(D)(O)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $EF\\parallel$平面$ADO$;\\\\\n(2) 若$\\angle POF=120^{\\circ}$, 求三棱锥$P-ABC$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466523,7 +470077,9 @@ "id": "018212", "content": "已知函数$f(x)=(\\dfrac{1}{x}+a) \\ln (x+1)$.\\\\\n(1) 当$a=-1$时, 求曲线$y=f(x)$在$(1, f(1))$处的切线方程;\\\\\n(2) 若函数$f(x)$在$(0,+\\infty)$单调递增, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466543,7 +470099,9 @@ "id": "018213", "content": "已知椭圆$C: \\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1$($a>b>0$)的离心率是$\\dfrac{\\sqrt{5}}{3}$, 点$A(-2,0)$在$C$上.\\\\\n(1) 求$C$的方程;\\\\\n(2) 过点$(2,3)$的直线交$C$于$P, Q$两点, 直线$AP, AQ$与$y$轴的交点分别为$M, N$, 证明: 线段$MN$的中点为定点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466563,7 +470121,9 @@ "id": "018214", "content": "在直角坐标系$x O y$中, 以坐标原点$O$为极点, $x$轴正半轴为极轴建立极坐标系, 曲线$C_1$的极坐标方程为$\\rho=2 \\sin \\theta$($\\dfrac{\\pi}{4} \\leq \\theta \\leq \\dfrac{\\pi}{2}$), $C_2: \\begin{cases}x=2 \\cos \\alpha, \\\\ y=2 \\sin \\alpha\\end{cases}$($\\alpha$为参数, $\\dfrac{\\pi}{2}<\\alpha<\\pi)$.\\\\\n(1) 写出$C_1$的直角坐标方程;\\\\\n(2) 若直线$y=x+m$既与$C_1$没有公共点, 也与$C_2$没有公共点, 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466583,7 +470143,10 @@ "id": "018215", "content": "已知$f(x)=2|x|+|x-2|$.\\\\\n(1) 求不等式$f(x) \\leq 6-x$的解集;\\\\\n(2) 在直角坐标系$x O y$中, 求不等式组$\\begin{cases}f(x) \\leq y, \\\\ x+y-6 \\leq 0\\end{cases}$所确定的平面区域的面积.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -466603,7 +470166,9 @@ "id": "018216", "content": "抛物线$y^2=4 x$的焦点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$(1,0)$", "solution": "", @@ -466636,7 +470201,9 @@ "id": "018217", "content": "抛掷一颗质地均匀的正方体骰子, 得点数$6$的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{1}{6}$", "solution": "", @@ -466669,7 +470236,9 @@ "id": "018218", "content": "半径为$1$厘米的球的表面积为\\blank{50}平方厘米.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$4\\pi$", "solution": "", @@ -466702,7 +470271,9 @@ "id": "018219", "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$中, 异面直线$AB$与$A_1C_1$所成角的大小是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (A_1)--(C_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$45^\\circ$", "solution": "", @@ -466735,7 +470306,9 @@ "id": "018220", "content": "双曲线$\\dfrac{x^2}{2}-\\dfrac{y^2}{4}=1$的两条渐近线方程分别是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$y=\\sqrt{2} x$和$y=-\\sqrt{2}x$", "solution": "", @@ -466768,7 +470341,9 @@ "id": "018221", "content": "以$C(1,1)$为圆心, 且经过$M(2,3)$的圆的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$(x-1)^2+(y-1)^2=5$", "solution": "", @@ -466801,7 +470376,9 @@ "id": "018222", "content": "如图, 靶子由一个中心圆面I和两个与I同心的圆环II、III构成, 射手命中I、 II及III的概率分别为$0.35$、$0.30$及$0.25$. 则不命中靶的概率为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle (0.3) node {I};\n\\draw (0,0) circle (0.7) (0.5,0) node {II};\n\\draw (0,0) circle (1.2) (0.95,0) node {III};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$0.10$", "solution": "", @@ -466834,7 +470411,9 @@ "id": "018223", "content": "``若直线$a\\parallel$平面$\\alpha$, 直线$b$在平面$\\alpha$上, 则直线$a\\parallel$直线$b$''是\\blank{50}命题 (填``真''或``假'').", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "假", "solution": "", @@ -466867,7 +470446,9 @@ "id": "018224", "content": "已知一个圆锥的体积为$3 \\pi$, 高为$3$, 则该圆锥的母线与底面所成角的大小是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{3}$", "solution": "", @@ -466900,7 +470481,9 @@ "id": "018225", "content": "已知$A$与$B$是独立事件, $P(A)=0.4$, $P(B)=0.3$, 给出下列式子: \\textcircled{1} $P(\\overline {A})=0.6$; \\textcircled{2} $P(A \\cap B)=0.12$; \\textcircled{3} $P(A \\cup B)=0.7$; \\textcircled{4} $P(A \\cap \\overline {B})=0.28$. 其中正确的式子是\\blank{50}. (填序号)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{4}", "solution": "", @@ -466933,7 +470516,9 @@ "id": "018226", "content": "如图, 正三棱柱$ABC-A_1B_1C_1$的各条棱长都相等, 线段$A_1B$、$B_1C$和$C_1A$是该正三棱柱的三条面对角线, 直线$l$与这三条面对角线所在直线所成的角大小相同, 则这个角的大小是\\blank{50}(写出所有可能的值).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [above right= 0.15 and 0] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\draw (B_1)--(C);\n\\draw [dashed] (A_1)--(B)(C_1)--(A);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{4}$或$\\arccos\\dfrac{3\\sqrt{34}}{34}$", "solution": "", @@ -466966,7 +470551,9 @@ "id": "018227", "content": "已知数列$a_1, a_2, a_3, \\cdots, a_{101}$的各项均为正整数, 其中$a_1=a_{101}=4999$, 对于每个正整数$i$($2 \\leq i \\leq 100$), $\\dfrac{a_{i-1}+a_{i+1}}{2}-a_i$为相同的正整数, 则$a_{100}$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$4900$", "solution": "", @@ -466999,7 +470586,9 @@ "id": "018228", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, 与$\\overrightarrow{AB}$相等的向量是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{3}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,1,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,1,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,1,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,1,0) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B_1) -- (C_1) -- (D_1) -- (A_1) -- cycle;\n\\draw (B) -- (B_1) (C) -- (C_1) (D) -- (D_1);\n\\draw [dashed] (A) -- (A_1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\overrightarrow{CD}$}{$\\overrightarrow{BA}$}{$\\overrightarrow{DC}$}{$\\overrightarrow{B_1A_1}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -467032,7 +470621,9 @@ "id": "018229", "content": "已知球$O$的半径为$5$, 球心$O$到平面$\\alpha$的距离为$3$, 则平面$\\alpha$截球$O$所得的小圆$O_1$的半径长是\\bracket{20}.\n\\fourch{$2$}{$3$}{$3 \\sqrt{2}$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -467065,7 +470656,9 @@ "id": "018230", "content": "下列命题:\\\\\n\\textcircled{1} 底面是正多边形的棱锥是正棱锥;\\\\\n\\textcircled{2} 各侧棱的长都相等的棱锥是正棱锥;\\\\\n\\textcircled{3} 各侧面是全等的等腰三角形的棱锥是正棱锥.\n\\\\\n其中真命题的个数是\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -467098,7 +470691,9 @@ "id": "018231", "content": "小李购买了一盒点心, 点心盒是长方体, 长、宽、高分别为$30$厘米、 $20$厘米和$10$厘米, 商家提供丝带捆扎服务, 有如图所示两种捆扎方案 (粗线表示丝带) 可供选择, 免去手工费, 但丝带需要按使用长度进行收费. 假设丝带紧贴点心盒表面, 且不计算丝带宽度以及重叠粘合打结的部分. 为了节约成本, 小李打算选择尽可能使用丝带较短的方案, 则小李需要购买的丝带长度至少是 \\bracket{20} .\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1]\n\\draw (0,0,0) coordinate (O);\n\\draw (O) --++ (3,0,0) --++ (0,0,-2) --++ (0,1,0) --++ (-3,0,0) --++ (0,0,2) --++ (0,-1,0);\n\\draw (O) ++ (3,0,0) --++ (0,1,0) --++ (0,0,-2) ++ (0,0,2) --++ (-3,0,0);\n\\draw [dashed] (O) --++ (0,0,-2) --++ (3,0,0) ++ (-3,0,0) --++ (0,1,0);\n\\draw (2,-0.5,0) node {点心盒(未捆扎)};\n\\draw (5,0,0) coordinate (O);\n\\draw (O) --++ (3,0,0) --++ (0,0,-2) --++ (0,1,0) --++ (-3,0,0) --++ (0,0,2) --++ (0,-1,0);\n\\draw (O) ++ (3,0,0) --++ (0,1,0) --++ (0,0,-2) ++ (0,0,2) --++ (-3,0,0);\n\\draw [dashed] (O) --++ (0,0,-2) --++ (3,0,0) ++ (-3,0,0) --++ (0,1,0);\n\\draw [ultra thick] (O) ++ ({3-5/6},0,0) --++ ({-4/3},1,0) --++ ({-5/6},0,{-5/8}) (O) ++ ({3-5/6},1,-2) --++ ({5/6},0,{5/8}) --++ (0,-1,{3/4});\n\\draw [ultra thick, dashed] (O) ++ ({3-5/6},0,0) --++ ({5/6},0,{-5/8}) (O) ++ (0,1,{-5/8}) --++ (0,-1,{-3/4}) --++ ({5/6},0,{-5/8}) --++ ({4/3},1,0);\n\\draw (7,-0.5,0) node {捆扎方案一};\n\\draw (10,0,0) coordinate (O);\n\\draw (O) --++ (3,0,0) --++ (0,0,-2) --++ (0,1,0) --++ (-3,0,0) --++ (0,0,2) --++ (0,-1,0);\n\\draw (O) ++ (3,0,0) --++ (0,1,0) --++ (0,0,-2) ++ (0,0,2) --++ (-3,0,0);\n\\draw [dashed] (O) --++ (0,0,-2) --++ (3,0,0) ++ (-3,0,0) --++ (0,1,0);\n\\draw [ultra thick] (O) ++ (1.5,0,0) --++ (0,1,0) --++ (0,0,-2) ++ (-1.5,0,1) --++ (3,0,0) --++ (0,-1,0);\n\\draw [ultra thick, dashed] (O) ++ (0,0,-1) --++ (0,1,0) ++ (0,-1,0) --++ (3,0,0) ++ (-1.5,0,1) --++ (0,0,-2) --++ (0,1,0);\n\\draw (12,-0.5,0) node {捆扎方案二};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$80$厘米}{$100$厘米}{$120$厘米}{$140$厘米}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -467131,7 +470726,9 @@ "id": "018232", "content": "设等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 已知$a_3=4$, $a_6=-32$.\\\\\n(1) 求公比$q$的值;\\\\\n(2) 求$S_5$的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) $-2$; (2) $11$", "solution": "", @@ -467164,7 +470761,9 @@ "id": "018233", "content": "已知$m \\in \\mathbf{R}$, 直线$l_1: 2 x+y-1=0$, 直线$l_2: m x+y+1=0$.\\\\\n(1) 若$l_1\\parallel l_2$, 求$l_1$与$l_2$之间的距离;\\\\\n(2) 若$l_1$与$l_2$的夹角大小为$\\arccos \\dfrac{\\sqrt{5}}{5}$, 求直线$l_2$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{2\\sqrt{5}}{5}$; (2) $4x-3y-3=0$或$y+1=0$", "solution": "", @@ -467197,7 +470796,9 @@ "id": "018234", "content": "某校高二年级共有学生$200$人, 其中男生$120$人, 女生$80$人. 为了了解全年级学生上学花费时间 (分) 的信息, 按照分层抽样的原则抽取了样本, 样本容量为 $20$, 并根据样本数据信息绘制了茎叶图和频率分布直方图. 由于保存不当, 茎叶图中有一个数据不小心被污染看不清了(如图), 频率分布直方图纵轴上的数据也遗失了.\n\\begin{center}\n\\begin{tikzpicture}\n\\foreach \\i/\\j/\\k in {1/1/1,1/2/5,1/3/8,1/4/9}\n{\\draw ({\\j*0.6},-\\i) node {$\\k$};};\n\\foreach \\i/\\j/\\k in {2/1/2,2/2/2,2/3/3,2/4/\\blacksquare,2/5/5,2/6/5,2/7/6,2/8/7,2/9/8}\n{\\draw ({\\j*0.6},-\\i) node {$\\k$};};\n\\foreach \\i/\\j/\\k in {3/1/3,3/2/1,3/3/1,3/4/2,3/5/4,3/6/6,4/1/4,4/2/2,4/3/7,5/1/5,5/2/1}\n{\\draw ({\\j*0.6},-\\i) node {$\\k$};};\n\\draw (0.9,-0.5) -- (0.9,-5.5);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 70]\n\\draw [->] (0,0) -- (80,0) node [below] {时间/分};\n\\draw [->] (0,0) -- (0,0.06) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {10/0.015,20/0.045,30/0.025,40/0.01,50/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {10/0.015/{},20/0.045/y,30/0.025/{},40/0.01/{},50/0.005/x}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (60,0) node [below] {$60$};\n\\end{tikzpicture}\n\\end{center}\n(1) 根据茎叶图提供的有限信息, 求频率分布直方图中$x$和$y$的值, 指出样本的``中位数、 平均数、众数、方差、极差''中, 哪些已经能确定, 并计算它们的值;\\\\\n(2) 通过对样本原始数据的计算, 得到男生上学花费时间的样本均值为$30$(分), 女生的样本均值为$27.75$(分), 试计算被污染的数值, 并根据样本估计该年级全体学生上学花费时间的``中位数、平均数、方差''.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) $x=0.005$, $y=0.045$, 中位数为$26.5$(分), 众数为$25$(分), 极差为$36$(分); (2) 估计中位数为$26$分, 平均数为$29.1$分, 方差为$82.39$", "solution": "", @@ -467230,7 +470831,9 @@ "id": "018235", "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$1$, 高为$2$, 点$M$是棱$CC_1$上一个动点(点$M$与$C, C_1$均不重合).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,{2*\\l},0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,{2*\\l},0) node [right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,{2*\\l},0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,{2*\\l},0) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B_1) -- (C_1) -- (D_1) -- (A_1) -- cycle;\n\\draw (B) -- (B_1) (C) -- (C_1) (D) -- (D_1);\n\\draw [dashed] (A) -- (A_1);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$M$} coordinate (M);\n\\draw (M)--(D_1)--(B_1)--cycle;\n\\draw [dashed] (A)--(M)(A)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 当点$M$是棱$CC_1$的中点时, 求证: 直线$AM \\perp$平面$B_1MD_1$;\\\\\n(2) 当$D_1M \\perp AB_1$时, 求点$D_1$到平面$AMB_1$的距离;\\\\\n(3) 当平面$AB_1M$将正四棱柱$ABCD-A_1B_1C_1D_1$分割成体积之比为$1: 2$的两个部分时, 求线段$MC$的长度.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{5\\sqrt{21}}{21}$; (3) $MC=-1+\\sqrt{5}$;", "solution": "", @@ -467263,7 +470866,9 @@ "id": "018236", "content": "如图, 已知点$A(\\sqrt{2}, 1)$是椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)上的一点, 顶点$C(-2,0)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,0) node [above left] {$C$} coordinate (C);\n\\filldraw (C) circle (0.03);\n\\draw [name path = elli] (0,0) ellipse (2 and {sqrt(2)});\n\\filldraw ({sqrt(2)},1) node [above] {$A$} coordinate (A) circle (0.03);\n\\path [name path = AB] (A) --++ (-3,{-3*0.7});\n\\path [name path = AD] (A) --++ (-2,{-2*1.3});\n\\path [name intersections = {of = AB and elli, by = B}] (B) node [below left] {$B$};\n\\path [name intersections = {of = AD and elli, by = D}] (D) node [below] {$D$};\n\\draw (A)--(B)(A)--(D)($(B)!-1!(D)$)--($(B)!2!(D)$);\n\\filldraw (B) circle (0.03) (D) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$\\Gamma$的离心率;\\\\\n(2) 直线$BD$交椭圆$\\Gamma$于$B$、$D$两点$(B$、$D$与$A$不重合), 若直线$AB$与直线$AD$的斜率之和为$2$, 直线$BD$是否过定点? 若是, 请求出该定点的坐标; 若不是, 请说明理由.\\\\\n(3) 点$E$、点$G$是椭圆$\\Gamma$上的两个点, 圆$I: (x-\\dfrac{2 \\sqrt{2}}{3})^2+y^2=r^2$($r>0$)是$\\triangle CEG$的内切圆, 过椭圆$\\Gamma$的顶点$M(0, b)$作圆$I$的两条切线, 分别交椭圆$\\Gamma$于点$P$和点$Q$, 判断直线$PQ$与圆$I$的位置关系并证明.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{2}}{2}$; (2) 过定点$P(-1+\\sqrt{2},-\\dfrac{\\sqrt{2}}2-1)$; (3) 直线$PQ$与圆相切", "solution": "", @@ -504372,7 +507977,9 @@ "id": "021441", "content": "判断下列命题是否正确:\\\\\n(1) 终边重合的两个角相等;\\blank{20}\\\\\n(2) 锐角是第一象限的角;\\blank{20}\\\\\n(3) 第二象限的角是钝角;\\blank{20}\\\\\n(4) 小于$90^{\\circ}$的角都是锐角.\\blank{20}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "错误, 正确, 错误, 错误", "solution": "", @@ -504399,7 +508006,9 @@ "id": "021442", "content": "下列各组的两个角中, 终边不重合的一组是\\bracket{20}.\n\\fourch{$-43^{\\circ}$与$677^{\\circ}$}{$900^{\\circ}$与$-1260^{\\circ}$}{$-120^{\\circ}$与$960^{\\circ}$}{$150^{\\circ}$与$630^{\\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -504426,7 +508035,9 @@ "id": "021443", "content": "如果$\\alpha$是锐角, 那么$2 \\alpha$是\\bracket{20}.\n\\fourch{第一象限角}{第二象限角}{小于$180^{\\circ}$的正角}{大于直角的正角}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -504453,7 +508064,9 @@ "id": "021444", "content": "如果$\\alpha$是钝角, 那么$\\dfrac{\\alpha}{2}$是\\bracket{20}.\n\\twoch{第一象限角}{第二象限角}{第二象限角或终边与坐标轴重合的角}{不小于直角的正角}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -504480,7 +508093,9 @@ "id": "021445", "content": "在平面直角坐标系中, 下列结论正确的是\\bracket{20}.\n\\twoch{小于$90^{\\circ}$的角一定是锐角}{第一象限角必定大于$0^{\\circ}$且小于$90^{\\circ}$}{始边重合且相等的角, 终边一定重合}{始边重合且终边也重合的角一定相等}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -504507,7 +508122,9 @@ "id": "021446", "content": "已知集合$A=\\{\\alpha|\\alpha\\text{是第一象限角}\\}$, $B=\\{\\alpha|\\alpha\\text{是锐角}\\}$, $C=\\{\\alpha|\\alpha\\text{是小于}90^\\circ\\text{的角}\\}$, 下列结论正确的是\\bracket{20}.\n\\fourch{$A=B=C$}{$A \\subset C$}{$A \\cap C=B$}{$B \\cup C=C$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -504534,7 +508151,9 @@ "id": "021447", "content": "平面内一条射线绕着其端点先逆时针旋转$60^{\\circ}$, 再顺时针旋转$450^{\\circ}$, 已原射线为始边, 所得射线为终边所形成的角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-390^\\circ$", "solution": "", @@ -504561,7 +508180,9 @@ "id": "021448", "content": "与角$1024^{\\circ}$终边重合的角中, 最小的正角为\\blank{50}, 最大的负角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$304^\\circ$, $-56^\\circ$", "solution": "", @@ -504588,7 +508209,9 @@ "id": "021449", "content": "与角$576^{\\circ}$终边重合的角中, 绝对值最小的角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-144^\\circ$", "solution": "", @@ -504615,7 +508238,9 @@ "id": "021450", "content": "若角$\\alpha$是第三象限的角, 则角$\\alpha+270^{\\circ}$是第\\blank{50}象限的角, 角$\\alpha-270^{\\circ}$是第\\blank{50}象限的角.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "二, 四", "solution": "", @@ -504642,7 +508267,9 @@ "id": "021451", "content": "写出与下列各角的终边重合的所有角组成的集合$S$, 并列举$S$中满足不等式$-360^{\\circ} \\leq \\alpha<720^{\\circ}$的所有元素$\\alpha$:\\\\\n(1) $60^{\\circ}$;\\\\\n(2) $-21^{\\circ}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\{\\alpha|\\alpha=60^\\circ+k\\cdot 360^\\circ, \\ k\\in \\mathbf{Z}\\}$, $-300^\\circ$, $60^\\circ$, $420^\\circ$; (2) $\\{\\alpha|\\alpha = -21^\\circ+k\\cdot 360^\\circ, \\ k \\in \\mathbf{Z}\\}$, $-21^\\circ$, $339^\\circ$, $699^\\circ$", "solution": "", @@ -504669,7 +508296,9 @@ "id": "021452", "content": "在平面直角坐标系中, 用阴影部分表示集合: $\\{\\alpha | 30^{\\circ}+k \\cdot 360^{\\circ} \\leq \\alpha \\leq 60^{\\circ}+k \\cdot 360^{\\circ},\\ k \\in \\mathbf{Z}\\}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "\\begin{tikzpicture}[>=latex]\n\\fill [pattern = north east lines] (30:2) arc (30:60:2) -- (0,0) -- cycle;\n\\draw (30:2) -- (0,0) -- (60:2);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}", "solution": "", @@ -504696,7 +508325,9 @@ "id": "021453", "content": "在同一天$10: 45$到$14: 20$期间, 时钟的分针转过的角度为\\blank{50}, 它是第\\blank{50}象限的角.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-1290^{\\circ}$;第二象限", "solution": "", @@ -504723,7 +508354,9 @@ "id": "021454", "content": "(1) 终边在第一象限角平分线上的所有角组成的集合为\\blank{50};\\\\\n(2) 终边在第二、四象限角平分线上的所有角组成的集合为\\blank{50};\\\\\n(3) 终边在直线$y=x$或$y=-x$上的所有角组成的集合为\\blank{50};\\\\\n(4) 终边位于第三象限的所有角组成的集合为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $ \\{\\alpha|\\alpha=45^{\\circ}+k\\cdot 360^{\\circ}, \\ k \\in \\mathbf{Z}\\}$;\\\\\n(2) $\\{\\alpha|\\alpha=135^{\\circ}+k\\cdot 180^{\\circ}, \\ k \\in \\mathbf{Z}\\}$;\\\\\n(3) $\\{\\alpha|\\alpha=45^{\\circ}+k\\cdot 90^{\\circ}, \\ k \\in \\mathbf{Z}\\}$;\\\\\n(4) $\\{\\alpha|180^{\\circ}+k\\cdot 360^{\\circ}<\\alpha<270^{\\circ}+k\\cdot 360^{\\circ}, \\ k \\in \\mathbf{Z}\\}$.", "solution": "", @@ -504750,7 +508383,9 @@ "id": "021455", "content": "已知角$\\alpha$, 写出满足下列条件的角$\\beta$的集合(用$\\alpha$表示$\\beta$):\\\\\n(1) $\\beta$的终边是$\\alpha$终边的反向延长线: \\blank{50};\\\\\n(2) $\\beta$的终边与$\\alpha$终边垂直: \\blank{50};\\\\\n(3) $\\beta$的终边与$\\alpha$终边关于$x$轴对称: \\blank{50};\\\\\n(4) $\\beta$的终边与$\\alpha$终边关于直线$y=x$对称: \\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $ \\{\\beta|\\beta=\\alpha+180^{\\circ}+k\\cdot 360^{\\circ}, \\ k \\in \\mathbf{Z}\\}$;\\\\\n(2) $\\{\\beta|\\beta=\\alpha+90^{\\circ}+k\\cdot 180^{\\circ}, \\ k \\in \\mathbf{Z}\\}$;\\\\\n(3) $\\{\\beta|\\beta=-\\alpha+k\\cdot 360^{\\circ}, \\ k \\in \\mathbf{Z}\\}$;\\\\\n(4) $\\{\\beta|\\beta=90^{\\circ}-\\alpha+k\\cdot 360^{\\circ}, \\ k \\in \\mathbf{Z}\\}$.", "solution": "", @@ -504777,7 +508412,9 @@ "id": "021456", "content": "如果存在角$\\beta \\in(-3 \\pi,-\\dfrac{5 \\pi}{2})$与角$\\alpha$的终边重合, 那么角$\\alpha$所在的象限是\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -504804,7 +508441,9 @@ "id": "021457", "content": "设$\\alpha=2019$, 则$\\alpha$的终边所在的象限为\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -504831,7 +508470,9 @@ "id": "021458", "content": "完成下列角度与弧度的换算:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline 角度数 &$15^{\\circ}$&$105^{\\circ}$&$225^{\\circ}$&\\blank{15} &\\blank{15} &\\blank{15} &\\blank{15} \\\\\n\\hline 弧度数 &\\blank{15} &\\blank{15} &\\blank{15} &$\\dfrac{5 \\pi}{3}$&$\\dfrac{9 \\pi}{5}$&$\\dfrac{7 \\pi}{4}$&$\\dfrac{3}{2}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{12}$; $\\dfrac{7\\pi}{12}$; $\\dfrac{5\\pi}{4}$; $300^{\\circ}$; $324^{\\circ}$; $315^{\\circ}$; $(\\dfrac{270}{\\pi})^{\\circ}$", "solution": "", @@ -504858,7 +508499,9 @@ "id": "021459", "content": "已知扇形的半径为$10$, 圆心角的大小为$\\dfrac{5 \\pi}{9}$, 则它的周长为\\blank{50}, 它的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1)$\\frac{50\\pi+180}{9}$;(2)$\\frac{250\\pi}{9}$", "solution": "", @@ -504885,7 +508528,9 @@ "id": "021460", "content": "若一圆弧长等于其所在圆的内接正三角形的边长, 则其圆心角的弧度数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\sqrt{3}$", "solution": "", @@ -504912,7 +508557,9 @@ "id": "021461", "content": "在定圆中, 长度等于半径的弦所对圆心角的弧度数为\\blank{50}, 长度等于半径的$\\sqrt{3}$倍的弦所对的圆心角的弧度数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1)$\\frac{\\pi}{3}$;(2)$\\frac{2\\pi}{3}$", "solution": "", @@ -504939,7 +508586,9 @@ "id": "021462", "content": "把下列各角化成$2 k \\pi+\\alpha$($0 \\leq \\alpha<2 \\pi$, $k \\in \\mathbf{Z}$)的形式, 并指出其终边所在的象限:\\\\\n(1) $\\dfrac{50 \\pi}{3}=$\\blank{50}, 第\\blank{50}象限;\\\\\n(2) $-\\dfrac{50 \\pi}{3}=$\\blank{50}, 第\\blank{50}象限;\\\\\n(3) $-108^{\\circ}=$\\blank{50}, 第\\blank{50}象限;\\\\\n(4) $-225^{\\circ}=$\\blank{50}, 第\\blank{50}象限.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1)$16\\pi+\\frac{2\\pi}{3}$,二;\\\\\n(2)$-18\\pi+\\frac{4\\pi}{3}$,三;\\\\\n(3)$-2\\pi+\\frac{7\\pi}{5}$,三;\\\\\n(4)$-2\\pi+\\frac{3\\pi}{4}$,二.", "solution": "", @@ -504966,7 +508615,9 @@ "id": "021463", "content": "已知扇形的周长为$10$, 面积为$4$, 求该扇形对应的圆心角的弧度数.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\frac{1}{2}$", "solution": "", @@ -504993,7 +508644,9 @@ "id": "021464", "content": "用弧度制表示下列角的集合:\\\\\n(1) 第四象限角的集合: \\blank{100};\\\\\n(2) 终边在坐标轴上的角的集合: \\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\{\\alpha|-\\frac{\\pi}{2}+2k\\pi<\\alpha<2k\\pi,\\ k \\in \\mathbf{Z}\\}$;\\\\\n(2) $\\{\\alpha|\\alpha=\\frac{k\\pi}{2},\\ k \\in \\mathbf{Z}\\}$.", "solution": "", @@ -505020,7 +508673,9 @@ "id": "021465", "content": "(1) 角$\\alpha$的终边与角$\\beta$的终边重合, 则$\\alpha$与$\\beta$的关系是\\blank{50};\\\\\n(2) 角$\\alpha$的终边与角$\\beta$的终边关于$x$轴对称, 则$\\alpha$与$\\beta$的关系是\\blank{50};\\\\\n(3) 角$\\alpha$的终边与角$\\beta$的终边关于$y$轴对称, 则$\\alpha$与$\\beta$的关系是\\blank{50};\\\\\n(4) 角$\\alpha$的终边与角$\\beta$的终边关于原点对称, 则$\\alpha$与$\\beta$的关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\beta=\\alpha+2k\\pi,\\ k \\in \\mathbf{Z}$;\\\\\n(2) $\\beta=-\\alpha+2k\\pi,\\ k \\in \\mathbf{Z}$;\\\\\n(3) $\\beta=-\\alpha+\\pi+2k\\pi,\\ k \\in \\mathbf{Z}$;\\\\\n(4) $\\beta=\\alpha+\\pi+2k\\pi,\\ k \\in \\mathbf{Z}$.", "solution": "", @@ -505047,7 +508702,9 @@ "id": "021466", "content": "用弧度制写出下图中的阴影部分表示的角的集合(包括边界).\n(1) \\blank{100};\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [gray!25] (0,0) --++ (-45:1.5) arc (-45:90:1.5);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$} coordinate (x);\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$} coordinate (y);\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,0) --++ (-45:1.5) coordinate (T);\n\\draw pic [draw, \"$45^\\circ$\", scale = 0.5, angle eccentricity = 2.5] {angle = T--O--x};\n\\end{tikzpicture}\n\\end{center}\n(2)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [gray!25] (0,0) --++ (30:1.5) arc (30:150:1.5);\n\\fill [gray!25] (0,0) --++ (-30:1.5) arc (-30:-150:1.5);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$} coordinate (x);\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$} coordinate (y);\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,0) --++ (30:1.5) coordinate (T);\n\\draw (0,0) --++ (-30:1.5) (0,0) --++ (-150:1.5);\n\\draw (0,0) --++ (150:1.5) coordinate (S);\n\\draw pic [draw, \"$30^\\circ$\", scale = 0.5, angle eccentricity = 3] {angle = x--O--T};\n\\draw (-2,0) coordinate (x1);\n\\draw pic [draw, \"$30^\\circ$\", scale = 0.5, angle eccentricity = 3] {angle = S--O--x1};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\{\\alpha|-\\frac{\\pi}{4}+2k\\pi \\le \\alpha \\le \\frac{\\pi}{2}+2k\\pi,\\ k \\in \\mathbf{Z}\\}$;\\\\\n(2) $\\{\\alpha|\\frac{\\pi}{6}+k\\pi \\le \\alpha \\le \\frac{5\\pi}{6}+k\\pi,\\ k \\in \\mathbf{Z}\\}$.", "solution": "", @@ -505074,7 +508731,9 @@ "id": "021467", "content": "已知角$\\alpha$是第三象限的角.\\\\\n(1) 指出角$\\pi-\\alpha$、$\\dfrac{\\pi}{2}+\\alpha$的终边位置;\\\\\n(2) 讨论角$\\dfrac{\\alpha}{2}$、$2 \\alpha$的终边位置.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 第四象限；第四象限；\\\\\n(2) 第二象限或者第四象限；第一象限或第二象限或者$y$轴正半轴.", "solution": "", @@ -505101,7 +508760,9 @@ "id": "021468", "content": "已知集合$A=\\{\\alpha | k \\pi-\\dfrac{\\pi}{6}<\\alpha0$, 则$\\alpha$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$ \\left( -2,\\frac{2}{3} \\right)$", "solution": "", @@ -505240,7 +508909,9 @@ "id": "021473", "content": "已知$-\\pi<\\alpha<-\\dfrac{\\pi}{2}$, $-\\dfrac{\\pi}{2}<\\beta<0$, 则$\\sin (\\alpha-\\beta)$\\blank{50}$0$.(选填``$>$''、``$<$''、``$\\geq$''、``$\\leq$'')", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$<$", "solution": "", @@ -505268,7 +508939,9 @@ "id": "021474", "content": "计算: $6 \\cos 270^{\\circ}+10 \\sin 0^{\\circ}-4 \\tan 180^{\\circ}+5 \\cos 360^{\\circ}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "5", "solution": "", @@ -505296,7 +508969,9 @@ "id": "021475", "content": "计算: $\\sin ^2 \\dfrac{\\pi}{3}-\\cos ^2 \\dfrac{\\pi}{6}+2 \\tan ^3 \\dfrac{\\pi}{4}$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "2", "solution": "", @@ -505324,7 +508999,9 @@ "id": "021476", "content": "已知常数$t \\in \\mathbf{R}$, 角$\\alpha$的终边经过点$P(-\\sqrt{3}, t)$, 且$\\sin \\alpha=\\dfrac{\\sqrt{2}}{4} t$, 求$\\cos \\alpha$和$\\tan \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "当$t=\\sqrt{5}$时, $\\cos \\alpha=- \\frac{\\sqrt{6}}{4}$, $\\tan \\alpha =- \\frac{\\sqrt{15}}{3}$;\\\\\n当$t=-\\sqrt{5}$时, $\\cos \\alpha=- \\frac{\\sqrt{6}}{4}$, $\\tan \\alpha = \\frac{\\sqrt{15}}{3}$;\\\\\n当$t=0$时, $\\cos \\alpha=-1$, $\\tan \\alpha = 0$.", "solution": "", @@ -505352,7 +509029,9 @@ "id": "021477", "content": "已知角$\\alpha$终边上一点$P$满足到$x$轴、$y$轴的距离之比为$4: 3$, 且$\\cos \\alpha<0$. 求$\\sin \\alpha$与$\\tan \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "当$\\alpha$在第二象限时，$ \\sin \\alpha =\\frac{4}{5}$, $\\tan \\alpha=-\\frac{4}{3}$;\\\\\n当$\\alpha$在第三象限时，$ \\sin \\alpha =-\\frac{4}{5}$, $\\tan \\alpha=\\frac{4}{3}$.", "solution": "", @@ -505380,7 +509059,9 @@ "id": "021478", "content": "若角$\\alpha$的终边在直线$y=-\\sqrt{3} x$上, 求$\\sin \\alpha \\cdot \\cos \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\frac{\\sqrt{3}}{4}$", "solution": "", @@ -505408,7 +509089,9 @@ "id": "021479", "content": "根据下列条件, 分别确定角$\\alpha$所在的象限:\\\\\n(1) $\\sin \\alpha<0$且$\\cos \\alpha>0$;\\\\\n(2) $\\dfrac{\\sin \\alpha}{\\tan \\alpha}>0$;\\\\\n(3) $\\tan \\alpha+\\cot \\alpha>0$;\\\\\n(4) $\\sin 2 \\alpha>0$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 第四象限； (2) 第一、四象限；(3)第一、三象限；(4)第一、三象限.", "solution": "", @@ -505436,7 +509119,9 @@ "id": "021480", "content": "已知集合$A=\\{y | y=\\dfrac{\\sin \\alpha}{|\\sin \\alpha|}+\\dfrac{|\\cos \\alpha|}{\\cos \\alpha}+\\dfrac{\\tan \\alpha}{|\\tan \\alpha|}+\\dfrac{|\\cot \\alpha|}{\\cot \\alpha}\\}$, 请用列举法表示集合$A$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$A=\\left\\{ -2,-0,4 \\right\\}$", "solution": "", @@ -505464,7 +509149,9 @@ "id": "021481", "content": "分别求下列式子有意义时, 角$x$的取值范围:\\\\\n(1) $\\sqrt{\\sin x}+\\sqrt{\\cos x}$;\\\\\n(2) $\\sqrt{\\sin x}+\\lg (9-x^2)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\{\\alpha|2k\\pi \\le \\alpha \\le \\frac{\\pi}{2}+2k\\pi,\\ k \\in \\mathbf{Z}\\}$;\\\\\n(2) $[0,3)$", "solution": "", @@ -505492,7 +509179,9 @@ "id": "021482", "content": "在下表填入相应的正弦、余弦、正切和余切值.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$\\alpha$&$\\dfrac{\\pi}{3}$&$\\dfrac{7 \\pi}{4}$&$\\dfrac{2021 \\pi}{2}$&$-\\dfrac{\\pi}{6}$&$-\\dfrac{22 \\pi}{3}$\\\\\n\\hline$\\sin \\alpha$&\\blank{30} &\\blank{30} & \\blank{30}&\\blank{30} &\\blank{30} \\\\\n\\hline$\\cos \\alpha$&\\blank{30} &\\blank{30} & \\blank{30}&\\blank{30} &\\blank{30} \\\\\n\\hline$\\tan \\alpha$&\\blank{30} &\\blank{30} & \\blank{30}&\\blank{30} &\\blank{30} \\\\\n\\hline$\\cot \\alpha$&\\blank{30} &\\blank{30} & \\blank{30}&\\blank{30} &\\blank{30} \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$\\alpha$&$\\dfrac{\\pi}{3}$&$\\dfrac{7 \\pi}{4}$&$\\dfrac{2021 \\pi}{2}$&$-\\dfrac{\\pi}{6}$&$-\\dfrac{22 \\pi}{3}$\\\\\n\\hline$\\sin \\alpha$& $\\frac{\\sqrt{3}}{2}$ &$-\\frac{\\sqrt{2}}{2}$ & $1$&$-\\frac{1}{2}$ &$\\frac{\\sqrt{3}}{2}$ \\\\\n\\hline$\\cos \\alpha$&$\\frac{1}{2}$ &$\\frac{\\sqrt{2}}{2}$ & $0$&$\\frac{\\sqrt{3}}{2}$ &$-\\frac{1}{2}$ \\\\\n\\hline$\\tan \\alpha$&$\\sqrt{3}$ &$-1$ & 不存在 &$-\\frac{\\sqrt{3}}{3}$ &$-\\sqrt{3}$\\\\\n\\hline$\\cot \\alpha$&$\\frac{\\sqrt{3}}{3}$ &$-1$ & $ 0$&$-\\sqrt{3}$ &$-\\frac{\\sqrt{3}}{3}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "solution": "", @@ -505520,7 +509209,9 @@ "id": "021483", "content": "已知$\\sin x=-\\dfrac{\\sqrt{3}}{2}$.\\\\\n(1) 当$x \\in \\mathbf{R}$时, 则满足条件的$x$的集合是\\blank{50};\\\\\n(2) 当$x \\in[-2 \\pi, 4 \\pi]$时, 则满足条件的$x$的集合是\\blank{50};\\\\\n(3) 当$x$是第三象限角时, 则满足条件的$x$的集合是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\{x|x=\\frac{4\\pi}{3}+2k \\pi$或$ x=\\frac{5\\pi}{3}+2k \\pi,\\ k \\in \\mathbf{Z} \\}$;\\\\\n(2) $\\{-\\frac{2\\pi}{3},-\\frac{\\pi}{3},\\frac{4\\pi}{3} ,\\frac{5\\pi}{3},\\frac{10\\pi}{3},\\frac{11\\pi}{3} \\}$", "solution": "", @@ -505548,7 +509239,9 @@ "id": "021484", "content": "若$\\alpha$为第三象限的角, $\\cos \\alpha=-\\dfrac{\\sqrt{5}}{5}$, 则$\\sin \\alpha=$\\blank{50}, $\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\frac{2\\sqrt{5}}{5}$;$2$", "solution": "", @@ -505576,7 +509269,9 @@ "id": "021485", "content": "给出以下四个命题:\\\\\n\\textcircled{1} 如果$\\alpha \\neq \\beta$, 则$\\sin \\alpha \\neq \\sin \\beta$;\\\\\n\\textcircled{2} 如果$\\sin \\alpha \\neq \\sin \\beta$, 则$\\alpha \\neq \\beta$;\\\\\n\\textcircled{3} 如果$\\sin \\alpha>0$, 则$\\alpha$是第一或第二象限角;\\\\\n\\textcircled{4}如果$\\alpha$是第一或第二象限角, 则$\\sin \\alpha>0$.\\\\\n在以上四个命题中, 所有真命题的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{2} \\textcircled{4}", "solution": "", @@ -505604,7 +509299,9 @@ "id": "021486", "content": "已知$\\cot \\alpha=\\dfrac{1}{3}$, 求$\\sin \\alpha$、$\\cos \\alpha$及$\\tan \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "当$\\alpha$在第一象限时，$ \\sin \\alpha =\\frac{3\\sqrt{10}}{10}$, $\\cos \\alpha =\\frac{\\sqrt{10}}{10}$,$\\tan \\alpha=3$;\\\\\n当$\\alpha$在第三象限时，$ \\sin \\alpha =-\\frac{3\\sqrt{10}}{10}$,$\\cos \\alpha =-\\frac{\\sqrt{10}}{10}$, $\\tan \\alpha=3$.", "solution": "", @@ -505632,7 +509329,9 @@ "id": "021487", "content": "分别求$\\sin k \\pi$($k \\in \\mathbf{Z}$)和$\\cos k \\pi$($k \\in \\mathbf{Z}$)的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\sin k\\pi =0$; $\\cos k\\pi=\\begin{cases}1, & k=2n, \\\\ -1, & k=2n-1\\end{cases}$($n \\in \\mathbf{Z}$).", "solution": "", @@ -505660,7 +509359,9 @@ "id": "021488", "content": "利用单位圆, 求满足下列条件的角$\\theta$的取值范围.\\\\\n(1) $\\sin \\theta>\\dfrac{\\sqrt{3}}{2}$;\\\\\n(2) $\\tan \\theta \\leq-\\dfrac{\\sqrt{3}}{3}$;\\\\\n(3) $\\dfrac{1}{2} \\leq \\cos \\theta \\leq \\dfrac{1}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\{\\theta | 2k \\pi+\\dfrac{\\pi}{3}<\\theta<2k \\pi+\\dfrac{2\\pi}{3},\\ k \\in \\mathbf{Z} \\}$;\\\\\n(2) $\\{\\theta | k \\pi-\\dfrac{\\pi}{2}<\\theta \\le k \\pi-\\dfrac{\\pi}{6},\\ k \\in \\mathbf{Z} \\}$;\\\\\n(3) $\\{\\theta | k \\pi+\\dfrac{\\pi}{3} \\le \\theta \\le k \\pi+\\dfrac{2\\pi}{3},\\ k \\in \\mathbf{Z} \\}$.", "solution": "", @@ -505688,7 +509389,9 @@ "id": "021489", "content": "已知$\\theta$是第三象限角且$\\cos \\dfrac{\\theta}{2}<0$, 问$\\dfrac{\\theta}{2}$是第几象限角?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "第二象限", "solution": "", @@ -505716,7 +509419,9 @@ "id": "021490", "content": "已知$\\alpha$是第三象限角.\\\\\n(1) 设点$P(\\sin \\dfrac{\\alpha}{2}, \\cos \\dfrac{\\alpha}{2})$, 判断点$P$所在象限;\\\\\n(2) 判断$\\sin (\\cos \\alpha) \\cdot \\cos (\\sin \\alpha)$的符号.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 当$\\dfrac{\\alpha}{2}$在第二象限时，点$P$在第四象限；\\\\\n当$\\dfrac{\\alpha}{2}$在第四象限时，点$P$在第二象限.\\\\\n(2) $\\sin (\\cos \\alpha) \\cdot \\cos (\\sin \\alpha)<0$", "solution": "", @@ -505744,7 +509449,9 @@ "id": "021491", "content": "设$\\alpha$、$m$满足$\\sin (\\alpha+105^{\\circ})=\\dfrac{\\sqrt{2}}{4} m$, 且角$\\alpha+1905^{\\circ}$的终边上有一点$P(-\\sqrt{3}, m)$. 求$\\cos (\\alpha+1905^{\\circ})$与$\\tan (\\alpha-615^{\\circ})$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "当$m=0$时，$ \\cos (\\alpha+1905^{\\circ})=-1$,$\\tan (\\alpha-615^{\\circ})=0$;\\\\\n当$m=\\sqrt{5}$时，$ \\cos (\\alpha+1905^{\\circ}) =-\\frac{\\sqrt{6}}{4}$,$\\tan (\\alpha-615^{\\circ})=-\\frac{\\sqrt{15}}{3}$;\\\\\n当$m=-\\sqrt{5}$时，$ \\cos (\\alpha+1905^{\\circ}) =-\\frac{\\sqrt{6}}{4}$,$\\tan (\\alpha-615^{\\circ})=\\frac{\\sqrt{15}}{3}$.", "solution": "", @@ -505772,7 +509479,9 @@ "id": "021492", "content": "已知$\\alpha$满足$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{2}$, 则$\\sin \\alpha \\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{3}{8}$", "solution": "", @@ -505800,7 +509509,9 @@ "id": "021493", "content": "已知$\\sin \\alpha=\\dfrac{3}{5}$, 且$\\dfrac{\\pi}{2}<\\alpha<\\dfrac{3 \\pi}{2}$, 则$\\cos \\alpha-\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{1}{20}$", "solution": "", @@ -505828,7 +509539,9 @@ "id": "021494", "content": "已知$\\cos \\alpha=-\\dfrac{2 \\sqrt{2}}{3}$, 且$\\sin \\alpha>0$, 则$\\tan \\alpha-\\cot \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{7\\sqrt{2}}{4}$", "solution": "", @@ -505856,7 +509569,9 @@ "id": "021495", "content": "已知$\\tan \\alpha=2$, 且$\\sin \\alpha=a^2$, 则$\\sin \\alpha+\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{3\\sqrt{5}}{5}$", "solution": "", @@ -505884,7 +509599,9 @@ "id": "021496", "content": "已知$\\tan \\alpha-\\cot \\alpha=3$, 则$\\tan ^2 \\alpha+\\cot ^2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$11$", "solution": "", @@ -505912,7 +509629,9 @@ "id": "021497", "content": "已知$\\tan \\alpha=-\\dfrac{1}{2}$, 求下列各式的值:\\\\\n(1) $\\dfrac{2 \\cos \\alpha-\\sin \\alpha}{\\sin \\alpha+\\cos \\alpha}=$\\blank{50};\\\\\n(2) $2 \\sin ^2 \\alpha+\\sin \\alpha \\cdot \\cos \\alpha-3 \\cos ^2 \\alpha=$\\blank{50};\\\\\n(3) $\\dfrac{2 \\sin ^2 \\alpha-\\sin \\alpha \\cos \\alpha}{1+\\cos ^2 \\alpha}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$5$;$-\\dfrac{12}{5}$;$\\dfrac{4}{9}$", "solution": "", @@ -505940,7 +509659,9 @@ "id": "021498", "content": "化简: $\\cot ^2 \\alpha(\\tan ^2 \\alpha-\\sin ^2 \\alpha)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\sin ^2 \\alpha$", "solution": "", @@ -505968,7 +509689,9 @@ "id": "021499", "content": "化简: $\\sin ^4 x+\\cos ^4 x+\\dfrac{2 \\sin x \\cos x}{\\tan x+\\cot x}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$1$", "solution": "", @@ -505996,7 +509719,9 @@ "id": "021500", "content": "证明: $\\tan ^2 \\alpha-\\sin ^2 \\alpha=\\tan ^2 \\alpha \\cdot \\sin ^2 \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -506024,7 +509749,9 @@ "id": "021501", "content": "证明: $\\dfrac{1-2 \\sin \\alpha \\cos \\alpha}{\\cos ^2 \\alpha-\\sin ^2 \\alpha}=\\dfrac{1-\\tan \\alpha}{1+\\tan \\alpha}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -506052,7 +509779,9 @@ "id": "021502", "content": "已知$\\sin \\theta=\\dfrac{k-3}{k+5}$, $\\cos \\theta=\\dfrac{4-2 k}{k+5}$, 若$\\theta$是第二象限角, 求$\\cot \\theta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{12}{5}$", "solution": "", @@ -506080,7 +509809,9 @@ "id": "021503", "content": "若$\\sin \\alpha \\cos \\alpha=\\dfrac{1}{8}$, $\\alpha \\in(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$, 则$\\cos \\alpha-\\sin \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{\\sqrt{3}}{2}$", "solution": "", @@ -506108,7 +509839,9 @@ "id": "021504", "content": "已知$\\alpha \\in(0, \\pi)$, 满足$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{2}$.\\\\ \n(1) 求$\\sin \\alpha-\\cos \\alpha$的值;\\\\\n(2) 求$\\sin ^4 \\alpha-\\cos ^4 \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\sqrt{7}}{2}$;$\\dfrac{\\sqrt{7}}{4}$", "solution": "", @@ -506136,7 +509869,9 @@ "id": "021505", "content": "已知$\\tan \\alpha$, $\\cot \\alpha$是关于$x$的一元二次方程$x^2+3 k x+k^2-8=0$的两个实数根, 且$3 \\pi<\\alpha<\\dfrac{7}{2} \\pi$, 求$\\sin \\alpha+\\cos \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{\\sqrt{11}}{3}$", "solution": "", @@ -506164,7 +509899,9 @@ "id": "021506", "content": "是否存在锐角$\\alpha$、$\\beta$满足$\\sin \\alpha=\\dfrac{7}{8} \\sin \\beta$, $\\cos \\alpha=\\dfrac{7}{2} \\cos \\beta$? 若存在, 求出$\\alpha$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\pi}{3}$", "solution": "", @@ -506192,7 +509929,9 @@ "id": "021507", "content": "已知$\\alpha \\in[0,2 \\pi)$, 且满足$\\sqrt{\\dfrac{1-\\cos \\alpha}{1+\\cos \\alpha}}=\\dfrac{\\sin \\alpha}{1+\\cos \\alpha}$, 求$\\alpha$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\left[ 0,\\pi \\right )$", "solution": "", @@ -506220,7 +509959,9 @@ "id": "021508", "content": "利用诱导公式求下列各值, 要求写出必要的步骤, 且不能使用计算器:\\\\\n(1) $\\sin (-\\dfrac{\\pi}{3})=$\\blank{200};\\\\\n(2) $\\cos (-\\dfrac{13 \\pi}{4})=$\\blank{200};\\\\\n(3) $\\tan \\dfrac{26 \\pi}{3}=$\\blank{200};\\\\\n(4) $\\cot (-\\dfrac{31 \\pi}{6})=$\\blank{200}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{\\sqrt{3}}{2}$;$-\\dfrac{\\sqrt{2}}{2}$;$-\\sqrt{3}$;$-\\sqrt{3}$", "solution": "", @@ -506250,7 +509991,9 @@ "id": "021509", "content": "分别根据下列条件, 找出满足条件的一个锐角$\\alpha$:\\\\\n(1) $\\sin \\alpha=\\sin 1551^{\\circ}, \\alpha=$\\blank{50};\\\\\n(2) $\\cos a=\\cos (-3312^{\\circ}), \\alpha=$\\blank{50};\\\\\n(3) $\\tan \\alpha=\\tan \\dfrac{190 \\pi}{9}, \\alpha=$\\blank{50};\\\\\n(4) $\\cot \\alpha=\\cot (-\\dfrac{158 \\pi}{15}), \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$69^{\\circ}$;$72^{\\circ}$;$\\dfrac{\\pi}{9}$;$\\dfrac{7 \\pi}{15}$", "solution": "", @@ -506280,7 +510023,9 @@ "id": "021510", "content": "化简: $\\dfrac{\\sin (\\alpha-\\pi) \\cot (\\alpha+\\pi)}{\\cos (\\alpha-\\pi) \\tan (\\alpha+\\pi)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\cot \\alpha$", "solution": "", @@ -506310,7 +510055,9 @@ "id": "021511", "content": "化简: $\\dfrac{\\cos (\\dfrac{7 \\pi}{10}+\\alpha)}{\\cos (\\alpha-\\dfrac{23 \\pi}{10})}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-1$", "solution": "", @@ -506340,7 +510087,9 @@ "id": "021512", "content": "化简: $\\dfrac{\\tan (177^{\\circ}-\\alpha)}{\\tan (543^{\\circ}+\\alpha)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-1$", "solution": "", @@ -506370,7 +510119,9 @@ "id": "021513", "content": "化简: $\\sqrt{1+2 \\sin (\\pi-2) \\cos (\\pi+2)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$ \\sin 2-\\cos 2$", "solution": "", @@ -506400,7 +510151,9 @@ "id": "021514", "content": "化简: $\\sin (\\dfrac{5 \\pi}{6}+\\alpha)+\\cos (\\alpha-\\dfrac{2 \\pi}{3})+\\tan (\\dfrac{3 \\pi}{4}+\\alpha)+\\sin (\\alpha-\\dfrac{13 \\pi}{6})+\\cos (\\dfrac{\\pi}{3}+\\alpha)+\\tan (\\dfrac{5 \\pi}{4}-\\alpha)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$0$", "solution": "", @@ -506430,7 +510183,9 @@ "id": "021515", "content": "不使用计算器, 求$\\tan \\dfrac{\\pi}{11}+\\tan \\dfrac{2 \\pi}{11}+\\cdots+\\tan \\dfrac{10 \\pi}{11}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$0$", "solution": "", @@ -506460,7 +510215,9 @@ "id": "021516", "content": "设$\\cos 130^{\\circ}=a$, 用含$a$的代数式表示$\\tan 230^{\\circ}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{\\sqrt{1-a^2}}{a}$", "solution": "", @@ -506490,7 +510247,9 @@ "id": "021517", "content": "已知$\\cos (\\dfrac{\\pi}{6}-\\alpha)=\\dfrac{\\sqrt{3}}{3}$, 求$\\cos (\\alpha+\\dfrac{5 \\pi}{6})-\\sin ^2(\\alpha-\\dfrac{\\pi}{6})$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{2+\\sqrt{3}}{3}$", "solution": "", @@ -506520,7 +510279,9 @@ "id": "021518", "content": "已知$\\cos (3 \\pi+\\alpha)=-\\dfrac{1}{2}$.\\\\\n(1) 若$\\alpha$为第四象限角, 求$\\sin (2 \\pi-\\alpha)$的值;\\\\\n(2) 设$n$是非零自然数, 求$\\dfrac{\\sin (n \\pi-\\alpha) \\cos [(n+1) \\pi+\\alpha]}{\\tan (\\alpha-n \\pi)}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{3}}{2}$;(2) $\\dfrac{1}{4}$.", "solution": "", @@ -506550,7 +510311,9 @@ "id": "021519", "content": "已知$\\sin \\alpha=\\dfrac{2}{3}, \\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, 则:\\\\\n(1) $\\sin (3 \\pi+\\alpha)=$\\blank{50};\\\\\n(2) $\\cos (\\dfrac{5 \\pi}{2}-\\alpha)=$\\blank{50};\\\\\n(3) $\\sin (-\\dfrac{7}{2} \\pi+\\alpha)=$\\blank{50};\\\\\n(4) $\\tan (\\alpha-\\dfrac{3}{2} \\pi)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $-\\dfrac{2}{3}$; \\\\\n(2) $\\dfrac{2}{3}$; \\\\\n(3) $-\\dfrac{\\sqrt{5}}{3}$;\\\\\n(4) $\\dfrac{\\sqrt{5}}{2}$.", "solution": "", @@ -506579,7 +510342,9 @@ "id": "021520", "content": "用锐角的正弦、余弦、正切或余切表示下列各值:\\\\\n(1) $\\sin 1731^{\\circ}=$\\blank{50};\\\\\n(2) $\\cos (-3412^{\\circ})=$\\blank{50};\\\\\n(3) $\\tan \\dfrac{188 \\pi}{9}=$\\blank{50};\\\\\n(4) $\\cot (-\\dfrac{158 \\pi}{15})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\sin 69^{\\circ}$ ; (2) $-\\cos 8^{\\circ}$ ; \n(3) $-\\tan \\dfrac{\\pi}{9}$; (4) $\\cot \\dfrac{7\\pi}{15}$.", "solution": "", @@ -506608,7 +510373,9 @@ "id": "021521", "content": "已知$\\tan (\\pi-\\alpha)=\\dfrac{1}{3}$, 则$1-2 \\sin \\alpha \\sin (\\alpha-\\dfrac{5 \\pi}{2})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{2}{5}$", "solution": "", @@ -506637,7 +510404,9 @@ "id": "021522", "content": "在平面直角坐标系中, 已知点$M(4,-3)$, 若将$OM$绕原点顺时针转$\\dfrac{3 \\pi}{2}$至$OM'$, 则点$M'$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$(3,4)$", "solution": "", @@ -506666,7 +510435,9 @@ "id": "021523", "content": "化简: $\\sin (21 \\pi-\\alpha)+\\cos (\\dfrac{9 \\pi}{2}-\\alpha)+\\tan (\\dfrac{9 \\pi}{4}-\\alpha)-\\sin (-\\alpha-19 \\pi)-\\cos (-\\alpha-\\dfrac{27 \\pi}{2})-\\tan (-\\alpha-\\dfrac{7 \\pi}{4})$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$0$", "solution": "", @@ -506695,7 +510466,9 @@ "id": "021524", "content": "化简: $\\dfrac{\\sin (\\pi-\\alpha) \\cos (\\dfrac{7 \\pi}{2}-\\alpha) \\cos (\\alpha-8 \\pi)}{\\sin (\\alpha-\\dfrac{5 \\pi}{2}) \\sin (-\\theta-5 \\pi)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\sin \\alpha$", "solution": "", @@ -506724,7 +510497,9 @@ "id": "021525", "content": "设$f(\\theta)=\\dfrac{\\cos ^2(2 \\pi-\\theta)+5 \\cos (-\\theta)-2 \\sin ^2 \\theta}{\\cos ^2(\\pi+\\theta)+\\sin (\\theta-\\dfrac{\\pi}{2})-6}$, 求$f(\\dfrac{\\pi}{3})$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{1}{5}$", "solution": "", @@ -506753,7 +510528,9 @@ "id": "021526", "content": "不使用计算器, 求下列各式的值:\\\\\n(1) $\\tan \\dfrac{5 \\pi}{6} \\cdot \\cos \\dfrac{3 \\pi}{4}+\\tan \\dfrac{2 \\pi}{3} \\cdot \\cot \\dfrac{21 \\pi}{4}=$\\blank{50};\\\\\n(2) $\\dfrac{\\sin (-1234^{\\circ}) \\cos (-495^{\\circ}) \\tan 909^{\\circ}}{\\sin 694^{\\circ} \\cos 1230^{\\circ} \\tan 711^{\\circ}}=$\\blank{50};\\\\\n(3) $\\dfrac{\\sin (31^{\\circ}+\\alpha)}{\\tan (27^{\\circ}+\\alpha)} \\cdot \\dfrac{\\tan (747^{\\circ}+\\alpha)}{\\cos (36^{\\circ}+\\alpha)} \\cdot \\dfrac{\\cos (1116^{\\circ}+\\alpha)}{\\sin (751^{\\circ}+\\alpha)}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\dfrac{\\sqrt{6}}{6}-\\sqrt{3}$;\\\\\n(2) $-\\dfrac{\\sqrt{6}}{3}$;\\\\\n(3) $1$", "solution": "", @@ -506782,7 +510559,9 @@ "id": "021527", "content": "已知角$\\alpha \\in[0,2 \\pi)$.\\\\\n(1) 若$\\alpha$的终边经过点$(\\cos (-\\dfrac{6 \\pi}{5}), \\sin \\dfrac{6 \\pi}{5})$, 则$\\alpha=$\\blank{50};\\\\\n(2) 若$\\alpha$的终边经过点$(\\cos \\dfrac{6 \\pi}{5}, \\sin (-\\dfrac{6 \\pi}{5}))$, 则$\\alpha=$\\blank{50};\\\\\n(3) 若$\\alpha$的终边经过点$(\\sin \\dfrac{6 \\pi}{5}, \\cos \\dfrac{6 \\pi}{5})$, 则$\\alpha=$\\blank{50};\\\\\n(4) 若$\\alpha$的终边经过点$(\\sin (-\\dfrac{6 \\pi}{5}), \\cos \\dfrac{6 \\pi}{5})$, 则$\\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\dfrac{6 \\pi}{5}$; (2) $\\dfrac{4 \\pi}{5}$; (3) $\\dfrac{13 \\pi}{10}$; (4) $\\dfrac{17 \\pi}{10}$.", "solution": "", @@ -506811,7 +510590,9 @@ "id": "021528", "content": "已知$\\cos (3 \\pi+\\alpha)=-\\dfrac{1}{2}$, $k \\in \\mathbf{Z}$.\\\\\n(1) 求$\\sin (2 \\pi-\\alpha)$;\\\\\n(2) 求$\\dfrac{1}{\\tan [\\dfrac{(2 k+1) \\pi}{2}+\\alpha]}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 当$\\alpha$在第一象限时, $\\sin (2 \\pi-\\alpha)=-\\dfrac{\\sqrt{3}}{2}$;\n当$\\alpha$在第三象限时, $\\sin (2 \\pi-\\alpha)=\\dfrac{\\sqrt{3}}{2}$.\\\\\n(2) 当$\\alpha$在第一象限时, $\\dfrac{1}{\\tan [\\dfrac{(2 k+1) \\pi}{2}+\\alpha]}=-\\sqrt{3}$;\n当$\\alpha$在第四象限时, $\\dfrac{1}{\\tan [\\dfrac{(2 k+1) \\pi}{2}+\\alpha]}=\\sqrt{3}$.", "solution": "", @@ -506840,7 +510621,9 @@ "id": "021529", "content": "根据下列条件, 分别写出角$x$的集合:\\\\\n(1) $\\sin x=\\dfrac{\\sqrt{2}}{2}$, $x$的集合为\\blank{100};\\\\\n(2) $\\cos x=-\\dfrac{1}{2}$, $x$的集合为\\blank{100};\\\\\n(3) $\\cot x=-\\sqrt{3}$, $x$的集合为\\blank{100};\\\\\n(4) $\\sin (x-\\dfrac{2 \\pi}{3})=\\dfrac{1}{2}$, $x$的集合为\\blank{100};\\\\\n(5) $\\tan (\\dfrac{\\pi}{4}-x)=\\dfrac{\\sqrt{3}}{2}$, $x$的集合为\\blank{100};\\\\\n(6) $\\cos (5 x+\\dfrac{\\pi}{4})=-\\dfrac{\\sqrt{3}}{2}$, $x$的集合为\\blank{100};\\\\\n(7) $\\sin (\\dfrac{4 x+7 \\pi}{2})=\\dfrac{\\sqrt{2}}{2}$, $x$的集合为\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\{x | x=k \\pi+ (-1)^k \\cdot \\dfrac{\\pi}{4},\\ k \\in \\mathbf{Z}\\}$;\\\\\n(2) $\\{x | x=2k \\pi \\pm \\dfrac{2\\pi}{3},\\ k \\in \\mathbf{Z}\\}$;\\\\\n(3) $\\{x | x=k \\pi + \\dfrac{5\\pi}{6},\\ k \\in \\mathbf{Z}\\}$;\\\\\n(4) $\\{x | x=2k \\pi + \\dfrac{5\\pi}{6}$ 或$x=2k \\pi + \\dfrac{3\\pi}{2} ,\\ k \\in \\mathbf{Z}\\}$;\\\\\n第二种写法： $\\{x | x=k \\pi+ (-1)^k \\cdot \\dfrac{\\pi}{6}+\\dfrac{2\\pi}{3},\\ k \\in \\mathbf{Z}\\}$;\\\\\n(5) $\\{x | x=k \\pi - \\arctan \\dfrac{\\sqrt{3}}{2}+ \\dfrac{\\pi}{4},\\ k \\in \\mathbf{Z}\\}$;\\\\\n(6) $\\{x | x=\\dfrac{2k \\pi}{5} + \\dfrac{7\\pi}{60}$ 或$ x=\\dfrac{2k \\pi}{5} - \\dfrac{13\\pi}{60} ,\\ k \\in \\mathbf{Z}\\}$;\\\\\n(7) $\\{x | x=k \\pi - \\dfrac{5\\pi}{8}$ 或$x=k \\pi - \\dfrac{3\\pi}{8} ,\\ k \\in \\mathbf{Z}\\}$;", "solution": "", @@ -506860,7 +510643,9 @@ "id": "021530", "content": "分别求满足下列条件的角$x$的集合:\\\\\n(1) $\\cos (x+\\dfrac{\\pi}{4})=\\dfrac{1}{2}$, $x \\in(0,2 \\pi)$;\\\\\n(2) $3 \\tan (x+\\dfrac{\\pi}{3})=\\sqrt{3}$, $x \\in(0, \\pi)$;\\\\\n(3) $2 \\sin 2 x-1=0$, $x \\in(0, \\dfrac{\\pi}{2})$;\\\\\n(4) $3 \\cot (\\dfrac{x}{2}-\\dfrac{\\pi}{12})=\\sqrt{3}$, $x \\in[0,2 \\pi)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\{ \\dfrac{\\pi}{12},\\dfrac{17\\pi}{12} \\}$;\\\\\n(2) $\\{ \\dfrac{5\\pi}{6} \\}$;\\\\\n(3) $\\{ \\dfrac{\\pi}{12},\\dfrac{5\\pi}{12} \\}$;\\\\\n(4) $\\{ \\dfrac{5\\pi}{6} \\}$.", "solution": "", @@ -506880,7 +510665,9 @@ "id": "021531", "content": "分别求满足下列条件的角$x$的集合:\\\\\n(1) $\\cos 2 x=\\cos 3 x$;\\\\\n(2) $\\sin x=\\cos 2 x$;\\\\\n(3) $2 \\sin ^2 x+\\sin x-1=0$;\\\\\n(4) $\\sin ^2 x-\\dfrac{2 \\sqrt{3}}{3} \\sin x \\cos x-\\cos ^2 x=0$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\{x | x= \\dfrac{2k \\pi}{5} ,\\ k \\in \\mathbf{Z}\\}$;\\\\\n(2) $\\{x | x= \\dfrac{2k \\pi}{3} +\\dfrac{ \\pi}{6},\\ k \\in \\mathbf{Z}\\}$;\\\\\n(3) $\\{x | x= 2k \\pi$ 或$x=k \\pi +(-1)^k \\cdot \\dfrac{ \\pi}{6},\\ k \\in \\mathbf{Z}\\}$;\\\\\n(4) $\\{x | x= k \\pi+\\dfrac{ \\pi}{3}$ 或$x=k \\pi -\\dfrac{ \\pi}{6},\\ k \\in \\mathbf{Z}\\}$.", "solution": "", @@ -506900,7 +510687,9 @@ "id": "021532", "content": "设第四象限角$\\alpha$满足$\\cos \\alpha=\\dfrac{3}{5}$, 则$\\cos (\\alpha+\\dfrac{\\pi}{3})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{3+4\\sqrt{3}}{10}$", "solution": "", @@ -506928,7 +510717,9 @@ "id": "021533", "content": "若$\\sin \\alpha \\cdot \\sin \\beta=1$, 则$\\cos (\\alpha+\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-1$", "solution": "", @@ -506956,7 +510747,9 @@ "id": "021534", "content": "已知$\\sin \\alpha+\\sin \\beta=\\dfrac{1}{5}$, $\\cos \\alpha+\\cos \\beta=\\dfrac{4}{5}$, 则$\\cos (\\alpha-\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{33}{50}$", "solution": "", @@ -506984,7 +510777,9 @@ "id": "021535", "content": "不用计算器, 求下列各式的值:\\\\\n(1) $\\cos \\dfrac{5 \\pi}{12}$;\\\\\n(2) $\\cos 345^{\\circ}$;\\\\\n(3) $\\cos \\dfrac{7 \\pi}{10} \\cos (-\\dfrac{2 \\pi}{10})+\\sin \\dfrac{3 \\pi}{10} \\sin \\dfrac{2 \\pi}{10}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{6}-\\sqrt{2}}{4}$;\n(2) $\\dfrac{\\sqrt{6}+\\sqrt{2}}{4}$;\n(3) $0$.", "solution": "", @@ -507012,7 +510807,9 @@ "id": "021536", "content": "化简下列各式:\\\\\n(1) $\\cos (\\dfrac{\\pi}{3}-\\alpha)-\\cos (\\dfrac{\\pi}{3}+\\alpha)$;\\\\\n(2) $\\sin (\\alpha-\\beta) \\sin \\beta+\\cos (\\alpha-\\beta) \\cos \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\sqrt{3} \\sin \\alpha$;\n(2) $\\cos(\\alpha-2\\beta)$.", "solution": "", @@ -507040,7 +510837,9 @@ "id": "021537", "content": "设$\\alpha, \\beta \\in(\\dfrac{\\pi}{2}, \\pi)$, 满足$\\sin \\alpha=\\dfrac{8}{17}$, $\\tan \\beta=-\\dfrac{5}{12}$, 求$\\cos (\\alpha+\\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{140}{221}$", "solution": "", @@ -507068,7 +510867,9 @@ "id": "021538", "content": "设$\\alpha \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 且满足$\\sin (\\alpha-\\dfrac{\\pi}{6})=\\dfrac{1}{3}$, 求$\\cos \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{2\\sqrt{6}-1}{6}$", "solution": "", @@ -507096,7 +510897,9 @@ "id": "021539", "content": "求证: $\\cos (\\alpha+\\beta) \\cos (\\beta-\\alpha)=\\cos ^2 \\alpha-\\sin ^2 \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "证明略", "solution": "", @@ -507124,7 +510927,9 @@ "id": "021540", "content": "若$\\alpha$、$\\beta$是锐角, 则下列不等式一定成立的是\\bracket{20}.\n\\twoch{$\\cos (\\alpha+\\beta)>0$}{$\\cos (\\alpha+\\beta)<0$}{$\\cos (\\alpha-\\beta)>\\cos (\\alpha+\\beta)$}{$\\cos (\\alpha-\\beta)<\\cos (\\alpha+\\beta)$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -507152,7 +510957,9 @@ "id": "021541", "content": "若$\\cos (\\alpha+\\beta) \\cos (\\alpha-\\beta)=\\dfrac{1}{4}$, 则$\\cos ^2 \\alpha+\\cos ^2 \\beta=$\\bracket{20}.\n\\fourch{$\\dfrac{5}{4}$}{$1$}{$\\dfrac{1}{4}$}{$\\dfrac{3}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -507180,7 +510987,9 @@ "id": "021542", "content": "已知$\\sin \\alpha=\\dfrac{2}{3}$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, $\\cos \\beta=-\\dfrac{3}{4}$, $\\beta \\in(\\dfrac{\\pi}{2}, \\pi)$, 求$\\cos (\\alpha-\\beta+\\dfrac{\\pi}{4})$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{3\\sqrt{10}+6\\sqrt{2}+2\\sqrt{14}-\\sqrt{70}}{24}$", "solution": "", @@ -507208,7 +511017,9 @@ "id": "021543", "content": "已知$\\sin \\theta+\\cos \\theta=\\dfrac{1}{5}$, $\\theta \\in(0, \\pi)$, 求$\\cos (\\theta-\\dfrac{\\pi}{3})+\\cot \\theta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{8\\sqrt{3}-21}{20}$", "solution": "", @@ -507236,7 +511047,9 @@ "id": "021544", "content": "已知锐角$\\alpha$、$\\beta$满足$\\sin \\alpha-\\sin \\beta=\\dfrac{\\sqrt{2}}{2}$, $\\cos \\alpha+\\cos \\beta=\\dfrac{\\sqrt{6}}{2}$, 求$\\alpha+\\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\pi}{2}$", "solution": "", @@ -507264,7 +511077,9 @@ "id": "021545", "content": "已知$\\sin \\alpha=\\dfrac{2}{3}$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, 则$\\sin (\\dfrac{\\pi}{3}-\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{2+\\sqrt{15}}{6}$", "solution": "", @@ -507294,7 +511109,9 @@ "id": "021546", "content": "不使用计算器, 求值: $\\sin 28^{\\circ} \\cos 73^{\\circ}-\\sin 62^{\\circ} \\cos 17^{\\circ}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{\\sqrt{2}}{2}$", "solution": "", @@ -507324,7 +511141,9 @@ "id": "021547", "content": "化简: $\\sin (\\alpha+\\beta) \\cdot \\cos (\\alpha-\\beta)-\\cos (\\alpha+\\beta) \\cdot \\sin (\\alpha-\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\sin 2\\beta$", "solution": "", @@ -507354,7 +511173,9 @@ "id": "021548", "content": "化简: $\\sin \\alpha+\\sin (\\alpha+\\dfrac{2 \\pi}{3})+\\sin (\\alpha+\\dfrac{4 \\pi}{3})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$0$", "solution": "", @@ -507384,7 +511205,9 @@ "id": "021549", "content": "不使用计算器, 求值: $\\dfrac{\\sin 9^{\\circ}+\\sin 6^{\\circ} \\cos 15^{\\circ}}{\\cos 9^{\\circ}-\\sin 6^{\\circ} \\sin 15^{\\circ}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2-\\sqrt{3}$", "solution": "", @@ -507414,7 +511237,9 @@ "id": "021550", "content": "在$\\triangle ABC$中, 若$\\sin A=\\dfrac{4}{5}$, $\\cos B=-\\dfrac{5}{13}$, 则$\\sin C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{16}{65}$", "solution": "", @@ -507444,7 +511269,9 @@ "id": "021551", "content": "已知$\\tan \\alpha=-\\dfrac{4}{3}$, $\\sin \\beta=\\dfrac{3}{5}$, 且$\\alpha$、$\\beta \\in(\\dfrac{\\pi}{2}, \\pi)$, 求$\\sin (\\alpha-\\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{7}{25}$", "solution": "", @@ -507474,7 +511301,9 @@ "id": "021552", "content": "若$\\alpha \\in(0, \\dfrac{\\pi}{2})$, $\\beta \\in(-\\dfrac{\\pi}{2}, 0)$, $\\sin (\\alpha+\\beta)=-\\dfrac{\\sqrt{2}}{4}$, $\\cos \\alpha=\\dfrac{3}{5}$, 求$\\sin \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{4\\sqrt{14}+3\\sqrt{2}}{20}$", "solution": "", @@ -507504,7 +511333,9 @@ "id": "021553", "content": "已知点$P(4,3)$, 将$P$绕坐标原点$O$顺时针方向旋转$\\dfrac{\\pi}{6}$至点$P'$, 求$P'$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$(\\dfrac{4\\sqrt{3}+3}{2},\\dfrac{3\\sqrt{3}-4}{2})$", "solution": "", @@ -507534,7 +511365,9 @@ "id": "021554", "content": "在$\\triangle ABC$中, 若$\\cos A=\\dfrac{12}{13}$, $\\sin B=\\dfrac{3}{5}$, 则$\\cos C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{33}{65}$或$\\dfrac{63}{65}$", "solution": "", @@ -507564,7 +511397,9 @@ "id": "021555", "content": "若$\\alpha$、$\\beta$为锐角, 则下列不等式中一定成立的是\\bracket{20}.\n\\twoch{$\\sin (\\alpha+\\beta)>\\sin \\alpha+\\sin \\beta$}{$\\sin (\\alpha+\\beta)<\\sin \\alpha+\\sin \\beta$}{$\\cos (\\alpha+\\beta)>\\cos \\alpha+\\cos \\beta$}{$\\cos (\\alpha+\\beta)>\\sin \\alpha+\\sin \\beta$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -507594,7 +511429,9 @@ "id": "021556", "content": "$\\triangle ABC$中, 若两内角$A$、$B$满足$\\cot A \\cdot \\cot B>1$, 则$\\triangle ABC$的形状为\\bracket{20}三角形.\n\\fourch{锐角}{直角}{钝角}{无法确定的}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -507624,7 +511461,9 @@ "id": "021557", "content": "已知$\\alpha, \\beta \\in(\\dfrac{3 \\pi}{4}, \\pi)$, $\\sin (\\alpha+\\beta)=-\\dfrac{3}{5}$, $\\sin (\\beta-\\dfrac{\\pi}{4})=\\dfrac{12}{13}$, 求$\\cos (\\alpha+\\dfrac{\\pi}{4})$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{56}{65}$", "solution": "", @@ -507654,7 +511493,9 @@ "id": "021558", "content": "若$\\sin \\alpha=\\dfrac{4}{5}, \\cot \\beta=3$, 且$\\alpha$是第二象限的角, 则$\\tan (\\alpha-\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-3$", "solution": "", @@ -507684,7 +511525,9 @@ "id": "021559", "content": "若$\\tan \\alpha, \\tan \\beta$是方程$x^2-3 x-3=0$的两个实数解, 则$\\tan (\\alpha+\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{3}{4}$", "solution": "", @@ -507714,7 +511557,9 @@ "id": "021560", "content": "若$\\tan (\\theta+\\dfrac{\\pi}{6})=3$, 则$\\tan \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{-6+5\\sqrt{3}}{3}$", "solution": "", @@ -507744,7 +511589,9 @@ "id": "021561", "content": "化简: $\\dfrac{\\tan (\\alpha-\\beta)+\\tan \\beta}{1-\\tan (\\alpha-\\beta) \\tan \\beta}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\tan \\alpha$", "solution": "", @@ -507774,7 +511621,9 @@ "id": "021562", "content": "不用计算器, 求值: $\\tan 36^{\\circ}+\\sqrt{3} \\tan 24^{\\circ} \\tan 36^{\\circ}+\\tan 24^{\\circ}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\sqrt{3}$", "solution": "", @@ -507804,7 +511653,9 @@ "id": "021563", "content": "不用计算器, 求值: $\\dfrac{1-\\tan 75^{\\circ}}{1+\\tan 75^{\\circ}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{\\sqrt{3}}{3}$", "solution": "", @@ -507834,7 +511685,9 @@ "id": "021564", "content": "甲: $\\tan \\alpha+\\tan \\beta=0$, 乙: $\\tan (\\alpha+\\beta)=0$, 则甲是乙的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -507864,7 +511717,9 @@ "id": "021565", "content": "已知$\\sin \\theta=-\\dfrac{7}{25}$, $\\theta \\in(\\pi, \\dfrac{3 \\pi}{2})$, 求$\\tan (\\theta-\\dfrac{\\pi}{4})$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{17}{31}$", "solution": "", @@ -507894,7 +511749,9 @@ "id": "021566", "content": "已知$(1+\\tan \\alpha)(1+\\tan \\beta)=2$, 且$\\alpha$、$\\beta$都是锐角, 求$\\alpha+\\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\pi}{4}$", "solution": "", @@ -507924,7 +511781,9 @@ "id": "021567", "content": "已知$\\tan (\\dfrac{\\pi}{4}+\\alpha)=2$, $\\tan \\beta=\\dfrac{1}{2}$. 求:\\\\\n(1) $\\tan \\alpha$;\\\\\n(2) $\\dfrac{\\sin (\\alpha+\\beta)-2 \\sin \\alpha \\cos \\beta}{2 \\sin \\alpha \\sin \\beta+\\cos (\\alpha+\\beta)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{1}{3}$;\n(2) $\\dfrac{1}{7}$", "solution": "", @@ -507954,7 +511813,9 @@ "id": "021568", "content": "已知$\\cos (\\alpha+\\beta)=\\dfrac{1}{2}$, $\\cos (\\alpha-\\beta)=\\dfrac{1}{3}$, 求$\\tan \\alpha \\tan \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{1}{5}$", "solution": "", @@ -507984,7 +511845,9 @@ "id": "021569", "content": "镜框$AB$的高为$0.96$米, 平挂在墙上, 与人眼的水平视线的高度差为$OB=1$米. 现人眼在点$C$处, 当人朝墙走去时, 在什么位置$\\tan \\angle ACB$最大?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [right] {$O$} coordinate (O);\n\\draw (0,1) node [right] {$B$} coordinate (B);\n\\draw (0,1.96) node [right] {$A$} coordinate (A);\n\\draw (-2,0) node [left] {$C$} coordinate (C);\n\\draw (C)--(O)--(A)--cycle(B)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "当$CD = 1.4$米时，$\\tan \\angle ACB$最大", "solution": "", @@ -508014,7 +511877,9 @@ "id": "021570", "content": "把下列各式化成$A \\sin (\\alpha+\\varphi)$($A>0$, $\\varphi \\in[0,2 \\pi)$)的形式:\\\\\n(1) $\\sqrt{3} \\sin \\alpha+\\cos \\alpha=$\\blank{50};\\\\\n(2) $\\sin x-\\cos x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $2 \\sin (\\alpha+\\dfrac{\\pi}{6})$;\n(2) $\\sqrt{2} \\sin (\\alpha+\\dfrac{7\\pi}{4})$.", "solution": "", @@ -508043,7 +511908,9 @@ "id": "021571", "content": "把$3 \\cos \\alpha-3 \\sqrt{3} \\sin \\alpha$化成$A \\cos (\\alpha+\\varphi)$($A>0$, $\\varphi \\in(-\\pi, \\pi)$)的形式, 则结果为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$6\\cos(\\alpha+\\dfrac{\\pi}{3})$", "solution": "", @@ -508072,7 +511939,9 @@ "id": "021572", "content": "若角$\\alpha$满足$\\dfrac{1}{2} \\cos \\alpha-\\dfrac{\\sqrt{3}}{2} \\sin \\alpha=1$, 则$\\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2k \\pi-\\dfrac{\\pi}{3}(k\\in \\mathbf{Z} )$", "solution": "", @@ -508103,7 +511972,9 @@ "id": "021573", "content": "下列关系中, 角$\\alpha$存在的是\\bracket{20}.\n\\twoch{$\\sin \\alpha+\\cos \\alpha=\\dfrac{3}{2}$}{$\\sin \\alpha+\\cos \\alpha=\\dfrac{4}{3}$}{$\\sin \\alpha=\\dfrac{1}{3}$且$\\cos \\alpha=\\dfrac{2}{3}$}{$\\cos \\alpha-\\sin \\alpha=-\\sqrt{3}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -508134,7 +512005,9 @@ "id": "021574", "content": "若$\\tan (\\alpha+\\beta)=\\dfrac{3}{4}$, $\\tan (\\beta+\\dfrac{\\pi}{4})=\\dfrac{1}{3}$, 求$\\tan (\\alpha-\\dfrac{\\pi}{4})$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{1}{3}$", "solution": "", @@ -508163,7 +512036,9 @@ "id": "021575", "content": "已知$\\alpha$是$\\triangle ABC$的一个内角, 且满足$\\sin \\alpha+\\cos \\alpha=\\dfrac{\\sqrt{6}}{2}$, 求$\\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\pi}{12}$或$\\dfrac{5\\pi}{12}$", "solution": "", @@ -508192,7 +512067,9 @@ "id": "021576", "content": "已知$\\sin (\\alpha+\\beta)=\\dfrac{1}{2}$, $\\sin (\\alpha-\\beta)=\\dfrac{1}{3}$, 求$\\tan \\alpha \\cot \\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$5$", "solution": "", @@ -508221,7 +512098,9 @@ "id": "021577", "content": "已知$8 \\cos (2 \\alpha+\\beta)+5 \\cos \\beta=0$, 且$\\cos (\\alpha+\\beta) \\cos \\alpha \\neq 0$, 求$\\tan (\\alpha+\\beta) \\cdot \\tan \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{13}{3}$", "solution": "", @@ -508250,7 +512129,9 @@ "id": "021578", "content": "已知关于$x$的方程$x^2+p x+q=0$的两根是$\\tan \\alpha$和$\\tan \\beta$, 求$\\dfrac{\\sin (\\alpha+\\beta)}{\\cos (\\alpha-\\beta)}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{p}{1+q}$", "solution": "", @@ -508279,7 +512160,9 @@ "id": "021579", "content": "已知$\\tan (\\alpha+\\beta)=-2$, $\\tan (\\alpha-\\beta)=\\dfrac{1}{2}$, 求$\\dfrac{\\sin 2 \\alpha}{\\sin 2 \\beta}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{3}{5}$", "solution": "", @@ -508308,7 +512191,9 @@ "id": "021580", "content": "若$\\tan \\dfrac{3 \\alpha}{2}=\\dfrac{3}{4}$, 则$\\tan 3 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{24}{7}$", "solution": "", @@ -508337,7 +512222,9 @@ "id": "021581", "content": "已知$\\sin 2 \\alpha=\\dfrac{4}{5}$, $\\alpha \\in(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$, 则$\\sin 4 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{24}{25}$", "solution": "", @@ -508366,7 +512253,9 @@ "id": "021582", "content": "若$\\alpha \\in(\\pi, \\dfrac{3}{2} \\pi)$, $\\tan \\alpha=4$, 则$\\cos 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{15}{17}$", "solution": "", @@ -508395,7 +512284,9 @@ "id": "021583", "content": "已知$\\cos \\varphi=-\\dfrac{1}{3}$, 且$\\pi<\\varphi<\\dfrac{3 \\pi}{2}$, 求$\\sin 2 \\varphi$、$\\cos 2 \\varphi$和$\\tan 2 \\varphi$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\sin 2 \\varphi=\\dfrac{4\\sqrt{2}}{9}$;\n$\\cos 2 \\varphi=-\\dfrac{7}{9}$;\n$\\tan 2 \\varphi=-\\dfrac{4\\sqrt{2}}{7}$.", "solution": "", @@ -508424,7 +512315,9 @@ "id": "021584", "content": "已知等腰三角形的底角的正弦值等于$\\dfrac{4}{5}$, 求这个三角形的顶角的正弦、余弦和正切的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{24}{25}$; $\\dfrac{7}{25}$; $\\dfrac{24}{7}$", "solution": "", @@ -508444,7 +512337,9 @@ "id": "021585", "content": "不用计算器, 求下列各式的值:\\\\\n(1) $\\dfrac{1}{1+\\tan 15^{\\circ}}-\\dfrac{1}{1-\\tan 15^{\\circ}}$;\\\\\n(2) $\\sin ^4 \\dfrac{\\pi}{8}+\\cos ^4 \\dfrac{\\pi}{8}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $-\\dfrac{\\sqrt{3}}{3}$;\\\\\n(2) $\\dfrac{3}{4}$.", "solution": "", @@ -508473,7 +512368,9 @@ "id": "021586", "content": "已知$\\sin \\alpha=\\dfrac{3}{5}$, $\\tan (\\pi-\\beta)=\\dfrac{1}{2}$, 求$\\tan (\\alpha-2 \\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{7}{24}$", "solution": "", @@ -508502,7 +512399,9 @@ "id": "021587", "content": "已知$\\sin 2 \\alpha=-\\dfrac{3}{5}$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, 求$\\cos \\alpha-\\sin \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{2\\sqrt{10}}{5}$", "solution": "", @@ -508531,7 +512430,9 @@ "id": "021588", "content": "化简: $\\cos ^2(\\theta+15^{\\circ})+\\sin ^2(\\theta-15^{\\circ})+\\sin (\\theta+180^{\\circ}) \\cos (\\theta-180^{\\circ})$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -508560,7 +512461,9 @@ "id": "021589", "content": "证明下列恒等式:\\\\\n(1) $1+\\sin \\alpha=(\\sin \\dfrac{\\alpha}{2}+\\cos \\dfrac{\\alpha}{2})^2$;\\\\\n(2) $\\dfrac{1+\\sin 2 \\alpha-\\cos 2 \\alpha}{1+\\sin 2 \\alpha+\\cos 2 \\alpha}=\\tan \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -508589,7 +512492,9 @@ "id": "021590", "content": "若$\\sin \\alpha=\\dfrac{8}{5} \\sin \\dfrac{\\alpha}{2}$, 求$\\cos \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$1$或$\\dfrac{7}{25}$", "solution": "", @@ -508618,7 +512523,9 @@ "id": "021591", "content": "若$\\theta \\in(-\\dfrac{3}{2} \\pi,-\\pi)$, 且$\\cos \\theta=a$, 则$\\sin \\dfrac{\\theta}{2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{\\sqrt{2-2a}}{2}$", "solution": "", @@ -508647,7 +512554,9 @@ "id": "021592", "content": "若$\\sin \\dfrac{\\alpha}{2}=-\\dfrac{5}{6}$, $\\cos \\dfrac{\\alpha}{2}=\\dfrac{\\sqrt{11}}{6}$, 则$\\alpha$的终边在第\\blank{50}象限.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "第三象限", "solution": "", @@ -508676,7 +512585,9 @@ "id": "021593", "content": "已知$\\cos \\theta=\\dfrac{1}{3}$, 且$\\theta$是第四象限的角, 求$\\sin \\dfrac{\\theta}{2}$、$\\cos \\dfrac{\\theta}{2}$和$\\tan \\dfrac{\\theta}{2}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "当$\\dfrac{\\theta}{2}$在第二象限时，\n$\\sin \\dfrac{\\theta}{2}=\\dfrac{\\sqrt{3}}{3}$,\n$\\cos \\dfrac{\\theta}{2}=-\\dfrac{\\sqrt{6}}{3}$,\n$\\tan \\dfrac{\\theta}{2}=-\\dfrac{\\sqrt{2}}{2}$;\\\\\n当$\\dfrac{\\theta}{2}$在第四象限时，\n$\\sin \\dfrac{\\theta}{2}=-\\dfrac{\\sqrt{3}}{3}$,\n$\\cos \\dfrac{\\theta}{2}=\\dfrac{\\sqrt{6}}{3}$,\n$\\tan \\dfrac{\\theta}{2}=-\\dfrac{\\sqrt{2}}{2}$.", "solution": "", @@ -508705,7 +512616,9 @@ "id": "021594", "content": "已知等腰三角形的顶角的余弦值等于$-\\dfrac{7}{25}$, 求这个三角形的底角的正弦、余弦的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{3}{5}$;$\\dfrac{4}{5}$", "solution": "", @@ -508734,7 +512647,9 @@ "id": "021595", "content": "证明下列恒等式:\\\\\n(1) $8 \\sin ^4 \\alpha=\\cos 4 \\alpha-4 \\cos 2 \\alpha+3$;\\\\\n(2) $\\dfrac{1+\\sin \\alpha}{\\cos \\alpha}=\\dfrac{1+\\tan \\dfrac{\\alpha}{2}}{1-\\tan \\dfrac{\\alpha}{2}}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -508763,7 +512678,9 @@ "id": "021596", "content": "已知$\\dfrac{\\cos 2 x}{1+\\sin 2 x}=\\dfrac{1}{5}$, 求$\\tan x$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{2}{3}$", "solution": "", @@ -508792,7 +512709,9 @@ "id": "021597", "content": "已知$\\alpha \\in(\\dfrac{5}{4} \\pi, \\dfrac{3}{2} \\pi)$, 化简$\\sqrt{1-\\sin 2 \\alpha}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\cos \\alpha-\\sin \\alpha$", "solution": "", @@ -508821,7 +512740,9 @@ "id": "021598", "content": "化简: $\\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2} \\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2} \\cos 2 \\alpha}},(\\pi<\\alpha<\\dfrac{3 \\pi}{2})$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\sin \\dfrac{ \\alpha}{2}$", "solution": "", @@ -508850,7 +512771,9 @@ "id": "021599", "content": "化简:\\\\\n(1) $\\dfrac{(\\sin \\theta+\\cos \\theta-1)(\\sin \\theta-\\cos \\theta+1)}{2 \\sin \\theta \\cos \\theta}$;\\\\\n(2) $\\dfrac{1+\\sin \\alpha}{2 \\cos ^2(\\dfrac{\\pi}{4}-\\dfrac{\\alpha}{2})}-2 \\sin ^2(\\dfrac{\\pi}{4}-\\dfrac{\\alpha}{2})$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\tan \\dfrac{\\theta}{2}$; (2) $\\sin \\alpha$.", "solution": "", @@ -508879,7 +512802,9 @@ "id": "021600", "content": "在$\\triangle ABC$中, 若$B=45^{\\circ}$, $C=60^{\\circ}$, $b=1$, 则边$c=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{6}}{2}$", "solution": "", @@ -508907,7 +512832,9 @@ "id": "021601", "content": "在$\\triangle ABC$中, 若$a=1$, $b=\\sqrt{3}$, $A=30^{\\circ}$, 则角$C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$30^{\\circ}$或$90^{\\circ}$", "solution": "", @@ -508935,7 +512862,9 @@ "id": "021602", "content": "在$\\triangle ABC$中, 若$B=45^{\\circ}$, $C=15^{\\circ}$, $b=2$, 则该三角形的最长边长等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\sqrt{6}$", "solution": "", @@ -508963,7 +512892,9 @@ "id": "021603", "content": "在$\\triangle ABC$中, 若$A=60^{\\circ}$, $b=16$, $S_{\\triangle ABC}=220 \\sqrt{3}$, 则边$c=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$55$", "solution": "", @@ -508991,7 +512922,9 @@ "id": "021604", "content": "在$\\triangle ABC$中, 若$\\sqrt{3} a=2 b \\sin A$, 则$B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{3}$或$\\dfrac{2\\pi}{3}$", "solution": "", @@ -509019,7 +512952,9 @@ "id": "021605", "content": "若三角形的三内角之比为$1: 2: 3$, 则三边长之比为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$1: \\sqrt{3}: 2$", "solution": "", @@ -509047,7 +512982,9 @@ "id": "021606", "content": "在$\\triangle ABC$中, 若$a^2+b^2=2 c^2$, 则$\\dfrac{\\sin ^2A+\\sin ^2B}{\\sin ^2C}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -509075,7 +513012,9 @@ "id": "021607", "content": "在$\\triangle ABC$中, 已知$a=5$, $b=4$, $A=2B$, 则$\\cos B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{5}{8}$", "solution": "", @@ -509103,7 +513042,9 @@ "id": "021608", "content": "在$\\triangle ABC$中, 满足$\\dfrac{a}{b}=\\dfrac{\\cos A}{\\cos B}$的三角形是\\blank{50}三角形.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "等腰", "solution": "", @@ -509131,7 +513072,9 @@ "id": "021609", "content": "在$\\triangle ABC$中, 三内角的度数比是$3: 4: 5$. 若最小边长为$3$, 则此三角形的外接圆的半径是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{3\\sqrt{2}}{2}$", "solution": "", @@ -509159,7 +513102,9 @@ "id": "021610", "content": "在$\\triangle ABC$中, 已知$\\cos A=\\dfrac{\\sqrt{2}}{2}$, $B-C=\\dfrac{\\pi}{12}$, $a=\\sqrt{2}$, 求$c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\sqrt{3}$", "solution": "", @@ -509187,7 +513132,9 @@ "id": "021611", "content": "在$\\triangle ABC$中, 若$a0$, 则$C$一定是锐角;\\\\\n\\textcircled{2} 在$\\triangle ABC$中, 若$a^2>b^2+c^2$, 则$A>B+C$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{1}；\\textcircled{2}", "solution": "", @@ -509387,7 +513346,9 @@ "id": "021618", "content": "若$a$、$a+1$、$a+2$是锐角三角形的三边长, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$a>3$", "solution": "", @@ -509416,7 +513377,9 @@ "id": "021619", "content": "在$\\triangle ABC$中, 已知$b=5$, $c=4$, $A=60^{\\circ}$, 求$a$和$\\sin B$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$a=\\sqrt{21}$和$\\sin B=\\dfrac{5\\sqrt{7}}{14}$", "solution": "", @@ -509445,7 +513408,9 @@ "id": "021620", "content": "在$\\triangle ABC$中, 已知$a, b, c$满足$(a+b+c)(a-b+c)=a c$, 求$B$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{2\\pi}{3}$", "solution": "", @@ -509474,7 +513439,9 @@ "id": "021621", "content": "在$\\triangle ABC$中, 已知$a=2 \\sqrt{3}$, $b=45^{\\circ}$, 面积$S=3+\\sqrt{3}$, 求$c$和$C$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$c=\\sqrt{6}+\\sqrt{2}$;$C=75^\\circ$.", "solution": "", @@ -509503,7 +513470,9 @@ "id": "021622", "content": "在$\\triangle ABC$中, $A=60^{\\circ}$, $b=3$, $c=2$, 求$BC$边上的中线长.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\sqrt{19}}{2}$", "solution": "", @@ -509532,7 +513501,9 @@ "id": "021623", "content": "已知三角形两边之和为$8$, 其夹角为$60^{\\circ}$, 分别求这个三角形周长的最小值和面积的最大值, 并指出面积最大时三角形的形状.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "周长的最小值为$12$,此时三角形为正三角形；\\\\\n面积最大值为$4\\sqrt{3}$,此时三角形为正三角形.", "solution": "", @@ -509561,7 +513532,9 @@ "id": "021624", "content": "在$\\triangle ABC$中, 若$\\sin B=\\dfrac{4}{5}$, $\\cos A=\\dfrac{\\sqrt{5}}{5}$, 则$\\cos C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}}{5}$", "solution": "", @@ -509590,7 +513563,9 @@ "id": "021625", "content": "在$\\triangle ABC$中, 若$\\sin B=\\dfrac{4}{5}$, $\\cos A=\\dfrac{2 \\sqrt{5}}{5}$, 则$\\cos C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{5}}{5}$或$-\\dfrac{2\\sqrt{5}}{25}$", "solution": "", @@ -509610,7 +513585,9 @@ "id": "021626", "content": "在$\\triangle ABC$中, 若$\\sin B=\\dfrac{4}{5}$, $\\sin A=\\dfrac{2 \\sqrt{5}}{5}$, 则$\\cos C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{5}}{5}$或$\\dfrac{11\\sqrt{5}}{25}$", "solution": "", @@ -509639,7 +513616,9 @@ "id": "021627", "content": "在$\\triangle ABC$中, 已知$a=2$, $A=45^{\\circ}$. 若此三角形有两解, 则$b$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\left ( 2,2\\sqrt{2} \\right )$", "solution": "", @@ -509668,7 +513647,9 @@ "id": "021628", "content": "根据下列条件, 判断$\\triangle ABC$的形状:\\\\\n(1) $\\cos ^2B-\\cos ^2C=\\sin ^2A$;\\\\\n(2) $a=2 b \\cos C$;\\\\\n(3) $\\tan B=\\dfrac{\\cos (B-C)}{\\sin A-\\sin (B-C)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 以$C$为直角的直角三角形；\\\\\n(2) 以$A$为顶角的等腰三角形；\\\\\n(3) 以$A$为直角的直角三角形.", "solution": "", @@ -509697,7 +513678,9 @@ "id": "021629", "content": "在$\\triangle ABC$中, $A=60^{\\circ}$, $b=1$, 且其面积为$\\sqrt{3}$, 求$a$和$\\triangle ABC$的外接圆半径$R$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$a=\\sqrt{13}$;$R=\\dfrac{\\sqrt{39}}{3}$.", "solution": "", @@ -509726,7 +513709,9 @@ "id": "021630", "content": "在$\\triangle ABC$中, 已知$\\sin A: \\sin C=5: 2$, $B=60^{\\circ}$, 且$S_{\\triangle ABC}=90 \\sqrt{3}$, 求$b$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$6\\sqrt{19}$", "solution": "", @@ -509755,7 +513740,9 @@ "id": "021631", "content": "求分别满足下列条件的角:\\\\\n(1) $\\sin x=\\dfrac{2}{5}$, $x \\in[0, \\pi]$;\\\\\n(2) $\\cos x=-\\dfrac{2}{3}$, $x \\in[0,2 \\pi]$;\\\\\n(3) $\\tan x=-\\dfrac{1}{2}$, $x \\in \\mathbf{R}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $x=\\arcsin \\dfrac{2}{5}$或$\\pi-\\arcsin \\dfrac{2}{5}$;\\\\\n(2) $x=\\pi-\\arccos \\dfrac{2}{3}$或$\\pi+\\arccos \\dfrac{2}{3}$;\\\\\n(3) $x=k\\pi- \\arctan \\dfrac{1}{2},k \\in \\mathbf{Z}$.", "solution": "", @@ -509784,7 +513771,9 @@ "id": "021632", "content": "在平行四边形$ABCD$中, 已知$AB=10 \\sqrt{3}$, $B=60^{\\circ}$, $AC=30$, 求平行四边形的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$300\\sqrt{3}$", "solution": "", @@ -509813,7 +513802,9 @@ "id": "021633", "content": "在$\\triangle ABC$中, 求证: $a \\cos ^2 \\dfrac{C}{2}+c \\cos ^2 \\dfrac{A}{2}=\\dfrac{1}{2}(a+b+c)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "证明略", "solution": "", @@ -509842,7 +513833,9 @@ "id": "021634", "content": "在地面某处测得塔顶的仰角为$\\theta$, 由此向塔底沿直线走$3$千米, 测得塔顶的仰角为\n$2 \\theta$, 再向塔底沿同一直线走$\\sqrt{3}$千米, 测得塔顶仰角为$4 \\theta$(三个测量点都在塔的同一\n侧). 试求$\\theta$与塔高.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\theta=\\dfrac{\\pi}{12}$;塔高为$1.5$千米.", "solution": "", @@ -509871,7 +513864,9 @@ "id": "021635", "content": "如图, 为了测定对岸$A, B$两点之间的距离, 在河的一岸定一条基线$CD$, 测得$CD=100$米, $\\angle ACD=80^{\\circ}$, $\\angle BCD=45^{\\circ}$, $\\angle BDC=70^{\\circ}$, $\\angle ADC=33^{\\circ}$, 求$A, B$间的距离. (结果精确到$0.01$米)\n\\begin{center}\n\\begin{tikzpicture}\n\\clip (-1.5,-1) rectangle (6.5,4);\n\\fill [gray!60] (-1,1.1) rectangle (6,2.2);\n\\draw (0,0) node [below left] {$C$} coordinate (C) ++ (-8:5) node [below right] {$D$} coordinate (D);\n\\path [name path = lineBC] (C) --++ (37:10);\n\\path [name path = lineBD] (D) --++(102:10);\n\\path [name intersections={of = lineBC and lineBD, by=B}];\n\\draw (C) -- (B) node [above right] {$B$} --(D);\n\\path [name path = lineAC] (C) --++ (72:10);\n\\path [name path = lineAD] (D) --++(139:10);\n\\path [name intersections={of = lineAC and lineAD, by=A}];\n\\draw (B) -- (A) -- (C) -- (D) -- (A) node [above left] {$A$};\n\\draw (C) ++ (-8:0.3) arc (-8:72:0.3) node [above right] {$80^\\circ$};\n\\draw (C) ++ (-8:0.4) arc (-8:37:0.4) node [right] {$45^\\circ$};\n\\draw (D) ++ (172:0.4) arc (172:139:0.4) node [left] {$33^\\circ$};\n\\draw (D) ++ (172:0.3) arc (172:102:0.3) node [above left] {$70^\\circ$};\n\\draw (-1,1.1) -- (6,1.1) (-1,2.2) -- (6,2.2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$64.81$米", "solution": "", @@ -509900,7 +513895,9 @@ "id": "021636", "content": "如图, 某市郊外景区内一条笔直的公路$a$经过三个景点$A$、$B$、$C$. 景区管委会又开发了风景优美的景点$D$. 经测量景点$D$位于景点$A$的北偏东$30^{\\circ}$方向$8$千米处, 且位于景点$B$的正北方向, 还位于景点$C$的北偏西$75^{\\circ}$方向上. 已知$AB=5$千米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.25]\n\\draw [->] (6,8) -- (10,8) node [right] {东};\n\\draw [->] (6,8) -- (6,12) node [above] {北};\n\\draw [->] (0,0) node [below] {$A$} coordinate (A) -- (0,8) node [left] {$N$} coordinate (N);\n\\draw (A) --++ (60:8) node [above] {$D$} coordinate (D);\n\\draw (4,3) node [below] {$B$} coordinate (B) -- (D);\n\\draw [name path = linea] (A) -- ($(A)!2.2!(B)$) node [right] {$a$} coordinate (a);\n\\path [name path = DC] (D) --++ (-15:4);\n\\path [name intersections = {of = linea and DC, by = C}];\n\\draw (D) -- (C) node [below] {$C$};\n\\draw (60:2) arc (60:90:2);\n\\draw (75:4) node {$30^\\circ$};\n\\end{tikzpicture}\n\\end{center}\n(1) 景区管委会准备由景点$D$向景点$B$修一条笔直的公路, 不考虑其他因素, 求出这条公路的长; (结果精确到$0.1$千米)\\\\\n(2) 求景点$C$与景点$D$之间的距离. (结果精确到$0.1$千米)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $3.9$千米;(2) $4.0$千米.", "solution": "", @@ -509929,7 +513926,9 @@ "id": "021637", "content": "如图所示, 某游客在$A$处望见一塔$B$在正北方向, 在北偏西$60^{\\circ}$方向的$C$处有一寺庙, 此游客乘车向西$1$千米后到达$D$处, 这时塔和寺庙分别在东北和西北方向.求塔与寺庙间的距离. (结果精确到$0.1$千米)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (-1,0) node [below] {$D$} coordinate (D);\n\\draw (0,1) node [right] {$B$} coordinate (B);\n\\path [name path = AC] (A)--++(150:3);\n\\path [name path = CD] (D)--++(135:2);\n\\path [name intersections = {of = AC and CD, by = C}];\n\\draw (C) node [above] {$C$};\n\\draw (C)--(D)--(A)--cycle (A)--(B)--(D);\n\\draw [->] (B)--($(B)!-1.5!(A)$) node [right] {北};\n\\draw pic [draw, \"$60^\\circ$\", scale = 0.5, angle eccentricity = 1.7] {angle = B--A--C};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$2.4$千米", "solution": "", @@ -509958,7 +513957,9 @@ "id": "021638", "content": "余弦函数$y=\\cos x$, $x \\in \\mathbf{R}$图像至少向右平移单位得到正弦函数$y=\\sin x$, $x \\in \\mathbf{R}$的图像.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\pi}{2}$", "solution": "", @@ -509987,7 +513988,9 @@ "id": "021639", "content": "函数$y=\\cos (x+\\dfrac{\\pi}{2})$, $x \\in[-\\pi, \\pi]$的图像是\\bracket{20}.\n\\twoch{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (-pi,0.1) -- (-pi,0) node [below left] {$-\\pi$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi:pi,samples = 100] plot (\\x,{sin(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi,0.1) -- (-pi,0) node [below] {$-\\pi$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi:pi,samples = 100] plot (\\x,{-sin(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi,0.1) -- (-pi,0) node [below] {$-\\pi$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [below left] {$-1$};\n\\draw [domain = -pi:pi,samples = 100] plot (\\x,{-cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi,0.1) -- (-pi,0) node [below] {$-\\pi$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.1,1) -- (0,1) node [below left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi:pi,samples = 100] plot (\\x,{cos(\\x/pi*180)});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -510016,7 +514019,9 @@ "id": "021640", "content": "作出下列函数的大致图像:\\\\\n(1) $y=1+\\sin x$, $x \\in[-\\pi, \\pi]$;\\\\\n(2) $y=-\\cos x$, $x \\in[0,2 \\pi]$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) \\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (-pi,0.1) -- (-pi,0) node [below left] {$-\\pi$};\n\\draw (-0.5*pi,0.1) -- (-0.5*pi,0) node [below] {$-\\frac{\\pi}{2}$};\n\\draw (0.5*pi,0.1) -- (0.5*pi,0) node [below] {$\\frac{\\pi}{2}$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi:pi,samples = 100] plot (\\x,{sin(\\x/pi*180)+1});\n\\end{tikzpicture}\\\\\n(2) \\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (0,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (pi/2,0.1) -- (pi/2,0) node [below] {$\\frac{\\pi}{2}$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (1.5*pi,0.1) -- (1.5*pi,0) node [below] {$\\frac{3\\pi}{2}$};\n\\draw (2*pi,0.1) -- (2*pi,0) node [below] {$2\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = 0:2*pi,samples = 100] plot (\\x,{-cos(\\x/pi*180)});\n\\end{tikzpicture}", "solution": "", @@ -510045,7 +514050,9 @@ "id": "021641", "content": "写出下列函数的定义域:\\\\\n(1) $y=\\dfrac{1}{1+\\sin x}$, 定义域为\\blank{100};\\\\\n(2) $y=\\sqrt{-2 \\cos x}$, 定义域为\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) 定义域为$\\left \\{x|x \\neq-\\dfrac{\\pi}{2}+2k\\pi,k \\in \\mathbf{Z} \\right \\}$;\\\\\n(2) 定义域为$\\left \\{x|\\dfrac{\\pi}{2}+2k\\pi \\leq x \\leq \\dfrac{3\\pi}{2}+2k\\pi,k \\in \\mathbf{Z} \\right \\}$.", "solution": "", @@ -510074,7 +514081,9 @@ "id": "021642", "content": "已知$\\sin \\alpha \\geq \\dfrac{1}{2}$, 则在$[0,2 \\pi]$中的$\\alpha$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\left \\{x|\\dfrac{\\pi}{6} \\leq x \\leq \\dfrac{5\\pi}{6},k \\in \\mathbf{Z} \\right \\}$", "solution": "", @@ -510103,7 +514112,9 @@ "id": "021643", "content": "已知函数$y=\\cos x$($0 \\leq x \\leq 2 \\pi$)的图像与直线$y=1$围成一个封闭的平面图形, 则该封闭图形的面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2\\pi$", "solution": "", @@ -510132,7 +514143,9 @@ "id": "021644", "content": "函数$y=\\sin |x|$, $x \\in[-2 \\pi, 2 \\pi]$的图像是\\bracket{20}.\n\\twoch{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-7,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw ({-2*pi},0.1) -- ({-2*pi},0) node [below] {$-2\\pi$};\n\\draw ({2*pi},0.1) -- ({2*pi},0) node [below] {$2\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = {-2*pi}:{2*pi},samples = 100] plot (\\x,{sin(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-7,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw ({-2*pi},0.1) -- ({-2*pi},0) node [below] {$-2\\pi$};\n\\draw ({2*pi},0.1) -- ({2*pi},0) node [below] {$2\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = {-2*pi}:{2*pi},samples = 100] plot (\\x,{abs(sin(\\x/pi*180))});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-7,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw ({-2*pi},0.1) -- ({-2*pi},0) node [below] {$-2\\pi$};\n\\draw ({2*pi},0.1) -- ({2*pi},0) node [below] {$2\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = {-2*pi}:{2*pi},samples = 100] plot (\\x,{sin(abs(\\x/pi*180))});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-7,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw ({-2*pi},0.1) -- ({-2*pi},0) node [below] {$-2\\pi$};\n\\draw ({2*pi},0.1) -- ({2*pi},0) node [below] {$2\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = {-2*pi}:{2*pi},samples = 100] plot (\\x,{-sin(abs(\\x/pi*180))});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -510161,7 +514174,9 @@ "id": "021645", "content": "下列各组函数中, 表示同一函数的是\\bracket{20} .\n\\twoch{$f(x)=\\sin x$, $g(x)=\\dfrac{x \\sin x}{x}$}{$f(x)=\\cos x$, $g(x)=\\sqrt{1-\\sin ^2 x}$}{$f(x)=1$, $g(x)=\\sin ^2 x+\\cos ^2 x$}{$f(x)=1$, $g(x)=\\tan x \\cdot \\cot x$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -510190,7 +514205,9 @@ "id": "021646", "content": "已知$a$为实常数, 作出相应的函数图像并讨论方程$f(x)=a$的实数解个数.\\\\\n(1) $f(x)=\\sin x$, $x \\in[0, \\dfrac{5}{4} \\pi]$;\\\\\n(2) $f(x)=\\begin{cases}-\\sin x,& x \\in(0, \\pi], \\\\ \\cos x,& x \\in[-\\pi, 0].\\end{cases}$", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 当$a \\in (-\\infty，-\\dfrac{\\sqrt{2}}{2})\\cup (1,+\\infty)$ 时，方程实数解个数为$0$个；\\\\\n当$a \\in [-\\dfrac{\\sqrt{2}}{2},0)\\cup \\{1\\}$ 时，方程实数解个数为$1$个；\\\\\n当$a \\in [0,1)$时，方程实数解个数为$2$个.\\\\\n(2) 当$a \\in (-\\infty，-1)\\cup (1,+\\infty)$ 时，方程实数解个数为$0$个；\\\\\n当$a \\in (0,1]$时，方程实数解个数为$1$个;\\\\\n当$a \\in \\{0,-1\\}$时，方程实数解个数为$2$个;\\\\\n当$a \\in (-1,0)$时，方程实数解个数为$3$个.", "solution": "", @@ -510219,7 +514236,9 @@ "id": "021647", "content": "写出下列函数的最小正周期:\\\\\n(1) $y=\\sin \\dfrac{x}{4}$的最小正周期为\\blank{50};\\\\\n(2) $y=2 \\cos (2 x+\\dfrac{\\pi}{3})$的最小正周期为\\blank{50};\\\\\n(3) $y=\\cos ^2 x$的最小正周期为\\blank{50};\\\\\n(4) $y=\\sin x+\\sqrt{3} \\cos x$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $8\\pi$;\n(2) $\\pi$;(3) $\\pi$;(4) $2\\pi$.", "solution": "", @@ -510248,7 +514267,9 @@ "id": "021648", "content": "若函数$y=f(x)$的周期为$3$, 则函数$y=f(x+1)$的一个周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -510277,7 +514298,9 @@ "id": "021649", "content": "``$\\omega=1$''是``函数$y=\\cos \\omega x$的最小正周期为$2 \\pi$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -510306,7 +514329,9 @@ "id": "021650", "content": "设常数$a \\neq 0$, 函数$y=\\sin (a \\pi x+1)$的周期为\\bracket{20}.\n\\fourch{$\\dfrac{2}{a}$}{$\\dfrac{a}{2}$}{$\\dfrac{2}{|a|}$}{$\\dfrac{2 \\pi}{|a|}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -510335,7 +514360,9 @@ "id": "021651", "content": "判断以下命题的真假:\\\\\n(1) 对于函数$f(x)$, 如果存在一个常数$T$($T \\neq 0$), 使得当$x$取定义域$D$内的某一个值$x_0$时, 有$f(x_0+T)=f(x_0)$成立, 那么这个函数$f(x)$叫做周期函数;\\\\\n(2) 周期函数的定义域一定是$\\mathbf{R}$;\\\\\n(3) 每个周期函数都有无数个周期.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 假;(1) 假;(3) 真.", "solution": "", @@ -510364,7 +514391,9 @@ "id": "021652", "content": "$x$为实数, $[x]$表示不超过$x$的最大整数, 则函数$f(x)=x-[x]$在$\\mathbf{R}$上为\\bracket{20}.\n\\fourch{奇函数}{偶函数}{增函数}{周期函数}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -510393,7 +514422,9 @@ "id": "021653", "content": "求下列函数的最小正周期:\\\\\n(1) $y=\\sin ^2 x$;\\\\\n(2) $y=\\sin ^2 x+2 \\sin x \\cos x$;\\\\\n(3) $y=\\sin ^4 x+\\cos ^4 x$;\\\\\n(4) $y=\\sin 2 a x$($a \\neq 0$).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\pi$; (2) $\\pi$; (3) $\\dfrac{\\pi}{2}$; (4) $\\dfrac{\\pi}{|a|}$.", "solution": "", @@ -510422,7 +514453,9 @@ "id": "021654", "content": "已知函数$y=f(x)$的最小正周期$4$. 若此函数的最大值为$2$, 最小值为$-6$, 则这个函数的一个可能的解析式是$f(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$4\\sin(\\dfrac{\\pi x}{2})-2$", "solution": "", @@ -510451,7 +514484,9 @@ "id": "021655", "content": "函数$y=|\\sin x|$的最小正周期为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{2}$}{$\\pi$}{$2 \\pi$}{$4 \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -510480,7 +514515,9 @@ "id": "021656", "content": "若$y=f(x) \\cdot \\cos x$是周期为$\\pi$, 且最大值为$1$, 则$y=f(x)$可能是\\bracket{20}.\n\\fourch{$y=\\cos x$}{$y=\\sin x$}{$y=-\\cos x$}{$y=-\\sin x$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -510509,7 +514546,9 @@ "id": "021657", "content": "已知函数$y=f(x)$满足$f(x+2)=-f(x)$.\\\\\n(1) 若当$x \\in[0,2)$时, $f(x)=x$, 求$f(3)$、$f(5)$、$f(7)$的值;\\\\\n(2) 求函数$y=f(x)$的一个周期, 并加以证明.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $f(3)=-1$; $f(5)=1$; $f(7)=-1$;\\\\\n(2) $T=4$.", "solution": "", @@ -510538,7 +514577,9 @@ "id": "021658", "content": "函数$y=\\cos x+3$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\left [2,4\\right] $", "solution": "", @@ -510567,7 +514608,9 @@ "id": "021659", "content": "函数$y=4 \\sin ^2 x-2$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\left [-2,2\\right] $", "solution": "", @@ -510596,7 +514639,9 @@ "id": "021660", "content": "函数$y=2 \\sin ^2 x+2 \\sin x-1$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$ [-\\dfrac{3}{2},3] $", "solution": "", @@ -510625,7 +514670,9 @@ "id": "021661", "content": "若$\\dfrac{\\pi}{6} \\leq \\theta<\\dfrac{4 \\pi}{3}$, 则$\\sin \\theta$的范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$ (-\\dfrac{\\sqrt{3}}{2},1] $", "solution": "", @@ -510654,7 +514701,9 @@ "id": "021662", "content": "函数$y=2-\\sin x$的最大值是\\blank{50}, 此时$x$的集合是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$3$; $\\left \\{x|x=-\\dfrac{\\pi}{2}+2k\\pi,k \\in \\bf{Z} \\right\\}$", "solution": "", @@ -510683,7 +514732,9 @@ "id": "021663", "content": "函数$y=3 \\sin (2 x-\\dfrac{\\pi}{3})$的最小值是\\blank{50}, 此时$x$的集合是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-3$; $\\left \\{x|x=-\\dfrac{\\pi}{12}+k\\pi,k \\in \\bf{Z} \\right\\}$", "solution": "", @@ -510712,7 +514763,9 @@ "id": "021664", "content": "求函数$y=2 \\sin x+3 \\cos x$的最大值与最小值, 并指出何时取到.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "当$x=\\dfrac{\\pi}{2}-\\arcsin \\dfrac{3\\sqrt{13}}{13}+2k\\pi,k \\in \\bf{Z}$时，函数的最大值为$\\sqrt{13}$;\\\\\n当$x=-\\dfrac{\\pi}{2}-\\arcsin \\dfrac{3\\sqrt{13}}{13}+2k\\pi,k \\in \\bf{Z}$时，函数的最小值为$-\\sqrt{13}$.", "solution": "", @@ -510741,7 +514794,9 @@ "id": "021665", "content": "若直角三角形$ABC$的两锐角为$\\alpha$、$\\beta$, 则$\\sin \\alpha \\sin \\beta$的值域是\\bracket{20}.\n\\fourch{$[0,1]$}{$(0,1)$}{$(0, \\dfrac{1}{2})$}{$(0, \\dfrac{1}{2}]$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -510770,7 +514825,9 @@ "id": "021666", "content": "若存在$\\alpha$使得$\\sin \\alpha+\\sqrt{3} \\cos \\alpha=\\dfrac{2 m-4}{m+3}$, 则$m$的取值范围是\\bracket{20}.\n\\fourch{$(-3,-\\dfrac{1}{2}]$}{$(-3,+\\infty)$}{$[-\\dfrac{1}{2},+\\infty)$}{$(-\\infty,-3) \\cup(-3,-\\dfrac{1}{2}]$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -510799,7 +514856,9 @@ "id": "021667", "content": "如图, 太阳光斜照地面, 光线与水平面所成的角为$\\theta$, 长为$l$的竹竿与地面所成的角为$\\alpha$(其中$\\theta, l$为常数). 问当$\\alpha$为多少时, 竹竿的影子最长?\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\filldraw [gray!30] (-1.5,-0.3) rectangle (1.5,0);\n\\draw (-1.5,0) -- (1.5,0);\n\\draw [very thick] (-1,0) coordinate (A) -- (-0.3,1) coordinate (B);\n\\draw (-0.3,1) -- (1,0) coordinate (C);\n\\draw (A) pic [\"$\\alpha$\", angle radius = 0.3cm, draw, angle eccentricity = 1.5] {angle = C--A--B};\n\\draw (C) pic [\"$\\theta$\", angle radius = 0.3cm, draw, angle eccentricity = 1.5] {angle = B--C--A};\n\\draw ($(B)!0.25!(A)$) coordinate (B1) ++ (-0.65,0.5) coordinate (D1);\n\\draw ($(B)!0.5!(A)$) coordinate (B2) ++ (-0.65,0.5) coordinate (D2);\n\\draw ($(B)!0.75!(A)$) coordinate (B3) ++ (-0.65,0.5) coordinate (D3);\n\\draw (B) ++ (-0.65,0.5) coordinate (D);\n\\draw [->] (D) -- (B);\n\\draw [->] (D1) -- (B1);\n\\draw [->] (D2) -- (B2);\n\\draw [->] (D3) -- (B3);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "当$\\alpha=\\dfrac{\\pi}{2}-\\theta$时，竹竿的影子最长，最长为$\\dfrac{\\sin(\\alpha+\\theta)}{\\sin \\theta}*l$.", "solution": "", @@ -510828,7 +514887,9 @@ "id": "021668", "content": "函数$y=\\sin (x^2+x+1)$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[-1,1]$", "solution": "", @@ -510857,7 +514918,9 @@ "id": "021669", "content": "函数$y=\\dfrac{\\cos x+1}{\\cos x-1}$的定义域是值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|x\\neq 2k\\pi,k \\in \\bf{Z}\\}$;$(-\\infty,0]$", "solution": "", @@ -510886,7 +514949,9 @@ "id": "021670", "content": "函数$y=k \\sin x+b$的最大值为 2 , 最小值为$-4$, 求$k$、$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$k=3$或$-3$;$b=-1$", "solution": "", @@ -510915,7 +514980,9 @@ "id": "021671", "content": "设$-\\dfrac{\\pi}{6} \\leq x \\leq \\dfrac{\\pi}{4}$, 求函数$y=\\log _2(1+\\sin x)+\\log _2(1-\\sin x)$的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "当$x=0$时，函数$y$取到最大值，最大值为$0$;\\\\\n当$x=\\dfrac{\\pi}{4}$时，函数$y$取到最小值，最小值为$-1$.", "solution": "", @@ -510944,7 +515011,9 @@ "id": "021672", "content": "已知函数$y=2+a^2-2 a \\sin x-\\cos ^2 x$的最小值为$f(a)$, $a$为实数, 求$f(a)$的表达式.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$f(a)=\\begin{cases}\na^2+2a+2, & a\\leq -1,\\\\\n1, & -1\\beta$, 则$\\sin \\alpha>\\sin \\beta$, \\blank{20};\\\\\n(4) 函数$y=\\sin (\\dfrac{7 \\pi}{2}+3 x)$是偶函数, \\blank{20}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) 真命题； (2) 真命题； (3) 假命题； (4) 真命题.", "solution": "", @@ -511031,7 +515104,9 @@ "id": "021675", "content": "若函数$y=\\sqrt{3} \\cos (3 x-\\theta)-\\sin (3 x-\\theta)$是奇函数, 则$\\theta$的一个可能值是\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{6}$}{$-\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{3}$}{$-\\dfrac{\\pi}{6}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -511060,7 +515135,9 @@ "id": "021676", "content": "函数$f(x)$具有下列性质: \\textcircled{1} $f(x)$是偶函数; \\textcircled{2} 对任意$x \\in \\mathbf{R}$, 都有$f(\\dfrac{\\pi}{4}-x)=f(\\dfrac{\\pi}{4}+x)$, 则函数$f(x)$的解析式可以是\\blank{50}. (写出一个答案即可)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$f(x)=\\cos4x$", "solution": "", @@ -511089,7 +515166,9 @@ "id": "021677", "content": "已知函数$f(x)=a x^3-b \\sin x-1$. 若$f(5)=5$, 则$f(-5)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-7$", "solution": "", @@ -511118,7 +515197,9 @@ "id": "021678", "content": "写出下列各函数的奇偶性:\\\\\n(1) 函数$y=\\cos x+\\sin x$的奇偶性为\\blank{50};\\\\\n(2) 函数$y=\\sin (x+\\dfrac{\\pi}{4})+\\cos (x+\\dfrac{\\pi}{4})$的奇偶性为\\blank{50};\\\\\n(3) 函数$y=\\sin x(|\\sin x-3|-3)$的奇偶性为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) 不是奇函数也不是偶函数；(2) 偶函数；(3) 偶函数.", "solution": "", @@ -511147,7 +515228,9 @@ "id": "021679", "content": "判断下列命题的真假:\\\\\n(1) 函数$y=|\\sin x|+|\\cos x|$的最小正周期为$\\pi$, \\blank{20};\\\\\n(2) 存在实数使$\\sin \\alpha-\\cos \\alpha=\\dfrac{3}{2}$成立, \\blank{20};\\\\\n(3) 存在实数使$\\sin \\alpha \\cos \\alpha=\\dfrac{1}{3}$成立, \\blank{20};\\\\\n(4) 函数$y=\\cos ^22 x-\\sin ^22 x$是偶函数, 且最小正周期为$\\dfrac{\\pi}{2}$, \\blank{20}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) 假命题；(2) 假命题；(3) 真命题； (4) 真命题.", "solution": "", @@ -511176,7 +515259,9 @@ "id": "021680", "content": "已知函数$f(x)=1+\\sin (x+\\theta)+\\sqrt{3} \\cos (x+\\theta)$, 是否存在实数$\\theta$, 使$f(x)$是奇函数? 若存在, 求出所有这样的$\\theta$, 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "不存在这样的$\\theta$,使得函数$f(x)=1+\\sin (x+\\theta)+\\sqrt{3} \\cos (x+\\theta)$为奇函数.", "solution": "", @@ -511205,7 +515290,9 @@ "id": "021681", "content": "若$\\alpha$、$\\beta$是锐角, $\\cos \\alpha<\\cos \\beta$, 则$\\alpha$、$\\beta$的大小关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$0<\\beta<\\alpha<\\dfrac{\\pi}{2}$", "solution": "", @@ -511234,7 +515321,9 @@ "id": "021682", "content": "判断下列命题的真假:\\\\\n(1) $y=\\sin x$在第一象限内为增函数, \\blank{20};\\\\\n(2) 直线$x=\\dfrac{\\pi}{8}$是$y=\\sin (2 x+\\dfrac{\\pi}{4})$图像的一条对称轴, \\blank{20};\\\\\n(3) 函数$y=\\cos x+\\sin x$的值域为$[-2,2]$, \\blank{20};\\\\\n(4) 函数$y=\\sqrt{2} \\sin 2 x \\cos 2 x$的最小正周期为$\\dfrac{\\pi}{2}$, \\blank{20}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) 假命题； (2) 真命题；(3) 假命题； (4)真命题.", "solution": "", @@ -511263,7 +515352,9 @@ "id": "021683", "content": "求函数$y=\\sin x$, $x \\in[-2 \\pi, 0]$的单调递减区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$[-\\dfrac{3\\pi}{2},-\\dfrac{\\pi}{2}]$", "solution": "", @@ -511292,7 +515383,9 @@ "id": "021684", "content": "求函数$y=\\sin 2 x$的单调递增区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$[-\\dfrac{\\pi}{4}+k\\pi,\\dfrac{\\pi}{4}+k\\pi],k \\in \\bf{Z}$", "solution": "", @@ -511321,7 +515414,9 @@ "id": "021685", "content": "已知$0 \\leq x \\leq 2 \\pi$, 写出适合下列条件的角$x$的区间:\\\\\n(1) 角$x$的正弦函数、余弦函数都是增函数, \\blank{100};\\\\\n(2) 角$x$的正弦函数是减函数, 余弦函数是增函数, \\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $[\\dfrac{3\\pi}{2},2\\pi]$;\n(2) $[\\pi,\\dfrac{3\\pi}{2}]$.", "solution": "", @@ -511350,7 +515445,9 @@ "id": "021686", "content": "求下列函数的单调递增区间:\\\\\n(1) $y=1-\\sin (x-\\dfrac{3 \\pi}{4})$;\\\\\n(2) $y=\\sqrt{3} \\cos x-\\sin x$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $[\\dfrac{5\\pi}{4}+2k\\pi,\\dfrac{9\\pi}{4}+2k\\pi],k \\in \\bf{Z}$;\\\\\n(2) $[-\\dfrac{7\\pi}{6}+2k\\pi,-\\dfrac{\\pi}{6}+2k\\pi],k \\in \\bf{Z}$.", "solution": "", @@ -511379,7 +515476,9 @@ "id": "021687", "content": "求函数$y=2 \\sin (\\dfrac{\\pi}{6}-4 x)+3$的递增区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$[\\dfrac{\\pi}{6}+\\dfrac{k\\pi}{2},\\dfrac{5\\pi}{12}+\\dfrac{k\\pi}{2}],k \\in \\bf{Z}$", "solution": "", @@ -511408,7 +515507,9 @@ "id": "021688", "content": "求函数$y=\\cos ^2 x+2$的递减区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$[k\\pi,k\\pi+\\dfrac{\\pi}{2}],k \\in \\bf{Z}$", "solution": "", @@ -511437,7 +515538,9 @@ "id": "021689", "content": "若$\\alpha$、$\\beta$是锐角, 且$\\sin \\alpha<\\cos \\beta$, 证明或否定: $\\alpha+\\beta$是锐角.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -511466,7 +515569,9 @@ "id": "021690", "content": "函数$y=3 \\sin (\\dfrac{x}{2}+\\dfrac{\\pi}{3})$的周期为\\blank{50}、振幅为\\blank{50}、初相为\\blank{50}、频率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$4\\pi$; $3$; $\\dfrac{\\pi}{3}$; $\\dfrac{1}{4\\pi}$.", "solution": "", @@ -511495,7 +515600,9 @@ "id": "021691", "content": "函数$y=\\dfrac{4}{3} \\sin 2 x$的周期为\\blank{50}, $y$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi$; $[-\\dfrac{4}{3},\\dfrac{4}{3}].$", "solution": "", @@ -511524,7 +515631,9 @@ "id": "021692", "content": "函数$y=b+a \\sin \\dfrac{x}{2}(a \\neq 0)$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$b-|a|$", "solution": "", @@ -511553,7 +515662,9 @@ "id": "021693", "content": "已知函数$y=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$)的振幅是$3$, 最小正周期是$\\dfrac{2 \\pi}{7}$, 初相是$\\dfrac{\\pi}{6}$, 写出这个函数的解析式: \\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$y=3\\sin(7x+\\dfrac{\\pi}{6})$", "solution": "", @@ -511582,7 +515693,9 @@ "id": "021694", "content": "作出下列函数在长度为一个周期的闭区间上的大致图像:\\\\\n(1) $y=\\sin (2 x+\\dfrac{\\pi}{6})$;\\\\\n(2) $y=2 \\sin \\dfrac{x}{2}$;\\\\\n(3) $y=\\sin x \\cdot \\cos x$;\\\\\n(4) $y=5 \\sin (2 x-\\dfrac{\\pi}{3})$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) \\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw ({-pi/12},0.1) -- ({-pi/12},0) node [below left] {$-\\frac{\\pi}{12}$};\n\\draw ({pi/6},0.1) -- ({pi/6},0) node [below] {$\\frac{\\pi}{6}$};\n\\draw ({5*pi/12},0.1) -- ({5*pi/12},0) node [below] {$\\frac{5\\pi}{12}$};\n\\draw ({2*pi/3},0.1) -- ({2*pi/3},0) node [above] {$\\frac{2\\pi}{3}$};\n\\draw ({11*pi/12},0.1) -- ({11*pi/12},0) node [below right] {$\\frac{11\\pi}{12}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = {-pi/12}:{11*pi/12},samples = 100] plot (\\x,{sin(2*\\x/pi*180+30)});\n\\end{tikzpicture}\\\\\n(2) \\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-1,0) -- (15,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2*pi,0.1) -- (2*pi,0) node [below] {$2\\pi$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (3*pi,0.1) -- (3*pi,0) node [below] {$\\frac{3\\pi}{2}$};\n\\draw (4*pi,0.1) -- (4*pi,0) node [below] {$4\\pi$};\n\\draw (0.1,2) -- (0,2) node [left] {$2$};\n\\draw (0.1,-2) -- (0,-2) node [left] {$-2$};\n\\draw [domain =0:4*pi,samples = 100] plot (\\x,{2*sin(0.5*\\x/pi*180)});\n\\end{tikzpicture}\\\\\n(3) \\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (0.25*pi,0.1) -- (0.25*pi,0) node [below] {$\\frac{\\pi}{4}$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.5*pi,0.1) -- (0.5*pi,0) node [below] {$\\frac{\\pi}{2}$};\n\\draw (0.75*pi,0.1) -- (0.75*pi,0) node [above] {$\\frac{3\\pi}{4}$};\n\\draw (0.1,0.5) -- (0,0.5) node [left] {$\\frac{1}{2}$};\n\\draw (0.1,-0.5) -- (0,-0.5) node [left] {$-\\frac{1}{2}$};\n\\draw [domain =0:pi,samples = 100] plot (\\x,{0.5*sin(2*\\x/pi*180)});\n\\end{tikzpicture}\\\\\n(4) \\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-1.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-5.5) -- (0,5.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw ({-pi/3},0.1) -- ({-pi/3},0) node [below left] {$-\\frac{\\pi}{3}$};\n\\draw ({-pi/12},0.1) -- ({-pi/12},0) node [above left] {$-\\frac{\\pi}{12}$};\n\\draw ({pi/6},0.1) -- ({pi/6},0) node [below right] {$\\frac{\\pi}{6}$};\n\\draw ({5*pi/12},0.1) -- ({5*pi/12},0) node [below] {$\\frac{5\\pi}{12}$};\n\\draw ({2*pi/3},0.1) -- ({2*pi/3},0) node [above right] {$\\frac{2\\pi}{3}$};\n\\draw ({11*pi/12},0.1) -- ({11*pi/12},0) node [below right] {$\\frac{11\\pi}{12}$};\n\\draw (0.1,5) -- (0,5) node [left] {$5$};\n\\draw (0.1,-5) -- (0,-5) node [below left] {$-5$};\n\\draw [domain = {-4*pi/12}:{2*pi/3},samples = 100] plot (\\x,{5*sin(2*\\x/pi*180-60)});\n\\end{tikzpicture}", "solution": "", @@ -511611,7 +515724,9 @@ "id": "021695", "content": "若函数$f(x)=2 a+b \\sin x$的最大值为$3$, 最小值为$1$, 则函数$g(x)=-4 a \\sin \\dfrac{b x}{2}$的周期是\\blank{50}, 最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$4\\pi$;$4$.", "solution": "", @@ -511640,7 +515755,9 @@ "id": "021696", "content": "已知函数$y=A \\sin (x+\\varphi)$($A>0$, $\\omega>0$, $0<\\varphi<\\pi$)在同一周期内, 当$x=\\dfrac{\\pi}{3}$时, 取到最大值$4$, 当$x=\\dfrac{4}{3} \\pi$时, 取到最小值$-4$. 求函数的解析式.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$f(x)=4\\sin(x+\\dfrac{\\pi}{6})$", "solution": "", @@ -511669,7 +515786,9 @@ "id": "021697", "content": "如图是某个定义在$\\mathbf{R}$上的函数$f(x)=A \\sin (\\omega x+\\varphi)+B$, ($A>0$, $\\omega>0$, $0<\\varphi<2 \\pi$)的一部分图像.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {pi/3}:pi] plot (\\x,{sqrt(3)/2+sqrt(3)/2*sin(3*\\x/pi*180-180)});\n\\draw [dashed] ({pi/3},0) -- ({pi/3},{sqrt(3)/2});\n\\draw [dashed] (pi,0) -- (pi,{sqrt(3)/2}) -- (0,{sqrt(3)/2});\n\\draw [dashed] ({pi/2},{sqrt(3)}) -- (0,{sqrt(3)});\n\\draw (pi,0) node [below] {$\\pi$};\n\\draw ({pi/3},0) node [below] {$\\dfrac \\pi 3$};\n\\draw (0,{sqrt(3)/2}) node [left] {$\\dfrac{\\sqrt{3}}2$};\n\\draw (0,{sqrt(3)}) node [left] {$\\sqrt{3}$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求$f(x)$的解析式;\\\\\n(2) 求该函数的单调递增区间;\\\\\n(3) 求该函数的最大值及取最大值时自变量的取值集合.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $f(x)=\\dfrac{\\sqrt{3}}{2}\\sin(3x+\\pi)+\\dfrac{\\sqrt{3}}{2};$\\\\\n(2) $[-\\dfrac{\\pi}{2}+\\dfrac{2k\\pi}{3},-\\dfrac{\\pi}{6}+\\dfrac{2k\\pi}{3}],k \\in \\bf{Z}$;\\\\\n(3) 函数最大值为$\\sqrt{3}$,此时$x$值为${x|x=-\\dfrac{\\pi}{6}+\\dfrac{2k\\pi}{3},k \\in \\bf{Z}}$", "solution": "", @@ -511698,7 +515817,9 @@ "id": "021698", "content": "函数$y=4 \\sin \\dfrac{x}{2}$的对称轴为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$x=\\pi+2k\\pi,k \\in \\bf{Z}$", "solution": "", @@ -511727,7 +515848,9 @@ "id": "021699", "content": "要得到函数$y=3 \\sin x$的图像, 只需要把函数$y=\\sin x$的图像上的对应点的坐标\\blank{30}(``伸长''或``缩短'')到原来的\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "纵;伸长; $3$.", "solution": "", @@ -511756,7 +515879,9 @@ "id": "021700", "content": "函数$f(x)=\\dfrac{1}{3} \\sin 2 x$的图像可由$y=\\sin x$的图像上所有点的横坐标\\blank{30}(``伸长''或\n``缩短'')到原来的\\blank{50}, 纵坐标\\blank{30}(``伸长''或``缩短'')到原来的\\blank{50}得到.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "缩短; $\\dfrac{1}{2}$; 缩短; $\\dfrac{1}{3}$.", "solution": "", @@ -511785,7 +515910,9 @@ "id": "021701", "content": "将函数$y=\\sin x$的图像向右平移$\\dfrac{\\pi}{3}$个单位, 再将所得图像上所有点的横坐标伸长到原来的$2$倍, 所得的曲线是$y=f(x)$的图像, 则$f(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$f(x)=\\sin(\\dfrac{1}{2}x-\\dfrac{\\pi}{3})$", "solution": "", @@ -511814,7 +515941,9 @@ "id": "021702", "content": "将函数$y=\\sin x$的图像上所有点的横坐标伸长到原来的$2$倍, 再向右平移$\\dfrac{\\pi}{3}$个单位, 所得的曲线是$y=f(x)$的图像, 则$f(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$f(x)=\\sin(\\dfrac{1}{2}x-\\dfrac{\\pi}{6})$", "solution": "", @@ -511843,7 +515972,9 @@ "id": "021703", "content": "将函数$y=f(x)$的图像上所有点的横坐标缩短到原来的$\\dfrac{1}{3}$倍, 再向右平移$\\dfrac{\\pi}{6}$个单位, 所得的曲线是$y=2 \\sin x$的图像, 则函数$f(x)$的解析式为$f(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$f(x)=2\\sin(\\dfrac{1}{3}x+\\dfrac{\\pi}{6})$", "solution": "", @@ -511872,7 +516003,9 @@ "id": "021704", "content": "函数$y=3 \\sin (\\dfrac{x}{2}+\\dfrac{\\pi}{3})$的所有垂直于$x$轴的对称轴是直线\\blank{50}, 对称中心是点\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$x=\\dfrac{\\pi}{3}+2k\\pi,k \\in \\bf{Z}$; $(-\\dfrac{2\\pi}{3}+2k\\pi,0),k \\in \\bf{Z}$.", "solution": "", @@ -511901,7 +516034,9 @@ "id": "021705", "content": "函数$y=3 \\sin (2 x+5 \\theta)$的图像关于$y$轴对称的充要条件是\\bracket{20}.\n\\fourch{$\\theta=\\dfrac{2 k \\pi}{5}+\\dfrac{\\pi}{10}$($k \\in\\mathbf{Z}$)}{$\\theta=\\dfrac{2 k \\pi}{5}+\\dfrac{\\pi}{5}$($k \\in\\mathbf{Z}$)}{$\\theta=\\dfrac{k \\pi}{5}+\\dfrac{\\pi}{10}$($k \\in\\mathbf{Z}$)}{$\\theta=\\dfrac{k \\pi}{5}+\\dfrac{\\pi}{5}$($k \\in\\mathbf{Z}$)}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -511930,7 +516065,9 @@ "id": "021706", "content": "若将函数$y=\\dfrac{1}{2} \\sin (2 x-\\dfrac{\\pi}{4})$的图像向\\blank{20}(左、右)平移\\blank{50}个单位, 则可得到函数$y=\\dfrac{1}{2} \\sin 2 x$的图像.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "左; $\\dfrac{\\pi}{8}$.", "solution": "", @@ -511959,7 +516096,9 @@ "id": "021707", "content": "已知函数$f(x)=\\sin (\\omega x+\\varphi)$($\\omega>0$, $0<\\varphi<\\pi$)的周期为$\\pi$, 图像的一个对称中心为$(\\dfrac{\\pi}{4}, 0)$. 将函数$f(x)$图像上所有点的横坐标伸长到原来的$2$倍(纵坐标不变), 再将所得到的图像向右平移$\\dfrac{\\pi}{2}$个单位长度后得到函数$g(x)$的图像, 求函数$f(x)$与$g(x)$的解析式.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$f(x)=\\sin(2x+\\dfrac{\\pi}{2})$,\n$g(x)=\\sin x$.", "solution": "", @@ -511988,7 +516127,9 @@ "id": "021708", "content": "已知函数$f(x)=\\sqrt{3} \\sin (\\omega x+\\varphi)-\\cos (\\omega x+\\varphi)$($\\omega>0$, $0<\\varphi<\\pi$)为偶函数, 且函数$y=f(x)$图像的两相邻对称轴间的距离为$\\dfrac{\\pi}{2}$.\\\\\n(1) 求$f(\\dfrac{\\pi}{8})$的值;\\\\\n(2) 将函数$y=f(x)$的图像向右平移$\\dfrac{\\pi}{6}$个单位后, 再将得到的图像上各点的横坐标伸长到原来的$4$倍, 纵坐标不变, 得到函数$y=g(x)$的图像, 求$g(x)$的单调递减区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\sqrt{2}$;\n(2) $g(x)=2\\cos(\\dfrac{1}{2}x-\\dfrac{\\pi}{3}) $, 单调递减区间为$[\\dfrac{2\\pi}{3}+4k\\pi,\\dfrac{8\\pi}{3}+4k\\pi],k \\in \\bf{Z}$.", "solution": "", @@ -512017,7 +516158,9 @@ "id": "021709", "content": "写出下列各函数的最小正周期:\\\\\n(1) $y=\\tan \\dfrac{x}{2}$的最小正周期为\\blank{50};\\\\\n(2) $y=\\tan \\pi x$的最小正周期为\\blank{50};\\\\\n(3) $y=\\tan (2 x-\\dfrac{\\pi}{4})$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $2\\pi$; (2) $1$; (3) $\\dfrac{\\pi}{2}$.", "solution": "", @@ -512046,7 +516189,9 @@ "id": "021710", "content": "已知$0 \\leq x \\leq 2 \\pi$, 写出适合下列条件的角$x$的区间:\\\\\n(1) 角$x$的正弦函数, 正切函数都是增函数: \\blank{100};\\\\\n(2) 角$x$的余弦函数是减函数, 正切函数是增函数: \\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $[0,\\dfrac{\\pi}{2})$, $(\\dfrac{3\\pi}{2},2\\pi]$; \\\\\n(2) $[0,\\dfrac{\\pi}{2})$, $(\\dfrac{\\pi}{2},\\pi]$.", "solution": "", @@ -512075,7 +516220,9 @@ "id": "021711", "content": "判断下列函数的奇偶性, 并说明理由.\\\\\n(1) $f(x)=-2 \\tan 3 x$;\\\\\n(2) $f(x)=x \\tan x$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 奇函数; (2) 偶函数.", "solution": "", @@ -512104,7 +516251,9 @@ "id": "021712", "content": "函数$y=\\tan ^2 x+4 \\tan x-1$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[-5,+\\infty)$", "solution": "", @@ -512133,7 +516282,9 @@ "id": "021713", "content": "比较下列各组数的大小 (填: ``$<$''``$>$''``$=$''), 不得使用计算器:\\\\\n(1) $\\tan \\dfrac{2 \\pi}{7}$\\blank{20}$\\tan \\dfrac{2 \\pi}{5}$;\\\\\n(2) $\\tan (-\\dfrac{2 \\pi}{7})$\\blank{20}$\\tan (-\\dfrac{2 \\pi}{5})$;\\\\\n(3) $\\cot 231^{\\circ}$\\blank{20}$\\cot 237^{\\circ}$;\\\\\n(4) $\\tan (k \\pi-\\dfrac{\\pi}{3})$\\blank{20}$\\tan (k \\pi+\\dfrac{\\pi}{3})$, $k \\in\\mathbf{Z}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $<$; (2) $>$; (3) $>$; (4)$<$.", "solution": "", @@ -512162,7 +516313,9 @@ "id": "021714", "content": "下列四组函数$f(x)$与$g(x)$表示同一函数的是\\blank{50}.\\\\\n\\textcircled{1}$f(x)=\\sin x$与$g(x)=\\tan x \\cos x$; \\textcircled{2}$f(x)=\\tan \\dfrac{x}{2}$与$g(x)=\\dfrac{1-\\cos x}{\\sin x}$; \\textcircled{3}$f(x)=\\tan \\dfrac{x}{2}$与$g(x)=\\dfrac{\\sin x}{1+\\cos x}$; \\textcircled{4}$f(x)=\\tan (x+\\dfrac{\\pi}{4})$与$g(x)=\\dfrac{1+\\tan x}{1-\\tan x}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{3}", "solution": "", @@ -512191,7 +516344,9 @@ "id": "021715", "content": "求函数$f(x)=\\tan (x+\\dfrac{\\pi}{6})$, $x \\in[-\\dfrac{\\pi}{3}, \\dfrac{\\pi}{3})$的最小值, 并指出使其取得最小值时$x$的所\n有值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "最小值为$-\\dfrac{\\sqrt{3}}{3}$，此时$x=-\\dfrac{\\pi}{3}$.", "solution": "", @@ -512220,7 +516375,9 @@ "id": "021716", "content": "已知函数$f(x)=\\tan x-\\cot x$.\\\\\n(1) 求函数$y=f(x)$的定义域;\\\\\n(2) 求函数$y=f(x)$的单调区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $ \\{x|x \\neq \\dfrac{k\\pi}{2},k \\in \\bf{Z}\\} $;\\\\\n(2) 单调增区间为$(-\\dfrac{\\pi}{2}+\\dfrac{k\\pi}{2},\\dfrac{k\\pi}{2}), k \\in \\bf{Z}$.", "solution": "", @@ -512249,7 +516406,9 @@ "id": "021717", "content": "函数$y=\\tan (2 x+1)$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|x\\neq \\dfrac{\\pi}{4}-\\dfrac{1}{2}+\\dfrac{k\\pi}{2},k \\in \\bf{Z} \\}$", "solution": "", @@ -512278,7 +516437,9 @@ "id": "021718", "content": "函数$y=\\tan (3 x+\\dfrac{\\pi}{4})$的单调递增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$(-\\dfrac{\\pi}{4}+\\dfrac{k\\pi}{3},\\dfrac{\\pi}{12}+\\dfrac{k\\pi}{3}), k \\in \\bf{Z}$", "solution": "", @@ -512307,7 +516468,9 @@ "id": "021719", "content": "已知$\\alpha$、$\\beta \\in(\\dfrac{\\pi}{2}, \\pi)$. 若$\\tan \\alpha<\\cot \\beta$, 则\\bracket{20}.\n\\fourch{$\\alpha<\\beta$}{$\\alpha+\\beta<\\dfrac{3 \\pi}{2}$}{$\\alpha>\\beta$}{$\\alpha+\\beta>\\dfrac{3 \\pi}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -512336,7 +516499,9 @@ "id": "021720", "content": "求函数$y=4 \\tan (\\dfrac{x}{2}-\\dfrac{\\pi}{5})$的定义域和单调区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "定义域为$ \\{x|x \\neq \\dfrac{7\\pi}{5}+2k\\pi,k \\in \\bf{Z}\\} $;\\\\\n严格增区间为$(-\\dfrac{3\\pi}{5}+2k\\pi,\\dfrac{7\\pi}{5}+2k\\pi), k \\in \\bf{Z}$.", "solution": "", @@ -512365,7 +516530,9 @@ "id": "021721", "content": "求函数$y=4 \\tan (\\dfrac{x}{2}-\\dfrac{\\pi}{5})$的零点.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "函数零点为$x=\\dfrac{2k\\pi}{5}+2k\\pi,k \\in \\bf{Z}$.", "solution": "", @@ -512394,7 +516561,9 @@ "id": "021722", "content": "判断下列命题的真假:\\\\\n(1) 函数$y=\\tan |x|$既是偶函数又是周期函数, \\blank{20};\\\\\n(2)函数$y=\\tan x$既是奇函数, 又是增函数, \\blank{20};\\\\\n(3)函数$y=\\dfrac{4 \\tan x}{2-\\sec x}$的最小正周期为$\\dfrac{\\pi}{2}$, \\blank{20};\\\\\n(4) 函数$y=\\lg [\\cos x(1+\\sqrt{3} \\tan x)]$的最大值为$\\lg 2$, \\blank{20}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) 假命题; (2) 假命题; (3) 假命题; (4) 真命题.", "solution": "", @@ -512423,7 +516592,9 @@ "id": "021723", "content": "求函数$y=\\tan ^2 x+4 \\tan x-1$, $x \\in[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}]$的值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$[-4,2+4\\sqrt{3}]$", "solution": "", @@ -512452,7 +516623,9 @@ "id": "021724", "content": "在墙壁上挂有一个表面直径为$0.3$米的时钟. 已知时钟距离地面的最近点到地面距离为$2$米, 某位学生的眼睛与地面距离为$1.7$米. 求该学生观察时钟表面的最大张角的正切值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-3,0) --++ (0,1.7) coordinate (A) --++ (3,0.3) coordinate (B) (-3,1.7)--++ (3,0.6) coordinate (C);\n\\filldraw [pattern = north east lines] (-3.5,0) -- (0,0) -- (0,2.5) -- (0.5,2.5) -- (0.5,-0.5) -- (-3.5,-0.5) -- cycle;\n\\draw pic [draw] {angle = B--A--C};\n\\draw (-2.5,2) node {$\\theta$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "最大张角的正切值为$\\dfrac{\\sqrt{2}}{4}$, 此时学生距离时钟$\\sqrt{0.18}$米.", "solution": "", @@ -512481,7 +516654,9 @@ "id": "021725", "content": "按要求, 分别以$A$、$B$、$C$为向量的起点, 在右图中画出以下向量. (图中每个小正方形的边长为$1$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale= 0.5]\n\\foreach \\i in {0,1,...,8}\n{\\draw [dashed] (0,\\i) -- (8,\\i) (\\i,0) -- (\\i,8);};\n\\filldraw (1,2) node [below left] {$A$} coordinate (A) circle (0.06);\n\\filldraw (7,3) node [below left] {$B$} coordinate (B) circle (0.06);\n\\filldraw (3,5) node [below left] {$C$} coordinate (C) circle (0.06);\n\\draw [->] (8.5,5) -- (8.5,7) node [right] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) 正北方向, 且模为$2$的向量$\\overrightarrow{AE}$;\\\\\n(2) 长度为$2 \\sqrt{2}$, 方向为北偏西$45^{\\circ}$的向量$\\overrightarrow{BF}$;\\\\\n(3) (2)中$\\overrightarrow{BF}$向量的负向量$\\overrightarrow{CG}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -512501,7 +516676,9 @@ "id": "021726", "content": "``$\\overrightarrow {a}=\\overrightarrow {b}$''是``$\\overrightarrow {a}\\parallel \\overrightarrow {b}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -512521,7 +516698,9 @@ "id": "021727", "content": "``$|\\overrightarrow {a}|=0$''是``$\\overrightarrow {a}=\\overrightarrow{0}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -512541,7 +516720,9 @@ "id": "021728", "content": "已知非零向量$\\overrightarrow {a}$和$\\overrightarrow {b}$, ``$|\\overrightarrow {a}|=|\\overrightarrow {b}|$''是``$\\overrightarrow {a}=\\overrightarrow {b}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -512561,7 +516742,9 @@ "id": "021729", "content": "把平面上一切单位向量放置到共同的始点, 那么这些向量的终点所构成的图形是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "单位圆", "solution": "", @@ -512581,7 +516764,9 @@ "id": "021730", "content": "现有以下命题:\n\\textcircled{1}向量的模是一个正实数;\n\\textcircled{2}所有的单位向量都相等;\n\\textcircled{3}零向量与任意非零向量平行. 其中真命题的个数是\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -512610,7 +516795,9 @@ "id": "021731", "content": "如图, 在四边形$ABCD$中, $\\overrightarrow{CB}+\\overrightarrow{AD}+\\overrightarrow{BA}=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0.2) node [right] {$D$} coordinate (D);\n\\draw (1,0.8) node [above] {$A$} coordinate (A);\n\\draw (1.3,-0.5) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle (A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\overrightarrow{CD}$", "solution": "", @@ -512639,7 +516826,9 @@ "id": "021732", "content": "向量$(\\overrightarrow{AB}+\\overrightarrow{MB})+(\\overrightarrow{BO}+\\overrightarrow{BC})+\\overrightarrow{OM}$化简后等于\\bracket{20}.\n\\fourch{$\\overrightarrow {BC}$}{$\\overrightarrow {AB}$}{$\\overrightarrow{AC}$}{$\\overrightarrow{AM}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "$\\overrightarrow{AC}$", "solution": "", @@ -512668,7 +516857,9 @@ "id": "021733", "content": "判断下列命题的真假:\\\\\n(1) 长度相等的向量都相等, \\blank{20};\\\\\n(2) 若$\\overrightarrow {a}=\\overrightarrow {b}, \\overrightarrow {c}=\\overrightarrow {b}$, 则$\\overrightarrow {a}=\\overrightarrow {c}$, \\blank{20};\\\\\n(3) 若四边形$ABCD$是平行四边形, 则$\\overrightarrow{AB}=\\overrightarrow{CD}$, \\blank{20};\\\\\n(4) 已知$A,B,C,D$四点两两不重合, 若$\\overrightarrow{AB}=\\overrightarrow{DC}$, 则$|\\overrightarrow{AB}|=|\\overrightarrow{CD}|$且直线$AB\\parallel CD$, \\blank{20}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "(1) 假命题; (2) 真命题; (3) 假命题; (4) 假命题.", "solution": "", @@ -512697,7 +516888,9 @@ "id": "021734", "content": "如图, 已知$D$、$E$、$F$分别是$\\triangle ABC$的$AB$、$BC$、$CA$边的中点. 以图中的点为始点和终点, 写出所有\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2,1) node [right] {$C$} coordinate (C);\n\\draw (1,2) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) node [above left] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [above right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--cycle (D)--(E)--(F)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 与$\\overrightarrow{AD}$相等的向量;\\\\\n(2) $\\overrightarrow{DE}$的负向量;\\\\\\\n(3) 与$\\overrightarrow{FE}$平行的非零向量.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) $\\overrightarrow{DB}$; $\\overrightarrow{FE}$.\\\\\n(2) $\\overrightarrow{ED}$; $\\overrightarrow{CF}$; $\\overrightarrow{FA}$.\\\\\n(3) $\\overrightarrow{EF}$; $\\overrightarrow{AD}$; $\\overrightarrow{DA}$; $\\overrightarrow{DB}$; $\\overrightarrow{BD}$; $\\overrightarrow{AB}$; $\\overrightarrow{BA}$.", "solution": "", @@ -512726,7 +516919,9 @@ "id": "021735", "content": "如图, 在$2 \\times 4$的矩形中, 起、终点都在小方格顶点、模与$|\\overrightarrow{AB}|$相等的向量共有几个?\n($\\overrightarrow{AB}$也算一个)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\foreach \\i in {0,1,...,4}\n{\\draw (\\i,0) -- (\\i,2);};\n\\foreach \\i in {0,1,2}\n{\\draw (0,\\i) -- (4,\\i);};\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (1,2) node [above] {$B$} coordinate (B);\n\\draw [->] (A)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$40$", "solution": "", @@ -512755,7 +516950,9 @@ "id": "021736", "content": "如图, 在$3 \\times 4$的矩形中, 起、终点都在小方格顶点、模与$|\\overrightarrow{AB}|$相等的向量共有几个?\n($\\overrightarrow{AB}$也算一个)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\foreach \\i in {0,1,...,4}\n{\\draw (\\i,0) -- (\\i,3);};\n\\foreach \\i in {0,1,2,3}\n{\\draw (0,\\i) -- (4,\\i);};\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (1,3) node [above] {$B$} coordinate (B);\n\\draw [->] (A)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$40$", "solution": "", @@ -512784,7 +516981,9 @@ "id": "021737", "content": "已知菱形$ABCD$的边长为$2$, 求向量$\\overrightarrow{AB}-\\overrightarrow{CB}+\\overrightarrow{CD}$的模.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$2$", "solution": "", @@ -512813,7 +517012,9 @@ "id": "021738", "content": "如图, 已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$, 分别以$O_1,O_2$为始点, 作出向量$\\overrightarrow {a}+\\overrightarrow {c}-\\overrightarrow {b}$和$\\overrightarrow {a}+(\\overrightarrow {c}-\\overrightarrow {b})$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1,0.7) node [midway, above] {$\\overline{a}$};\n\\draw [->] (1.2,0) -- (2,0) node [midway, above] {$\\overline{b}$};\n\\draw [->] (3,0) -- (2.4,0.6) node [midway, above] {$\\overline{c}$};\n\\filldraw (6,0) node [below] {$O_1$} coordinate (O_1) circle (0.03);\n\\filldraw (9,0) node [below] {$O_2$} coordinate (O_2) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -512842,7 +517043,9 @@ "id": "021739", "content": "化简: $\\dfrac{1}{2}(2 \\overrightarrow {a}+8 \\overrightarrow {b})-(4 \\overrightarrow {a}-2 \\overrightarrow {b})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-3\\overrightarrow {a}+6 \\overrightarrow {b}$", "solution": "", @@ -512871,7 +517074,9 @@ "id": "021740", "content": "化简: $3(\\overrightarrow {a}-2 \\overrightarrow {b}+\\overrightarrow {c})-4(-\\overrightarrow {a}-\\overrightarrow {b}+\\overrightarrow {c})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$7 \\overrightarrow {a}-2 \\overrightarrow {b}- \\overrightarrow {c}$", "solution": "", @@ -512900,7 +517105,9 @@ "id": "021741", "content": "判断题:(填``真''、``假'')\\\\\n(1) $0 \\overrightarrow {a}=0$, \\blank{20};\\\\\n(2) 对于实数$m$和向量$\\overrightarrow {a}$、$\\overrightarrow {b}$, 恒有$m(\\overrightarrow {a}-\\overrightarrow {b})=m \\overrightarrow {a}-m \\overrightarrow {b}$, \\blank{20};\\\\\n(3) 若$m \\overrightarrow {a}=m \\overrightarrow {b}(m \\in \\mathbf{R})$, 则$\\overrightarrow {a}=\\overrightarrow {b}$, \\blank{20};\\\\\n(4) 若$m \\overrightarrow {a}=n \\overrightarrow {a}$($m$、$n \\in \\mathbf{R}$且$\\overrightarrow {a} \\neq \\overrightarrow{0}$), 则$m=n$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "(1) 假命题; (2) 真命题; (3) 假命题; (4) 真命题.", "solution": "", @@ -512929,7 +517136,9 @@ "id": "021742", "content": "已知四边形$ABCD$是平行四边形, $\\overrightarrow{AC}=\\overrightarrow {a}$, $\\overrightarrow{BD}=\\overrightarrow {b}$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$表示$\\overrightarrow {AB}$、$\\overrightarrow {BC}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) $\\overrightarrow {AB}=\\dfrac{1}{2}\\overrightarrow {a}-\\dfrac{1}{2}\\overrightarrow {b}$;\\\\\n(2) $\\overrightarrow {BC}=\\dfrac{1}{2}\\overrightarrow {a}+\\dfrac{1}{2}\\overrightarrow {b}$.", "solution": "", @@ -512958,7 +517167,9 @@ "id": "021743", "content": "已知$\\overrightarrow{AB}=-\\dfrac{3}{4} \\overrightarrow{BC}$, $\\overrightarrow{BC} \\neq \\overrightarrow{0}$, 记$\\overrightarrow{AC}=\\lambda \\overrightarrow{BA}$, 求实数$\\lambda$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\lambda=\\dfrac{1}{3}$", "solution": "", @@ -512987,7 +517198,9 @@ "id": "021744", "content": "设$\\overrightarrow {a}$、$\\overrightarrow {b}$是两个不平行的非零向量, $x(2 \\overrightarrow {a}+\\overrightarrow {b})+y(3 \\overrightarrow {a}-2 \\overrightarrow {b})=7 \\overrightarrow {a}$, 其中$x$、$y \\in \\mathbf{R}$, 求$x$、$y$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$x=2$; $y=1$.", "solution": "", @@ -513016,7 +517229,9 @@ "id": "021745", "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$是两个不平行的非零向量.\\\\\n(1) 若向量$\\overrightarrow{AB}=\\overrightarrow {a}-\\overrightarrow {b}, \\overrightarrow{BC}=2 \\overrightarrow {a}-8 \\overrightarrow {b}, \\overrightarrow {C} \\overrightarrow {D}=3 \\overrightarrow {a}+3 \\overrightarrow {b}$, 证明: $A$、$B$、$D$三点共线;\\\\\n(2) 若向量$m \\overrightarrow {a}-\\overrightarrow {b}$与$\\overrightarrow {a}-m \\overrightarrow {b}$平行, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(2) $m=1$或$-1$.", "solution": "", @@ -513045,7 +517260,9 @@ "id": "021746", "content": "梯形$ABCD$中, $AB\\parallel CD$, 且$AB=2CD, M$、$N$分别是$DC$、$AB$的中点, 已知$\\overrightarrow{AB}=\\overrightarrow {a}, \\overrightarrow{AD}=\\overrightarrow {b}$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$分别表示$\\overrightarrow{DC}$、$\\overrightarrow{BC}$、$\\overrightarrow{MN}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\overrightarrow{DC}=\\dfrac{1}{2}\\overrightarrow{a}$;\\\\ $\\overrightarrow{DC}=-\\dfrac{1}{2}\\overrightarrow{a}+\\overrightarrow{b}$;\\\\\n$\\overrightarrow{MN}=-\\dfrac{1}{4}\\overrightarrow{a}-\\overrightarrow{b}$.", "solution": "", @@ -513074,7 +517291,9 @@ "id": "021747", "content": "设$G$是$\\triangle ABC$的重心, $AB$、$BC$、$CA$的中点分别为$D$、$E$、$F$, 则$\\overrightarrow{GD}+\\overrightarrow{GE}+\\overrightarrow{GF}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\overrightarrow{0}$", "solution": "", @@ -513103,7 +517322,9 @@ "id": "021748", "content": "在平行四边形$ABCD$中, $AC$与$BD$交于点$O$, $E$是线段$OD$的中点, $AE$的延长线与$CD$交于点$F$, 若$\\overrightarrow{AC}=\\overrightarrow {a}$, $\\overrightarrow{BD}=\\overrightarrow {b}$, 则$\\overrightarrow{AF}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\dfrac{2}{3}\\overrightarrow{a}+\\dfrac{1}{3}\\overrightarrow{b}$", "solution": "", @@ -513145,7 +517366,9 @@ "id": "021749", "content": "如果$|\\overrightarrow {a}|=2,|\\overrightarrow {b}|=\\dfrac{1}{2}$, $\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$60^{\\circ}$, 那么$\\overrightarrow {a}$与$\\overrightarrow {b}$的数量积等于\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{1}{4}$}{$1$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -513174,7 +517397,9 @@ "id": "021750", "content": "已知$|\\overrightarrow {b}|=3$. 如果$\\overrightarrow {a}$在$\\overrightarrow {b}$方向上的投影是$\\dfrac{1}{2}\\overrightarrow{b}$, 那么$\\overrightarrow {a} \\cdot \\overrightarrow {b}$为\\bracket{20}.\n\\fourch{$3$}{$\\dfrac{9}{2}$}{$2$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -513203,7 +517428,9 @@ "id": "021751", "content": "如果$\\overrightarrow {a}$、$\\overrightarrow {b}$是两个非零向量, 那么``$(\\overrightarrow {a}+\\overrightarrow {b})^2=\\overrightarrow {a}^2+\\overrightarrow {b}^2$''是``$\\overrightarrow {a} \\perp \\overrightarrow {b}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -513232,7 +517459,9 @@ "id": "021752", "content": "已知$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=1$. 若$\\overrightarrow {a}$与$\\overrightarrow {b}$之间的夹角为$\\dfrac{\\pi}{3}$, 则向量$\\overrightarrow {m}=\\overrightarrow {a}-\\overrightarrow {b}$的模为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\sqrt{3}$", "solution": "", @@ -513261,7 +517490,9 @@ "id": "021753", "content": "在$\\triangle ABC$中, 已知$\\angle A$、$\\angle B$、$\\angle C$的对边长分别为$a$、$b$、$c$. 若$a=3$, $b=1$, $\\angle C=30^{\\circ}$, 则$\\overrightarrow{BC} \\cdot \\overrightarrow{CA}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-\\dfrac{3\\sqrt{3}}{2}$", "solution": "", @@ -513290,7 +517521,9 @@ "id": "021754", "content": "若$|\\overrightarrow{AB}|=|\\overrightarrow{AC}|=6$, $\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=18$, 则$\\triangle ABC$的形状是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "等边三角形", "solution": "", @@ -513319,7 +517552,9 @@ "id": "021755", "content": "已知$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=\\sqrt{2}$. 若$(\\overrightarrow {a}-\\overrightarrow {b}) \\perp \\overrightarrow {a}$, 求$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\pi}{4}$", "solution": "", @@ -513348,7 +517583,9 @@ "id": "021756", "content": "已知$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=3$, $|2 \\overrightarrow {a}+\\overrightarrow {b}|=\\sqrt{7}$, 求$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\dfrac{2\\pi}{3}$", "solution": "", @@ -513377,7 +517614,9 @@ "id": "021757", "content": "在$\\triangle ABC$中, 若$BC=5$, $AC=4$, $\\angle C=45^{\\circ}$, 则$\\overrightarrow{BC} \\cdot \\overrightarrow{CA}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-10\\sqrt{2}$", "solution": "", @@ -513406,7 +517645,9 @@ "id": "021758", "content": "已知$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=1$. 若$(\\overrightarrow {a}+k \\overrightarrow {b}) \\perp(\\overrightarrow {a}-3 \\overrightarrow {b})$, $\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则实数$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\dfrac{4}{3}$", "solution": "", @@ -513435,7 +517676,9 @@ "id": "021759", "content": "已知$|\\overrightarrow {a}|=3$, $|\\overrightarrow {b}|=4$. 若$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$120^{\\circ}$, 则$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-\\dfrac{2}{3}\\overrightarrow {a}$", "solution": "", @@ -513464,7 +517707,9 @@ "id": "021760", "content": "下列等式中一定成立的是\\bracket{20}.\n\\twoch{$|\\overrightarrow {a}| \\cdot \\overrightarrow {a}=(\\overrightarrow {a})^2$}{$\\overrightarrow {a}(\\overrightarrow {b} \\cdot \\overrightarrow {b})=\\overrightarrow {a}(\\overrightarrow {b})^2$}{$\\overrightarrow {a}(\\overrightarrow {a} \\cdot \\overrightarrow {b})=(\\overrightarrow {a})^2 \\cdot \\overrightarrow {b}$}{$(\\overrightarrow {a} \\cdot \\overrightarrow {b})^2=(\\overrightarrow {a})^2(\\overrightarrow {b})^2$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -513493,7 +517738,9 @@ "id": "021761", "content": "在边长为$1$的正三角形$ABC$中, 若$\\overrightarrow{BC}=\\overrightarrow {a}$, $\\overrightarrow{CA}=\\overrightarrow {b}$, $\\overrightarrow{AB}=\\overrightarrow {c}$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}+\\overrightarrow {b} \\cdot \\overrightarrow {c}+\\overrightarrow {c} \\cdot \\overrightarrow {a}=$\\bracket{20}.\n\\fourch{$\\dfrac{3}{2}$}{$-\\dfrac{3}{2}$}{$0$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -513522,7 +517769,9 @@ "id": "021762", "content": "若$\\overrightarrow{AB} \\cdot \\overrightarrow{BC}+\\overrightarrow{AB}^2=0$, 则$\\triangle ABC$为\\bracket{20}.\n\\fourch{直角三角形}{钝角三角形}{锐角三角形}{等腰直角三角形}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -513551,7 +517800,9 @@ "id": "021763", "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$|\\overrightarrow {a}+\\overrightarrow {b}|=8$, $|\\overrightarrow {a}-\\overrightarrow {b}|=6$, 求$\\overrightarrow {a} \\cdot \\overrightarrow {b}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$7$", "solution": "", @@ -513580,7 +517831,9 @@ "id": "021764", "content": "已知$|\\overrightarrow {a}|=4$, $|\\overrightarrow {b}|=5$, $|2 a-3 \\overrightarrow {b}|=7$, 求$|2 \\overrightarrow {b}-\\overrightarrow{3 a}|$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$2$", "solution": "", @@ -513609,7 +517862,9 @@ "id": "021765", "content": "已知平面上的三点$A$、$B, C$与$BD$满足$(\\overrightarrow{BC} \\cdot \\overrightarrow{CA}): (\\overrightarrow{AB} \\cdot \\overrightarrow{CA}): (\\overrightarrow{BC} \\cdot \\overrightarrow{AB})=3: 4: 5$, 则\n$A$、$B$、$C$这三点的关系是\\bracket{20}.\n\\twoch{是一个直角三角形的三个顶点}{是一个钝角三角形的三个顶点}{是一个锐角三角形的三个顶点}{三点共线}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -513638,7 +517893,9 @@ "id": "021766", "content": "已知$O$、$N$、$P$在$\\triangle ABC$所在的平面内, 且$|\\overrightarrow{OA}|=|\\overrightarrow{OB}|=|\\overrightarrow{OC}|$, $\\overrightarrow{NA}+\\overrightarrow{NB}+\\overrightarrow{NC}=\\overrightarrow{0}$, $\\overrightarrow{PA} \\cdot \\overrightarrow{PB}=\\overrightarrow{PB} \\cdot \\overrightarrow{PC}=\\overrightarrow{PA} \\cdot \\overrightarrow{PC}$, 则点$O$、$N$、$P$依次是$\\triangle ABC$的\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "外心; 重心; 垂心.", "solution": "", @@ -513667,7 +517924,9 @@ "id": "021767", "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$为非零向量, 且$(\\overrightarrow {a}-4 \\overrightarrow {b}) \\perp(7 \\overrightarrow {a}-2 \\overrightarrow {b})$, $(\\overrightarrow {a}+3 \\overrightarrow {b}) \\perp(7 \\overrightarrow {a}-5 \\overrightarrow {b})$. 求向量$\\overrightarrow {a}$、$\\overrightarrow {b}$的夹角.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\pi}{3}$", "solution": "", @@ -513696,7 +517955,9 @@ "id": "021768", "content": "已知$\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$, 且$|\\overrightarrow {a}|=4$, $|\\overrightarrow {b}|=3$, $|\\overrightarrow {c}|=5$. 求$\\overrightarrow {a} \\cdot \\overrightarrow {b}+\\overrightarrow {b} \\cdot \\overrightarrow {c}+\\overrightarrow {c} \\cdot \\overrightarrow {a}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$-25$", "solution": "", @@ -513725,7 +517986,9 @@ "id": "021769", "content": "已知向量$\\overrightarrow{AB}$、$\\overrightarrow{AC}$的夹角为$120^{\\circ}$, 且$|\\overrightarrow{AB}|=3$, $|\\overrightarrow{AC}|=2$, 若$\\overrightarrow{AP}=\\lambda \\overrightarrow{AB}+\\overrightarrow{AC}$, 且\n$\\overrightarrow{AP} \\perp \\overrightarrow{BC}$, 求实数$\\lambda$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\lambda=\\dfrac{7}{12}$", "solution": "", @@ -513754,7 +518017,9 @@ "id": "021770", "content": "在平行四边形$ABCD$中, $AD=1$, $\\angle BAD=60^{\\circ}$, $E$为$CD$的中点, 若$\\overrightarrow{AD} \\cdot \\overrightarrow{BE}=1$, 求$AB$的长.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$AB=8$", "solution": "", @@ -513783,7 +518048,9 @@ "id": "021771", "content": "已知$\\overrightarrow {a} \\overrightarrow {b}$是两个不共线的向量, 若它们起点相同, $\\overrightarrow {a}$、$\\dfrac{\\overrightarrow {b}}{2}$、$t(\\overrightarrow {a}+\\overrightarrow {b})$三向量的终点在一条直线上, 求实数$t$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$t=\\dfrac{1}{3}$", "solution": "", @@ -513812,7 +518079,9 @@ "id": "021772", "content": "已知平面上三个向量$\\overrightarrow {a},\\overrightarrow {b},\\overrightarrow {c}$的模均为 1 , 它们相互之间的夹角均为$\\dfrac{2 \\pi}{3}$.\\\\\n(1) 求证: $(\\overrightarrow {a}-\\overrightarrow {b}) \\perp \\overrightarrow {c}$;\\\\\n(2) 若$|k \\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}|>1$, $k \\in \\mathbf{R}$, 求$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) $(\\overrightarrow {a}-\\overrightarrow {b}) \\cdot \\overrightarrow {c}=\\overrightarrow {a} \\cdot \\overrightarrow {c}- \\overrightarrow {b} \\cdot \\overrightarrow {c}=1*1*(-\\dfrac{1}{2})-1*1*(-\\dfrac{1}{2})=0;\\\\$\n(2) $k<0$或$k>2$.", "solution": "", @@ -513841,7 +518110,9 @@ "id": "021773", "content": "在平行四边形$ABCD$中, $A=\\dfrac{\\pi}{3}$, 边$AB$、$AD$的长分别为$2$和$1$, 若$M$、$N$分别是边$BC$、$CD$上的点, 且$\\dfrac{|\\overrightarrow{BM}|}{|\\overrightarrow{BC}|}=\\dfrac{|\\overrightarrow{CN}|}{|\\overrightarrow{CD}|}$, 求$\\overrightarrow{AM} \\cdot \\overrightarrow{AN}$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$[2,5]$", "solution": "", @@ -513870,7 +518141,9 @@ "id": "021774", "content": "在等腰$\\triangle ABC$中, 两条腰上的中线$BD$, $CE$互相垂直, 求$\\angle A$的大小.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\arccos \\dfrac{4}{5}$", "solution": "", @@ -513899,7 +518172,9 @@ "id": "021775", "content": "如图, 设$\\overrightarrow {a}=\\overrightarrow{OA}=3 \\overrightarrow{OC}$, $\\overrightarrow {b}=\\overrightarrow{OB}=4 \\overrightarrow{OD}$, 且$\\overrightarrow {a}$、$\\overrightarrow {b}$不共线, $AD$、$BC$交于点$P$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$表示$\\overrightarrow{OP}$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (4,1) node [below] {$A$} coordinate (A);\n\\draw (2,3) node [right] {$B$} coordinate (B);\n\\draw ($(O)!{1/3}!(A)$) node [below] {$C$} coordinate (C);\n\\draw ($(O)!{1/4}!(B)$) node [left] {$D$} coordinate (D);\n\\draw (B)--(O)--(A)(B)--(C)(A)--(D);\n\\draw ($(C)!{2/11}!(B)$) node [above right] {$P$} coordinate (P);\n\\draw (O)--(P);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\overrightarrow{OP}=\\dfrac{3}{11}\\overrightarrow {a}+\\dfrac{2}{11}\\overrightarrow {b}$", "solution": "", @@ -513928,7 +518203,9 @@ "id": "021776", "content": "用坐标表示下列向量:\\\\\n(1) $-\\overrightarrow {i}=$\\blank{50};\\\\\n(2) $2 \\overrightarrow {i}+\\dfrac{1}{2} \\overrightarrow {j}=$\\blank{50};\\\\\n(3) 与$x$轴平行、模为$2$的向量的坐标为\\blank{50};\\\\\n(4) 向东南方向前进$3$个单位长度, 对应向量的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "(1) $(-1,0)$; (2) $(2,\\dfrac{1}{2})$; (3) $(2,0)$或 $(-2,0)$; (4) $(\\dfrac{3\\sqrt{2}}{2},-\\dfrac{3\\sqrt{2}}{2})$.", "solution": "", @@ -513957,7 +518234,9 @@ "id": "021777", "content": "已知向量$\\overrightarrow {a}=(-3,1)$、$\\overrightarrow {b}=(-1,-3)$.\\\\\n(1) 求$|3 \\overrightarrow {a}-\\overrightarrow {b}|$;\\\\\n(2) 求$3 \\overrightarrow {a}-\\overrightarrow {b}$的单位向量的坐标.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) 10; (2) $(-\\dfrac{4}{5},\\dfrac{3}{5})$.", "solution": "", @@ -513986,7 +518265,9 @@ "id": "021778", "content": "已知$\\overrightarrow {a}=(x, 3)$, $\\overrightarrow {b}=(1, y)$, $\\overrightarrow {a}-2 \\overrightarrow {b}=(2,5)$, 求实数$x$、$y$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$x=4$, $y=1$.", "solution": "", @@ -514015,7 +518296,9 @@ "id": "021779", "content": "若$\\overrightarrow {a}=3 \\overrightarrow {i}-4 \\overrightarrow {j}$, 则$\\overrightarrow {a}$的单位向量是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(\\dfrac{3}{5},-\\dfrac{4}{5})$", "solution": "", @@ -514044,7 +518327,9 @@ "id": "021780", "content": "若$\\overrightarrow {a}=\\overrightarrow {i}-3 \\overrightarrow {j}, \\overrightarrow {b}=-2 \\overrightarrow {i}+2 \\overrightarrow {j}$, 则$2 \\overrightarrow {a}-\\overrightarrow {b}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(4,-8)$", "solution": "", @@ -514073,7 +518358,9 @@ "id": "021781", "content": "若向量$\\overrightarrow{AB}=(2-x) \\overrightarrow {i}+(1-x) \\overrightarrow {j}$的坐标所表示的点在第四象限内, 则$x$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(1,2)$", "solution": "", @@ -514102,7 +518389,9 @@ "id": "021782", "content": "化简$\\overrightarrow{AB}-\\overrightarrow{AC}-\\overrightarrow{BC}$等于\\bracket{20}.\n\\fourch{$2 \\overrightarrow{BC}$}{$2 \\overrightarrow{AC}$}{$-2 \\overrightarrow{BC}$}{$\\overrightarrow{0}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -514131,7 +518420,9 @@ "id": "021783", "content": "若$A(0,-3)$, $B(3,3)$, $C(x, 1)$, 且$\\overrightarrow{AB}\\parallel 2 \\overrightarrow{BC}$, 则$x=$\\bracket{20}.\n\\fourch{$2$}{$1$}{$-1$}{$-2$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -514160,7 +518451,9 @@ "id": "021784", "content": "已知$\\overrightarrow {m}=(-1, a)$, $\\overrightarrow {n}=(2 a, 4)$. 若$\\overrightarrow {p}=\\overrightarrow {m}+\\dfrac{1}{2} \\overrightarrow {n}$, 且$|\\overrightarrow {p}|=3$, 则实数$a$的值等于\\bracket{20}.\n\\fourch{$1$或$2$}{$-2$或$1$}{$\\pm 1$}{$\\pm 2$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -514189,7 +518482,9 @@ "id": "021785", "content": "已知$O(0,0)$、$A(1,2)$、$B(4,6)$、$C(3,4)$, 求证: 四边形$OABC$为平行四边形.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -514218,7 +518513,9 @@ "id": "021786", "content": "已知$\\overrightarrow {a}=(\\sin 30^{\\circ}, \\cos 30^{\\circ})$, $\\overrightarrow {b}=(\\cos 30^{\\circ}, \\sin 30^{\\circ}), \\overrightarrow {m}=(1,1)$. 是否存在非零实数$\\lambda$、$\\mu$, 使得$\\overrightarrow {m}\\parallel(\\lambda \\overrightarrow {a}+\\mu \\overrightarrow {b})$? 若存在, 分别求出$\\lambda$、$\\mu$的值; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\lambda=\\mu$ 且$\\lambda$和$\\mu$非零.", "solution": "", @@ -514247,7 +518544,9 @@ "id": "021787", "content": "已知点$O$、$A$、$B$的坐标分别为$(0,0)$、$(1,2)$、$(4,5)$, 且$\\overrightarrow{OP}=\\overrightarrow{OA}+t \\overrightarrow{AB}$.\\\\\n(1) 当$t$分别为何值时, 点$P$在$x$轴上? 点$P$在$y$轴上? 点$P$在第二象限?\\\\\n(2) 四边形$OABP$能否为平行四边形? 若能, 求出相应的$t$值; 若不能, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) 当$t=\\dfrac{3}{2}$时,点$P$在$x$轴上; 当$t=\\dfrac{1}{3}$时,点$P$在$y$轴上;当$-\\dfrac{2}{3}=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (3,0) node [above] {$B$} coordinate (B);\n\\draw (2,-2) node [below] {$O$} coordinate (O);\n\\draw ($(A)!{1/3}!(B)$) node [above] {$M$} coordinate (M);\n\\draw [->] (A)--(M);\n\\draw [->] (M)--(B);\n\\draw [->] (O)--(M);\n\\draw (O)--(A)(O)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\overrightarrow {c}=\\dfrac{2}{3}\\overrightarrow {a}+\\dfrac{1}{3}\\overrightarrow {b}$", "solution": "", @@ -515244,7 +519609,9 @@ "id": "021821", "content": "已知$\\overrightarrow{e_1}$与$\\overrightarrow{e_2}$不平行, 实数$x$、$y$满足: $3 x \\overrightarrow{e_1}+(10-y) \\overrightarrow{e_2}=(4 y+7) \\overrightarrow{e_1}+2 x \\overrightarrow{e_2}$, 求$5 x-3 y$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$17$", "solution": "", @@ -515273,7 +519640,9 @@ "id": "021822", "content": "如图, 已知$\\triangle AOB$的重心为$G$, 过点$G$的直线与边$OA$、$OB$分别交于点$P$、$Q$, 设$\\overrightarrow{OP}=h \\overrightarrow{OA}$, $\\overrightarrow{OQ}=k \\overrightarrow{OB}$, $\\triangle AOB$、$\\triangle POQ$的面积分别为$S$、$T$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0) node [below] {$B$} coordinate (B);\n\\draw (2,2.5) node [above] {$O$} coordinate (O);\n\\filldraw ($1/3*(A)+1/3*(B)+1/3*(O)$) node [below] {$G$} coordinate (G) circle (0.03);\n\\draw (0,0.2) coordinate (S);\n\\draw ($(S)!2!(G)$) coordinate (T);\n\\path [name path = PQ, draw] (S)--(T);\n\\path [name path = AOB, draw] (A)--(O)--(B)--cycle;\n\\path [name intersections = {of = PQ and AOB, by = {P,Q}}];\n\\draw (P) node [above] {$P$};\n\\draw (Q) node [above] {$Q$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $\\dfrac{1}{h}+\\dfrac{1}{k}=3$;\\\\\n(2) 求证: $\\dfrac{4}{9} S \\leq T \\leq \\dfrac{1}{2} S$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -515302,7 +519671,9 @@ "id": "021823", "content": "两块斜边长相等的直角三角板如图拼在一起, 若$\\overrightarrow{AD}=x \\overrightarrow{AB}+y \\overrightarrow{AC}$, 求$x, y$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$C$} coordinate (C);\n\\draw (B) ++ (45:{sqrt(6)}) node [right] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(B)$) node [above] {$E$} coordinate (E);\n\\draw (C)--(A)--(B)--(D)--(E)(C)--(B);\n\\draw pic [draw,scale = 0.5] {right angle = B--A--C};\n\\draw pic [draw,scale = 0.5,\"$45^\\circ$\", angle eccentricity = 2.5] {angle = A--C--B};\n\\draw pic [draw,scale = 0.5,\"$60^\\circ$\", angle eccentricity = 2.5] {angle = B--E--D};\n\\draw (A)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$x=1+\\dfrac{\\sqrt{3}}{2}$,$y=\\dfrac{\\sqrt{3}}{2}$.", "solution": "", @@ -515331,7 +519702,9 @@ "id": "021824", "content": "用向量法证明: 平行四边形的对角线的平方和等于四边的平方和.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -515361,7 +519734,9 @@ "id": "021825", "content": "用向量法证明: 在正方形$ABCD$中, $E$、$F$分别为边$AB$、$BC$的中点, 有$AF \\perp DE$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -515391,7 +519766,9 @@ "id": "021826", "content": "已知三角形$\\triangle ABC$的三个顶点坐标分别为$A(2,5)$、$B(3,1)$、$C(-1,4)$, 求该三角形面积.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\dfrac{13}{2}$", "solution": "", @@ -515421,7 +519798,9 @@ "id": "021827", "content": "已知$\\triangle ABC$的重心为$G$, $O$为三角形外的任一点, $\\overrightarrow{OA}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, $\\overrightarrow{OC}=\\overrightarrow {c}$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$表示$\\overrightarrow{OG}$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\overrightarrow{OG}=\\dfrac{1}{3}(\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c})$", "solution": "", @@ -515451,7 +519830,9 @@ "id": "021828", "content": "试用向量法解决下列问题:\\\\\n(1) 求函数$y=2 \\sqrt{3-x}+5 \\sqrt{x+8}$的最大值;\\\\\n(2) 求函数$y=\\sqrt{x^2-2 x+2}+\\sqrt{x^2-10 x+34}$的最小值;\\\\\n(3) 求函数$y=\\sqrt{x^2-12 x+52}-\\sqrt{x^2-4 x+5}$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) 当$x=\\dfrac{43}{29}$时，最大值为$\\sqrt{319}$;\\\\\n(2) 当$x=2$时，最小值为$4*\\sqrt{2}$;\\\\\n(3) 当$x=\\dfrac{2}{3}$时，最大值为$5$.", "solution": "", @@ -515481,7 +519862,9 @@ "id": "021829", "content": "$O$是平面上一点, $A$、$B$、$C$是平面上不共线的三个点, 动点$P$满足$\\overrightarrow{OP}=\\overrightarrow{OA}+\\lambda(\\dfrac{\\overrightarrow{AB}}{|\\overrightarrow{AB}|}+\\dfrac{\\overrightarrow{AC}}{|\\overrightarrow{AC}|})$, $\\lambda \\in[0,+\\infty)$, 则$P$点轨迹一定通过$\\triangle ABC$的\\bracket{20}.\n\\fourch{外心}{内心}{重心}{垂心}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -515511,7 +519894,9 @@ "id": "021830", "content": "计算:\n(1) $(\\dfrac{1}{4}-\\dfrac{13}{5} \\mathrm{i})+(\\dfrac{2}{3}+\\dfrac{5}{2} \\mathrm{i})=$\\blank{50};\\\\\n(2) $-1-\\mathrm{i}-(2+3 \\mathrm{i})+4 \\mathrm{i}=$\\blank{50};\\\\\n(3) $[(a+b)+(a-b) \\mathrm{i}]-[(a-b)-(a+b) \\mathrm{i}]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "(1) $\\dfrac{11}{12}-\\dfrac{1}{10} \\mathrm{i}$; (2) $-3$; (3) $2b+2a \\mathrm{i}$.", "solution": "", @@ -515541,7 +519926,9 @@ "id": "021831", "content": "计算:\n(1) $(\\dfrac{\\sqrt{2}}{2}-\\dfrac{\\sqrt{2}}{2} \\mathrm{i})^2=$\\blank{50};\\\\\n(2) $(2-5 \\mathrm{i})(1+2 \\mathrm{i})(12+\\mathrm{i})=$\\blank{50};\\\\\n(3) $(\\dfrac{1-\\mathrm{i}}{1+\\mathrm{i}})^3=$\\blank{50};\\\\\n(4) $\\dfrac{-2+2 \\sqrt{3} \\mathrm{i}}{(\\sqrt{3}+\\mathrm{i})^2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "(1) $-\\mathrm{i}$; (2) $145$; (3) $\\mathrm{i}$; (4) $\\dfrac{1}{2}+\\dfrac{\\sqrt{3}}{2} \\mathrm{i}$.", "solution": "", @@ -515571,7 +519958,9 @@ "id": "021832", "content": "计算:\\\\\n(1) $\\mathrm{i}+\\mathrm{i}^2+\\mathrm{i}^3+\\cdots+\\mathrm{i}^{200}$;\\\\\n(2) $(1+\\mathrm{i})^{10}-(1-\\mathrm{i})^{10}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) $0$; (2) $64\\mathrm{i}$.", "solution": "", @@ -515601,7 +519990,9 @@ "id": "021833", "content": "已知$z=1-\\mathrm{i}$, 求$\\dfrac{z^2-z+1}{z^2+z+1}$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\dfrac{3}{13}-\\dfrac{2}{13}\\mathrm{i}$", "solution": "", @@ -515631,7 +520022,9 @@ "id": "021834", "content": "已知$z_1=5+10 \\mathrm{i}$, $z_2=3-4 \\mathrm{i}$, 复数$z$满足$\\dfrac{1}{z}=\\dfrac{1}{z_1}+\\dfrac{1}{z_2}$, 求$z$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$5-\\dfrac{5}{2}\\mathrm{i}$", "solution": "", @@ -515661,7 +520054,9 @@ "id": "021835", "content": "若复数$z=x^2-y^2-7+(x-y-3) \\mathrm{i}$等于$-2 \\mathrm{i}$, 求实数$x$、$y$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$x=4,y=3$.", "solution": "", @@ -515691,7 +520086,9 @@ "id": "021836", "content": "已知实数$x$、$y$满足$\\dfrac{x}{1-\\mathrm{i}}+\\dfrac{y}{1-2 \\mathrm{i}}=\\dfrac{5}{1-3 \\mathrm{i}}$, 求$x$、$y$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$x=-1,y=5$.", "solution": "", @@ -515721,7 +520118,9 @@ "id": "021837", "content": "从很久以前开始, 我们先后学习了整数、有理数、实数, 到了高中, 我们分别用记号$\\mathbf{Z}, \\mathbf{Q}, \\mathbf{R}$表示相应的数的集合, 现在, 我们学习了复数集合, 并用记号$\\mathbf{C}$表示, 用集合中的真子集的关系表示这四个集合的关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -515750,7 +520149,9 @@ "id": "021838", "content": "若复数$z=a+b \\mathrm{i}$($a, b \\in \\mathbf{R}$)是虚数, 则$a, b$应满足的条件是\\bracket{20}.\n\\fourch{$a=0$且$b \\neq 0$}{$a \\neq 0$}{$a \\neq 0$且$b \\neq 0$}{$b \\neq 0$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -515779,7 +520180,9 @@ "id": "021839", "content": "设$z_1=a+b \\mathrm{i}$($a, b \\in \\mathbf{R}$), $z_2=c+d\\mathrm{i}$($c, d \\in \\mathbf{R}$), 则``$a=c$''是``$z_1=z_2$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分又非必要}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -515808,7 +520211,9 @@ "id": "021840", "content": "下列有关复数的描述中, 正确的是\\bracket{20}.\n\\twoch{$\\mathrm{i}$是$-1$的一个平方根}{$-2 \\mathrm{i}<-\\mathrm{i}$}{$b \\mathrm{i}(b \\in \\mathbf{R})$表示纯虚数}{若$z=3-4 \\mathrm{i}$, 则$\\mathrm{Im} z=4$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -515837,7 +520242,9 @@ "id": "021841", "content": "复数$z$与$\\overline {z}$在复平面$xOy$上所对应的点关于\\blank{50}对称.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -515866,7 +520273,9 @@ "id": "021842", "content": "当且仅当$m \\in$\\blank{50}时, $(m^2+3 m-4)+(m^2+5 m-6) i$($m \\in \\mathbf{R}$)是纯虚数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -515895,7 +520304,9 @@ "id": "021843", "content": "已知复数$(3 m+2 n-5)+(-m+4 n+7) \\mathrm{i}$是纯虚数, 复数$(2 m-n-1)+(m+n+1) \\mathrm{i}$是实数, 求实数$m, n$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -515924,7 +520335,9 @@ "id": "021844", "content": "判断是否存在实数$m$, 使复数$z=m^2-2 m-15+\\dfrac{m^2+5 m+6}{m^2-25} \\mathrm{i}$分别满足下列条件. 若存在, 求出$m$的值; 若不存在, 请说明理由.\\\\\n(1) $z$是实数;\\\\\n(2) $z$是虚数;\\\\\n(3) $z$是纯虚数;\\\\\n(4) $z$是零.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -515953,7 +520366,9 @@ "id": "021845", "content": "设$z_1=2 a(a-1)+a^2 \\mathrm{i}$, $z_2=(a-1)+a(a-1) \\mathrm{i}$, 其中$a \\in \\mathbf{R}$. 若$\\overline{z_1}=z_2$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -515982,7 +520397,9 @@ "id": "021846", "content": "求分别满足下列各等式的实数$x$与$y$的值.\\\\\n(1) $(x+y)-x y \\mathrm{i}=-5+24 \\mathrm{i}$;\\\\\n(2) $2 x^2-5 x+2+(y^2+y-2) \\mathrm{i}=0$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516011,7 +520428,9 @@ "id": "021847", "content": "设$z$是复数, 求证: ``$z$是纯虚数''的一个充要条件是``$z+\\overline {z}=0$且$z \\neq 0$''.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516040,7 +520459,9 @@ "id": "021848", "content": "下列命题中, 是真命题的命題的序号是\\blank{50}.\\\\\n\\textcircled{1} 在复平面上, 表示实数的点都在实轴上, 表示纯虚数的点都在虚轴上;\\\\\n\\textcircled{2} 在复平面上, 表示虚数的点都落在四个象限内;\\\\\n\\textcircled{3} 复数的模表示该复数在复平面上所对应的点到原点的距离.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516070,7 +520491,9 @@ "id": "021849", "content": "复数$-4+3 \\mathrm{i}$、$4-\\mathrm{i}$在复平面上对应点的象限分别为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516100,7 +520523,9 @@ "id": "021850", "content": "复数$-3 \\mathrm{i}$、$-4-\\mathrm{i}$在复平面上对应点的坐标分别为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516130,7 +520555,9 @@ "id": "021851", "content": "复数$-4+3 \\mathrm{i}$、$-6 \\mathrm{i}$、$2-\\mathrm{i}$的模分别为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516160,7 +520587,9 @@ "id": "021852", "content": "若$\\overrightarrow{OA}=(5,-1)$, $\\overrightarrow{OB}=(3,2)$, 则$\\overrightarrow{AB}$在复平面上所对应的复数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516190,7 +520619,9 @@ "id": "021853", "content": "复平面上向量$\\overrightarrow{OC}$, $\\overrightarrow{CD}$对应的复数分别为$-1-2 \\mathrm{i}$, $2-\\mathrm{i}$, 则$D$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516220,7 +520651,9 @@ "id": "021854", "content": "在复平面上, 平行于$y$轴的非零向量所对应的复数的集合是\\bracket{20}.\n\\twoch{实数集}{虚数集}{纯虚数集}{实数集与纯虚数集的并集}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -516250,7 +520683,9 @@ "id": "021855", "content": "在复平面上, 复数$z$对应点$Z$落在以原点为圆心的单位圆上, 下列复数中, 其对应点总落在以原点为圆心, 半径为$2$的圆上的是\\blank{50}(填上正确的序号).\\\\\n\\textcircled{1} $1+z$; \\textcircled{2} $2 z$; \\textcircled{3} $\\dfrac{2}{z}$; \\textcircled{4} $z+\\overline {z}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516280,7 +520715,9 @@ "id": "021856", "content": "求实数$m$取何值时, 复数$z=(m^2-8 m+15)+(m^2-m-6) \\mathrm{i}$在复平面上所对应的点$Z$分别满足下列条件.\\\\\n(1) 点$Z$在实轴上;\\\\\n(2) 点$Z$在虚轴上;\\\\\n(3) 点$Z$在第四象限.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516310,7 +520747,9 @@ "id": "021857", "content": "设复数$z_1=m+8 \\mathrm{i}$, $m \\in \\mathbf{R}$, $z_2+\\mathrm{i} z_1=0$, 在复平面$xOy$上, $z_1, z_2$所对应的点分别为$Z_1, Z_2$.\\\\\n(1) 用$m$表示$z_2$;\\\\\n(2) 求$\\angle Z_1OZ_2$;\\\\\n(3) 若三角形$Z_1OZ_2$的面积为$50$, 求$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516340,7 +520779,9 @@ "id": "021858", "content": "已知复数$z_1=-4+3 \\mathrm{i}$, $z_2=4 \\mathrm{i}$. 设复数$z_k$($k=1,2,3$)在复平面$xOy$上所对应的点$Z_k$.\\\\\n(1) 若四边形$OZ_1Z_2Z_3$是一个平行四边形, 求$z_3$;\\\\\n(2) 若$O, Z_1, Z_2, Z_3$是一个平行四边形的四个顶点, 求$z_3$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516370,7 +520811,9 @@ "id": "021859", "content": "已知复数$z=m+(2 m-1) \\mathrm{i}$, $m \\in \\mathbf{R}$. 设$z$在复平面上对应的点为$Z$.\\\\\n(1) 若点$Z$到原点的距离为$2$, 求$m$的值;\\\\\n(2) 问: 无论$m$取何值, 点$Z$总不落在第几象限? 为什么?", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516400,7 +520843,9 @@ "id": "021860", "content": "若复数$z$满足$z-3+\\mathrm{i}=5-\\mathrm{i}$, 则$|\\mathrm{i} z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516430,7 +520875,9 @@ "id": "021861", "content": "若复数$z$满足$z+\\dfrac{2}{z}=0$, 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516460,7 +520907,9 @@ "id": "021862", "content": "若复数$z$满足$|z|=3$, $\\mathrm{Re} z=2$, 则$\\mathrm{Im} z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516490,7 +520939,9 @@ "id": "021863", "content": "计算: $|\\mathrm{i} \\cdot \\mathrm{i}^3 \\cdot \\mathrm{i}^5 \\cdot \\cdots \\cdot \\mathrm{i}^{2021} \\cdot \\mathrm{i}^{2022}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516520,7 +520971,9 @@ "id": "021864", "content": "计算: $|(\\dfrac{1+\\mathrm{i}}{1-\\mathrm{i}})^{2022}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516550,7 +521003,9 @@ "id": "021865", "content": "计算: $|(1+\\mathrm{i})^2(\\mathrm{i}-2 \\sqrt{2})^3|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516580,7 +521035,9 @@ "id": "021866", "content": "计算: $|\\dfrac{(\\sqrt{5}-2 \\mathrm{i})(1+\\sqrt{3} \\mathrm{i})^2}{\\sqrt{13}+\\sqrt{23} \\mathrm{i}}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516610,7 +521067,9 @@ "id": "021867", "content": "复数$z$分别满足下列条件, 复数$z$在复平面上对应点$Z$, 画出点$Z$的集合对应的图形.\\\\\n(1) $|z|=3$;\\\\\n(2) $1<\\mathrm{Re} z<2$且$1<\\mathrm{Im} z<2$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516640,7 +521099,9 @@ "id": "021868", "content": "已知复数$z_1=2+\\mathrm{i}$, $|z_2|=5$, $z_2 z_1$是负实数, 求复数$z_2$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516670,7 +521131,9 @@ "id": "021869", "content": "已知复数$z=m+(3 m-5) \\mathrm{i}$, $m \\in \\mathbf{R}$.\\\\\n(1) 若$|z| \\leq 5$, 求$m$的取值范围;\\\\\n(2) 求$z$的模的最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516700,7 +521163,9 @@ "id": "021870", "content": "``复数$z_1, z_2$互为共轭''是``$|z_1|=|z_2|$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分又非必要}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -516730,7 +521195,9 @@ "id": "021871", "content": "设$z_1, z_2, z$是复数, 有下列五个命题: \\textcircled{1} 若$z_1^2+z_2^2=0$, 则$z_1=z_2=0$.\n\\textcircled{2} 若$|z_1|=|z_2|$, 则$z_1^2=z_2^2$. \\textcircled{3} 若$z_1^2=z_2^2$, 则$|z_1|=|z_2|$.\n\\textcircled{4} 若$|z|=1$, 则$z=1$或$z=-1$或$z=\\mathrm{i}$或$z=-\\mathrm{i}$. \\textcircled{5} 若$|z+\\mathrm{i}|=|z-\\mathrm{i}|$, 则$z$为实数. 其中, 正确的命题的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516760,7 +521227,9 @@ "id": "021872", "content": "设$z_1, z_2$是复数, 有下列五个条件:\\\\\n\\textcircled{1} $|z_1-\\overline{z_2}|=0$; \\textcircled{2} $|z_1|=|z_2|$; \\textcircled{3} $z_1=z_2$; \\textcircled{4} $\\overline {z}_1=z_2$; \\textcircled{5} $z_1^2=\\overline{z_2^2}$.\\\\\n则可以成为$z_1=\\overline{z_2}$的必要非充分条件的是\\blank{50}(填正确的条件的序号).", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516790,7 +521259,9 @@ "id": "021873", "content": "已知$z \\in \\mathbf{C}$, $z+\\dfrac{4}{z} \\in \\mathbf{R}$, $|z-2|=2$, 求$z$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516820,7 +521291,9 @@ "id": "021874", "content": "复数$-1$的平方根是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516849,7 +521322,9 @@ "id": "021875", "content": "设$a<0$. 则$a$的平方根是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -516878,7 +521353,9 @@ "id": "021876", "content": "若复数$z$满足$z^2 \\in[0,+\\infty)$, 则$z$的集合为\\bracket{20}.\n\\twoch{实数集}{虚数集}{纯虚数集}{实数集与纯虚数集的并集}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -516907,7 +521384,9 @@ "id": "021877", "content": "一个非零复数的平方根恰有两个, 求下列复数的平方根.\\\\\n(1) $5\\mathrm{i}$;\\\\\n(2) $5-12\\mathrm{i}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516936,7 +521415,9 @@ "id": "021878", "content": "已知复数$z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$)满足$|z-1|=1$.\\\\\n(1) 求$x, y$满足的关系式;\\\\\n(2) 求复数$z$的模的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516965,7 +521446,9 @@ "id": "021879", "content": "设$z$为复数. 若$\\dfrac{z-3}{z+3}$为纯虚数, 求$|z|$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -516994,7 +521477,9 @@ "id": "021880", "content": "已知复数$z_1, z_2$满足$|z_1|=|\\overline{z_2}|=1$, 且$z_1+z_2=-\\mathrm{i}$, 求$z_1, z_2$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517023,7 +521508,9 @@ "id": "021881", "content": "已知复数$z$的平方等于$8+6 \\mathrm{i}$, 求$z^3-16 z-\\dfrac{100}{z}$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517052,7 +521539,9 @@ "id": "021882", "content": "设$z_1, z_2 \\in \\mathbf{C}$, 且$|z_1|=|z_2|=1$, $|z_1+z_2|=\\sqrt{2}$, 求$|z_1-z_2|$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517081,7 +521570,9 @@ "id": "021883", "content": "下列命题中, 是真命题的命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 在复数范围内, 方程$a x^2+b x+c=0$($a, b, c \\in \\mathbf{R}$, $a \\neq 0$)总有两个根;\\\\\n\\textcircled{2} 设$p, q \\in \\mathbf{C}$. 若$3+2 \\mathrm{i}$是方程$x^2+p x+q=0$的一个根, 则该方程的另一个根是$3-2 \\mathrm{i}$;\\\\\n\\textcircled{3} 设$p, q \\in \\mathbf{C}$. 若方程$x^2+p x+q=0$有两个共轭虚数根, 则$p, q$均为实数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517107,7 +521598,9 @@ "id": "021884", "content": "若实系数方程$x^2+b x+c=0$的一个根是$\\dfrac{1}{3}+\\dfrac{\\sqrt{7}}{3} \\mathrm{i}$, 则实数数对$(b, c)$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517133,7 +521626,9 @@ "id": "021885", "content": "若复系数方程$x^2+m x+n=0$的根为$-\\mathrm{i}$与$1+\\mathrm{i}$, 则复数数对$(m, n)$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517159,7 +521654,9 @@ "id": "021886", "content": "设$a \\in \\mathbf{R}$. 若方程$x^2-a x+2=0$有虚数根$z$, 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517185,7 +521682,9 @@ "id": "021887", "content": "设$m \\in \\mathbf{R}$. 若关于$x$的方程$x^2+m x+m=0$有虚数根, 则$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517211,7 +521710,9 @@ "id": "021888", "content": "设$k \\in \\mathbf{R}$. 若关于$x$的方程$x^2+(k+2 \\mathrm{i}) x+2+k \\mathrm{i}=0$有实数根, 则$k$值的集合为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517237,7 +521738,9 @@ "id": "021889", "content": "在复数集中解下列一元二次方程:\\\\\n(1) $4 x^2+9=0$;\\\\\n(2) $x^2+4 x+12=0$;\\\\\n(3) $x^2+x+1=0$;\\\\\n(4) $x^2-(1+\\mathrm{i}) x+\\mathrm{i}=0$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517263,7 +521766,9 @@ "id": "021890", "content": "在复数集中分解因式:\\\\\n(1) $x^2+2 x y+3 y^2$;\\\\\n(2) $x^3+y^3$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517289,7 +521794,9 @@ "id": "021891", "content": "已知两个数的和等于$\\sqrt{3}$, 它们的积等于$3$, 求这两个数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517315,7 +521822,9 @@ "id": "021892", "content": "设$k \\in \\mathbf{R}$. 若关于$x$的方程$x^2+k x+3=0$有两个虚根$\\alpha$和$\\beta$, 且$|\\alpha-\\beta|=2 \\sqrt{2}$, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517341,7 +521850,9 @@ "id": "021893", "content": "设$p \\in \\mathbf{R}$, 关于$x$的方程$2 x^2-p x+p=0$的两个根为$x_1$和$x_2$.\\\\\n(1) 若$p=3$, 求$x_1^2+x_2^2$的值;\\\\\n(2) 若$x_1^2+x_2^2=3$, 求$p$的值;\\\\\n(3) 若$|x_1|+|x_2|=3$, 求$p$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517367,7 +521878,9 @@ "id": "021894", "content": "在复数集中解下列方程:\\\\\n(1) $(x-3)(x-5)+2=0$;\\\\\n(2) $x^4-16=0$;\\\\\n(3) $\\dfrac{1}{x+3}-\\dfrac{1}{x}=1$;\\\\\n(4) $x^2-(2+2 \\mathrm{i}) x+4 \\mathrm{i}=0$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517393,7 +521906,9 @@ "id": "021895", "content": "在复数集中分解因式:\\\\\n(1) $a^2+2 a b+b^2+c^2$;\\\\\n(2) $x^4+3 x^2-10$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517419,7 +521934,9 @@ "id": "021896", "content": "设$p \\in \\mathbf{R}$. 若关于$x$的方程$x^2+(4+\\mathrm{i}) x+3+p \\mathrm{i}=0$有实数根, 求$p$的值, 并解这个方程.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517445,7 +521962,9 @@ "id": "021897", "content": "复数$w$满足$w-4=(3-2 w) \\mathrm{i}$, $z=\\dfrac{5}{w}+|w-2|$. 写出一个以$z$为根的实系数一元二次方程.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517471,7 +521990,9 @@ "id": "021898", "content": "设$k \\in \\mathbf{R}$. 若关于$x$的方程$x^2+k x+5=0$有两根$\\alpha$和$\\beta$, 且$|\\alpha-\\beta|=2$, 求$k$的值的集合.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517497,7 +522018,9 @@ "id": "021899", "content": "已知虚数$z_1, z_2$满足$z_1^2=z_2$.\\\\\n(1) 设$z_1, z_2$是一个实系数一元二次方程的两个根, 求$z_1, z_2$;\\\\\n(2) 设$z_1=1+m \\mathrm{i}$, $m>0$, $|z_1| \\leq \\sqrt{2}$, 复数$\\omega=z_2+3$, 求$|\\omega|$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517523,7 +522046,9 @@ "id": "021900", "content": "设$k \\in \\mathbf{R}$. 若关于$x$的方程$x^2+k x+k^2-2 k=0$有一个模为$1$的虚根, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517549,7 +522074,9 @@ "id": "021901", "content": "把下列复数化为三角形式:(辐角用主值)\\\\\n(1) $-2 \\mathrm{i}=$\\blank{100};\\\\\n(2) $-1=$\\blank{100};\\\\\n(3) $-1+\\mathrm{i}=$\\blank{100};\\\\\n(4) $-1-\\sqrt{3} \\mathrm{i}=$\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517575,7 +522102,9 @@ "id": "021902", "content": "把下列复数化为三角形式:(辐角用主值)\\\\\n(1) $3-4 \\mathrm{i}$;\\\\\n(2) $\\cos \\dfrac{\\pi}{5}-\\mathrm{i} \\sin \\dfrac{\\pi}{5}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517601,7 +522130,9 @@ "id": "021903", "content": "设复数$\\sqrt{3}-\\mathrm{i}$在复平面上对应的向量为$\\overrightarrow{OA}$, 将$\\overrightarrow{OA}$绕原点$O$逆时针旋转$120^{\\circ}$, 且模缩小到原来$\\dfrac{1}{2}$得到向量$\\overrightarrow{OB}$, 求点$B$对应的复数的代数形式.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517627,7 +522158,9 @@ "id": "021904", "content": "设复数$z_1=2+\\mathrm{i}$与复数$z_2=1+3 \\mathrm{i}$.\\\\\n(1) 将$\\dfrac{z_2}{z_1}$表示为复数三角形式;\\\\\n(2) 在复平面$xOy$上, $z_1$、$z_2$所对应的点为$Z_1$、$Z_2$, 求$\\angle Z_1OZ_2$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517653,7 +522186,9 @@ "id": "021905", "content": "计算(结果用代数形式).\\\\\n(1) $\\sqrt{6}(\\cos \\dfrac{\\pi}{8}+\\mathrm{i} \\sin \\dfrac{\\pi}{8}) \\cdot \\sqrt{10}(\\cos \\dfrac{\\pi}{12}+\\mathrm{i} \\sin \\dfrac{\\pi}{12}) \\cdot \\sqrt{15}(\\cos \\dfrac{\\pi}{24}+\\mathrm{i} \\sin \\dfrac{\\pi}{24})$;\\\\\n(2) $\\dfrac{\\cos \\dfrac{8\\pi}{15}+\\mathrm{i}\\sin\\dfrac{8\\pi}{15}}{\\sqrt{3}(\\cos \\dfrac\\pi 5+\\mathrm{i}\\sin\\dfrac\\pi 5)}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517679,7 +522214,9 @@ "id": "021906", "content": "计算(结果用代数形式).\\\\\n(1) $(1+\\mathrm{i})^{20}$;\\\\\n(2) $(-\\sqrt{3}+\\mathrm{i})^{11}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517705,7 +522242,9 @@ "id": "021907", "content": "将向量$\\overrightarrow{OP}$按逆时针方向旋转$\\dfrac{2 \\pi}{3}$后再将其模伸长至原来的$3$倍, 所得向量$\\overrightarrow{OQ}$对应的复数$-6 \\mathrm{i}$, 则向量$\\overrightarrow{OP}$对应的复数为\\blank{50}(结果用代数形式表示).", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517731,7 +522270,9 @@ "id": "021908", "content": "设$m, n$是正整数, 求满足$(\\sqrt{3}+\\mathrm{i})^m=(1+\\mathrm{i})^n$的$m+n$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -517757,7 +522298,9 @@ "id": "021909", "content": "判断下列命题的真假:\\\\\n(1) 若一条直线的倾斜角为$90^{\\circ}$, 则这条直线与$y$轴平行, \\blank{20};\\\\\n(2) 若直线经过$(x_1, y_1),(x_2, y_2)$, 则该直线的斜率$k=\\dfrac{y_2-y_1}{x_2-x_1}$, \\blank{20};\\\\\n(3) 直线的倾斜角的变化范围是$[0, \\dfrac{\\pi}{2}) \\cup(\\dfrac{\\pi}{2}, \\pi)$, \\blank{20};\\\\\n(4) 倾斜角为$0$的直线是$x$轴, \\blank{20}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517786,7 +522329,9 @@ "id": "021910", "content": "经过两个点$P(2,1)$, $Q(0,2)$的直线的斜率是\\blank{50}, 倾斜角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517815,7 +522360,9 @@ "id": "021911", "content": "经过两个点$P(2,1)$, $Q(a,-2)$(其中实常数$a>2$)的直线的斜率是\\blank{50}, 倾斜角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517844,7 +522391,9 @@ "id": "021912", "content": "经过两个点$P(2,1)$, $Q(a,-2)$(其中实常数$a<2$)的直线的斜率是\\blank{50}, 倾斜角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517873,7 +522422,9 @@ "id": "021913", "content": "若直线$l$的倾斜角的取值范围是$[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}]$, 则$l$的斜率的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517902,7 +522453,9 @@ "id": "021914", "content": "已知直线$l$的倾斜角的取值范围是$[0, \\arccos \\dfrac{3}{5}] \\cup[\\pi-\\arccos \\dfrac{3}{5}, \\pi)$, 则$l$的斜率的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517931,7 +522484,9 @@ "id": "021915", "content": "若直线$l$经过$P(0,0)$, $Q(\\cos \\dfrac{7 \\pi}{5}, \\sin \\dfrac{7 \\pi}{5})$, 则直线$l$的倾斜角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517960,7 +522515,9 @@ "id": "021916", "content": "若直线$l$经过$P(0,0)$, $Q(\\sin \\dfrac{7 \\pi}{5}, \\cos \\dfrac{7 \\pi}{5})$, 则直线$l$的倾斜角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -517989,7 +522546,9 @@ "id": "021917", "content": "已知斜率为$2$的直线过点$(2,2)$和$(x, 3)$, 则实数$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518018,7 +522577,9 @@ "id": "021918", "content": "经过点$P(2,0)$的直线$l$的倾斜角为$60^{\\circ}$. 若$l$绕$P$沿顺时针方向转过$90^{\\circ}$后所得直线$l'$的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518047,7 +522608,9 @@ "id": "021919", "content": "经过点$P(2,0)$的直线$l$的倾斜角为$30^{\\circ}$. 若$l$绕$P$沿顺时针方向转过$60^{\\circ}$后所得直线$l'$的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518076,7 +522639,9 @@ "id": "021920", "content": "已知$3$个不同点$A(2, a+1)$、$B(a+2,2 a+3)$、$C(-4,-a)$在同一条直线上, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518105,7 +522670,9 @@ "id": "021921", "content": "平面直角坐标系中有一个边长为$1$的正方形$OABC$, 其中点$O$为坐标原点, 点$A$、$C$分别在$x$轴和$y$轴上.\\\\\n(1) 若点$B$在第一象限, 求直线$OB$和$AC$的斜率;\\\\\n(2) 若点$B$不在第一象限, 求直线$AC$的斜率的所有可能值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518134,7 +522701,9 @@ "id": "021922", "content": "设一次函数$y=k x+b$($k \\neq 0$)的图像表示直线$l$.\\\\\n(1) 利用符号$\\arctan$, 试着用$k$表示$l$的倾斜角$\\alpha$;\\\\\n(2) 设$l$经过点$A(x_1, y_1)$、$B(x_2, y_2)$, 向量$\\overrightarrow {d}=(1, k)$, 求证: $\\overrightarrow {d}\\parallel \\overrightarrow{AB}$;\\\\\n(3) 设点$D$为$(0, b)$, 点$P$为$(x_0, y_0)$($x_0 \\neq 0$). 若直线$DP$的斜率为$k$, 求证: $y_0=k x_0+b$;\\\\\n(4) 设点$D$为$(0, b)$, 点$P$为$(x_0, y_0)$($x_0 \\neq 0$). 若$y_0=k x_0+b$, 求证: 直线$DP$的斜率为$k$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518163,7 +522732,9 @@ "id": "021923", "content": "在平面直角坐标系内, 非零向量$\\overrightarrow {d}=(a, b)$在直线$l$上, $a \\neq 0$.\\\\\n(1) 求证: $\\overrightarrow {d}$在$x$轴上投影为$(a, 0)$;\\\\\n(2) 求证: $l$的斜率为$\\dfrac{b}{a}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518192,7 +522763,9 @@ "id": "021924", "content": "已知直线$l$倾斜角$\\theta$的取值范围是$(\\dfrac{\\pi}{2}, \\dfrac{2 \\pi}{3})$, 则$l$的斜率的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518222,7 +522795,9 @@ "id": "021925", "content": "已知三个点$A(-1,-1)$、$B(1,1), C(0,3)$. 若点$M$是线段$BC$上任意一点, 则直线$AM$的斜率$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518252,7 +522827,9 @@ "id": "021926", "content": "设$\\theta \\in[\\pi, 2 \\pi)$. 若直线$l$经过$P(0,0)$, $Q(\\cos \\theta, \\sin \\theta)$, 则用$\\theta$表示$l$的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518282,7 +522859,9 @@ "id": "021927", "content": "已知直线$l$经过$P(0,0)$, $Q(\\cos \\theta, \\sin \\theta)$. 若$\\theta$的取值范围为$[-\\dfrac{\\pi}{3}, 0)$, 则$l$的倾斜角取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518312,7 +522891,9 @@ "id": "021928", "content": "设$\\theta \\in(\\pi, \\dfrac{3 \\pi}{2})$. 若直线$l$经过$P(0,0)$, $Q(\\sin \\theta, \\cos \\theta)$, 则用$\\theta$表示$l$的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518342,7 +522923,9 @@ "id": "021929", "content": "已知$A(0,1)$、$B(a, 2)$、$C(2 a, 4)$是某三角形的三个顶点, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518372,7 +522955,9 @@ "id": "021930", "content": "若直线$l$的倾斜角$\\alpha$满足$-2 \\leq \\tan \\alpha \\leq 1$, 则$\\alpha$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518402,7 +522987,9 @@ "id": "021931", "content": "已知两点$A(\\tan ^2 \\alpha, 0)$、$B(1,2 \\tan \\alpha)$.\\\\\n(1) 求证: $A$、$B$是两个不同的点, 并用$\\alpha$表示$|AB|$;\\\\\n(2) 若$0<\\alpha<\\dfrac{\\pi}{2}$, 求用$\\alpha$表示直线$AB$的倾斜角.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518432,7 +523019,9 @@ "id": "021932", "content": "已知$A(a+2, a)$、$B(1,-a)$、$C(a-4, a-1)$是某三角形的三个顶点, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518462,7 +523051,9 @@ "id": "021933", "content": "已知直线$l$经过点$P(1, a)$, 其中常数$a>0$, 且直线$l$与$x$轴、$y$轴分别交于$A$、$B$两个不同的点. 若$P$恰为$AB$中点, 求直线$l$的斜率和倾斜角\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518492,7 +523083,9 @@ "id": "021934", "content": "已知直线$l_1$经过点$P$, 斜率为$k_1$($k_1\\ne 1$). 若$l_1$绕着点$P$沿逆时针方向转过$45^\\circ$后与直线$l$重合, 则$l$的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518522,7 +523115,9 @@ "id": "021935", "content": "已知直线$l_2$经过点$P$, 斜率为$k_2$($k_2\\ne -\\sqrt{3}$). 若直线$l$绕着点$P$沿逆时针方向转过$30^\\circ$后与直线$l_2$重合, 则$l$的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518552,7 +523147,9 @@ "id": "021936", "content": "已知三个点$A(-1,-1)$、$B(1,1), C(0,3)$, 点$M$在直线$BC$上, 设直线$AM$的斜率为$k$. 求证: 当且仅当$k \\in[1,4]$时, 存在$\\lambda \\in[0,1]$, 使得$\\overrightarrow{BM}=\\lambda \\overrightarrow{BC}$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518582,7 +523179,9 @@ "id": "021937", "content": "设直线$l$在平面直角坐标系内的斜率是$k$, 非零向量$\\overrightarrow {d}$在$l$上, 试用$k$、$|\\overrightarrow {d}|$表示$\\overrightarrow {d}$在$x$轴上投影与数量投影.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518612,7 +523211,9 @@ "id": "021938", "content": "若直线$l$经过两点$A(1,-2)$、倾斜角为$\\dfrac{\\pi}{3}$, 则直线$l$的点斜式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518642,7 +523243,9 @@ "id": "021939", "content": "若直线$l$经过两点$A(1,-2)$、$B(3,1)$, 则直线$l$的两点式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518672,7 +523275,9 @@ "id": "021940", "content": "若直线$l$在$x$轴上的截距是$-2$, $y$轴上的截距是$5$, 则直线$l$的斜截式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518702,7 +523307,9 @@ "id": "021941", "content": "已知直线$l: y+1=\\dfrac{2}{3}(x-2)$, 则它在$x$轴上的截距是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518732,7 +523339,9 @@ "id": "021942", "content": "已知直线$l$的倾斜角为$\\alpha$, $\\sin \\alpha=\\dfrac{3}{5}$, 且经过点$P(3,5)$, 则直线$l$的点斜式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518762,7 +523371,9 @@ "id": "021943", "content": "过点$M(-2,3)$, 且垂直于$x$轴的直线的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518792,7 +523403,9 @@ "id": "021944", "content": "已知直线$l: y=k x+2$经过点$(1,-3)$, 则$l$的倾斜角的大小是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518822,7 +523435,9 @@ "id": "021945", "content": "若直线$x-2 y+c=0$与两坐标轴围成的三角形的面积不大于$3$, 则实数$c$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -518852,7 +523467,9 @@ "id": "021946", "content": "已知$\\triangle ABC$的两个顶点的坐标分别是$A(2,2)$、$B(3,0)$, 此三角形的重心坐标为$(3,1)$.\\\\\n(1) 求此三角形的三边所在直线的方程;\\\\\n(2) 求此三角形的三条中线所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518882,7 +523499,9 @@ "id": "021947", "content": "已知平行四边形$ABCD$中, 三个顶点的坐标分别为$A(1,2)$、$B(3,4)$、$C(2,6)$, 求$AD$与$CD$边所在的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518912,7 +523531,9 @@ "id": "021948", "content": "已知梯形$ABCD$的三个顶点的坐标分别为$A(2,3)$、$B(-2,1)$、$C(4,5)$, 求此梯形中位线所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518942,7 +523563,9 @@ "id": "021949", "content": "已知直线$l$过点$(1,2)$, 且$M(2,3)$、$N(4,-5)$两点到直线$l$的距离相等, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -518972,7 +523595,9 @@ "id": "021950", "content": "直线$l: y=k x+b$($k, b \\in \\mathbf{R}$)与线段$AB$相交, 其中$A(4,2)$, $B(1,5)$.\\\\\n(1) 当$k=1$时, 求$l$的取值范围;\\\\\n(2) 当$b=-1$时, 求$l$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519002,7 +523627,9 @@ "id": "021951", "content": "若直线$l$过点$(2,-3)$, 它的一个法向量为$\\overrightarrow {n}=(3,4)$, 则直线$l$的点法式方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519032,7 +523659,9 @@ "id": "021952", "content": "已知点$A(2,1)$、$B(5,3)$, 则$AB$的垂直平分线的点法式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519062,7 +523691,9 @@ "id": "021953", "content": "若直线$l: (2-m) x+m y+3=0$的纵截距为$3$, 则$m=$\\blank{50}; 若直线$l$的倾斜角为$\\dfrac{\\pi}{3}$, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519092,7 +523723,9 @@ "id": "021954", "content": "已知直线$l_1$与直线$l_2: 2 x-3 y+4=0$有相同法向量, 且直线$l_1$在$y$轴上的截距为$-2$, 则直线$l_1$的一般式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519122,7 +523755,9 @@ "id": "021955", "content": "求过点$M(1,-2)$, 且与两坐标轴围成等腰直角三角形的直线$l$的一般式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519152,7 +523787,9 @@ "id": "021956", "content": "直线$l: 2 x-y-4=0$围绕它与$x$轴的交点$M$逆时针方向旋转$45^{\\circ}$, 则得到的直线的一般式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519182,7 +523819,9 @@ "id": "021957", "content": "直线$l$的方程是$3 x-4 y+5=0$, 则$l$的一个法向量是\\bracket{20} .\n\\fourch{$(3,4)$}{$(-4,3)$}{$(3,-4)$}{$(4,3)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -519212,7 +523851,9 @@ "id": "021958", "content": "直线$x=-2$的一个法向量$\\overrightarrow {n}$的坐标是\\bracket{20}.\n\\fourch{$(4,0)$}{$(0,3)$}{$(-4,-2)$}{$(0,-2)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -519242,7 +523883,9 @@ "id": "021959", "content": "直线$x-a y+2=0(a<0)$的倾斜角是\\bracket{20}.\n\\fourch{$\\arctan \\dfrac{1}{a}$}{$-\\arctan \\dfrac{1}{a}$}{$\\pi-\\arctan \\dfrac{1}{a}$}{$\\pi+\\arctan \\dfrac{1}{a}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -519272,7 +523915,9 @@ "id": "021960", "content": "已知四边形$ABCD$是平行四边形, $AB$边所在直线的方程是$x+y-1=0$, $AD$边所在直线的方程是$3 x-y+4=0$, 顶点$C$的坐标是$(3,3)$, 求这个平行四边形其他两条边所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519302,7 +523947,9 @@ "id": "021961", "content": "已知$\\triangle ABC$的三个顶点的坐标分别$A(4,0)$、$B(6,7)$、$C(0,3)$, 求此三角形的三条高所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519332,7 +523979,9 @@ "id": "021962", "content": "求证: 直线$2 x+(1-\\cos 2 \\theta) y-\\sin \\theta=0$($\\theta \\in \\mathbf{R}$, 且$\\theta$不是$\\pi$的整数倍)与两坐标轴围成的图形面积是定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519362,7 +524011,9 @@ "id": "021963", "content": "已知直线$l_1: 3 k x-(k+2) y+6=0$, 直线$l_2: k x+(2 k-3) y+2=0$, 若这两条直线的倾斜角互补, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519392,7 +524043,9 @@ "id": "021964", "content": "已知直线$l: (a-1) x+(3-2 a) y+a+1=0$.\\\\\n(1) 若直线$l$的斜率$k \\in[-1,2]$, 求实数$a$的取值范围;\\\\\n(2) 证明: 对任意实数$a$, 直线$l$都经过一个确定的点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519422,7 +524075,9 @@ "id": "021965", "content": "已知向量$\\overrightarrow {n}=(5,-1)$是直线$l$的一个法向量, 在下列条件下分别求直线$l$的方程:\n(1) 在$x$轴、$y$轴上的截距之和为$4$;\\\\\n(2) 与$x$轴、$y$轴围成的三角形面积为$20$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519452,7 +524107,9 @@ "id": "021966", "content": "已知直线$l_1: a x-2 y-1=0$和直线$l_2: 6 x-4 y-b=0$. 若直线$l_1$与直线$l_2$平行, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519481,7 +524138,9 @@ "id": "021967", "content": "已知直线$l_1: m x+y-m-1=0$和直线$l_2: x+m y-2 m=0$. 当且仅当$m=$\\blank{50}时, $l_1$平行于$l_2$; 当且仅当$m=$\\blank{50}时, $l_1$与$l_2$重合, 当且仅当$m \\in$\\blank{50}时.$l_1$与$l_2$相交.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519510,7 +524169,9 @@ "id": "021968", "content": "已知过点$A(-2, m)$, $B(m, 4)$的直线$l_1$与直线$l_2: 2 x+y-1=0$平行, 则实数$m$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519539,7 +524200,9 @@ "id": "021969", "content": "``直线$l_1$与直线$l_2$平行''是``直线$l_1$与直线$l_2$的斜率相等''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -519568,7 +524231,9 @@ "id": "021970", "content": "直线$3 x-2 y+m=0$和直线$6 x-4 y+5=0$的位置关系为\\bracket{20}.\n\\fourch{平行}{平行或重合}{相交}{相交、平行和重合都有可能}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -519597,7 +524262,9 @@ "id": "021971", "content": "若直线$l_1: a x+2 y+6=0$与直线$l_2: x+(a-1) y+a^2-1=0$平行, 则$a$的值为\\bracket{20}.\n\\fourch{$-1$或$2$}{$-1$}{$2$}{$\\dfrac{2}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -519626,7 +524293,9 @@ "id": "021972", "content": "已知集合$A=\\{(x, y) | 2 x-a(a+1) y-1=0\\}$, $B=\\{(x, y) | a x-y+1=0\\}$, 且$A \\cap B=\\varnothing$, 求实数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519655,7 +524324,9 @@ "id": "021973", "content": "已知四边形$ABCD$的四个顶点分别为$A(-1,2)$、$B(3,4)$、$C(3,2)$、$D(1,1)$, 求证: 四边形$ABCD$是梯形.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519684,7 +524355,9 @@ "id": "021974", "content": "已知直线$l$经过直线$l_1: 2 x-5 y-1=0$和直线$l_2: x+4 y-7=0$的交点, 且直线$l$与线段$AB$的交点$P$, 满足$|AP|: |BP|=2: 3$, 其中点$A(4,-3)$、$B(-1,2)$, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519713,7 +524386,9 @@ "id": "021975", "content": "已知直线$l_1: y=a x+b$, 直线$l_2: y=b x-a$, 若$l_1$的倾斜角为$\\dfrac{3 \\pi}{4}$, 且与$l_2$的交点落在第二象限, 求实数$b$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519742,7 +524417,9 @@ "id": "021976", "content": "如图, 在一本打开的书的封面上有一只蚂蚁, 在封底有一小块饼干, 蚂蚁想爬过书脊到达饼干处. 若蚂蚁和饼干离书脊的距离分别是$4 \\text{cm}$和$3 \\text{cm}$, 书脊的长度是$20 \\text{cm}$, 求蚂蚁爬行的最短路线和最短距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,3) ++ (2,0.5) coordinate (T) (0,3) ++ (-2,0.5) coordinate (S);\n\\filldraw [gray!30] (S) --++ (0.2,0.2) -- (0,3.3) -- ($(T)+(-0.2,0.2)$) -- (T) -- (0,3);\n\\draw (0,0) coordinate (O) -- (0,3) coordinate (O1);\n\\draw (0,3) --++ (2,0.5) --++ (0,-2.7) coordinate (A) -- (0,0);\n\\draw (0,3) --++ (-2,0.5) coordinate (B) --++ (0,-2.7) -- (0,0);\n\\draw [dashed] ($(O)!0.3!(A)$) node [below] {蚂蚁} -- (0,0.7) -- ($(O1)!0.4!(B)$) node [above] {饼干};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519771,7 +524448,9 @@ "id": "021977", "content": "判断下列各组直线的位置关系, 若相交, 求出它们的夹角.\\\\\n(1) $l_1: 2 x-3 y-1=0$, $l_2: 4 x-6 y-2=0$;\\\\\n(2) $l_1: y=\\dfrac{1}{3} x+1$, $l_2: x-6 y-2=0$;\\\\\n(3) $l_1: x+2=0$, $l_2: 2 x-3 y+1=0$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519800,7 +524479,9 @@ "id": "021978", "content": "已知直线$l_1: (a-2) x+a y-2=0$与$l_2: (1-a) x+(a+1) y+1=0$垂直, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519829,7 +524510,9 @@ "id": "021979", "content": "``两条直线的斜率的乘积等于$-1$''是``这两条直线互相垂直''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -519858,7 +524541,9 @@ "id": "021980", "content": "若直线$l$过点$(3,4)$, 且与直线$x+2 y-1=0$的夹角为$\\arctan \\dfrac{1}{2}$, 则直线$l$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -519887,7 +524572,9 @@ "id": "021981", "content": "已知直线$l_1: \\sqrt{3} x+y=0$与直线$l_2: k x-y+1=0$. 若直线$l_1$和直线$l_2$的夹角为$60^{\\circ}$, 则$k$的值为\\bracket{20}.\n\\fourch{$\\sqrt{3}$或$0$}{$-\\sqrt{3}$或$0$}{$\\sqrt{3}$}{$-\\sqrt{3}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -519916,7 +524603,9 @@ "id": "021982", "content": "若直线$l_1$和$l_2$的斜率是方程$6 x^2+x-1=0$的两根, 则$l_1$与$l_2$的夹角等于\\bracket{20}.\n\\fourch{$15^{\\circ}$}{$30^{\\circ}$}{$45^{\\circ}$}{$60^{\\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -519945,7 +524634,9 @@ "id": "021983", "content": "已知等腰直角三角形$ABC$的斜边$AB$所在直线的方程为$3 x-y-5=0$, 直角顶点为$C(4,-1)$, 求两条直角边所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -519974,7 +524665,9 @@ "id": "021984", "content": "一直线$l$被两直线$4 x+y+6=0$和$3 x-5 y-6=0$截得的线段中点恰好是坐标原点, 则直线$l$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520003,7 +524696,9 @@ "id": "021985", "content": "在$\\triangle ABC$中, 已知点$A(3,-1)$和点$B(10,5)$, $\\angle B$的平分线所在直线$BD$的方程为$x-4 y+10=0$, 求边$BC$所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520032,7 +524727,9 @@ "id": "021986", "content": "一条光线从点$M(5,3)$射出, 被直线$l: x+y+1=0$反射, 入射光线与$l$的夹角为$\\arctan 2$. 求入射光线和反射光线分别所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520061,7 +524758,9 @@ "id": "021987", "content": "已知直线$l_1: 6 x+(t-1) y-8=0$, 直线$l_2: (t+4) x+(t+6) y-16=0$, 当且仅当$t=$时, $l_1$与$l_2$平行, 当且仅当$t=$时, $l_1$与$l_2$重合, 当且仅当$t \\in$\\blank{50}时, $l_1$与$l_2$相交.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520090,7 +524789,9 @@ "id": "021988", "content": "已知两条直线$l_1: m x+8 y+n=0$和$l_2: 2 x+m y-1=0$, 且它们的交点为$(m,-1)$, 则$(m, n)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520119,7 +524820,9 @@ "id": "021989", "content": "若$a$、$b$、$c$分别为$\\triangle ABC$中$\\angle A$、$\\angle B$、$\\angle C$所对边的长, 则直线$l: x \\sin A-a y+2 c=0$与直线$m: b x+y \\sin B+2 \\sin C=0$的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520148,7 +524851,9 @@ "id": "021990", "content": "已知三条直线$l_1: 2 x+1=0$, 直线$l_2: m x+y=0$, 直线$l_3: x+m y-1=0$, 若这三条直线中有且仅有两条直线平行, 则实数$m$所有可能的值的个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -520177,7 +524882,9 @@ "id": "021991", "content": "设$f(x, y)=a x+b y+c$, 方程$f(x, y)=0$表示定直线$l, M(x_0, y_0)$为不在直线$l$上的定点, 则方程$f(x, y)-f(x_0, y_0)=0$一定是\\bracket{20}.\n\\twoch{经过点$M$且与直线$l$斜交的直线}{经过点$M$且与直线$l$平行的直线}{经过点$M$且与直线$l$垂直的直线}{不经过点$M$的直线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -520206,7 +524913,9 @@ "id": "021992", "content": "分别求经过直线$l_1: 5 x+2 y-3=0$和$l_2: 3 x-5 y-8=0$的交点, 且与直线$x+4 y-7=0$垂直或平行时的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520235,7 +524944,9 @@ "id": "021993", "content": "与一条直线平行的非零向量称为它的方向向量.\\\\\n(1) 写出直线$a x+b y+c=0$($a^2+b^2 \\neq 0$)的一个方向向量;\\\\\n(2) 用直线的方向向量推导两直线夹角的余弦公式.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520264,7 +524975,9 @@ "id": "021994", "content": "已知$a, b \\in \\mathbf{R}$, 且$2 a+4 b=5$. 当$a, b$变化时, 直线$a x+b y-1=0$是否一定过平面上的一个定点? 若是, 求出这个定点的坐标; 若否, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520293,7 +525006,9 @@ "id": "021995", "content": "给定直线$l_1: 2 x+y-4=0$和$l_2: x-y+2=0$, $\\lambda$是任意实数, 求证: 无论$\\lambda$取何值, 直线$l: 2 x+y-4+\\lambda(x-y+2)=0$一定经过平面上的定点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520322,7 +525037,9 @@ "id": "021996", "content": "已知定点$A(0,3)$, 动点$B$在直线$l_1: y=1$上移动, 动点$C$在直线$l_2: y=-1$上移动, 且$\\angle BAC=90^{\\circ}$, 求$\\triangle ABC$的面积的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520351,7 +525068,9 @@ "id": "021997", "content": "一束光线经过点$(-2,1)$, 由直线$l: y=x$反射后经过点$(3,5)$射出, 求反射光线所在的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520380,7 +525099,9 @@ "id": "021998", "content": "点$P(3,2)$到直线$l: 3 x-2 y=13$的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520410,7 +525131,9 @@ "id": "021999", "content": "直线$3 x-y+4=0$与直线$6 x-2 y-1=0$的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520440,7 +525163,9 @@ "id": "022000", "content": "与直线$6 x-8 y+3=0$垂直、且与原点距离等于$1$的直线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520470,7 +525195,9 @@ "id": "022001", "content": "平行于直线$x-y-2=0$、且与它的距离等于$\\sqrt{2}$的直线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520500,7 +525227,9 @@ "id": "022002", "content": "若点$(1, \\cos \\theta)$到直线$x \\sin \\theta+y \\cos \\theta=1$($0 \\leq \\theta \\leq \\dfrac{\\pi}{2}$)的距离等于$\\dfrac{1}{4}$, 则$\\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520530,7 +525259,9 @@ "id": "022003", "content": "若实数$x$、$y$满足关系式$4 x+3 y-12=0$, 则$\\sqrt{(x-2)^2+(y+3)^2}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520560,7 +525291,9 @@ "id": "022004", "content": "若点$A(-1,6)$与点$B(13,6)$到直线$l$的距离都等于$7$, 则直线$l$的不同位置有\\bracket{20}.\n\\fourch{$1$种}{$2$种}{$3$种}{$4$种}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -520590,7 +525323,9 @@ "id": "022005", "content": "``$a=b$''是``点$(a, b)$到直线$y=x+2$的距离是$\\sqrt{2}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -520620,7 +525355,9 @@ "id": "022006", "content": "已知正方形$ABCD$的中心的坐标为点$P(1,1), AB$边所在直线的方程为$x+2 y+3=0$. 求这个正方形的其他三边所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520650,7 +525387,9 @@ "id": "022007", "content": "已知直线$l_1: 2 x-y+a=0$与直线$l_2: -4 x+2 y+1=0$, 且直线$l_1$与直线$l_2$的距离为$\\dfrac{7 \\sqrt{5}}{10}$, 求实数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520680,7 +525419,9 @@ "id": "022008", "content": "证明: 点$A(-2,2)$到直线$(m+2) x-(m+1) y-2(2 m+3)=0$($m \\in \\mathbf{R}$)的距离$d$恒小于$4 \\sqrt{2}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520710,7 +525451,9 @@ "id": "022009", "content": "已知点$A(-4,5)$、$B(2,1)$试在$x$轴上求一点$M$, 使得$|MA|+|MB|$最小.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520740,7 +525483,9 @@ "id": "022010", "content": "已知直线$l$经过点$P(1,2)$, 且被两平行直线$l_1: 4 x+3 y+1=0$与$l_2: 4 x+3 y+6=0$截得的线段长$|AB|=\\sqrt{2}$, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -520770,7 +525515,9 @@ "id": "022011", "content": "已知两相互平行的直线分别过点$A(6,2)$与$B(3,-1)$. 它们以相同的角速度旋转, 在旋转过程中, 则这两条平行直线间的距离$d$的取值范围是 , 当$d$取到最大值时, 过点$A$直线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520799,7 +525546,9 @@ "id": "022012", "content": "已知$A(2,3)$、$B(-4,8)$两点, 直线$l$经过原点, 且$A$、$B$两点到直线$l$的距离相等, 则直线$l$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520828,7 +525577,9 @@ "id": "022013", "content": "已知平行直线$l_1$与$l_2$的距离为$\\sqrt{5}$, 且直线$l_1$经过原点, 直线$l_2$经过点$(1,3)$, 则直线$l_2$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520857,7 +525608,9 @@ "id": "022014", "content": "已知直线$l$经过点$P(0,-1)$, 且它与以$A(3,2)$、$B(2,-3)$为端点的线段$AB$有交点, 求直线$l$斜率的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520886,7 +525639,9 @@ "id": "022015", "content": "点$P(-2,-1)$关于点$Q(3,5)$的对称点是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520915,7 +525670,9 @@ "id": "022016", "content": "点$P(-2,-1)$关于直线$x+2 y-2=0$的对称点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520944,7 +525701,9 @@ "id": "022017", "content": "直线$l: x+2 y-11=0$关于点$(-1,1)$对称的直线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -520973,7 +525732,9 @@ "id": "022018", "content": "直线$y=2 x-3$关于$x$轴对称的直线方程为\\blank{50}, 直线$y=2 x-3$关于直线$y=x$对称的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -521002,7 +525763,9 @@ "id": "022019", "content": "点到直线的距离是该点到直线上任意一点距离的最小值. 如果把一个给定点到线段上任意一点的距离的最小值定义为该点到该线段的距离. 试求点$P(1,1)$到线段$l: x-y-3=0$($3 \\leq x \\leq 5$)的距离.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521031,7 +525794,9 @@ "id": "022020", "content": "已知直线$l: 2 x-3 y+1=0$.\\\\\n(1) 求与直线$l$关于$x$轴对称的直线的方程;\\\\\n(2) 求与直线$l$关于$y$轴对称的直线的方程;\\\\\n(3) 求与直线$l$关于原点对称的直线的方程;\\\\\n(4)求与直线$l$关于$y=x$对称的直线的方程;\\\\\n(5) 求与直线$l$关于$y=-x$对称的直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521060,7 +525825,9 @@ "id": "022021", "content": "如图, $\\angle BAC$为伸入江中的半岛, $AB$和$AC$为两江岸, $M$处为水文站, $N$处为电讯局, 现欲在两江岸$AB$和$AC$上各建一个水文观测点$P$、$Q$. 现测得$\\angle BAC=45^{\\circ}$, 当直角坐标系以点$A$为坐标原点且以直线$BA$为$x$轴时, 测得$M(-4,1)$、$N(-3,2)$. $P$、$Q$两点应建在何处才能使路程$MPQN$最短?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-6,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$A$} node [below right] {($O$)};\n\\draw (-6,0) node [below] {$B$} coordinate (B);\n\\draw (0,0) -- (-6,6) node [left] {$C$} coordinate (C);\n\\draw (-4,1) node [left] {$M$} coordinate (M);\n\\draw (-3,2) node [left] {$N$} coordinate (N);\n\\draw (-3,0) node [below] {$P$} coordinate (P);\n\\draw (-2,2) node [above right] {$Q$} coordinate (Q);\n\\draw (M)--(P)--(Q)--(N);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521089,7 +525856,9 @@ "id": "022022", "content": "借助函数图像, 判断下列导数的正负:(用铅笔作图)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}\n\\end{center}\n(1) $f'(-\\dfrac{\\pi}{4})$, 其中$f(x)=\\cos x$;\\\\\n(2) $f'(3)$, 其中$f(x)=\\ln x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521109,7 +525878,9 @@ "id": "022023", "content": "过曲线$y=x^2$上某点$P$的切线满足下列条件, 分别求出$P$点.\\\\\n(1) 平行于直线$y=4 x-5$;\\\\\n(2) 垂直于直线$2 x-6 y+5=0$;\\\\ \n(3) 切线的倾斜角为$135^{\\circ}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521129,7 +525900,9 @@ "id": "022024", "content": "证明: 函数$f(x)=\\ln x$与函数$g(x)=e^x$没有驻点.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521149,7 +525922,9 @@ "id": "022025", "content": "求余弦函数$y=\\cos x$的所有驻点.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521169,7 +525944,9 @@ "id": "022026", "content": "设$f_0(x)=\\sin x , f_1(x)=f_0'(x) , f_2(x)=f_1'(x), \\ldots, f_{n+1}(x)=f_n'(x), n \\in \\mathbf{N}$, 则$f_{2022}(x)=()$\\bracket{20}.\n\\fourch{$\\sin x$}{$-\\sin x$}{$\\cos x$}{$-\\cos x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -521189,7 +525966,9 @@ "id": "022027", "content": "求下列函数的导数:\\\\\n(1) $y=\\sqrt{x}-\\ln x$;\\\\\n(2) $y=(x^2+1)(x-1)$;\\\\\n(3) $y=x^2 \\mathrm{e}^x$;\\\\\n(4) $y=\\sqrt{x} \\sin x$;\\\\\n(5) $y=x \\ln x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521209,7 +525988,9 @@ "id": "022028", "content": "求下列函数的导数:\\\\\n(1) $y=\\dfrac{\\sin x}{x}$;\\\\\n(2) $y=\\dfrac{x^2}{\\ln x}$;\\\\\n(3) $y=\\lg x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521229,7 +526010,9 @@ "id": "022029", "content": "曲线$y=\\mathrm{e}^{-2 x}+1$在点$(0,2)$处的切线与直线$y=0$和$y=x$围成的三角形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -521249,7 +526032,9 @@ "id": "022030", "content": "求下列函数的导数.\\\\\n(1) $y=(5 x-3)^4$;\\\\\n(2) $y=(3 x+2)^5$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521269,7 +526054,9 @@ "id": "022031", "content": "求下列函数的导数.\\\\\n(1) $y=\\dfrac{1}{(1-3 x)^4}$;\\\\\n(2) $y=\\sqrt[4]{\\dfrac{1}{3 x+1}}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521289,7 +526076,9 @@ "id": "022032", "content": "求下列函数的导数.\\\\\n(1) $y=\\sin (3 x-\\dfrac{\\pi}{6})$;\\\\\n(2) $y=\\cos ^2 x-\\sin ^2 x$;\\\\\n(3) $y=\\ln \\sqrt{1+2 x}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521309,7 +526098,9 @@ "id": "022033", "content": "求下列曲线在点$P$处的切线方程.\\\\\n(1) $y=\\sin 2 x$, $P(\\dfrac{\\pi}{3}, \\dfrac{\\sqrt{3}}{2})$;\\\\\n(2) $y=2^{1-3 x}$, $P(0,2)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521329,7 +526120,9 @@ "id": "022034", "content": "求下列函数的导数.\\\\\n(1) $y=\\mathrm{e}^{2 x} \\sin 3 x$;\\\\\n(2) $y=\\ln \\sqrt{\\dfrac{1+x}{1-x}}$;\\\\\n(3) $y=\\dfrac{x^2}{(2 x+1)^3}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521349,7 +526142,9 @@ "id": "022035", "content": "利用导数研究下列函数的单调性, 并说明所得结果与你之前的认识是否一致.\\\\\n(1) $y=2^x$;\\\\\n(2) $y=x-\\dfrac{1}{x}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521369,7 +526164,9 @@ "id": "022036", "content": "利用导数研究下列函数的单调性.\\\\\n(1) $y=x+\\dfrac{1}{x}$;\\\\\n(2) $y=x+\\dfrac{1}{x^2}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521389,7 +526186,9 @@ "id": "022037", "content": "研究函数$y=x^3-9 x^2+24 x+1$的单调性.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521409,7 +526208,9 @@ "id": "022038", "content": "已知$f(x)=x^3-a x-1$.\\\\\n(1) 若函数$y=f(x)$在$\\mathbf{R}$上为严格增函数, 求实数$a$的取值范围;\\\\\n(2) 若函数$y=f(x)$在$(-1,1)$上为严格减函数, 求实数$a$的取值范围;\\\\\n(3) 若函数$y=f(x)$的单调递减区间为$[-1,1]$, 求实数$a$的值;\\\\\n(4) 若函数$y=f(x)$在区间$(-1,1)$上不单调, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521429,7 +526230,9 @@ "id": "022039", "content": "研究下列函数的单调性.\\\\\n(1) $y=\\dfrac{\\ln (1+x)}{x}$($x>0$);\\\\\n(2) $y=\\dfrac{\\sin x}{x}$($0=latex]\n\\draw (0,0,0) node [below] {$D_1$} coordinate (D_1);\n\\draw (2,0,0) node [below] {$A_1$} coordinate (A_1);\n\\draw (0.5,0,-2) node [below] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)+(C_1)-(D_1)$) node [right] {$B_1$} coordinate (B_1);\n\\draw (1,2,0) node [left] {$D$} coordinate (D);\n\\draw ($(D)+(A_1)-(D_1)$) node [above] {$A$} coordinate (A);\n\\draw ($(A)+(C_1)-(D_1)$) node [above] {$B$} coordinate (B);\n\\draw ($(D)+(B)-(A)$) node [above] {$C$} coordinate (C);\n\\draw ($(D_1)!0.5!(A_1)$) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(D)$) node [above left] {$N$} coordinate (N);\n\\draw (D_1)--(A_1)--(B_1)--(B)--(C)--(D)--cycle(A)--(B)(A)--(A_1)(A)--(D);\n\\draw [dashed] (C_1)--(C)(C_1)--(B_1)(C_1)--(D_1)(M)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 写出以该平行六面体的顶点为起点与终点, 且与$\\overrightarrow{AB}$相等的向量;\\\\\n(2) 写出以该平行六面体的顶点为起点与终点的$\\overrightarrow{AA_1}$的负向量;\\\\\n(3) 写出以该平行六面体的顶点为起点与终点, 且与$\\overrightarrow{AD}$平行的向量;\\\\\n(4) 设$M$、$N$分别是$A_1D_1$和$DC$的中点, 用$\\overrightarrow{AB}$、$\\overrightarrow{AA_1}$、$\\overrightarrow{AD}$表示向量$\\overrightarrow{MN}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521637,7 +526458,9 @@ "id": "022049", "content": "对于平行六面体$ABCD-A_1B_1C_1D_1$, 求证: $\\overrightarrow{AB_1}+\\overrightarrow{AC}+\\overrightarrow{AD_1}=2 \\overrightarrow{AC_1}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521665,7 +526488,9 @@ "id": "022050", "content": "在三棱锥$O-ABC$中, $G$是三角形$ABC$的重心, 用向量$\\overrightarrow{OA}$、$\\overrightarrow{OB}$、$\\overrightarrow{OC}$表示向量$\\overrightarrow{OG}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521693,7 +526518,9 @@ "id": "022051", "content": "已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$两两垂直, 且$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=2$, $|\\overrightarrow {c}|=3$, $\\overrightarrow {m}=\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}$.\\\\\n(1) 求$|\\overrightarrow {m}|$;\\\\\n(2) 分别求$\\overrightarrow {m}$与$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$的夹角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521721,7 +526548,9 @@ "id": "022052", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $P$、$Q$分别是$A_1B_1$、$CD$的中点, $R$、$S$分别是棱$AA_1$、棱$CC_1$上的点, 且$AR=2RA_1$, $C_1S=2SC$, 求证: $PS\\parallel RQ$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521749,7 +526578,9 @@ "id": "022053", "content": "已知正方体$ABCD-A_1B_1C_1D_1$的边长为$1$. 求:\\\\\n(1) $\\overrightarrow{AC} \\cdot \\overrightarrow{AA_1}$;\\\\\n(2) $\\overrightarrow{AC} \\cdot \\overrightarrow{A_1C_1}$;\\\\\n(3) $\\overrightarrow{AC} \\cdot \\overrightarrow{AC_1}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521779,7 +526610,9 @@ "id": "022054", "content": "在长方体$ABCD-A' B' C' D'$中, $A' C'$和$B' D'$相交于$O'$, 求证$DO'\\parallel$平面$ACB'$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521809,7 +526642,9 @@ "id": "022055", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $G$是三角形$ACD_1$的重心. 求证: $3 \\overrightarrow{DG}=\\overrightarrow{DB_1}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521839,7 +526674,9 @@ "id": "022056", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 已知$AB=6$, $AD=2$, $AA_1=1$, $P$是棱$AB$上的点且$PB=2AP$, $M$是棱$DC$上的点, 且$DM=2MC$, $N$是$B_1C_1$的中点, 求直线$PD_1$与$MN$所成的角$\\theta$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521869,7 +526706,9 @@ "id": "022057", "content": "已知棱长为$1$的正四面体$A-BCD$中, $E$、$F$分别在$AB$、$CD$上, 且$\\overrightarrow{AE}=\\dfrac{1}{4} \\overrightarrow{AB}$, \n$\\overrightarrow{CF}=\\dfrac{1}{3} \\overrightarrow{CD}$.\\\\\n(1) 求直线$DE$和$BF$所成的角的大小;\\\\\n(2) 求$|\\overrightarrow{EF}|$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521899,7 +526738,9 @@ "id": "022058", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的高为$h$, 上、下底面是边长为$a$的正方形, 坐标原点$O$设在下底面的中心, $x$轴、$y$轴分别与下底面的对角线重合, $z$轴垂直于底面(如图). 写出下列点的坐标以及向量的坐标:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) $A$的坐标: \\blank{50}; (2) $D$的坐标: \\blank{50}; (3) $B_1$的坐标: \\blank{50};\\\\\n(4) $\\overrightarrow{OA}$的坐标: \\blank{50}; (5) $\\overrightarrow{D_1A_1}$的坐标: \\blank{50}; (6) $\\overrightarrow{B_1D}$的坐标: \\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -521927,7 +526768,9 @@ "id": "022059", "content": "已知$\\overrightarrow {a}=(1,-5,4)$, $\\overrightarrow {b}=(2,1,7)$.\\\\\n(1) 求$3 \\overrightarrow {a}+2 \\overrightarrow {b}$的坐标;\\\\\n(2) 求$|\\overrightarrow {a}+\\overrightarrow {b}|$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521955,7 +526798,9 @@ "id": "022060", "content": "已知$\\overrightarrow {a}=(2,1,-2)$, $\\overrightarrow {b}=(5,-4,3)$, $\\overrightarrow {c}=(-8,4,1)$.\\\\\n(1) 求证: $\\overrightarrow {a} \\perp \\overrightarrow {b}$;;\n(2) 设$\\overrightarrow {a}$与$\\overrightarrow {c}$的夹角为$\\theta$, 求$\\cos \\theta$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -521983,7 +526828,9 @@ "id": "022061", "content": "已知$P_1(2,5,4)$, $P_2(6,4,7)$, 设$\\overrightarrow {a}=\\overrightarrow{P_1P_2}$, 求$\\overrightarrow {a}$、$-\\overrightarrow {a}$和单位向量$\\overrightarrow{a_0}$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522011,7 +526858,9 @@ "id": "022062", "content": "已知$P_1(2,5,-6)$, 在$y$轴上求一点$P_2$, 使$|P_1P_2|=7$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522039,7 +526888,9 @@ "id": "022063", "content": "已知$P_1(1,2,3), P_2(5,4,7)$, 在$y$轴上求一点$Q$, 使$|P_1Q|=|P_2Q|$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522067,7 +526918,9 @@ "id": "022064", "content": "已知向量$\\overrightarrow {a}=(1,-3,2)$, $\\overrightarrow {b}=(2,0,-8)$, 求单位向量$\\overrightarrow {c}$, 使$\\overrightarrow {c}$与向量$\\overrightarrow {a}$、$\\overrightarrow {b}$都垂直.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522095,7 +526948,9 @@ "id": "022065", "content": "已知平面$\\alpha$经过点$A(3,1,-1)$、$B(1,-1,0)$, 且平行于向量$\\overrightarrow {a}=(-1,0,2)$, 求平面$\\alpha$的一个法向量.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522123,7 +526978,9 @@ "id": "022066", "content": "已知点$A$、$B$、$C$的坐标分别为$(x_1, y_1, z_1)$、$(x_2, y_2, z_2)$、$(x_3, y_3, z_3)$, $G$是$\\triangle ABC$的重心, 求点$G$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522151,7 +527008,9 @@ "id": "022067", "content": "已知正方体$ABCD-A_1B_1C_1D_1$, 求证: $BD_1 \\perp C_1D$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522179,7 +527038,9 @@ "id": "022068", "content": "正三棱柱$ABC-A_1B_1C_1$中, $AB=2AA_1=\\dfrac{\\sqrt{6}}{2}$.\\\\\n(1) $P$点在棱$A_1B_1$上什么位置时, 异面直线$AP$与$A_1C$互相垂直?\\\\\n(2) $P$点在棱$A_1B_1$上什么位置时, 直线$AP$与平面$A_1BC$成$30^{\\circ}$角?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522207,7 +527068,9 @@ "id": "022069", "content": "如图, 平面$ABEF \\perp$平面$ABCD$, 四边形$ABEF$与$ABCD$都是直角梯形, \n$\\angle BAD=\\angle FAB=90^{\\circ}$, $BC =\\dfrac 12 AD$且$BC\\parallel AD$, $BE = \\dfrac{1}{2} AF$且$BE\\parallel AF$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,3,0) node [above] {$F$} coordinate (F);\n\\draw (0,0,1.5) node [below] {$B$} coordinate (B);\n\\draw (B) ++ (1.5,0,0) node [below] {$C$} coordinate (C);\n\\draw (B) ++ (0,1.5,0) node [left] {$E$} coordinate (E);\n\\draw (B)--(C)--(D)--(F)--(E)--cycle(E)--(C);\n\\draw [dashed] (A)--(D)(A)--(F)(A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $C, D, F, E$四点共面;\\\\\n(2) 设$AB=BC=BE$, 求二面角$A-ED-B$的大小;", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522235,7 +527098,9 @@ "id": "022070", "content": "如图, 已知四棱锥$P-ABCD$的底面$ABCD$为等腰梯形, $AB\\parallel DC$, $AC \\perp BD$, $AC$与$BD$相交于点$O$, 且顶点$P$在底面上的射影恰为$O$点, 又$BO=2$, $PO=\\sqrt{2}$, $PB \\perp PD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$O$} coordinate (O);\n\\draw (-2,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,2) node [right] {$B$} coordinate (B);\n\\draw ($(O)!-0.5!(B)$) node [left] {$D$} coordinate (D);\n\\draw ($(O)!-0.5!(A)$) node [right] {$C$} coordinate (C);\n\\draw (O) ++ (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle(B)--(P);\n\\draw [dashed] (P)--(O)(P)--(D)--(B)(A)--(C)(A)--(D)--(C);\n\\draw (O) pic [draw, scale = 0.3] {right angle = B--O--A};\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$PD$与$BC$所成角的余弦值;\\\\\n(2) 求二面角$P-AB-C$的大小;\\\\\n(3) 设点$M$在棱$PC$上, 且$\\dfrac{PM}{MC}=\\lambda$, 问$\\lambda$为何值时, $PC \\perp$平面$BMD$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522263,7 +527128,9 @@ "id": "022071", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系, 求下列直线的一个方向向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) $AD_1$;\\\\\n(2) $AA_1$;\\\\\n(3) $AC_1$;\\\\\n(4) $AB_1$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522289,7 +527156,9 @@ "id": "022072", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系. 下列向量是图中哪些经过两个顶点的直线的一个方向向量?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) $\\overrightarrow {a}=(1,0,0)$;\\\\\n(2) $\\overrightarrow {b}=(0,1,0)$;\\\\\n(3) $\\overrightarrow {c}=(3 \\sqrt{2}, 0,4)$;\\\\\n(4) $\\overrightarrow {d}=(0,3 \\sqrt{2}, 8)$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522315,7 +527184,9 @@ "id": "022073", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系, 求下列平面的一个法向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) 平面$AA_1D_1D$;\\\\\n(2) 平面$BB_1D_1D$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522341,7 +527212,9 @@ "id": "022074", "content": "已知点$A(0,-7,0)$、$B(2,-1,1)$、$C(2,2,2)$, 求平面$ABC$的一个法向量.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522367,7 +527240,9 @@ "id": "022075", "content": "已知点$S(0,6,4)$、$A(3,5,3)$、$B(-2,11,-5)$、$C(1,-1,4)$, 求点$S$到平面$ABC$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522393,7 +527268,9 @@ "id": "022076", "content": "已知平面$\\alpha$的一个法向量$\\overrightarrow {n}=(3,-2,6)$, 且经过点$A(0,7,0)$, 求原点到平面$\\alpha$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522419,7 +527296,9 @@ "id": "022077", "content": "已知三棱锥$A-BCD$的三条侧棱$AB$、$AC$、$AD$两两垂直, 且$AB=1$, $AC=2$, $AD=3$, 求顶点$A$到平面$BCD$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522445,7 +527324,9 @@ "id": "022078", "content": "正三棱柱$ABC-A_1B_1C_1$中, $AB=2AA_1=\\dfrac{\\sqrt{6}}{2}$.\\\\\n(1) $P$点在棱$A_1B_1$上什么位置时, 异面直线$AP$与$A_1C$互相垂直?\\\\\n(2) $P$点在棱$A_1B_1$上什么位置时, 直线$AP$与平面$A_1BC$成$30^{\\circ}$角?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522471,7 +527352,9 @@ "id": "022079", "content": "已知正方体$ABCD-A_1B_1C_1D_1$, 求二面角$B-AC-D_1$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522503,7 +527386,9 @@ "id": "022080", "content": "已知$ABCD-A_1B_1C_1D_1$为正方体.\\\\\n(1) 求直线$AC$与$B_1D$所成的角的大小;\\\\\n(2) 求直线$B_1D$与平面$ACD_1$所成的角的大小;\\\\\n(3) 求平面$ACD_1$与平面$B_1CD_1$所成的二面角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522535,7 +527420,9 @@ "id": "022081", "content": "已知正三棱锥的底面边长和高都为$a$. 求侧面与底面所成的二面角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522567,7 +527454,9 @@ "id": "022082", "content": "在三棱锥$P-ABC$中, 已知底面$ABC$是以$C$为直角的直角三角形, $PC \\perp$平面$ABC$, $AC=18$, $PC=6$, $BC=9$, $G$是$\\triangle PAB$的重心, $M$是棱$AC$的中点, 求直线$CG$与直线$BM$所成的角$\\theta$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522599,7 +527488,9 @@ "id": "022083", "content": "已知矩形$ABCD$, 且$PD \\perp$平面$ABCD$, 若$PB=2$, $PB$与平面$PCD$所成的角为$45^{\\circ}$. $PB$与平面$ABD$所成的角为$30^{\\circ}$, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.3]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(B)--(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) $CD$的长;\\\\\n(2) 求$PB$与$CD$所成的角;\\\\\n(3) 求二面角$C-PB-D$的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522631,7 +527522,9 @@ "id": "022084", "content": "用二项式定理展开下列两式:\\\\\n(1) $(a+2 b)^6$;\\\\\n(2) $(1-\\dfrac{1}{x})^5$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522661,7 +527554,9 @@ "id": "022085", "content": "化简:\\\\ \n(1) $(1+\\sqrt{x})^5+(1-\\sqrt{x})^5$;\\\\\n(2) $(2 x+y)^4-(2 x-y)^4$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522691,7 +527586,9 @@ "id": "022086", "content": "分别写出$(x-1)^{15}$的二项展开式中的前$4$项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522721,7 +527618,9 @@ "id": "022087", "content": "求$(2 a^3-3 b^2)^{10}$的二项展开式中的第$8$项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522751,7 +527650,9 @@ "id": "022088", "content": "$(x-1)^n$的二项展开式中第$m$项($1 \\leq m \\leq n$且$m$、$n \\in \\mathbf{N}$)的二项式的系数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_n^{m-1}$}{$(-1)^{m-1} \\mathrm{C}_n^m$}{$\\mathrm{C}_n^m$}{$(-1)^m \\mathrm{C}_n^m$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -522781,7 +527682,9 @@ "id": "022089", "content": "求$(3 x^3-\\dfrac{1}{3 x^3})^{10}$的二项展开式中的常数项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522811,7 +527714,9 @@ "id": "022090", "content": "已知$(1+x)^n$的二项展开式中第$4$项与第$8$项的系数相等, 求这两项的系数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522841,7 +527746,9 @@ "id": "022091", "content": "在$(\\sqrt[3]{x}-\\dfrac{2}{\\sqrt{x}})^{11}$的二项展开式中,\\\\\n(1) 求含$x^2$项的二项式系数;\\\\\n(2) 含$x^{\\frac{1}{3}}$的项是第几项? 并写出这项的系数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522871,7 +527778,9 @@ "id": "022092", "content": "已知$(x \\sin \\theta+1)^6$的二项展开式$x^2$项的系数与$(x-\\dfrac{15}{2} \\cos \\theta)^4$的二项展开式中$x^3$项的系数相等, 求$\\cos \\theta$的值.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522901,7 +527810,9 @@ "id": "022093", "content": "求证: 当$n$为正整数时, $2^n-\\mathrm{C}_n^1 \\cdot 2^{n-1}+\\mathrm{C}_n^2 \\cdot 2^{n-2}+\\cdots+\\mathrm{C}_n^{n-1} \\cdot 2+(-1)^n=1$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522931,7 +527842,9 @@ "id": "022094", "content": "求$(1+2 x)^3(1-x)^4$展开式中$x^6$的系数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -522961,7 +527874,9 @@ "id": "022095", "content": "在$(3 x-2 y)^9$的展开式中, 二项式系数的和是\\blank{50}, 各项系数的和是各项系数的绝对值之和是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -522989,7 +527904,9 @@ "id": "022096", "content": "$\\mathrm{C}_n^1+3\\mathrm{C}_n^2+9\\mathrm{C}_n^3+\\cdots+3^{n-1} \\mathrm{C}_n^n$等于\\bracket{20}.\n\\fourch{$4^n$}{$\\dfrac{4^n}{3}$}{$\\dfrac{4^n}{3}-1$}{$\\dfrac{4^n-1}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -523017,7 +527934,9 @@ "id": "022097", "content": "求$(\\dfrac{\\sqrt{x}}{2}-\\dfrac{2}{\\sqrt{x}})^{10}$的二项展开式的正中间一项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -523045,7 +527964,9 @@ "id": "022098", "content": "求$(x \\sqrt{y}-y \\sqrt{x})^{11}$的二项展开式的正中间两项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -523065,7 +527986,9 @@ "id": "022099", "content": "用二项式定理证明: $99^{10}-1$能被$1000$整除.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -523093,7 +528016,9 @@ "id": "022100", "content": "求$77^{77}-15$除以$19$的余数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -523121,7 +528046,9 @@ "id": "022101", "content": "求$(1+2 x+x^2)^{10}(1-x)^6$的展开式中各项系数之和.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -523149,7 +528076,9 @@ "id": "022102", "content": "在$(x^2-\\dfrac{3}{x})^n$的二项展开式中, 有且只有第五项的二项式系数最大, 求:\n$\\mathrm{C}_n^0-\\dfrac{1}{2} \\mathrm{C}_n^1+\\dfrac{1}{4} \\mathrm{C}_n^2-\\cdots+(-1)^n \\cdot \\dfrac{1}{2^n} \\mathrm{C}_n^n$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -523177,7 +528106,9 @@ "id": "022103", "content": "在$(1+3 x)^n$的二项展开式中, 末三项的二项式系数之和等于$631$.\\\\\n(1) 求二项展开式中二项式系数最大的项;\\\\\n(2) 求二项展开式中系数最大的项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -523205,7 +528136,9 @@ "id": "022104", "content": "已知$(x+1)^n=x^n+\\cdots+a x^3+b x^2+c x+1$($n \\geq 1$, $n \\in \\mathrm{N}$), 且$a: b=3: 1$, 求$c$的值.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -523233,7 +528166,9 @@ "id": "022105", "content": "已知$n$为大于$1$的自然数, 用二项式定理证明: $(1+\\dfrac{1}{n})^n>2$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -523261,7 +528196,9 @@ "id": "022106", "content": "半径为$2$, 弧长为$2$的扇形的圆心角为\\blank{50}弧度.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -523294,7 +528231,9 @@ "id": "022107", "content": "函数$y=\\tan x$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi$", "solution": "", @@ -523327,7 +528266,9 @@ "id": "022108", "content": "向量$\\overrightarrow {b}=(3,4)$的单位向量$\\overrightarrow{b_0}$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(\\dfrac{3}{5}, \\dfrac{4}{5})$", "solution": "", @@ -523360,7 +528301,9 @@ "id": "022109", "content": "若角$\\alpha$的终边过点$P(4,-3)$, 则$\\sin (\\dfrac{3}{2} \\pi+\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{4}{5}$", "solution": "", @@ -523393,7 +528336,9 @@ "id": "022110", "content": "已知复数$z=1+2 \\mathrm{i}$(其中$\\mathrm{i}$为虚数单位), 则$z$除以$\\overline {z}$的商$\\dfrac{z}{\\overline {z}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-\\dfrac{3}{5}+\\dfrac{4}{5} \\mathrm{i}$", "solution": "", @@ -523426,7 +528371,9 @@ "id": "022111", "content": "已知直角坐标平面上两点$P_1(-1,1)$、$P_2(2,3)$, 若$P$满足$\\overrightarrow{P_1P}=2 \\overrightarrow{PP_2}$, 则点$P$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(1, \\dfrac{7}{3})$", "solution": "", @@ -523459,7 +528406,9 @@ "id": "022112", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边为$a$、$b$、$c$. 若$a=4$, $b=6$, $c=9$, 则$\\cos C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{29}{48}$", "solution": "", @@ -523492,7 +528441,9 @@ "id": "022113", "content": "直线$l: y=2 x-1$绕着点$A(1,1)$逆时针旋转$\\dfrac{\\pi}{4}$与直线$l_1$重合, 则$l_1$的斜截式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$y=-3 x+4$", "solution": "", @@ -523525,7 +528476,9 @@ "id": "022114", "content": "已知函数$y=1-\\sin x-\\cos ^2 x$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -523558,7 +528511,10 @@ "id": "022115", "content": "直角三角形$ABC$中, $AB=3$, $AC=4$, $BC=5$, 点$M$是三角形$ABC$外接圆上任意一点, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AM}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第五单元" + ], "genre": "填空题", "ans": "$12$", "solution": "", @@ -523591,7 +528547,9 @@ "id": "022116", "content": "已知常数$m \\in \\mathbf{R}$, 若关于$x$的方程$x+\\sqrt{4-x^2}=m$有且仅有一个实数解, 则$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$[-2,2) \\cup\\{2 \\sqrt{2}\\}$", "solution": "", @@ -523624,7 +528582,9 @@ "id": "022117", "content": "已知常数$t \\in \\mathbf{R}$, 集合$S=\\{z|| z-1 | \\leq 3,\\ z \\in \\mathbf{C}\\}, T=\\{z | z=\\dfrac{w+2}{3} \\mathrm{i}+t,\\ w \\in S\\}$, 若$S \\cup T=S$, 则$t$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$[-\\sqrt{3}+1, \\sqrt{3}+1]$", "solution": "", @@ -523657,7 +528617,9 @@ "id": "022118", "content": "已知常数$a \\in \\mathbf{R}$, 直线$l_1: x+a y-2=0$, $l_2: a x+y+1=0$, 则$a=1$是$l_1\\parallel l_2$的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -523690,7 +528652,9 @@ "id": "022119", "content": "已知常数$\\varphi \\in \\mathbf{R}$, 如果函数$y=\\cos (2 x+\\varphi)$的图像关于点$(\\dfrac{4 \\pi}{3}, 0)$中心对称, 那么$|\\varphi|$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{4}$}{$\\dfrac{\\pi}{6}$}{$\\dfrac{\\pi}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -523723,7 +528687,9 @@ "id": "022120", "content": "已知常数$a, b \\in \\mathbf{R}$, 且$a, b$不全为零, 若直线$a x+b y=1$与圆$C: x^2+y^2=1$相交, 则点$P(a, b)$与圆$C$的位置关系是\\bracket{20}.\n\\twoch{点在圆内}{点在圆上}{点在圆外}{随$a$、$b$取值的变化而变化}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -523756,7 +528722,9 @@ "id": "022121", "content": "在平面直角坐标系中, $\\triangle ABC$的顶点坐标分别为$A(1,-2)$, $B(-7,0)$, 点$C$在直线$y=5$上运动, $O$为坐标原点, $G$为$\\triangle ABC$的重心, 则$\\overrightarrow{OG} \\cdot \\overrightarrow{OA}$、$\\overrightarrow{OG} \\cdot \\overrightarrow{OB}$、$\\overrightarrow{OG} \\cdot \\overrightarrow{OC}$中正数的个数为$n$, 则$n$的值的集合为\\bracket{20}.\n\\fourch{$\\{1,2\\}$}{$\\{1,3\\}$}{$\\{2,3\\}$}{$\\{1,2,3\\}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -523789,7 +528757,9 @@ "id": "022122", "content": "已知直线$l: x-2 y+1=0$.\\\\\n(1) 若直线$l_1: 2 x+y+1=0$, 求直线$l$与直线$l_1$的夹角;\\\\\n(2) 若直线$l_2$与直线$l$的距离等于$1$, 求直线$l_2$的一般式方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{\\pi}{2}$; (2) $x-2y+\\sqrt{5}+1=0$或$x-2y-\\sqrt{5}+1=0$", "solution": "", @@ -523822,7 +528792,9 @@ "id": "022123", "content": "设常数$p \\in \\mathbf{R}$, 已知关于$x$的方程$x^2+p x+2=0$.\\\\\n(1) 若$p=2$, 求该方程的复数根;\\\\\n(2) 若方程的两个复数根为$\\alpha$、$\\beta$, 且$|\\alpha-\\beta|=1$, 求$p$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) $-1\\pm \\mathrm{i}$; (2) $\\pm \\sqrt{7}$或$\\pm 3$", "solution": "", @@ -523855,7 +528827,9 @@ "id": "022124", "content": "记$f(x)=2 \\sin 2 x+4 \\sin ^2 x$.\\\\\n(1) 求关于$x$的方程$f(x)=0$的解集;\\\\\n(2) 求函数$y=f(x)$的单调减区间.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\{x|x=k\\pi \\text{或}-\\dfrac{\\pi}{4}+k\\pi, \\ k\\in \\mathbf{Z}\\}$; (2) $[\\dfrac{3\\pi}{8}+k\\pi,\\dfrac{7\\pi}{8}+k\\pi ]$, $k\\in \\mathbf{Z}$", "solution": "", @@ -523888,7 +528862,9 @@ "id": "022125", "content": "如图, 设$ABCDEF$是半径为$1$的圆$O$的内接正六边形, $M$是圆$O$上的动点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) circle (2);\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\draw (90:2) node [above] {$A$} coordinate (A);\n\\draw (150:2) node [left] {$B$} coordinate (B);\n\\draw (210:2) node [left] {$C$} coordinate (C);\n\\draw (270:2) node [below] {$D$} coordinate (D);\n\\draw (330:2) node [right] {$E$} coordinate (E);\n\\draw (30:2) node [right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(D)--(E)--(F)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求$|\\overrightarrow{AB}+\\overrightarrow{BC}-\\overrightarrow{AM}|$的最大值;\\\\\n(2) 求证: $\\overrightarrow{MA}^2+\\overrightarrow{MD}^2$为定值;\\\\\n(3) 对于平面中的点$P$, 存在实数$x$与$y$, 使得$\\overrightarrow{OP}=x \\overrightarrow{OE}+y \\overrightarrow{OF}$, 若点$P$是正六边形$ABCDEF$内的动点 (包含边界), 求$x-y$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) $2$; (2) 值为$4$, 证明略; (3) $-2$", "solution": "", @@ -523921,7 +528897,9 @@ "id": "022126", "content": "设$f(z)$是一个关于复数$z$的表达式, 若$f(x+y \\mathrm{i})=x_1+y_1 \\mathrm{i}$(其中$x, y, x_1, y_1 \\in\\mathbf{R}$, $\\mathrm{i}$为虚数单位), 就称$f$将点$P(x, y)$``$f$对应''到点$Q(x_1, y_1)$. 例如: $f(z)=\\dfrac{1}{z}$将点$(0,1)$``$f$对应''到点$(0,-1)$.\\\\\n(1) 若$f(z)=z+1$($z \\in \\mathbf{C}$), 点$P_1(1,1)$``$f$对应''到点$Q_1$, 点$P_2$``$f$对应''到点$Q_2(1,1)$, 求点$Q_1$、$P_2$的坐标;\\\\\n(2) 设常数$k, t \\in \\mathbf{R}$, 若直线$l: y=k x+t$, $f(z)=z^2$($z \\in \\mathbf{C}$), 是否存在一个有序实数对$(k, t)$, 使得直线$l$上的任意一点$P(x, y)$``$f$对应''到点$Q(x_1, y_1)$后, 点$Q$仍在直线$l$上? 若存在, 试求出所有的有序实数对$(k, t)$; 若不存在, 请说明理由;\\\\\n(3) 设常数$a, b \\in \\mathbf{R}$, 集合$D=\\{z | z \\in \\mathbf{C}$且$\\mathrm{Re} z>0\\}$和$A=\\{w | w \\in \\mathbf{C}$且$|w|<1\\}$, 若$f(z)=\\dfrac{a z+b}{z+1}$满足: \\textcircled{1} 对于集合$D$中的任意一个元素$z$, 都有$f(z) \\in A$; \\textcircled{2} 对于集合$A$中的任意一个元素$w$, 都存在集合$D$中的元素$z$使得$w=f(z)$. 请写出满足条件的一个有序实数对$(a, b)$, 并论证此时的$f(z)$满足条件.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) $Q_1(2,1)$, $P_2(0,1)$; (2) 存在, $k=0$, $t=0$, 证明略; (3) $f(z)=\\dfrac{z-1}{z+1}$满足题意, 证明略", "solution": "", @@ -557134,7 +562112,9 @@ "id": "031244", "content": "设$z=1+\\mathrm{i}$, 则$z^2-\\mathrm{i}=$\\bracket{20}.\n\\fourch{$\\mathrm{i}$}{$-\\mathrm{i}$}{$1$}{$-1$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -557154,7 +562134,9 @@ "id": "031245", "content": "设集合$A=\\{2,3, a^2-2 a-3\\}$, $B=\\{0,3\\}$, $C=\\{2, a\\}$. 若$B \\subseteq A$, $A \\cap C=\\{2\\}$, 则$a=$\\bracket{20}.\n\\fourch{$-3$}{$-1$}{$1$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -557174,7 +562156,9 @@ "id": "031246", "content": "甲、乙、丙、丁四名教师带领学生参加校园植树活动, 教师随机分成三组, 每组至少一人, 则甲、乙在同一组的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{6}$}{$\\dfrac{1}{4}$}{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -557194,7 +562178,9 @@ "id": "031247", "content": "平面向量$\\overrightarrow{a}$与$\\overrightarrow{b}$相互垂直, 已知$\\overrightarrow{a}=(6,-8)$, $|\\overrightarrow{b}|=5$, 且$\\overrightarrow{b}$与向量$(1,0)$的夹角是钝角, 则$\\overrightarrow{b}=$\\bracket{20}.\n\\fourch{$(-3,-4)$}{$(4,3)$}{$(-4,3)$}{$(-4,-3)$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -557214,7 +562200,9 @@ "id": "031248", "content": "已知点$A, B, C$为椭圆$D$的三个顶点, 若$\\triangle ABC$是正三角形, 则$D$的离心率是\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{2}{3}$}{$\\dfrac{\\sqrt{6}}{3}$}{$\\dfrac{\\sqrt{3}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -557234,7 +562222,9 @@ "id": "031249", "content": "三棱锥$A-BCD$中, $AC \\perp$平面$BCD$, $BD \\perp CD$. 若$AB=3, BD=1$, 则该三棱锥体积的最大值为\\bracket{20}.\n\\fourch{$2$}{$\\dfrac{4}{3}$}{$1$}{$\\dfrac{2}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -557254,7 +562244,9 @@ "id": "031250", "content": "设函数$f(x), g(x)$在$\\mathbf{R}$的导函数存在, 且$f'(x)g(x)$}{$f(x)+g(a)=latex,scale = 3]\n\\draw (0,0,0) node [left] {$O$} coordinate (O);\n\\draw ({sqrt(2)/2},0,{sqrt(2)/2}) node [below] {$A$} coordinate (A);\n\\draw ({sqrt(6)/4},{1/2},{sqrt(6)/4}) node [below right] {$C$} coordinate (C);\n\\draw ({sqrt(6)/4},0,{sqrt(6)/4}) node [above left] {$K$} coordinate (K);\n\\draw ({sqrt(6)/4},0,0) node [above right] {$N$} coordinate (N);\n\\draw ($(N)+(C)-(K)$) node [above] {$M$} coordinate (M);\n\\draw (1,0,0) node [right] {$B$} coordinate (B);\n\\draw ($(O)!{4/sqrt(10)}!(M)$) node [right] {$D$} coordinate (D);\n\\draw (O)--(A)(O)--(C)(O)--(D)(K)--(C)--(M);\n\\draw [dashed] (O)--(B)(M)--(N)--(K)(O)--++(0,1,0)(O)--++(0,0,1);\n\\draw [domain = 0:45] plot ({cos(\\x)},0,{sin(\\x)});\n\\draw [dashed, domain = 45:90] plot ({cos(\\x)},0,{sin(\\x)});\n\\draw [domain = 0:{acos(sqrt(6)/sqrt(10))}] plot ({cos(\\x)},{sin(\\x)},0);\n\\draw [dashed,domain = {acos(sqrt(6)/sqrt(10))}:90] plot ({cos(\\x)},{sin(\\x)},0);\n\\draw [dashed, domain = 0:90] plot (0,{cos(\\x)},{sin(\\x)});\n\\draw [domain = 0:30] plot ({sqrt(2)/2*cos(\\x)},{sin(\\x)},{sqrt(2)/2*cos(\\x)});\n\\draw [dashed, domain = 30:90] plot ({sqrt(2)/2*cos(\\x)},{sin(\\x)},{sqrt(2)/2*cos(\\x)});\n\\draw [domain = 0:{acos(sqrt(10)/4)}] plot ({sqrt(6)/sqrt(10)*cos(\\x)},{2/sqrt(10)*cos(\\x)},{sin(\\x)});\n\\draw [dashed, domain = {acos(sqrt(10)/4)}:90] plot ({sqrt(6)/sqrt(10)*cos(\\x)},{2/sqrt(10)*cos(\\x)},{sin(\\x)});\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557374,7 +562376,9 @@ "id": "031256", "content": "某工厂生产的产品的质量指标服从正态分布$N(100, \\sigma^2)$. 质量指标介于$99$至$101$之间''的产品为良品, 为使这种产品的良品率达到$95.45 \\%$, 则需调整生产工艺, 使得$\\sigma$至多为\\blank{50}. (若$X \\sim N(\\mu, \\sigma^2)$, 则$P\\{|X-\\mu|<2 \\sigma\\}=0.9545$)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557394,7 +562398,9 @@ "id": "031257", "content": "若$P, Q$分别是抛物线$x^2=y$与圆$(x-3)^2+y^2=1$上的点, 则$|PQ|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557414,7 +562420,9 @@ "id": "031258", "content": "数学家祖冲之曾给出圆周率$\\pi$的两个近似值: ``约率''$\\dfrac{22}{7}$与``密率''$\\dfrac{355}{113}$、它们可用``调日法''得到: 称小于$3.1415926$的近似值为弱率, 大于$3.1415927$的近似值为强率. 由$\\dfrac{3}{1}<\\pi<\\dfrac{4}{1}$, 取$3$为弱率, $4$为强率, 得$a_1=\\dfrac{3+4}{1+1}=\\dfrac{7}{2}$, 故$a_1$为强率, 与上一次的弱率$3$计算得$a_2=\\dfrac{3+7}{1+2}=\\dfrac{10}{3}$, 故$a_2$为强率, 继续计算, $\\cdots \\cdots$. 若某次得到的近似值为强率, 与上一次的弱率继续计算得到新的近似值; 若某次得到的近似值为弱率, 与上一次的强率继续计算得到新的近似值, 依此类推. 已知$a_m=\\dfrac{22}{7}$, 则$m=$\\blank{50}; $a_{8}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557434,7 +562442,9 @@ "id": "031259", "content": "如图为一个开关阵列, 每个开关只有``开''和``关''两种状态, 按其中一个开关$1$次, 将导致自身和所有相邻的开关改变状态. 例如, 按$(2,2)$将导致$(1,2),(2,1),(2,2),(2,3),(3,2)$改变状态. 如果要求只改变$(1,1)$的状态, 则需按开关的最少次数为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline$(1,1)$&$(1,2)$&$(1,3)$\\\\\n\\hline$(2,1)$&$(2,2)$&$(2,3)$\\\\\n\\hline$(3,1)$&$(3,2)$&$(3,3)$\\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557454,7 +562464,9 @@ "id": "031260", "content": "如图, 四边形$ABCD$是圆柱底面的内接四边形, $AC$是圆柱的底面直径, $PC$是圆柱的母线, $E$是$AC$与$BD$的交点, $AB=AD$, $\\angle BAD=60^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,0) arc (180:360:2 and 0.5);\n\\draw [dashed] (-2,0) arc (180:0:2 and 0.5);\n\\draw (0,4) ellipse (2 and 0.5);\n\\draw (-2,0) -- (-2,4) (2,0) -- (2,4);\n\\draw (150:2 and 0.5) node [above] {$B$} coordinate (B);\n\\draw (30:2 and 0.5) node [below] {$D$} coordinate (D);\n\\draw (80:2 and 0.5) node [above right] {$C$} coordinate (C);\n\\draw (260:2 and 0.5) node [below] {$A$} coordinate (A);\n\\draw (C)++(0,4) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.25!(A)$) node [left] {$F$} coordinate (F);\n\\draw [dashed] (P)--(A)(P)--(B)(P)--(C)(P)--(D)(A)--(C)(B)--(D)(C)--(F)(D)--(F)(A)--(B)--(C)--(D)--cycle;\n\\path [name path = AC] (A)--(C);\n\\path [name path = BD] (B)--(D);\n\\path [name intersections = {of = AC and BD, by = E}];\n\\draw (E) node [below right] {$E$};\n\\end{tikzpicture}\n\\end{center}\n(1) 记圆柱的体积为$V_1$, 四棱锥$P-ABCD$的体积为$V_2$, 求$\\dfrac{V_1}{V_2}$;\\\\\n(2) 设点$F$在线段$AP$上, $PA=4PF$, $PC=4CE$, 求二面角$F-CD-P$的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -557474,7 +562486,9 @@ "id": "031261", "content": "已知函数$f(x)=\\sin (\\omega x+\\varphi)$在区间$(\\dfrac{\\pi}{6}, \\dfrac{\\pi}{2})$单调, 其中$\\omega$为正整数, $|\\varphi|<\\dfrac{\\pi}{2}$, 且$f(\\dfrac{\\pi}{2})=f(\\dfrac{2 \\pi}{3})$.\\\\\n(1) 求$y=f(x)$图像的一条对称轴;\\\\\n(2) 若$f(\\dfrac{\\pi}{6})=\\dfrac{\\sqrt{3}}{2}$, 求$\\varphi$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -557494,7 +562508,9 @@ "id": "031262", "content": "记数列$\\{a_n\\}$的前$n$项和为$T_n$, 且$a_1=1$, $a_n=T_{n-1}$($n \\geq 2$).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式:\\\\\n(2) 设$m$为整数, 且对任意$n \\in \\mathbf{N}$, $n \\ge 1$, $m \\geq \\dfrac{1}{a_1}+\\dfrac{2}{a_2}+\\cdots+\\dfrac{n}{a_n}$, 求$m$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -557514,7 +562530,10 @@ "id": "031263", "content": "一个池塘里的鱼的数目记为$N$, 从池塘里捞出$200$尾鱼, 并给鱼作上标识, 然后把鱼放回池塘里, 过一小段时间后再从池塘里拱出$500$尾鱼, $X$表示捞出的$500$尾鱼中有标识的鱼的数目.\\\\\n(1) 若$N=5000$, 求$X$的数学期望;\\\\\n(2) 已知捞出的$500$尾鱼中$15$尾有标识, 试给出$N$的估计值(以使得$P(X=15)$取大的$N$的值作为$N$的估计值).", "objs": [], - "tags": [], + "tags": [ + "第九单元", + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -557534,7 +562553,9 @@ "id": "031264", "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)过点$A(4 \\sqrt{2}, 3)$, 且焦距为$10$.\\\\\n(1) 求$C$的方程;\\\\\n(2) 已知点$B(4 \\sqrt{2},-3)$, $D(2 \\sqrt{2}, 0)$, $E$为线段$AB$上一点, 且直线$DE$交$C$于$G$, $H$两点. 证明: $\\dfrac{|GD|}{|GE|}=\\dfrac{|HD|}{|HE|}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -557554,7 +562575,9 @@ "id": "031265", "content": "椭圆曲线加密算法运用于区块链.\\\\\n椭圆曲线$C=\\{(x, y) | y^2=x^3+a x+b,\\ 4 a^3+27 b^2 \\neq 0\\}$, $P \\in C$关于$x$轴的对称点记为 $\\tilde{C}$. $C$在点$P(x, y)$($y \\neq 0$)处的切线是指曲线$y=\\pm \\sqrt{x^3+a x+b}$在点$P$处的切线. 定义``$\\oplus$\" 运算满足:\\\\\n\\textcircled{1} 若$P \\in C$, $Q \\in C$, 且直线$PQ$与$C$有第三个交点$R$, 则$P \\oplus Q=\\tilde{R}$;\\\\\n\\textcircled{2} 若$P \\in C$, $Q \\in C$, 且$PQ$为$C$的切线, 切点为$P$, 则$P \\oplus Q=\\tilde{P}$;\\\\\n\\textcircled{3} 若$P \\in C$, 规定$P \\oplus \\tilde{P}=0^{\\circ}$, 且$P \\oplus 0^{\\circ}=0^{\\circ} \\oplus P=P$.\\\\\n(1) 当$4 a^3+27 b^2=0$时, 讨论函数$h(x)=x^3+a x+b$零点的个数;\\\\\n(2) 已知``$\\oplus$''运算满足交换律、结合律, 若$P \\in C$, $Q \\in C$, 且$PQ$为$C$的切线, 切点为$P$, 证明: $P \\oplus P=\\tilde{Q}$;\\\\(3) 已知$P(x_1, y_1) \\in C$, $Q(x_2, y_2) \\in C$, 且直线$PQ$与$C$有第三个交点, 求$P \\oplus Q$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -557609,7 +562632,9 @@ "id": "031267", "content": "半径为$1$的球的表面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557642,7 +562667,9 @@ "id": "031268", "content": "双曲线$x^2-y^2=1$的两条渐近线的夹角大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557675,7 +562702,9 @@ "id": "031269", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, 直线$BD_1$与平面$ABCD$所成角的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (B)--(D1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557708,7 +562737,9 @@ "id": "031270", "content": "从$1,2,3,4,5,6,7,8,9$这$9$个数中任取$3$个不同的数, 则中位数为$5$的取法有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557741,7 +562772,9 @@ "id": "031271", "content": "已知圆锥的底面半径为$1$, 侧面积为$4 \\pi$, 则该圆锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557774,7 +562807,9 @@ "id": "031272", "content": "两个实习生每人加工一个零件, 加工成一等品的概率分别为$\\dfrac{4}{5}$和$\\dfrac{9}{10}$, 两个零件是否加工成一等品相互独立, 则这两个零件中恰有一个一等品的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557807,7 +562842,9 @@ "id": "031273", "content": "已知双曲线方程为$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$, 点$P$是该双曲线上的点, $F_1$、$F_2$分别是它的左、右焦点, 若$|PF_1| \\cdot|PF_2|=40$, 则$\\angle F_1PF_2$的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557840,7 +562877,9 @@ "id": "031274", "content": "设直线$a x-y+3=0$与圆$(x+1)^2+(y+2)^2=4$相交于$A$、$B$两点, 且弦$AB$长为$2 \\sqrt{3}$, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557873,7 +562912,9 @@ "id": "031275", "content": "已知样本容量为$5$的样本的平均数为$3$, 方差为$\\dfrac{18}{5}$, 现把一个新数据$15$加入原样本从而得到一个样本容量为$6$的新样本, 则该新样本的方差为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557906,7 +562947,10 @@ "id": "031276", "content": "若方程$a x^2+b y^2=c$的系数$a, b, c$是从$-1,2,3,5,7,11,13$这$7$个数中任取$3$个不同的数而得到, 则这样的方程表示焦点在$x$轴上的椭圆的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元", + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557939,7 +562983,9 @@ "id": "031277", "content": "已知三棱锥$P-ABC$中, $PA$、$PB$、$PC$两两垂直, 且$|PA|=|PB|=|PC|$, 若$P$、$A$、$B$、$C$这四点都落在某个半径为$\\sqrt{3}$的球的球面上, 则该球的球心到平面$ABC$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -557972,7 +563018,9 @@ "id": "031278", "content": "点$P(m, n)$是椭圆$4 x^2+y^2=1$上的动点且点$P$不在坐标轴上, 动点$Q(\\dfrac{1}{m}, \\dfrac{1}{n})$构成的轨迹为曲线$\\Gamma$. 若圆$x^2+y^2=r^2(r>0)$与曲线$\\Gamma$无公共点, 则实数$r$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -558005,7 +563053,9 @@ "id": "031279", "content": "已知直线$l$是平面$\\alpha$的一条斜线, 则\\bracket{20}.\n\\onech{在平面$\\alpha$上存在唯一的一条直线$m$, 使得$l \\perp m$}{在平面$\\alpha$上存在唯一的一条直线$m$, 使得$l\\parallel m$}{在平面$\\alpha$上存在无数多条直线$m$, 使得$l \\perp m$}{在平面$\\alpha$上存在无数多条直线$m$, 使得$l\\parallel m$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -558038,7 +563088,9 @@ "id": "031280", "content": "从$6$名男医生、$4$名女医生中选$3$名医生组成一个医疗小分队, 要求其中男、女医生都有, 则不同的组队方案共有\\bracket{20}.\n\\fourch{$192$种}{$96$种}{$576$种}{$384$种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -558071,7 +563123,9 @@ "id": "031281", "content": "在梯形$ABCD$中, $\\angle ABC=\\dfrac{\\pi}{2}$, $AD\\parallel BC$, $AB=2$, $BC=4$, $AD=2$. 将梯形$ABCD$绕直线$AD$旋转一周而形成的曲面所围成的几何体的体积为\\bracket{20}.\n\\fourch{$16 \\pi$}{$\\dfrac{40}{3} \\pi$}{$\\dfrac{32}{3} \\pi$}{$\\dfrac{16}{3} \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -558151,7 +563205,9 @@ "id": "031283", "content": "某电子商务公司对$20000$名网络购物者$2022$年度的消费情况进行统计, 发现所有购物者的消费金额(单位: 万元)都在区间$[0.3,0.9]$内, 其频率分布直方图如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 7]\n\\draw [->] (0,0) -- (1.1,0) node [below] {金额/万元};\n\\draw [->] (0,0) -- (0,3.5) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {0.3/1.5/1.5,0.4/2.5/2.5,0.5/3/a,0.6/2/2,0.7/0.8/0.8,0.8/0.2/0.2}\n{\\draw (\\i,0) node [below] {$\\i$};\n\\draw (\\i,0) -- (\\i,\\j) --++ (0.1,0) --++ (0,-\\j);\n\\draw [dashed] (0,\\j) node [left] {$\\k$} -- (\\i,\\j);\n};\n\\draw (0.9,0) node [below] {$0.9$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求频率分布直方图中$a$的值, 并求在这些购物者中, 消费金额在区间$[0.6,0.9]$内的购物者的人数;\\\\\n(2) 对消费金额在区间$[0.6,0.7)$、$[0.7,0.8)$、$[0.8,0.9]$内的三组购物者中用分层随机抽样的方法共抽取$90$名购物者, 则消费金额在区间$[0.7,0.8)$内的购物者应抽取多少名?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -558184,7 +563240,9 @@ "id": "031284", "content": "如图, 在Rt$\\triangle AOB$中, $\\angle AOB$为直角, $AO=4$, $BO=2$. Rt$\\triangle AOC$可以通过Rt$\\triangle AOB$以直线$AO$为轴旋转得到, 且二面角$B-AO-C$是直二面角. 动点$D$在线段$AB$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0,0) node [above right] {$O$} coordinate (O);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [below left] {$C$} coordinate (C);\n\\draw (0,4,0) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.4!(B)$) node [right] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--cycle(C)--(D);\n\\draw [dashed] (A)--(O)--(B)(O)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$D$为$AB$的中点时, 求异面直线$AO$与$CD$所成角的大小;\\\\\n(2) 求$CD$与平面$AOB$所成角的最大值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -558217,7 +563275,9 @@ "id": "031285", "content": "已知椭圆$C$的方程为$\\dfrac{x^2}{2}+y^2=1$, $F_1$、$F_2$分别是它的左、右焦点.\\\\\n(1) 求椭圆$C$的长轴长以及离心率;\\\\\n(2) 过点$F_2$的直线$l$与椭圆$C$相交于$P$、$Q$两点, $O$为坐标原点, 若直线$l$的斜率为$k$且$\\overrightarrow{OP} \\perp \\overrightarrow{OQ}$, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -558250,7 +563310,9 @@ "id": "031286", "content": "我国古代数学名著《九章算术》中记载: ``刍甍者, 下有袤有广, 而上有袤无广. 刍, 草也. 甍, 屋盖也. ''翻译为``底面有长有宽为矩形, 顶部只有长没有宽为一条棱. 刍甍的字面意思为茅草屋顶. ''\n现有一个``刍甍''如图所示, 四边形$ABCD$为正方形, 四边形$ABFE$、$CDEF$为两个全等的等腰梯形, $EF\\parallel AB$, $AB=4$, $EF=2$, $EA=ED=FB=FC=\\sqrt{10}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (-2,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,2) node [right] {$B$} coordinate (B);\n\\draw (2,0,-2) node [right] {$C$} coordinate (C);\n\\draw (-2,0,-2) node [below] {$D$} coordinate (D);\n\\draw (-1,{sqrt(5)},0) node [above] {$E$} coordinate (E);\n\\draw (1,{sqrt(5)},0) node [above] {$F$} coordinate (F);\n\\draw (E)--(A)--(B)--(C)--(F)--cycle(B)--(F);\n\\draw [dashed] (E)--(D)--(A)(D)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 设过点$F$且与直线$EF$垂直的平面为平面$\\alpha$, 且平面$\\alpha$与直线$AB$、$CD$分别交于$P$、$Q$两点, 求$\\triangle FPQ$的周长;\\\\\n(2) 求四面体$ABDE$的体积;\\\\\n(3) 点$N$在线段$AD$上且满足$\\dfrac{AN}{AD}=\\dfrac{1}{3}$. 试问: 在线段$CF$上是否存在点$M$, 使$NF\\parallel$平面$BDM$? 若存在, 求出$\\dfrac{CM}{CF}$的值; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -558283,7 +563345,9 @@ "id": "031287", "content": "已知椭圆$\\Gamma$的方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{9}=1$(常数$a>3$), 点$A$、$B$分别为椭圆短轴的上端点、 下端点, 点$P$是椭圆$\\Gamma$上异于点$A$的一个动点. 若动点$P$到定点$A$的距离的最大值仅在点$P$运动到点$B$时取到, 则称此椭圆为``圆椭圆''.\\\\\n(1) 若$a=5$, 判断椭圆$\\Gamma$是否为``圆椭圆'';\\\\\n(2) 若椭圆$\\Gamma$是``圆椭圆'', 求$a$的取值范围;\\\\\n(3) 已知椭圆$\\Gamma$是``圆椭圆'', 且$a$取最大值. 点$P$关于原点$O$的对称点为点$Q$(点$Q$也异于点$A$), 且直线$AP$、$AQ$分别与$x$轴交于$M$、$N$两点. 试问以线段$MN$为直径的圆是否过定点? 证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -558316,7 +563380,9 @@ "id": "031288", "content": "已知全集$U=\\{x | 20\\end{cases}$ 若$f(a)=9$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$-9$或$3$", "solution": "", @@ -558514,7 +563590,9 @@ "id": "031294", "content": "若数列$\\{a_n\\}$满足$a_{n+1}=-\\dfrac{1}{3} a_n$, 前$5$项和为$\\dfrac{61}{27}$, 则$a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$\\dfrac{1}{27}$", "solution": "", @@ -558547,7 +563625,9 @@ "id": "031295", "content": "若定义在$\\mathbf{R}$上的奇函数$f(x)$在$[0,+\\infty)$上的图像如图所示, 则不等式$x \\cdot f(x) \\geq 0$的解集是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw [->] (-5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3.8] plot (\\x,{\\x*(3-\\x)/2.25*3});\n\\draw [dashed] (0,3) -- (1.5,3);\n\\draw (0,3) node [left] {$3$} (3,0) node [below] {$3$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$[-3,3]$", "solution": "", @@ -558580,7 +563660,9 @@ "id": "031296", "content": "已知$a_n=\\dfrac{n-\\sqrt{2022}}{n-\\sqrt{2023}}$($n$为正整数), 且数列$\\{a_n\\}$共有$100$项, 则此数列中最大项为第\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$45$", "solution": "", @@ -558613,7 +563695,9 @@ "id": "031297", "content": "已知函数$f(x)=\\begin{cases}\\log _a x, & 01\\end{cases}$在$(0,+\\infty)$上严格单调增, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(1,\\dfrac 32]$", "solution": "", @@ -558646,7 +563730,9 @@ "id": "031298", "content": "已知函数$f(x)=a^2 \\cdot x+3 a$, $x \\in[-\\dfrac{1}{2}, 1]$, 与函数$g(x)=(\\dfrac{1}{5})^x-1$, $x \\in[-1,0]$. 对任意的$x_1 \\in[-\\dfrac{1}{2}, 1]$, 总存在$x_2 \\in[-1,0]$, 使得$f(x_1)=g(x_2)$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$[0,1]$", "solution": "", @@ -558679,7 +563765,9 @@ "id": "031299", "content": "已知函数$f(x)=x^3+\\lg \\dfrac{5+x}{5-x}+2$, 若实数$a$、$b$满足$f(3 a^2)+f(b^2-1)=4$, 则$a \\sqrt{1+2 b^2}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{6}}{4}$", "solution": "", @@ -558712,7 +563800,9 @@ "id": "031300", "content": "已知$a, b \\in \\mathbf{R}$, 则$a b(a-b)<0$成立的一个充要条件是\\bracket{20}.\n\\fourch{$\\dfrac{1}{a}>\\dfrac{1}{b}>0$}{$\\dfrac{1}{a}<\\dfrac{1}{b}$}{$0<\\dfrac{1}{a}<\\dfrac{1}{b}$}{$\\dfrac{1}{a}>\\dfrac{1}{b}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -558747,7 +563837,9 @@ "id": "031301", "content": "若函数$y=\\log _2(k x^2+4 k x+3)$的定义域为$\\mathbf{R}$, 则$k$的取值范围是\\bracket{20}.\n\\fourch{$(0, \\dfrac{3}{4})$}{$[0, \\dfrac{3}{4})$}{$[0, \\dfrac{3}{4}]$}{$(-\\infty, 0] \\cup(\\dfrac{3}{4},+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -558780,7 +563872,9 @@ "id": "031302", "content": "等差数列$\\{a_n\\}$中, 首项为$a_1$公差$d$不为零, 前$n$项和为$S_n$. 若$S_8$是$S_4$的 3倍, 则$a_1$与$d$的比为\\bracket{20}.\n\\fourch{$5: 2$}{$2: 5$}{$5: 1$}{$1: 5$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -558813,7 +563907,10 @@ "id": "031303", "content": "已知非空集合$A, B$满足: $A \\cup B=\\mathbf{R}$, $A \\cap B=\\varnothing$. 已知函数$f(x)=\\begin{cases}-x^2, & x \\in A, \\\\ -2 x+1, & x \\in B,\\end{cases}$ 对于下列两个命题: \\textcircled{1} 存在无穷多非空集合对$(A, B)$, 使得方程$f(x)=-2$无解; \\textcircled{2} 存在唯一的非空集合对$(A, B)$, 使得$f(x)$为偶函数. 下面判断正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}正确, \\textcircled{2}错误}{\\textcircled{1}错误, \\textcircled{2}正确}{\\textcircled{1}、\\textcircled{2}都正确}{\\textcircled{1}、 \\textcircled{2}都错误}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第一单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -558846,7 +563943,9 @@ "id": "031304", "content": "已知等差数列$\\{a_n\\}$中, 首项$a_1=16$, 公差$d \\neq 0$, 且$a_1, a_5, a_6$是等比数列$\\{b_n\\}$的前三项.\\\\\n(1) 求数列$\\{a_n\\}$与$\\{b_n\\}$的通项公式;\\\\\n(2) 设数列$\\{a_n\\}$的前$n$项和为$S_n$, 且记$T_n=\\log _4 b_n^n$, 试比较$S_n$与$T_n$的大小.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "$(1)a_n=-3n+19,b_n=4^{3-n}\\\\\n(2)1\\le n \\le 28,S_n>T_n;n=29,S_n=T_n;n \\ge 30,S_n0$), 需另外投入生产成本$p(x)$万元: 当产量不足$60$万箱时, $p(x)=\\dfrac{1}{2} x^2+50 x$; 当产量不小于$60$万箱时, $p(x)=101 x+\\dfrac{6400}{x}-1860$. 若每箱口罩售价$100$元, 通过市场分析, 该口罩厂生产的口罩可以全部销售完.\\\\\n(1) 求口罩销售利润$y$(万元)关于产量$x$(万箱)的函数关系式;(定义: 销售利润$=$销售总价固定成本$-$生产成本)\\\\\n(2) 当产量为多少万箱时, 该口罩生产厂所获得利润最大, 最大利润值是多少(万元)?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -558945,7 +564048,9 @@ "id": "031307", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n$满足: $S_n=\\dfrac{m}{m-1}(a_n-1)$($m$为常数, 且$m \\neq 0$, $m \\neq 1$).\\\\\n(1) 求$\\{a_n\\}$的通项公式 (结果用含$m$的式子表示); \\\\\n(2) 设$b_n=\\dfrac{2S_n}{a_n}+1$, 若数列$\\{b_n\\}$为等比数列, 求$m$的值;\\\\\n(3) 若数列$\\{b_n\\}$是 (2) 中的等比数列, 记数列$\\{c_n\\}$的第$n$项$c_n=(n-1) b_n$, 求数列$\\{c_n\\}$的前$n$项和$T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "$(1)a_n=m^n;(2)m=\\dfrac 13;(3)T_n=\\dfrac {2n-3}4 \\cdot 3^{n+1}+ \\dfrac 94$", "solution": "", @@ -558978,7 +564083,9 @@ "id": "031308", "content": "已知函数$f(x)=2^x$($x \\in \\mathbf{R}$), 记$g(x)=f(x)-f(-x)$, $h(x)=f(x)+f(-x)$.\\\\\n(1) 求不等式的解集: $f(2 x)-2 f(x) \\leq 8$;\\\\\n(2) 设$t$为实数, 若存在实数$x_0 \\in[1,2]$, 使得$h(2 x_0)=t \\cdot g(x_0)-1$成立, 求$t$的取值范围;\\\\\n(3) 记$H(x)=8 \\cdot f(2 x)+2 a \\cdot f(x)+2 b$(其中$a$、$b$均为实数), 若对于任意的$x \\in[0,1]$, 均有$|H(x)| \\leq 1$, 求$a$、$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "$(1)(-\\infty,2];(2)[2\\sqrt{3},\\dfrac{91}{20}];(3)a=-12,b=\\dfrac{17}2$", "solution": "", @@ -559851,7 +564958,9 @@ "id": "031332", "content": "若集合$A=\\{x | x \\leq 2\\}$, $B=\\{x | x \\geq a\\}$满足$A \\cap B=\\{2\\}$, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -559884,7 +564993,9 @@ "id": "031333", "content": "在空间直角坐标系$O-x y z$中, 点$A(2,-1,3)$关于$y O z$平面对称的点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$(-2,-1,3)$", "solution": "", @@ -559917,7 +565028,9 @@ "id": "031334", "content": "方程$2^x+\\log _4 x=17$的解为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -559950,7 +565063,9 @@ "id": "031335", "content": "已知复数$z_1=1+\\sqrt{3} \\mathrm{i}$($\\mathrm{i}$为虚数单位), $|z_2|=2$, 若$z_1 z_2$是正实数, 则$z_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$1-\\sqrt{3}\\mathrm{i}$", "solution": "", @@ -559983,7 +565098,9 @@ "id": "031336", "content": "已知随机变量$X$服从二项分布$B(n, p)$, 若$E[X]=30$, $D[X]=20$, 则$p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 13$", "solution": "", @@ -560016,7 +565133,9 @@ "id": "031337", "content": "在$\\triangle ABC$中, 若$BC=3$, $AC=2 \\sqrt{6}$, $B=2A$, 则$B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\arccos \\dfrac 13$", "solution": "", @@ -560049,7 +565168,9 @@ "id": "031338", "content": "已知甲、乙两组数据如茎叶图所示, 其中$m$、$n \\in \\mathbf{N}$. 若这两组数据的中位数相等, 平均数也相等, 则$\\dfrac{m}{n}=$\\blank{50}.\n\\begin{center}\n\\begin{tabular}{cc|c|ccc}\n\\multicolumn{2}{c|}{甲} & & \\multicolumn{3}{c}{乙}\\\\\\hline\n& 7 & 2 & $n$\\\\ \n9 & $m$ & 3 & 2 & 4 & 8\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$\\dfrac 38$", "solution": "", @@ -560083,7 +565204,9 @@ "id": "031339", "content": "已知单位向量$\\overrightarrow {a}$与$\\overrightarrow {b}$, 向量$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影向量为$\\overrightarrow {b_0}$, 且$\\overrightarrow{b_0}=\\lambda \\overrightarrow {a}$($\\lambda \\in \\mathbf{R}$), 若$\\langle\\overrightarrow {a}, \\overrightarrow {b}\\rangle$的取值范围是$[\\dfrac{\\pi}{3}, \\dfrac{5 \\pi}{6}]$, 则$\\lambda$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$[-\\dfrac{\\sqrt{3}}2,\\dfrac 12]$", "solution": "", @@ -560116,7 +565239,9 @@ "id": "031340", "content": "若关于$x$的不等式$x^2+b x+c \\geq 0$($b>1$)的解集为$\\mathbf{R}$, 则$\\dfrac{1+2 b+4 c}{b-1}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$8$", "solution": "", @@ -560149,7 +565274,10 @@ "id": "031341", "content": "如图, $ABCDEF-A'B'C'D'E'F'$为正六棱柱, 若从该正六棱柱的$6$个侧面的$12$条面对角线中, 随机选取两条, 则它们共面的概率是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({1/2},0,{sqrt(3)/2}) node [below] {$B$} coordinate (B);\n\\draw ({3/2},0,{sqrt(3)/2}) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw ($(A)+(D)-(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)+(D)-(C)$) node [below] {$F$} coordinate (F);\n\\draw (A) ++ (0,1) node [left] {$A'$} coordinate (A');\n\\draw (B) ++ (0,1) node [above] {$B'$} coordinate (B');\n\\draw (C) ++ (0,1) node [above] {$C'$} coordinate (C');\n\\draw (D) ++ (0,1) node [right] {$D'$} coordinate (D');\n\\draw (E) ++ (0,1) node [above] {$E'$} coordinate (E');\n\\draw (F) ++ (0,1) node [above] {$F'$} coordinate (F');\n\\draw (A)--(B)--(C)--(D)--(D')--(E')--(F')--(A')--cycle(B)--(B')(C)--(C')(A')--(B')--(C')--(D');\n\\draw [dashed] (A)--(F)--(E)--(D)(F)--(F')(E)--(E');\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac 6{11}$", "solution": "", @@ -560182,7 +565310,9 @@ "id": "031342", "content": "设$a>b>0$, 椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$的离心率为$e_1$, 双曲线$\\dfrac{x^2}{b^2}-\\dfrac{y^2}{a^2-2 b^2}=1$的离心率为$e_2$, 若$e_1 e_2<1$, 则$\\dfrac{e_2}{e_1}$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$(\\sqrt{2},\\dfrac{\\sqrt{5}+1}2)$", "solution": "", @@ -560215,7 +565345,9 @@ "id": "031343", "content": "在$\\triangle ABC$中, $AB=3$, $BC=4$, $AC=5, P$为$\\triangle ABC$内部一动点(含边界), 在空间中, 若到点$P$的距离不超过$1$的点的轨迹为$L$, 则几何体$L$的体积等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{22\\pi}3+12$", "solution": "", @@ -560248,7 +565380,9 @@ "id": "031344", "content": "设$m \\in \\mathbf{R}$, 若幂函数$y=x^{m^2-2 m+1}$定义域为$\\mathbf{R}$, 且其图像关于$y$轴成轴对称, 则$m$的值可以为\\bracket{20}.\n\\fourch{$1$}{$4$}{$7$}{$10$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -560281,7 +565415,9 @@ "id": "031345", "content": "已知$x \\in \\mathbf{R}$, 下列不等式中正确的是\\bracket{20}.\n\\twoch{$\\dfrac{1}{2^x}>\\dfrac{1}{3^x}$}{$\\dfrac{1}{x^2-x+1}>\\dfrac{1}{x^2+x+1}$}{$\\dfrac{1}{2|x|}>\\dfrac{1}{x^2+1}$}{$\\dfrac{1}{x^2+1}>\\dfrac{1}{x^2+2}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -560314,7 +565450,9 @@ "id": "031346", "content": "设$a$、$b$、$c$、$d \\in \\mathbf{R}$, 若函数$y=a x^3+b x^2+c x+d$的部分图像如图所示, 则下列结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.1:1.5] plot (\\x,{(\\x+0.5)*(\\x-0.4)*(\\x-1.1)+1.5});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$b>0$, $c>0$}{$b>0$, $c<0$}{$b<0$, $c>0$}{$b<0$, $c<0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -560347,7 +565485,9 @@ "id": "031347", "content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 若对任意正整数$n$, 总存在正整数$m$, 使得$S_n=a_m$, 有结论: \\textcircled{1} $\\{a_n\\}$可能为等差数列; \\textcircled{2} $\\{a_n\\}$可能为等比数列. 关于以上两个结论, 正确的判断是\\bracket{20}.\n\\fourch{\\textcircled{1}成立, \\textcircled{2}成立}{\\textcircled{1}成立, \\textcircled{2}不成立}{\\textcircled{1}不成立, \\textcircled{2}成立}{\\textcircled{1}不成立, \\textcircled{2}不成立}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -560380,7 +565520,9 @@ "id": "031348", "content": "深入实施科教兴国战略是中华人民伟大复兴的必由之路, 2020 年第七次全国人口普查对$6$岁及以上人口的受教育程度进行统计 (未包括中国香港、澳门特别行政区和台湾省的人口数据), 我国$31$个省级行政区具有初中及以上文化程度人口比例情况经统计得到如下的频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 8, yscale = 0.7]\n\\draw [->] (0,0) -- (1.1,0) node [below] {比例};\n\\draw [->] (0,0) -- (0,7) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.35/0.323,0.45/0.323,0.55/1.613,0.65/5.161,0.75/1.935,0.85/0.645}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (0.1,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0.35/0.323,0.55/1.613,0.65/5.161,0.75/1.935/a,0.85/0.645}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (0.95,0) node [below] {$0.95$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求具有初中及以上文化程度人口比例在区间$[0.75,0.85)$内的省级行政区有几个?\\\\\n(2) 已知上海具有初中及以上文化程度人口比例是这组数据的第$41$百分位数, 求该比例落在哪个区间内?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) $6$个; (2) 落在区间$[0.65,0.75)$内", "solution": "", @@ -560413,7 +565555,9 @@ "id": "031349", "content": "如图, $AB$为圆$O$的直径, 点$E$、$F$在圆$O$上, $AB\\parallel EF$, 矩形$ABCD$所在平面和圆$O$所在的平面互相垂直. 已知$AB=2$, $EF=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(225:0.5cm)}]\n\\draw [domain = -30:{180+atan(sqrt(2)*7/8)}, samples = 100] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\draw [domain = -30:{-180+atan(sqrt(2)*7/8)}, samples = 100, dashed] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (0,0,-2) node [above left] {$B$} coordinate (B);\n\\draw ({sqrt(3)},0,1) node [below] {$F$} coordinate (F);\n\\draw ({sqrt(3)},0,-1) node [above] {$E$} coordinate (E);\n\\draw (A) --++ (0,2) node [above] {$D$} coordinate (D);\n\\draw [dashed] (B) --++ (0,2) node [above] {$C$} coordinate (C);\n\\filldraw (0,0) node [right] {$O$} coordinate (O) circle (0.03);\n\\draw (A)--(F)--(C) (D)--(C);\n\\draw [dashed] (A)--(B)--(E)(B)--(F);\n\\draw (D)--(F)(C)--(E);\n\\draw (E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$DAF \\perp$平面$CBF$;\\\\\n(2) 当$AD$的长为何值时, 二面角$C-EF-B$的大小为$60^{\\circ}$?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac 32$", "solution": "", @@ -560446,7 +565590,9 @@ "id": "031350", "content": "设$a \\in \\mathbf{R}$, $f(x)=\\sin 2 x+a \\cos x$.\\\\\n(1) 是否存在$a$使得$y=f(x)$为奇函数? 说明理由;\\\\\n(2) 当$a<-4$时, 求证: 函数$y=f(x)$在区间$(0, \\dfrac{\\pi}{2})$上是严格增函数.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 存在, $a=0$; (2) 证明略", "solution": "", @@ -560479,7 +565625,9 @@ "id": "031351", "content": "已知圆$C: x^2+y^2=4$, 点$P(2,2)$.\\\\\n(1) 直线$l$过点$P$且与圆$C$相交于$A$、$B$两点, 若$\\overrightarrow{CA} \\cdot \\overrightarrow{CB}=0$, 求直线$l$的方程;\\\\\n(2) 若动圆$D$经过点$P$且与圆$C$外切, 求动圆的圆心$D$的轨迹方程;\\\\\n(3) 是否存在异于点$P$的点$Q$, 使得对于圆$C$上任意一点$M$, 均有$\\dfrac{|MP|}{|MQ|}=\\lambda$? 若存在, 求出$Q$点坐标和常数$\\lambda$的值; 若不存在, 也请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $y-2=(2\\pm \\sqrt{3})(x-2)$; (2) $2xy-2x-2y+1=0$($x>1$); (3) 存在, $Q(1,1)$, $\\lambda=2$", "solution": "", @@ -560512,7 +565660,9 @@ "id": "031352", "content": "定义在$\\mathbf{R}$上的函数$y=f(x)$、$y=g(x)$, 若$|f(x_1)-f(x_2)| \\geq|g(x_1)-g(x_2)|$对任意的$x_1$、$x_2 \\in \\mathbf{R}$成立, 则称函数$y=g(x)$是函数$y=f(x)$的``从属函数''.\\\\\n(1) 若函数$y=g(x)$是函数$y=f(x)$的``从属函数''且$y=f(x)$是偶函数, 求证: $y=g(x)$是偶函数;\\\\\n(2) 若$f(x)=a x+\\mathrm{e}^x$, $g(x)=\\sqrt{x^2+1}$, 求证: 当$a \\geq 1$时, 函数$y=g(x)$是函数$y=f(x)$的``从属函数'';\\\\\n(3) 设定义在$\\mathbf{R}$上的函数$y=f(x)$与$y=g(x)$, 它们的图像各为一条连续的曲线, 且函数$y=g(x)$是函数$y=f(x)$的``从属函数''. 设$\\alpha: $``函数$y=f(x)$在$\\mathbf{R}$上是严格增函数或严格减函数''; $\\beta$: ``函数$y=g(x)$在$\\mathbf{R}$上为严格增函数或严格减函数''. 试判断$\\alpha$是$\\beta$的什么条件? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) 证明略; (3) $\\alpha$是$\\beta$的必要非充分条件", "solution": "", @@ -560741,7 +565891,9 @@ "id": "031358", "content": "若直线$x+y-c=0$, $x-y+2=0$, $y+2=0$无法围成三角形, 则实数$c$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$-6$", "solution": "", @@ -560769,7 +565921,9 @@ "id": "031359", "content": "已知$2^x<3^x$, 则$x$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(0,+\\infty)$", "solution": "", @@ -560832,7 +565986,9 @@ "id": "031361", "content": "已知复数$z$满足$z(1+\\mathrm{i})=2 \\mathrm{i}$, $\\mathrm{i}$为虚数单位, 则$|z|$值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\sqrt{2}$", "solution": "", @@ -560853,7 +566009,9 @@ "id": "031362", "content": "在等差数列$\\{a_n\\}$中, $2 a_9-a_{12}=6$, 则数列$\\{a_n\\}$的前$11$项和$S_{11}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$66$", "solution": "", @@ -560873,7 +566031,9 @@ "id": "031363", "content": "已知向量$\\overrightarrow {a}$和$\\overrightarrow {b}$满足$\\overrightarrow {a}=(1,2)$, $\\overrightarrow {b}=(-2,0)$, 则$\\overrightarrow {a}$在$\\overrightarrow {b}$方向上的数量投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-1$", "solution": "", @@ -560893,7 +566053,9 @@ "id": "031364", "content": "某单位为了解该单位党员开展学习党史知识活动情况, 随机抽取了部分党员, 对他们一周的党史学习时间进行了统计, 统计数据如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 党史学习时间 (小时) & 7 & 8 & 9 & 10 & 11 \\\\\n\\hline 党员人数 & 6 & 10 & 9 & 8 & 7 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n则该单位党员一周学习党史时间的第$40$百分位数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$8.5$", "solution": "", @@ -560913,7 +566075,9 @@ "id": "031365", "content": "二项式$(2 x+\\dfrac{1}{\\sqrt[3]{x}})^6$的展开式中, 含$x^2$的项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$160$", "solution": "", @@ -560933,7 +566097,9 @@ "id": "031366", "content": "已知圆锥的底面半径为$2$, 底面圆心到某条母线的距离为$1$, 则该圆锥的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{8\\sqrt{3}}3\\pi$", "solution": "", @@ -560953,7 +566119,9 @@ "id": "031367", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分別为$a$、$b$、$c$, 且$a=5$, $b=7$, $B=60^{\\circ}$, 则$\\triangle ABC$的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$10\\sqrt{3}$", "solution": "", @@ -560973,7 +566141,9 @@ "id": "031368", "content": "若$x=1$是函数$f(x)=\\dfrac{1}{3} x^3+(a+1) x^2-(a^2+3 a-3) x$的极大值点, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$-3$", "solution": "", @@ -560993,7 +566163,9 @@ "id": "031369", "content": "非空集合$A$中所有元素乘积记为$T(A)$. 已知集合$M=\\{1,4,5,8\\}$, 从集合$M$的所有非空子集中任选一个子集$A$, 则$T(A)$为偶数的概率是\\blank{50}(结果用最简分数表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 45$", "solution": "", @@ -561013,7 +566185,9 @@ "id": "031370", "content": "已知函数$f(x)=\\sin (2 x+\\dfrac{\\pi}{6})$($0 \\leq x \\leq \\pi$), 且$f(\\alpha)=f(\\beta)=\\dfrac{1}{3}$($\\alpha \\neq \\beta$), 则$\\alpha+\\beta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{4\\pi}3$", "solution": "", @@ -561033,7 +566207,9 @@ "id": "031371", "content": "已知函数$f(x)=\\begin{cases}x+2, x1$''是``$a^3>1$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分也非必要}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -561093,7 +566273,9 @@ "id": "031374", "content": "已知盒中装有形状完全相同的$4$个黑球与$2$个白球, 现从中有放回的摸取$4$次, 每次都是从盒子中随机摸出$1$个球, 设摸得白球个数为$X$, 则$E[X]$为\\bracket{20}.\n\\fourch{$\\dfrac{4}{3}$}{$\\dfrac{5}{3}$}{$2$}{$\\dfrac{4}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -561113,7 +566295,9 @@ "id": "031375", "content": "《九章算术》中所述``羡除'', 是指如图所示五面体$ABCDEF$, 其中$AB\\parallel DC\\parallel EF$, ``羡除''形似``楔体''. ``广''是指``羡除''的三条平行侧棱之长$a$、$b$、$c$,``深''是指一条侧棱到另两条侧棱所在平面的距离$m$, ``袤''是指这两条侧棱所在平行直线之问的距离$n$(如图). 羡除的体积公式为$V=\\dfrac{(a+b+c) m n}{6}$, 过线段$AD, BC$的中点$G, H$及直线$EF$作该羡除的一个截面$\\alpha$, 已知$\\alpha$刚好将羡除分成体积比为$5: 4$的两部分. 若$AB=4$, $DC=2$, 则$EF$的长为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (-1.5,0,2) node [left] {$A$} coordinate (A);\n\\draw (2.5,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,2,0.5) node [above] {$E$} coordinate (E);\n\\draw (2,2,0.5) node [above] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(F)--(E)--cycle(B)--(F);\n\\draw [dashed] (E)--(D)--(A)(D)--(C);\n\\draw [dashed] (E) ++ (1,0,0) --++ (0,-2,0) node [midway, right] {$m$} coordinate (G);\n\\draw [dashed] (G) ++ (0,0,1.5) --++ (0,0,-2) node [midway, right] {$n$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$2$}{$3$}{$4$}{$6$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -561133,7 +566317,9 @@ "id": "031376", "content": "考虑这样的等腰三角形: 它的三个顶点都在椭圆$\\Gamma: \\dfrac{x^2}{9}+y^2=1$上, 且其中恰有两个顶点为$\\Gamma$的顶点. 这样的等腰三角形的个数为\\bracket{20}\n\\fourch{$8$}{$12$}{$16$}{$20$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -561153,7 +566339,9 @@ "id": "031377", "content": "如图, $AB$是圆柱底面圆的一条直径, $AB=2, PA$是圆柱的母线, $PA=3$, 点$C$是圆柱底面圆周上的点, $\\angle ABC=30^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (1,3) ellipse (1 and 0.25);\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (1,0) ++ (-120:1 and 0.25) node [below] {$C$} coordinate (C);\n\\draw (A) ++ (0,1) node [left] {$E$} coordinate (E);\n\\draw (A) ++ (0,3) node [left] {$P$} coordinate (P);\n\\draw (A) arc (180:360:1 and 0.25);\n\\draw [dashed] (A) arc (180:0:1 and 0.25);\n\\draw (A)--(P)(B)--++(0,3);\n\\draw [dashed] (P)--(B)(P)--(C)(E)--(B)(E)--(C)(A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp$平面$PAC$;\\\\\n(2) 若点$E$在$PA$上且$EA=\\dfrac{1}{3} PA$, 求$BE$与平面$PAC$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arcsin \\dfrac{\\sqrt{15}}5$", "solution": "", @@ -561173,7 +566361,9 @@ "id": "031378", "content": "已知$f_a(x)=|x|+|x-a|$, 其中$a \\in \\mathbf{R}$.\\\\\n(1) 判断函数$y=f_a(x)$的奇偶性, 并说明理由;\\\\\n(2) 当$a=4$时, 对任意非零实数$c$, 不等式$f_a(t) \\leq 2|c+\\dfrac{1}{c}|$均成立, 求实数$t$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 当$a=0$时, $f_a(x)$是偶函数, 当$a\\ne 0$时, $f_a(x)$既不是奇函数又不是偶函数; (2) $[0,4]$", "solution": "", @@ -561193,7 +566383,9 @@ "id": "031379", "content": "某公园为了美化环境和方便顾客, 计划建造一座圆弧形拱桥, 已知该桥的剖面如图所示, 共包括一段圆弧形桥面$ACB$和两段长度相等的直线型桥面$AD, BE$, 拱桥$ACB$所在圆的半径为$3$米, 圆心$O$在$DE$上, 且$AD$和$BE$所在直线与圆$O$分别在连结点$A$和$B$处相切. 根据空间限制及桥面坡度的限制, 桥面跨度$DE$的长要求不大于$18$米, 不小于$12$米. 己知直线型桥面的修建费用是每米$0.6$万元, 弧形桥面的修建费用是每米$2.5$万元. 设$\\angle ADO=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (0,1) node [above] {$C$} coordinate (C);\n\\draw (70:1) node [above right] {$B$} coordinate (B);\n\\draw (110:1) node [above left] {$A$} coordinate (A);\n\\draw ({-1/cos(70)},0) node [below] {$D$} coordinate (D);\n\\draw ({1/cos(70)},0) node [below] {$E$} coordinate (E);\n\\draw (B)--(E)(A)--(D)(B) arc (70:110:1);\n\\draw [dashed] (-4,0) -- (4,0) (B) arc (70:-250:1);\n\\filldraw (O) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 若桥面(线段$AD, BE$和弧$ACB$)的修建总费用为$W$万元, 求$W$关于\n$\\theta$的函数表达式, 并写出$\\theta$的取值范围;\\\\\n(2) 当$\\theta$为何值时, 桥面修建总费用$W$最低?(角$\\theta$的取值精确到$0.1^{\\circ}$)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $[arcsin \\dfrac 13,\\dfrac{\\pi}6]$; (2) 约为$29.3^\\circ$", "solution": "", @@ -561214,7 +566406,9 @@ "id": "031380", "content": "如图, 已知椭圆$\\Gamma_1: \\dfrac{x^2}{8}+\\dfrac{y^2}{4}=1$的两个焦点为$F_1, F_2$, 且$F_1, F_2$为双曲线$\\Gamma_2$的顶点, 双曲线$\\Gamma_2$的一条浙近线方程为$y=-x$, 设$P$为该双曲线$\\Gamma_2$上异于顶点的任意一点, 直线$PF_1, PF_2$的斜率分别的$k_1, k_2$, 且直线$PF_1$和$PF_2$与椭圆$\\Gamma_1$的交点分别为$A, B$和$C, D$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [name path = elli, draw] (0,0) ellipse ({2*sqrt(2)} and 2);\n\\filldraw (-2,0) node [below right] {$F_1$} coordinate (F_1) circle (0.03) (2,0) node [below right] {$F_2$} coordinate (F_2) circle (0.03);\n\\draw [domain = -3:3, samples = 100] plot ({sqrt(\\x*\\x+4)},\\x);\n\\draw [domain = -3:3, samples = 100] plot ({-sqrt(\\x*\\x+4)},\\x);\n\\draw (3,{sqrt(5)}) node [above] {$P$} coordinate (P);\n\\path [name path = l1] (P) -- ($(P)!1.3!(F_1)$);\n\\path [name path = l2] (P) -- ($(P)!1.9!(F_2)$);\n\\path [name intersections = {of = elli and l1, by = {A,B}}];\n\\draw (P) -- (A) node [above] {$A$} -- (B) node [below left] {$B$};\n\\path [name intersections = {of = elli and l2, by = {C,D}}];\n\\draw (P) -- (C) node [right] {$C$} -- (D) node [below] {$D$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求双曲线$\\Gamma_2$的标准方程;\\\\\n(2) 证明: 直线$PF_1, PF_2$的斜率之积$k_1 \\cdot k_2$为定值;\\\\\n(3) 求$\\dfrac{|AB|}{|CD|}$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{4}-\\dfrac{y^2}{4}=1$; (2) $k_1k_2=1$; (3) $(\\dfrac 12,1)\\cup (1,2)$", "solution": "", @@ -561234,7 +566428,9 @@ "id": "031381", "content": "在一个有穷数列的每相邻两项之间插入这两项的和, 形成新的数列, 我们把这样的操作称为该数列的一次``和扩充''. 如数列$1,2$第$1$次``和扩充''后得到数列$1,3,2$, 第$2$次``和扩充''后得到数列$1,4,3,5,2$. 设数列$a, b, c$经过第$n$次``和扩充''后所得数列的项数记为$P_n$, 所有项的和记为$S_n$.\\\\\n(1) 若$a=1$, $b=2$, $c=3$, 求$P_2, S_2$;\\\\\n(2) 设满足$P_n \\geq 2023$的$n$的最小值为$n_0$, 求$n_0$及$S_{[\\frac{n_0}{3}]}$(其中$[x]$是指不超过$x$的最大整数, 如$[1.2]=1$, $[-2.6]=-3)$;\\\\\n(3) 是否存在实数$a, b, c$, 使得数列$\\{S_n\\}$为等比数列? 若存在, 求$a, b, c$满足的条件; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) $P_2=9$, $S_2=38$; (2) $14a+27b+14c$; (3) 存在, 满足的条件为$\\begin{cases} a+c=0, \\\\ b\\ne 0,\\end{cases}$或$\\begin{cases} a+2b+c=0, \\\\ b\\ne 0.\\end{cases}$", "solution": "", @@ -561804,7 +567000,9 @@ "id": "031397", "content": "对于非零向量$\\overrightarrow {a}$、$\\overrightarrow {b}$, 定义一种向量的运算: $\\overrightarrow {a} \\otimes \\overrightarrow {b}=\\dfrac{\\overrightarrow {a} \\cdot \\overrightarrow {b}}{\\overrightarrow {b} \\cdot \\overrightarrow {b}}$. 设集合$P=\\{\\dfrac{n}{2} | n \\in \\mathbf{N}\\}$, 若非零向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$\\overrightarrow {a} \\otimes \\overrightarrow {b} \\in P$, $\\overrightarrow {b} \\otimes \\overrightarrow {a} \\in P$, 且其夹角$\\theta \\in(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$, 求$\\overrightarrow {a} \\otimes \\overrightarrow {b}$的所有可能的值组成的集合.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\{\\dfrac 12\\}$", "solution": "", @@ -561833,7 +567031,9 @@ "id": "031398", "content": "已知函数$f(x)=2^x+x$, 若$a,b,c\\in \\mathbf{R}$, 满足$2f(b)=f(a)+f(c)$, 求证: $2b\\ge a+c$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "证明略", "solution": "", @@ -561862,7 +567062,9 @@ "id": "031399", "content": "(1) 是否存在第一象限的角$\\alpha$和第三象限的角$\\beta$, 使得$\\tan \\alpha \\tan \\beta=\\tan (\\alpha-\\beta)$? 请说明理由;\\\\\n(2) 是否存在第二象限的角$\\alpha$和第四象限的角$\\beta$, 使得$\\tan \\alpha \\tan \\beta=\\tan (\\alpha-\\beta)$? 请说明理由;\\\\\n(3) 是否存在第一象限的角$\\alpha$和第三象限的角$\\beta$, 使得$\\sin \\alpha \\sin \\beta=\\sin (\\alpha-\\beta)$? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 存在, 理由略; (2) 存在, 理由略; (3) 存在, 理由略", "solution": "", @@ -561893,7 +567095,9 @@ "id": "031400", "content": "若方程$x^4+a x-4=0$的各个实根$x_1, x_2, \\cdots, x_k$($k \\leq 4$)所对应的点$(x_i, \\dfrac{4}{x_i})$($i=1,2, \\cdots, k$)均在直线$y=x$的同侧, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-\\infty,-6)\\cup (6,+\\infty)$", "solution": "", @@ -561921,7 +567125,9 @@ "id": "031401", "content": "如图, 棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $E$为棱$CC_1$的中点, 点$P$、$Q$分别为面$A_1B_1C_1D_1$和线段$B_1C$上的动点, 则$\\triangle PEQ$周长的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(C)!0.5!(C1)$) node [right] {$E$} coordinate (E);\n\\draw ($(B1)!0.3!(C)$) node [below] {$Q$} coordinate (Q);\n\\draw ($1/3*(A1)+1/3*(B1)+1/3*(C1)$) node [left] {$P$} coordinate (P);\n\\draw (B1)--(C)(E)--(Q);;\n\\draw [dashed] (Q)--(P)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\sqrt{10}$", "solution": "", @@ -561949,7 +567155,9 @@ "id": "031402", "content": "已知$P$点是$\\triangle ABC$所在平面内一点, 若$\\overrightarrow{AB} \\perp \\overrightarrow{AC}$, $|\\overrightarrow{AB}| \\cdot|\\overrightarrow{AC}|=1$, 且$\\overrightarrow{AP}=\\dfrac{1}{|\\overrightarrow{AB}|} \\overrightarrow{AB}+\\dfrac{9}{|\\overrightarrow{AC}|} \\overrightarrow{AC}$, 则$\\overrightarrow{PB} \\cdot \\overrightarrow{PC}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -561972,7 +567180,9 @@ "id": "031403", "content": "设$a>0, b$、$c \\in \\mathbf{R}, f(x)=a x^3+b x^2+x$, 若$y=f'(x)$在区间$(-\\infty, 1]$上是严格减函数, 且$b-a^2+2 a+2 \\geq 0$, 则$\\dfrac{b-3}{a-2}$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -561995,7 +567205,9 @@ "id": "040001", "content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\ y=t^2-2\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{一支双曲线}{线段}{圆弧}{射线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -562023,7 +567235,9 @@ "id": "040002", "content": "将参数方程$\\begin{cases}x=1+2 \\cos \\theta, \\\\ y=2 \\sin \\theta\\end{cases}$($\\theta$为参数)化为普通方程, 所得方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562051,7 +567265,9 @@ "id": "040003", "content": "下列参数($t$为参数)方程中, 与$x^2-y=0$表示同一曲线的是\\bracket{20}.\n\\fourch{$\\begin{cases}x=t^2, \\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sqrt{|t|}, \\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sin t, \\\\ y=\\sin ^2 t\\end{cases}$}{$\\begin{cases}x=\\tan t, \\\\ y=\\dfrac{1-\\cos 2 t}{1+\\cos 2 t}\\end{cases}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -562091,7 +567307,9 @@ "id": "040004", "content": "参数方程$\\begin{cases}x=t+\\dfrac{1}{t}, \\\\ y=t-\\dfrac{1}{t}\\end{cases}$表示的曲线是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562119,7 +567337,9 @@ "id": "040005", "content": "曲线$\\begin{cases}x=1+2 \\cos ^2 \\theta, \\\\ y=\\sqrt{2} \\sin \\theta\\end{cases}$($\\theta$为参数, $\\theta \\in \\mathbf{R}$)与直线$y=x$的交点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562159,7 +567379,9 @@ "id": "040006", "content": "将参数方程$\\begin{cases}x=\\sin \\theta+\\cos \\theta, \\\\ y=\\sin \\theta-\\cos \\theta,\\end{cases}$ $\\theta \\in[\\dfrac{3 \\pi}{4}, \\dfrac{5 \\pi}{4}]$($\\theta$为参数)化为普通方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562189,7 +567411,9 @@ "id": "040007", "content": "经过点$P(2,1)$, 且倾斜角为$\\dfrac{2 \\pi}{3}$的直线$l$的参数方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562217,7 +567441,9 @@ "id": "040008", "content": "已知直线$l$的参数方程为: $\\begin{cases}x=1+\\dfrac{1}{2} t, \\\\ y=2-\\dfrac{\\sqrt{3}}{2} t\\end{cases}$($t$为参数), 则直线$l$的倾斜角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562245,7 +567471,9 @@ "id": "040009", "content": "已知$A(3,1), F$是抛物线$y^2=4 x$的焦点, $P$是抛物线上的一个动点, 则$\\triangle APF$周长的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562286,7 +567514,9 @@ "id": "040010", "content": "已知长度为$7$的线段$AB$的两个端点在抛物线$x^2=4 y$上运动, 则线段$AB$的中点$G$到$x$轴的距离的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562306,7 +567536,9 @@ "id": "040011", "content": "过抛物线$C: y^2=4 x$的焦点$F$的直线交$C$于$A$、$B$两点, 过$A$、$B$两点分别作$C$的准线的垂线, 垂足为$A_1$、$B_1$, 以线段$A_1B_1$为直径的圆$E$过点$M(-2,3)$, 则圆$E$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562334,7 +567566,9 @@ "id": "040012", "content": "在平面直角坐标系$x O y$中, $O$为坐标原点, 定点$A(-2,3)$, 动点$B$在曲线$x^2+4 y^2=4$上运动, 以$OA$、$OB$为两边作平行四边形$OACB$, 则动点$C$的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562362,7 +567596,9 @@ "id": "040013", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+y^2=1$($a>1$)的左、右焦点分别是$F_1$、$F_2$, 点$P$是椭圆$C$上的一点且在第一象限, $\\triangle PF_1F_2$的周长为$4+2 \\sqrt{3}$. 过点$P$作椭圆$C$的切线$l$, 分别与$x$轴和$y$轴交于$A$、$B$两点, $O$为坐标原点. 当点$P$在椭圆$C$上移动时, $\\triangle AOB$面积的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562390,7 +567626,9 @@ "id": "040014", "content": "已知椭圆$C: \\dfrac{x^2}{2}+y^2=1$, 过点$A(0,2)$的直线$l$交椭圆$C$于不同的两点$P$、$Q$. 若$\\overrightarrow{AQ}=\\lambda \\overrightarrow{AP}$, 则实数$\\lambda$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562418,7 +567656,9 @@ "id": "040015", "content": "在平面直角坐标系$x O y$中, 若直线$y=k x+1$与抛物线$x^2=2 y$相交于$A$、$B$两点.\\\\\n(1) 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值;\\\\\n(2) 若$\\triangle AOB$的面积为$2$, 求实数$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -562446,7 +567686,9 @@ "id": "040016", "content": "已知两圆$C_1: (x-2)^2+y^2=54$, $C_2: (x+2)^2+y^2=6$, 动圆$M$在圆$C_1$内部且和圆$C_1$内切、和圆$C_2$外切.\\\\\n(1) 求动圆圆心$M$的轨迹$C$的方程;\\\\\n(2) 过点$A(3,0)$的直线与(1)中的曲线$C$交于$P$、$Q$两点, 点$P$关于$x$轴对称的点为$R$, 求$\\triangle ARQ$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -562476,7 +567718,9 @@ "id": "040017", "content": "已知斜率为$k$的直线$l$经过抛物线$C: y^2=4 x$的焦点$F$, 且与抛物线$C$交于不同的两点$A(x_1, y_1)$、$B(x_2, y_2)$.\\\\\n(1) 若点$A$和$B$到抛物线准线的距离分别为$\\dfrac{3}{2}$和$3$, 求$|AB|$;\\\\\n(2) 若$|AF|+|AB|=2|BF|$, 求$k$的值;\\\\\n(3) 点$M(t, 0), t>0$, 对任意确定的实数$k$, 若$\\triangle AMB$是以$AB$为斜边的直角三角形, 判断符合条件的点$M$有几个, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -562506,7 +567750,9 @@ "id": "040018", "content": "请将下列的角的单位从角度制化为弧度制:\\\\\n(1) $45^{\\circ}=$\\blank{50};\n(2) $30^{\\circ}=$\\blank{50};\n(3) $18^{\\circ}=$\\blank{50};\n(4) $60^{\\circ}=$\\blank{50};\n(5) $75^{\\circ}=$\\blank{50};\n(6) $12^{\\circ}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\dfrac{\\pi}{4}$; (2) $\\dfrac{\\pi}{6}$; (3) $\\dfrac{\\pi}{10}$; (4) $\\dfrac{\\pi}{3}$; (5) $\\dfrac{5\\pi}{12}$; (6) $\\dfrac{\\pi}{15}$", "solution": "", @@ -562526,7 +567772,9 @@ "id": "040019", "content": "请将下列的角的单位从弧度制化为角度制:\\\\\n(1) $\\dfrac{\\pi}{3}=$\\blank{50};\n(2) $\\dfrac{\\pi}{5}=$\\blank{50};\n(3) $\\dfrac{\\pi}{4}=$\\blank{50};\n(4) $\\dfrac{5 \\pi}{12}=$\\blank{50};\n(5) $\\dfrac{2 \\pi}{9}=$\\blank{50};\n(6) $\\dfrac{3 \\pi}{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $60^{\\circ}$; (2) $36^{\\circ}$; (3) $45^{\\circ}$; (4) $75^{\\circ}$; (5) $40^{\\circ}$; (6) $54^{\\circ}$", "solution": "", @@ -562546,7 +567794,9 @@ "id": "040020", "content": "请将下列的角的单位从角度制化为弧度制:\\\\\n(1) 设$k \\in \\mathbf{Z}$, 则角$k \\times 360^{\\circ}+90^{\\circ}=$\\blank{50};\\\\\n(2) 设$k \\in \\mathbf{Z}$, 则角$k \\times 360^{\\circ}+270^{\\circ}=$\\blank{50};\\\\\n(3) 设$k \\in \\mathbf{Z}$, 则角$k \\times 360^{\\circ}+210^{\\circ}=$\\blank{50};\\\\\n(4) 设$k \\in \\mathbf{Z}$, 则角$k \\times 180^{\\circ}+45^{\\circ}=$\\blank{50};\\\\\n(5) 设$k \\in \\mathbf{Z}$, 则角$k \\times 90^{\\circ}+30^{\\circ}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $2k\\pi+\\dfrac{\\pi}{2}$; (2) $2k\\pi+\\dfrac{3\\pi}{2}$; (3) $2k\\pi+\\dfrac{7\\pi}{6}$; (4) $k\\pi+\\dfrac{\\pi}{4}$; (5) $\\dfrac{k\\pi}{2}+\\dfrac{\\pi}{6}$", "solution": "", @@ -562566,7 +567816,9 @@ "id": "040021", "content": "请将下列的角的单位从弧度制化为角度制:\\\\\n(1) 设$k \\in \\mathbf{Z}$, 则角$2 k \\pi+\\dfrac{\\pi}{3}=$\\blank{50};\\\\\n(2) 设$k \\in \\mathbf{Z}$, 则角$2 k \\pi+\\dfrac{11 \\pi}{6}=$\\blank{50};\\\\\n(3) 设$k \\in \\mathbf{Z}$, 则角$2 k \\pi-\\dfrac{7 \\pi}{6}=$\\blank{50};\\\\\n(4) 设$k \\in \\mathbf{Z}$, 则角$k \\pi-\\dfrac{\\pi}{4}=$\\blank{50};\\\\\n(5) 设$k \\in \\mathbf{Z}$, 则角$k \\cdot \\dfrac{\\pi}{2}+\\dfrac{5 \\pi}{18}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $k \\times 360^{\\circ}+60^{\\circ}$;\\\\\n(2) $k \\times 360^{\\circ}+330^{\\circ}$; \\\\\n(3) $k \\times 360^{\\circ}-210^{\\circ}$; \\\\\n(4) $k \\times 180^{\\circ}-45^{\\circ}$; \\\\\n(5) $k \\times 90^{\\circ}+50^{\\circ}$", "solution": "", @@ -562586,7 +567838,9 @@ "id": "040022", "content": "下面的各个角$\\beta$与角$\\alpha(0^{\\circ} \\leq \\alpha<360^{\\circ})$的终边重合, 请你写出相应的角$\\alpha$.\\\\\n(1) 设$\\beta=1410^{\\circ}$, 则角$\\alpha=$\\blank{100};\\\\\n(2) 设$\\beta=-120^{\\circ}$, 则角$\\alpha=$\\blank{100};\\\\\n(3) 设$\\beta=2010^{\\circ}$, 则角$\\alpha=$\\blank{100};\\\\\n. (4) 设$\\beta=-420^{\\circ}$, 则角$\\alpha=$\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $330^{\\circ}$; (2) $240^{\\circ}$; (3) $210^{\\circ}$; (4) $300^{\\circ}$", "solution": "", @@ -562606,7 +567860,9 @@ "id": "040023", "content": "下面的各个角与角$\\alpha(\\alpha \\in[0,2 \\pi))$的终边重合, 请你写出相应的角$\\alpha$.\\\\0\n(1) 设$\\beta=\\dfrac{22}{3} \\pi$, 则角$\\alpha=$\\blank{100};\\\\\n(2) 设$\\beta=-\\dfrac{13}{6} \\pi$, 则角$\\alpha=$\\blank{100};\\\\\n(3) 设$\\beta=10$, 则角$\\alpha=$\\blank{100};\\\\\n(4) 设$\\beta=-10$, 则角$\\alpha=$\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $\\dfrac{4\\pi}{3}$; (2) $\\dfrac{11\\pi}{6}$; (3) $10-2\\pi$; (4) $-10+4\\pi$", "solution": "", @@ -562626,7 +567882,9 @@ "id": "040024", "content": "在等差数列$\\{a_n\\}$中, $a_5=6, a_{10}=12$, 则$a_{15}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$18$", "solution": "", @@ -562646,7 +567904,9 @@ "id": "040025", "content": "若数列$\\{a_n\\}$为等差数列, $a_5=9, a_{11}=-3$, 则$a_8=$\\blank{50}, 公差$d=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$3$,$-2$", "solution": "", @@ -562666,7 +567926,9 @@ "id": "040026", "content": "等差数列$\\{a_n\\}$中, $a_1=51, a_2=49$.\\\\\n(1) 设$-2021$是数列$\\{a_n\\}$的的第$m$项, 则$m=$\\blank{50};\\\\\n(2) 数列$\\{a_n\\}$中的偶数项依次构成数列$\\{b_n\\}$, 则$\\{b_n\\}$的第$k$项$b_k=$\\blank{50};\\\\\n(3) 设数列$\\{a_n\\}$在区间$[-999,0]$内共有$t$项, 则$t=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "(1) $1037$; (2) $-4k+53$; (3) $500$", "solution": "", @@ -562686,7 +567948,9 @@ "id": "040027", "content": "等差数列$\\{a_n\\}$的公差小于 0 , 且有$a_2 \\cdot a_4=12, a_2+a_4=8$, 则通项$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$-2n+10$", "solution": "", @@ -562706,7 +567970,9 @@ "id": "040028", "content": "等差数列$\\{a_n\\}$中, $a_3+a_4+a_{10}+a_{11}=20$, 则$a_5+a_7+a_9=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "15", "solution": "", @@ -562726,7 +567992,9 @@ "id": "040029", "content": "在首项为 40 , 公差为$-7$的等差数列$\\{a_n\\}$中, 绝对值最小的项的序数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$7$", "solution": "", @@ -562746,7 +568014,9 @@ "id": "040030", "content": "设常数$d \\in \\mathbf{R}$. 已知等差数列$\\{a_n\\}$的公差是$d$, 首项$a_1=1$. 若$a_8$是第一个比$29$大的项, 则$d$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$(4,\\dfrac{14}{3}]$", "solution": "", @@ -562766,7 +568036,9 @@ "id": "040031", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 根据$S_n$, 求$\\{a_n\\}$的通项公式.\n(1) 若$S_n=n^2$, 则$a_n=$\\blank{50};\\\\\n(2) 若$S_n=n^2+1$, 则$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$2n-1$", "solution": "", @@ -562786,7 +568058,10 @@ "id": "040032", "content": "设常数$m, n \\in \\mathbf{R}$. 已知关于$x$的方程$(x^2-4 x+m)(x^2-4 x+n)=0$的四个根组成一个首项为$1$的等差数列, 则数对$(m, n)$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第四单元" + ], "genre": "填空题", "ans": "$(3,\\dfrac{35}{9})$或$(\\dfrac{35}{9},3)$", "solution": "", @@ -562806,7 +568081,9 @@ "id": "040033", "content": "数列$\\{a_n\\}$对于任意正整数$p, q$, 恒有$a_p+a_q=a_{p+q}$, 若$a_1=2$, 则$a_{100}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$200$", "solution": "", @@ -562826,7 +568103,9 @@ "id": "040034", "content": "已知数列$\\{a_n\\}$中, $a_n=3^n-n$, 求证: 数列$\\{a_n\\}$是严格增数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "略", "solution": "", @@ -562846,7 +568125,9 @@ "id": "040035", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n, S_n=\\begin{cases}n^2,& n=2 k-1, \\\\ n^2+1,& n=2 k,\\end{cases}$ ($k \\in \\mathbf{N}$, $k\\ge 1$), 求$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "$a_n=\\begin{cases}1, & n=1,\\\\ 2n, & n=2k, \\\\ 2n-2, & n=2k+1\\end{cases}$($k\\in \\mathbf{N}$, $k\\ge 1$)", "solution": "", @@ -562866,7 +568147,9 @@ "id": "040036", "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$的通项公式分别是$a_n=2 n+1, b_n=3 n$, $n \\in \\mathbf{N}$, $n\\ge 1$. 将集合$\\{x | x=a_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\} \\cap \\{x | x=b_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$中的元素从小到大依次排列, 构成数列$c_1, c_2, \\cdots, c_n, \\cdots$, 求数列$\\{c_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "$6n-3$", "solution": "", @@ -562886,7 +568169,9 @@ "id": "040037", "content": "已知点$M$的极坐标是$(-2,-\\dfrac{\\pi}{6})$, 它关于极轴的对称点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562914,7 +568199,9 @@ "id": "040038", "content": "在极坐标系中, 若点$A(1, \\dfrac{3 \\pi}{4})$、$B(2, \\dfrac{\\pi}{4})$, 则$A$、$B$两点间的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562942,7 +568229,9 @@ "id": "040039", "content": "若点$M$的直角坐标为$(-3,-3 \\sqrt{3})$, 若$\\rho>0$, $0 \\leq \\theta<2 \\pi$, 则点$M$的极坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562970,7 +568259,9 @@ "id": "040040", "content": "极坐标系中, 若点$A$的极坐标是$(3, \\dfrac{\\pi}{6})$, 则:\\\\\n(1) 点$A$关于极轴对称的点极坐标是\\blank{50};\\\\\n(2) 点$A$关于极点对称的点极坐标是\\blank{50};\\\\\n(3) 点$A$关于直线$\\theta=\\dfrac{\\pi}{2}$的对称点的极坐标是\\blank{50}.(规定$\\rho>0$, $\\theta \\in[0,2 \\pi)$)", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -562998,7 +568289,9 @@ "id": "040041", "content": "若直线$l$的极坐标方程为$2 \\rho \\sin (\\theta-\\dfrac{\\pi}{4})=\\sqrt{2}$, 点$A$的极坐标为$(2 \\sqrt{2}, \\dfrac{7 \\pi}{4})$, 则点$A$到直线$l$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -563026,7 +568319,9 @@ "id": "040042", "content": "若曲线$C$的极坐标方程为$\\rho=3 \\sin \\theta$, 则曲线$C$的直角坐标方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -563054,7 +568349,9 @@ "id": "040043", "content": "在极坐标系中, 圆$\\rho=8 \\sin \\theta$上的点到直线$\\theta=\\dfrac{\\pi}{3}$($\\rho \\in \\mathbf{R}$)距离的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -563082,7 +568379,9 @@ "id": "040044", "content": "若双曲线的渐近线方程为$y=\\pm 3 x$, 它的一个焦点是$(\\sqrt{10}, 0)$, 则双曲线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -563110,7 +568409,9 @@ "id": "040045", "content": "已知椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{7}=1$的右焦点为$F$, 过原点$O$作直线交椭圆于$A$、$B$两点, 点$A$在$x$轴的上方. 若三角形$ABF$的面积为$2$, 则点$A(p, q)$的纵坐标$q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -563138,7 +568439,9 @@ "id": "040046", "content": "直线$y=x+3$与曲线$\\dfrac{y^2}{9}-\\dfrac{x|x|}{4}=1$公共点的个数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -563166,7 +568469,9 @@ "id": "040047", "content": "若实数$x$、$y$满足$\\dfrac{y}{2}=\\sqrt{1-x^2}$, 则$y-2 x$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -563194,7 +568499,9 @@ "id": "040048", "content": "直线$y=k(x-3)$($k \\in \\mathbf{R}$)与双曲线$\\dfrac{x^2}{m}-\\dfrac{y^2}{27}=1$, 某学生作如下变形: 由$\\begin{cases}y=k(x-3), \\\\ \\dfrac{x^2}{m}-\\dfrac{y^2}{27}=1,\\end{cases}$消去$y$后得到形如: $a x^2+b x+c=0$的方程, 当$a=0$时该方程恒有一解; 当$a \\neq 0$时$\\Delta=b^2-4 a c \\geq 0$对$k \\in \\mathbf{R}$恒成立, 假设该学生的演算过程是正确的, 根据该学生的演算过程提供的信息, 则实数$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -563222,7 +568529,9 @@ "id": "040049", "content": "若点$A(-2,-\\dfrac{\\pi}{2})$、$B(\\sqrt{2}, \\dfrac{3 \\pi}{4})$、$0(0,0)$, 则$\\triangle ABO$为\\bracket{20}.\n\\fourch{正三角形}{直角三角形}{锐角等腰三角形}{直角等腰三角形}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -563250,7 +568559,9 @@ "id": "040050", "content": "已知动点$P(x, y)$满足$13 \\sqrt{(x+2)^2+(y-3)^2}=|12 x+5 y-1|$, 则$P$点的轨迹是\\bracket{20}.\n\\fourch{两条相交直线}{拋物线}{双曲线}{椭圆}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -563278,7 +568589,9 @@ "id": "040051", "content": "若动点$(x, y)$在曲线$\\dfrac{x^2}{4}+\\dfrac{y^2}{b^2}=1(0=latex,scale = 0.3]\n\\draw [->] (-8,0) -- (8,0) node [below] {$x$};\n\\draw [->] (0,-6) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (2,0) circle (0.06) node [below] {$M$} coordinate (M);\n\\filldraw (4,0) circle (0.06) node [below] {$F$} coordinate (F);\n\\draw (-6,0) node [below left] {$A$} coordinate (A);\n\\draw (6,0) node [below right] {$B$} coordinate (B);\n\\draw (1.5,{sqrt(18.75)}) node [above] {$P$} coordinate (P);\n\\draw (A)--(P)--(F);\n\\draw (0,0) ellipse (6 and {sqrt(20)});\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$P$的坐标;\\\\\n(2) 设$\\mathrm{M}$是椭圆长轴$\\mathrm{AB}$上的一点, $\\mathrm{M}$到直线$\\mathrm{A} P$的距离等于$|MB|$, 求椭圆上的点到点$\\mathrm{M}$的距离$d$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -563390,7 +568709,9 @@ "id": "040055", "content": "已知抛物线$y^2=2 p x$($p>0$), $a$是实数. 过动点$M(a, 0)$且斜率为$1$的直线$l$与该抛物线交于不同两点$A$、$B$, 且$|AB| \\leq 2 p$.\\\\\n(1) 求$a$的取值范围;\\\\\n(2) 若线段$AB$的垂直平分线交$x$轴于点$N$, 求三角形$NAB$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -563418,7 +568739,9 @@ "id": "040056", "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{2}-\\dfrac{y^2}{4}=1$的右顶点为$A$, 点$B$的坐标为$(1, \\sqrt{2})$.\\\\\n(1) 设双曲线$\\Gamma$的两条渐近线的夹角为$\\theta$, 求$\\cos \\theta$;\\\\\n(2) 设点$D$是双曲线$\\Gamma$上的动点, 若点$N$满足$\\overrightarrow{BN}=\\overrightarrow{ND}$, 求点$N$的轨迹方程;\\\\\n(3) 过点$B$的动直线$l$交双曲线$\\Gamma$于$P$、$Q$两个不同的点, $M$为线段$PQ$的中点, 求直线$AM$的斜率的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -563450,7 +568773,9 @@ "id": "040057", "content": "若$\\alpha$是锐角, $\\sin \\alpha=\\dfrac{3}{4}$, 则$\\cos \\alpha+\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{19}{28}\\sqrt{7}$", "solution": "", @@ -563470,7 +568795,9 @@ "id": "040058", "content": "若$\\alpha \\in(2 k \\pi+\\dfrac{\\pi}{2}, 2 k \\pi+\\pi)$, $k \\in \\mathbf{Z}$, $|\\cos \\alpha|=\\dfrac{5}{13}$, 则$\\sin \\alpha+\\cot \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{79}{156}$", "solution": "", @@ -563490,7 +568817,9 @@ "id": "040059", "content": "若$\\tan \\alpha=3$, 则$\\dfrac{\\sin \\alpha+\\cos \\alpha}{\\sin \\alpha-\\cos \\alpha}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -563510,7 +568839,9 @@ "id": "040060", "content": "设$m$是常数, 且$-1-\\dfrac{2 \\pi}{3}$. 当$a=\\dfrac{1}{2}$时, 方程$\\cos x=a$在$[-\\dfrac{2 \\pi}{3}, m]$内恰有一个$x$, 则$m$的取值范围为\\blank{50};\\\\\n(3) 常数$a$使得: 对于任意常数$m>-\\dfrac{2 \\pi}{3}$, 方程$\\cos x=a$在$[-\\dfrac{2 \\pi}{3}, m]$内总是至少有一个$x$存在, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "(1) $[-\\dfrac{1}{2},\\dfrac{1}{2})\\cup\\{1\\}$; (2) $[-\\dfrac{\\pi}{3},\\dfrac{\\pi}{3})$; (3) $\\{-\\dfrac{1}{2}\\}$", "solution": "", @@ -563750,7 +569103,9 @@ "id": "040072", "content": "设$\\dfrac{3 \\pi}{2}<\\alpha<2 \\pi$, 化简:\\\\\n(1) $\\sqrt{\\tan ^2 \\alpha+\\cot ^2 \\alpha+2}$;\\\\\n(2) $\\sqrt{\\dfrac{1}{1-\\cos \\alpha}+\\dfrac{1}{1+\\cos \\alpha}}(\\dfrac{3 \\pi}{2}<\\alpha<2 \\pi)$;\\\\\n(3) $\\tan (\\dfrac{9 \\pi}{4}+\\theta) \\tan (\\dfrac{3 \\pi}{4}+\\theta)$;\\\\\n(4) $\\dfrac{1+\\sin (\\pi-x)}{\\cos (\\pi-x)}-\\dfrac{\\cos (\\pi+x)}{1+\\sin (\\pi+x)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $-\\tan \\alpha-\\cot \\alpha$; (2) $-\\dfrac{\\sqrt{2}}{\\sin \\alpha}$; (3) $-1$; (4) $0$", "solution": "", @@ -563770,7 +569125,9 @@ "id": "040073", "content": "证明下列恒等式:\\\\\n(1) $(\\sin \\alpha+\\cos \\alpha)^2=1+2 \\sin \\alpha \\cos \\alpha$;\\\\\n(2) $(\\sin \\alpha-\\cos \\alpha)^2=1-2 \\sin \\alpha \\cos \\alpha$;\\\\\n(3) $(\\sin \\alpha+\\cos \\alpha)^2+(\\sin \\alpha-\\cos \\alpha)^2=2$;\\\\\n(4) $(\\sin \\alpha+\\cos \\alpha)^2-(\\sin \\alpha-\\cos \\alpha)^2=4 \\sin \\alpha \\cos \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "略", "solution": "", @@ -563790,7 +569147,9 @@ "id": "040074", "content": "已知$\\sin \\alpha$、$\\cos \\alpha$是方程$8 x^2+6 k x+2 k+1=0$的两个实数根, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{10}{9}$", "solution": "", @@ -563810,7 +569169,9 @@ "id": "040075", "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_n=\\dfrac{a_{n-1}}{3 a_{n-1}+1}$, 那么它的通项公式是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$a_n=\\dfrac{1}{3n-2}$", "solution": "", @@ -563830,7 +569191,9 @@ "id": "040076", "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n-1} a_n=a_{n-1}-a_n$, 那么它的通项公式是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$a_n=\\dfrac{1}{n}$", "solution": "", @@ -563850,7 +569213,9 @@ "id": "040077", "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_n=5 a_{n-1}+5^n$, 那么它的通项公式是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$(n-\\dfrac{4}{5})5^n$", "solution": "", @@ -563870,7 +569235,9 @@ "id": "040078", "content": "已知数列$\\{a_n\\}$满足$a_1=1$且$a_{n+1}=2 a_n+3$, 则$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$2^{n+1}-3$", "solution": "", @@ -563890,7 +569257,9 @@ "id": "040079", "content": "已知数列$\\{a_n\\}$满足: $a_1=1$, $a_{n+1}-a_n \\in\\{a_1, a_2, \\cdots, a_n\\}$($n \\geq 1$), 记数列$\\{a_n\\}$的前$n$项和为$S_n$, 若对所有满足条件的数列$\\{a_n\\}$, $S_{10}$的最大值为$M$. 最小值为$m$, 则$M+m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$1078$", "solution": "", @@ -563910,7 +569279,9 @@ "id": "040080", "content": "数列$\\{a_n\\}$中, $a_n=\\begin{cases}2^n, & n=2 k-1, \\\\ 2 n, & n=2 k\\end{cases}$($k \\geq 1$). 求$\\{a_n\\}$的前$n$项和$S_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "$S_n=\\begin{cases}\\dfrac{n^2}{2}+n-\\dfrac 23+\\dfrac 23\\cdot 2^n, & n\\text{为偶数},\\\\ \\dfrac{n^2}{2}-\\dfrac 76+\\dfrac 23\\cdot 2^{n+1}, & n\\text{为奇数} \\end{cases}$", "solution": "", @@ -563930,7 +569301,9 @@ "id": "040081", "content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, 当$n \\geq 1$时$a_{n+1}>a_n$, 且满足$(a_{n+1}+a_n-1)^2=4 a_{n+1} a_n$, 设$b_n=\\sqrt{a_n}$.\\\\\n(1) 求证: 数列$\\{b_n\\}$是等差数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) 略； (2) $n^2$", "solution": "", @@ -563950,7 +569323,9 @@ "id": "040082", "content": "数列$\\{a_n\\}$中, $a_n=2 n-1$, $n$为正整数.\\\\\n(1) 是否存在等差数列$\\{b_n\\}$, 其前$n$项和$S_n$满足$\\{x | x=S_n, n$为正整数$\\} \\subseteq\\{x | x=a_n, n$为正整数$\\}$? 若存在, 找出一个这样的等差数列$\\{b_n\\}$; 若不存在, 说明理由;\\\\\n(2) 是否存在等比数列$\\{c_n\\}$, 其前$n$项和$T_n$满足$\\{x | x=T_n, n$为正整数$\\} \\subseteq\\{x | x=a_n, n$为正整数$\\}$? 若存在, 找出一个这样的等比数列$\\{c_n\\}$; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) 不存在； (2) 存在, 如$c_n=2^{n-1}$", "solution": "", @@ -563970,7 +569345,9 @@ "id": "040083", "content": "化简: $\\sin (\\alpha-30^{\\circ}) \\cdot \\sin \\alpha+\\cos (\\alpha-30^{\\circ}) \\cdot \\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}{2}$", "solution": "", @@ -563990,7 +569367,9 @@ "id": "040084", "content": "化简: $\\cos x+\\cos (120^{\\circ}-x)+\\cos (120^{\\circ}+x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$0$", "solution": "", @@ -564010,7 +569389,9 @@ "id": "040085", "content": "方程$\\cos x=1$, $x \\in[-2 \\pi, 0]$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{0,-2\\pi\\}$", "solution": "", @@ -564030,7 +569411,9 @@ "id": "040086", "content": "已知$\\tan x=-\\dfrac{\\sqrt{3}}{3}$, $x \\in[-\\pi, \\pi)$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{\\pi}6,\\dfrac 56\\pi$", "solution": "", @@ -564050,7 +569433,9 @@ "id": "040087", "content": "化简: $\\dfrac{\\cos (\\alpha+\\beta)+\\cos (\\alpha-\\beta)}{\\sin (\\alpha+\\beta)+\\sin (\\alpha-\\beta)}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\cot \\alpha$", "solution": "", @@ -564070,7 +569455,9 @@ "id": "040088", "content": "若$\\tan \\alpha=2$, 则$\\dfrac{\\sin (\\alpha+\\dfrac{\\pi}{3})}{\\sin (\\alpha-\\dfrac{\\pi}{3})}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$7+4\\sqrt{3}$", "solution": "", @@ -564090,7 +569477,9 @@ "id": "040089", "content": "若$\\sin (\\alpha-\\dfrac{\\pi}{4})=\\dfrac{\\sqrt{6}-\\sqrt{2}}{4}$, 则$\\cos (\\alpha+\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{2}-\\sqrt{6}}{4}$", "solution": "", @@ -564110,7 +569499,9 @@ "id": "040090", "content": "若$\\alpha$是锐角, 且$\\sin (\\alpha-\\dfrac{\\pi}{6})=\\dfrac{1}{6}$, 则$\\sin \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}+\\sqrt{35}}{12}$", "solution": "", @@ -564130,7 +569521,9 @@ "id": "040091", "content": "已知$\\cos \\alpha=\\dfrac{1}{7}$, $\\cos (\\alpha+\\beta)=-\\dfrac{11}{14}$, 且$\\alpha$、$\\beta$均为锐角, 则$\\cos \\beta$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 12$", "solution": "", @@ -564150,7 +569543,9 @@ "id": "040092", "content": "若$\\dfrac{\\sin (A-B)}{\\sin (A+B)}=\\dfrac{2}{3}$, 则$\\dfrac{\\tan A}{\\tan B}$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -564170,7 +569565,9 @@ "id": "040093", "content": "若$\\cos A+\\cos B+\\cos C=0$, $\\sin A+\\sin B+\\sin C=0$, 则$\\cos (A-B)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac 12$", "solution": "", @@ -564190,7 +569587,9 @@ "id": "040094", "content": "满足方程$\\sin (2 x+\\dfrac{\\pi}{4})=\\cos (\\dfrac{\\pi}{6}-x)$的最小的正角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{12}$", "solution": "", @@ -564225,7 +569624,9 @@ "id": "040095", "content": "方程$2 \\sin ^2 x-5 \\cos x-4=0$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|x=\\pm\\frac 23 \\pi+2k\\pi,k \\in \\mathbf{Z}\\}$", "solution": "", @@ -564245,7 +569646,9 @@ "id": "040096", "content": "若$x=\\dfrac{\\pi}{3}$是方程$2 \\cos (x+\\theta)=1$的解, 其中$\\theta \\in(0,2 \\pi)$, 则$\\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 43 \\pi$", "solution": "", @@ -564265,7 +569668,9 @@ "id": "040097", "content": "若$\\alpha$、$\\beta \\in(0, \\dfrac{\\pi}{2})$, 则下列各式中成立的序号为\\blank{50}.\\\\\n\\textcircled{1} $\\cos (\\alpha+\\beta)>0$; \\textcircled{2} $\\cos (\\alpha+\\beta)<0$; \\textcircled{3} $\\cos (\\alpha-\\beta)<\\cos (\\alpha+\\beta)$; \\textcircled{4} $\\cos (\\alpha-\\beta)>\\cos (\\alpha+\\beta)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\textcircled{4}$", "solution": "", @@ -564285,7 +569690,9 @@ "id": "040098", "content": "若$S_n=\\sin \\dfrac{\\pi}{7}+\\sin \\dfrac{2 \\pi}{7}+\\sin \\dfrac{3 \\pi}{7}+\\ldots+\\sin \\dfrac{n \\pi}{7}$($n \\geq 1$, $n \\in \\mathbf{N}$), 则在$S_1, S_2, S_3, \\ldots, S_{100}$中, 正数的个数是\\bracket{20}.\n\\fourch{$16$}{$72$}{$86$}{$100$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -564305,7 +569712,9 @@ "id": "040099", "content": "若$\\sin \\alpha=\\dfrac{1}{3}$, $\\cos \\beta=-\\dfrac{1}{2}$, 且$\\alpha \\in(0, \\dfrac{\\pi}{2})$, $\\beta \\in(\\dfrac{\\pi}{2}, \\pi)$, 求$\\cos (\\alpha+\\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{-2\\sqrt{2}-\\sqrt{3}}6$", "solution": "", @@ -564325,7 +569734,9 @@ "id": "040100", "content": "已知$\\cos (\\alpha+\\beta)=\\dfrac{4}{5}$, $\\cos (\\alpha-\\beta)=-\\dfrac{4}{5}$, $\\alpha+\\beta \\in(\\dfrac{7 \\pi}{4}, 2 \\pi)$, $\\alpha-\\beta \\in(\\dfrac{3 \\pi}{4}, \\pi)$, 求$\\cos 2 \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac 7{25}$", "solution": "", @@ -564345,7 +569756,9 @@ "id": "040101", "content": "已知锐角$\\alpha, \\beta, \\gamma$满足$\\sin \\alpha+\\sin \\gamma=\\sin \\beta$, $\\cos \\alpha-\\cos \\gamma=\\cos \\beta$, 求$\\alpha-\\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac {\\pi}3$", "solution": "", @@ -564367,7 +569780,9 @@ "id": "040102", "content": "在直角坐标系$x O y$中, 线段$OM$绕着原点$O$逆时针旋转大小为$\\dfrac{\\pi}{2}$的角至$OM'$. 设点$M$的坐标为$(\\dfrac{5}{13}, \\dfrac{12}{13})$, 求点$M'$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$(-\\dfrac {12}{13}, \\dfrac{5}{13})$", "solution": "", @@ -564387,7 +569802,9 @@ "id": "040103", "content": "在直角坐标系$x O y$中, 线段$OM$绕着原点$O$顺时针旋转大小为$\\dfrac{\\pi}{3}$的角至$OM'$. 设点$M$的坐标为$(5,12)$, 求点$M'$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$(\\dfrac {5-12\\sqrt{3}}{2}, \\dfrac{12-5\\sqrt{3}}{2})$", "solution": "", @@ -564407,7 +569824,9 @@ "id": "040104", "content": "已知$\\sin A=4 \\sin (A+B)$, 求证: $\\tan (A+B)=\\dfrac{\\sin B}{\\cos B-4}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "略", "solution": "", @@ -564427,7 +569846,9 @@ "id": "040105", "content": "若$\\sin \\alpha=\\dfrac{8}{17}$, $\\cos \\beta=-\\dfrac{5}{13}$, $\\alpha, \\beta \\in(\\dfrac{\\pi}{2}, \\pi)$, 则$\\cos (\\alpha-\\beta)=\\cos (\\alpha+\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac {171} {221}, -\\dfrac {21} {221}$", "solution": "", @@ -564447,7 +569868,9 @@ "id": "040106", "content": "方程$\\cos x=-1$, $x \\in[-2 \\pi, 0]$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{-\\pi\\}$", "solution": "", @@ -564467,7 +569890,9 @@ "id": "040107", "content": "已知$\\alpha \\in(0, \\dfrac{\\pi}{2})$, $\\beta \\in(0, \\dfrac{\\pi}{2})$.如果$\\cos \\beta=\\dfrac{2 \\sqrt{2}}{3}$, $\\cos (\\alpha+\\beta)=\\dfrac{3}{5}$, 则$\\sin \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{8\\sqrt{2}-3}{15}$", "solution": "", @@ -564487,7 +569912,9 @@ "id": "040108", "content": "化简$\\dfrac{\\sin (\\theta-5 \\pi)}{\\tan (3 \\pi-\\theta)} \\cdot \\dfrac{\\cot (\\dfrac{\\pi}{2}-\\theta)}{\\tan (\\theta-\\dfrac{3 \\pi}{2})} \\cdot \\dfrac{\\cos (8 \\pi-\\theta)}{\\sin (-\\theta-4 \\pi)}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\sin \\theta$", "solution": "", @@ -564509,7 +569936,9 @@ "id": "040109", "content": "已知$\\alpha, \\beta \\in(\\dfrac{3 \\pi}{4}, \\pi)$, $\\sin (\\alpha+\\beta)=-\\dfrac{3}{5}$, $\\sin (\\beta-\\dfrac{\\pi}{4})=\\dfrac{12}{13}$, 则$\\cos (\\alpha+\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{56}{65}$", "solution": "", @@ -564529,7 +569958,9 @@ "id": "040110", "content": "已知$\\sin \\alpha=\\dfrac{\\sqrt{5}}{5}$, $\\sin \\beta=\\dfrac{\\sqrt{10}}{10}$, $\\alpha$、$\\beta$都是锐角, 求$\\alpha+\\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac {\\pi}4$", "solution": "", @@ -564551,7 +569982,9 @@ "id": "040111", "content": "求证: $\\dfrac{1-2 \\sin \\alpha \\cos \\alpha}{\\tan \\alpha-1}=\\dfrac{1}{\\tan \\alpha+\\cot \\alpha}-\\cos ^2 \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "略", "solution": "", @@ -564571,7 +570004,9 @@ "id": "040112", "content": "已知$\\dfrac{\\cos ^4 \\alpha}{\\cos ^2 \\beta}+\\dfrac{\\sin ^4 \\alpha}{\\sin ^2 \\beta}=1$. 求证: $\\dfrac{\\sin ^4 \\beta}{\\cos ^2 \\alpha}+\\dfrac{\\sin ^4 \\beta}{\\sin ^2 \\alpha}=1$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "略", "solution": "", @@ -564591,7 +570026,9 @@ "id": "040113", "content": "已知$\\sin x=\\dfrac{3}{5}$, $\\cos y=\\dfrac{4}{5}$, 其中$x, y$是第一象限角, 则$\\cos (x+y)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564613,7 +570050,9 @@ "id": "040114", "content": "已知$\\sin x=\\dfrac{3}{5}$, 则$\\cos 2 x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564633,7 +570072,9 @@ "id": "040115", "content": "已知$\\cos (\\alpha+\\dfrac{\\pi}{6})=\\dfrac{3}{5}$, $\\alpha \\in[0, \\pi)$, 则$\\sin \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564655,7 +570096,9 @@ "id": "040116", "content": "已知$\\tan \\alpha=-\\dfrac{1}{7}$, $\\tan (2 \\alpha-\\beta)=2$, 则$\\tan (\\alpha-\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564677,7 +570120,9 @@ "id": "040117", "content": "化简: $(\\cos ^4 \\theta-\\sin ^4 \\theta) \\dfrac{\\tan \\theta}{1-\\tan ^2 \\theta}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564699,7 +570144,9 @@ "id": "040118", "content": "$\\cos 12^{\\circ} \\cos 24^{\\circ} \\cos 48^{\\circ} \\cos 96^{\\circ}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564719,7 +570166,9 @@ "id": "040119", "content": "$\\cos ^2 \\alpha+\\cos ^2(\\dfrac{\\pi}{3}+\\alpha)+\\cos ^2(\\dfrac{\\pi}{3}-\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564739,7 +570188,9 @@ "id": "040120", "content": "$\\sin \\dfrac{\\alpha}{2} \\cos \\dfrac{\\alpha}{2}=\\dfrac{1}{6}$, 则$\\sin \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564759,7 +570210,9 @@ "id": "040121", "content": "若$\\cos (\\dfrac{\\pi}{4}-\\theta) \\cdot \\cos (\\dfrac{\\pi}{4}+\\theta)=\\dfrac{1}{3}$, 则$\\sin ^4 \\theta+\\cos ^4 \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564782,7 +570235,9 @@ "id": "040122", "content": "已知$\\dfrac{3 \\pi}{2}<\\alpha<2 \\pi$, 化简: $\\sqrt{\\dfrac{1}{2}-\\dfrac{1}{2} \\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2} \\cos 2 \\alpha}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564804,7 +570259,9 @@ "id": "040123", "content": "若$x$为锐角, 则$\\sqrt{1+\\sin x}+\\sqrt{1-\\sin x}-\\sqrt{2+2 \\cos x}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564826,7 +570283,9 @@ "id": "040124", "content": "已知$\\dfrac{\\pi}{2}<\\alpha<\\pi$, $-\\pi<\\beta<0$, $\\tan \\alpha=-\\dfrac{1}{3}$, $\\tan \\beta=-\\dfrac{1}{7}$, 则$2 \\alpha+\\beta$的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564846,7 +570305,9 @@ "id": "040125", "content": "已知$x, y \\in(0, \\dfrac{\\pi}{2})$, $\\sin x=\\dfrac{\\sqrt{2}}{10}$, $\\sin y=\\dfrac{\\sqrt{10}}{10}$, 则$2 x+4 y$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564866,7 +570327,9 @@ "id": "040126", "content": "已知$\\dfrac{\\pi}{2}<\\alpha<\\pi$, 且$\\sin \\alpha=\\dfrac{3}{5}$, $\\tan (\\alpha-\\beta)=-1$, 则$2 \\cos ^2 \\beta-\\dfrac{4}{5} \\tan \\dfrac{\\alpha}{2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -564886,7 +570349,9 @@ "id": "040127", "content": "设$\\alpha, \\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, $\\tan \\alpha$、$\\tan \\beta$是一元二次方程$x^2+3 \\sqrt{3} x-8=0$的两个根, 求$\\alpha+\\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -564908,7 +570373,9 @@ "id": "040128", "content": "化简$\\sin ^2 \\alpha \\cdot \\sin ^2 \\beta+\\cos ^2 \\alpha \\cdot \\cos ^2 \\beta-\\dfrac{1}{2} \\cdot \\cos 2 \\alpha \\cdot \\cos 2 \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -564930,7 +570397,9 @@ "id": "040129", "content": "已知$\\dfrac{\\pi}{4}<\\alpha<\\dfrac{\\pi}{2}$, 且$\\dfrac{2 \\sin ^2 \\alpha+\\sin 2 \\alpha}{1+\\tan \\alpha}=k$, 试用$k$表示$\\sin \\alpha-\\cos \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -564952,7 +570421,9 @@ "id": "040130", "content": "已知: $\\cos (\\alpha-\\dfrac{\\beta}{2})=-\\dfrac{1}{9}$, $\\sin (\\dfrac{\\alpha}{2}-\\beta)=\\dfrac{2}{3}$, 且$\\dfrac{\\pi}{2}<\\alpha<\\pi, 0<\\beta<\\dfrac{\\pi}{2}$. 求$\\cos (\\alpha+\\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -564974,7 +570445,9 @@ "id": "040131", "content": "若$\\cos \\alpha=\\dfrac{4}{5}$, 且$\\alpha$是第四象限角, 则$\\tan \\alpha+\\cot \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{25}{12}$", "solution": "", @@ -564994,7 +570467,9 @@ "id": "040132", "content": "若$\\sin 2 \\alpha=\\dfrac{4}{5}$, 则$\\tan \\alpha+\\cot \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 52$", "solution": "", @@ -565014,7 +570489,9 @@ "id": "040133", "content": "若$\\sin \\alpha=\\dfrac{\\sqrt{5}}{5}, \\alpha \\in(0, \\dfrac{\\pi}{2}), \\cos \\beta=\\dfrac{\\sqrt{10}}{10}, \\beta \\in(0, \\dfrac{\\pi}{2})$, 则$\\alpha-\\beta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{\\pi}4$", "solution": "", @@ -565034,7 +570511,9 @@ "id": "040134", "content": "计算: $\\cos 76^{\\circ} \\cos 44^{\\circ}+\\cos 166^{\\circ} \\cos 46^{\\circ}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac 12$", "solution": "", @@ -565054,7 +570533,9 @@ "id": "040135", "content": "若$\\tan \\alpha=3$, 则$\\dfrac{\\sin ^2 \\alpha-\\sin \\alpha \\cos \\alpha}{\\sin ^2 \\alpha+1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 6{19}$", "solution": "", @@ -565074,7 +570555,9 @@ "id": "040136", "content": "若$\\cos (\\dfrac{\\pi}{6}-\\alpha)=\\dfrac{\\sqrt{3}}{3}$, 则$\\cos (\\dfrac{5 \\pi}{6}+\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac {\\sqrt{3}}3$", "solution": "", @@ -565096,7 +570579,9 @@ "id": "040137", "content": "若$\\tan (\\alpha+\\beta)=\\dfrac{2}{5}, \\tan (\\alpha-\\dfrac{\\pi}{4})=\\dfrac{1}{4}$, 则$\\tan (\\beta+\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 3{22}$", "solution": "", @@ -565116,7 +570601,9 @@ "id": "040138", "content": "已知$\\sin (\\alpha+\\beta)=\\dfrac{2}{3}, \\sin (\\alpha-\\beta)=\\dfrac{2}{5}$, 求$\\dfrac{\\tan \\alpha}{\\tan \\beta}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$4$", "solution": "", @@ -565136,7 +570623,9 @@ "id": "040139", "content": "已知$\\dfrac{\\pi}{2}<\\beta<\\alpha<\\dfrac{3 \\pi}{4}, \\cos (\\alpha-\\beta)=\\dfrac{12}{13}, \\sin (\\alpha+\\beta)=-\\dfrac{3}{5}$, 求$\\cos 2 \\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{63}{65}$", "solution": "", @@ -565156,7 +570645,9 @@ "id": "040140", "content": "直线$y=2 x+b$被圆$x^2+y^2=4$所截得的弦长为$2 \\sqrt{2}$, 则实数$b$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565188,7 +570679,9 @@ "id": "040141", "content": "过$P(4,3)$的直线$l$与抛物线$y^2=4 x$相交于$A$、$B$两点, 且$P$为线段$AB$的中点, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565220,7 +570713,9 @@ "id": "040142", "content": "若过点$(0,2)$且斜率为$k$的直线$l$与双曲线$x^2-y^2=1$的右支有两个不同的公共点, 则$k$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565252,7 +570747,9 @@ "id": "040143", "content": "设直线$l$过点$A(0,2)$且与抛物线$y^2=4 x$只有一个公共点, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565284,7 +570781,9 @@ "id": "040144", "content": "椭圆$x^2+\\dfrac{y^2}{4}=1$上的点$P(x, y)$到定点$(0,1)$的最远距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565316,7 +570815,9 @@ "id": "040145", "content": "已知动圆$P$过点$A(3,0)$, 且与圆$(x+3)^2+y^2=4$相外切, 则动圆圆心$P$的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565348,7 +570849,9 @@ "id": "040146", "content": "已知$A(3,1), F$是抛物线$y^2=4 x$的焦点, $P$是抛物线上的一个动点, 则$\\triangle APF$周长的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565389,7 +570892,9 @@ "id": "040147", "content": "设两条不同的直线$l_1: a_1 x+b_1 y+1=0$和$l_2: a_2 x+b_2 y+1=0$交于点$(2,3)$, 则过点$P(a_1, b_1)$和$Q(a_2, b_2)$的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565421,7 +570926,9 @@ "id": "040148", "content": "设双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0, b>0)$的左、右焦点分别为$F_1$、$F_2$, 其渐近线方程为$y=\\pm \\dfrac{4}{3} x$. 若双曲线$C$的实轴长为$6$, 点$P$为$C$的右支上一点, 且$|PF_2|=|F_1F_2|$, 则$\\triangle PF_1F_2$的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565453,7 +570960,9 @@ "id": "040149", "content": "若$a, b, c$成等差数列, 则直线$a x+b y+c=0$被椭圆$\\dfrac{x^2}{2}+\\dfrac{y^2}{8}=1$截得线段的中点的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565485,7 +570994,9 @@ "id": "040150", "content": "巳知圆$C$和圆$\\begin{cases}x=4+4 \\cos \\theta, \\\\ y=5+4 \\sin \\theta\\end{cases}$($\\theta$为参数)关于直线$\\begin{cases}x=\\dfrac{1}{\\sqrt{10}} t,\\\\ y=3-\\dfrac{1}{\\sqrt{10}} t\\end{cases}$($t$为参数)对称, 则圆$C$的普通方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565519,7 +571030,9 @@ "id": "040151", "content": "已知双曲线方程$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$, 点$P$是双曲线上的点, $F_1$、$F_2$是它的左、右焦点, 且$|PF_1| \\cdot|PF_2|=32$, 则$\\overrightarrow{PF_1} \\cdot \\overrightarrow{PF_2}$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565551,7 +571064,9 @@ "id": "040152", "content": "$P$是椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$上的点, $F_1, F_2$是两个焦点, 则$|PF_1| \\cdot|PF_2|$的最大值与最小值之差是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565583,7 +571098,9 @@ "id": "040153", "content": "已知双曲线$x^2-\\dfrac{y^2}{3}=1$的右焦点为$F$, $P$是双曲线坐支上的一点, $M(0,2)$, 则$\\triangle PFM$周长的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565615,7 +571132,9 @@ "id": "040154", "content": "已知斜率为$\\dfrac{1}{2}$的直线$l$与抛物线$y^2=2 p x$($p>0$)交于$x$轴上方的不同两点$A, B$, 记直线$OA, OB$的斜率分别为$k_1, k_2$, 则$k_1+k_2$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565647,7 +571166,9 @@ "id": "040155", "content": "过抛物线$y^2=16 x$的焦点作一条直线与抛物线相交于$A$、$B$两点, 它们的横坐标之和等于$15$, 则这样的直线\\bracket{20}.\n\\fourch{有且仅有一条}{有且仅有两条}{有无穷多条}{不存在}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -565679,7 +571200,9 @@ "id": "040156", "content": "方程$\\dfrac{x^2}{2 \\cos \\theta+3}+\\dfrac{y^2}{\\sin \\theta-2}=1$表示的曲线是\\bracket{20}.\n\\twoch{焦点在$x$轴上的椭圆}{焦点在$y$轴上的椭圆}{焦点在$x$轴上的双曲线}{焦点在$y$轴上的双曲线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -565711,7 +571234,9 @@ "id": "040157", "content": "过点$A(1,3)$的直线$l$与$x$轴、$y$轴分别交于$P, Q$两点, 求使$|AP| \\cdot|AQ|$最小的直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -565743,7 +571268,9 @@ "id": "040158", "content": "设$k \\in \\mathbf{R}$, 直线$l: a x-y+1=0$, 双曲线$C: 3 x^2-y^2=1$, 若直线$l$与双曲线$C$有两个公共点$A$、$B$.\\\\\n(1) 若以线段$AB$为直径的圆过坐标原点, 求$k$的值;\\\\\n(2) 若$\\angle AOB$为钝角(其中$O$为坐标原点), 求$k$的取值范围;\\\\\n(3) 是否存在这样的实数$k$, 使得$A$、$B$关于直线$y=2 x$对称? 若存在, 求出$k$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -565775,7 +571302,9 @@ "id": "040159", "content": "椭圆$\\dfrac{x^2}{m^2}+\\dfrac{y^2}{144}=1$($m>0$)与双曲线$\\dfrac{x^2}{n^2}-\\dfrac{y^2}{9}=1$($n>0$)有公共的焦点$F_1$和$F_2, P$为它们的一个交点, 试求$\\angle F_1PF_2$及$\\triangle F_1PF_2$的面积.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -565795,7 +571324,9 @@ "id": "040160", "content": "在平面直角坐标系$x O y$中, 已知椭圆$\\Gamma: \\dfrac{x^2}{m^2+1}+\\dfrac{y^2}{m^2}=1$($m>0$)的左、右焦点分别为$F_1$、$F_2$, $Q$是椭圆上任意一点, 且$\\triangle F_1QF_2$的周长为$2+2 \\sqrt{2}$. 过$F_2$作两条互相垂直的弦$AB$、$CD$, 设$AB$、$CD$中点分别为$M$、$N$.\\\\\n(1) 求椭圆$\\Gamma$的标准方程;\\\\\n(2) 证明: 直线$MN$必过定点, 并求出此定点坐标;\\\\\n(3) 若弦$AB$、$CD$的斜率均存在, 求$\\triangle FMN$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -565827,7 +571358,9 @@ "id": "040161", "content": "函数$f(x)=\\lg (4-2 x)$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565859,7 +571392,9 @@ "id": "040162", "content": "已知函数$f(x)=\\begin{cases}2 \\sin x,& 0 \\leq x \\leq 2 \\pi, \\\\ x^2, & x<0 .\\end{cases}$若$f(f(x_0))=3$, 则$x_0=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565891,7 +571426,9 @@ "id": "040163", "content": "若函数$f(x)=a^x$($a>0$, $a \\neq 1$)的反函数图像过点$(2,-1)$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565923,7 +571460,9 @@ "id": "040164", "content": "若函数$f(x)=x^2+a x+1$是偶函数, 则函数$y=\\dfrac{f(x)}{|x|}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565955,7 +571494,9 @@ "id": "040165", "content": "设$a$为常数, 函数$f(x)=x^2-4 x+3$. 若$f(x)$在$[a,+\\infty)$上是增函数, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -565987,7 +571528,9 @@ "id": "040166", "content": "已知函数$y=f(x)$和函数$y=\\log _2(x+1)$的图像关于直线$x-y=0$对称, 则函数$y=f(x)$的解析式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -566019,7 +571562,9 @@ "id": "040167", "content": "若关于$x,y$的二元一次方程组$\\begin{cases}m x-y+3=0, \\\\ (2 m-1) x+y-4=0\\end{cases}$有唯一一组解, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -566051,7 +571596,9 @@ "id": "040168", "content": "函数$y=f(x)$的定义域为$[-1,0) \\cup(0,1]$, 其图像上任一点$P(x, y)$满足$x^2+y^2=1$.\\\\\n\\textcircled{1} 函数$y=f(x)$一定是偶函数;\\\\\n\\textcircled{2} 函数$y=f(x)$可能既不是偶函数, 也不是奇函数;\\\\\n\\textcircled{3} 函数$y=f(x)$可以是奇函数;\\\\\n\\textcircled{4} 函数$y=f(x)$如果是偶函数, 则值域是$[0,1)$或$(-1,0]$;\\\\\n\\textcircled{5} 函数$y=f(x)$值域是$(-1,1)$, 则$y=f(x)$一定是奇函数.\\\\\n其中正确命题的序号是\\blank{50}(填上所有正确的序号).", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -566083,7 +571630,9 @@ "id": "040169", "content": "如果$M$是函数$y=f(x)$图像上的点, $N$是函数$y=g(x)$图像上的点, 且$M, N$两点之间的距离$|MN|$能取到最小值$d$, 那么将$d$称为函数$y=f(x)$与$y=g(x)$之间的距离. 按这个定义, 函数$f(x)=x$和$g(x)=\\sqrt{-x^2+4 x-3}$之间的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -566115,7 +571664,9 @@ "id": "040170", "content": "已知$f(x)=4-\\dfrac{1}{x}$, 若存在区间$[a, b] \\subseteq(0,+\\infty)$, 使得$\\{y | y=f(x), x \\in[a, b]\\}=[m a, m b]$, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -566147,7 +571698,9 @@ "id": "040171", "content": "在导数定义中``当$h \\to 0$时, $\\dfrac{f(x_0+h)-f(x)}{h} \\to f'(x_0)$'', $h$\\bracket{20}.\n\\twoch{恒取正值}{恒取正值或恒取负值}{有时可取$0$}{可取正值可取负值, 但不能取零}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -566179,7 +571732,9 @@ "id": "040172", "content": "自由落体运动公式为$s(t)=\\dfrac{1}{2} g t^2$($g=10 \\text{m} / \\text{s}^2$). 若$v=\\dfrac{s(1+\\Delta t)-s(1)}{\\Delta t}$, 则下列说法正确的是\\bracket{20}.\n\\onech{$v$是在$0-1$秒这段时间内的速度}{$v$是$1$秒到$(1+\\Delta t)$秒这段时间内的速度}{$5 \\Delta t+10$是物体在$t=1$秒这一时刻的速度}{$5 \\Delta t+10$是物体从$1$秒到$(1+\\Delta t)$秒这段时间内的平均速度}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -566211,7 +571766,9 @@ "id": "040173", "content": "若$f(x)=x^2-x+a$, $f(-m)<0$, 则$f(m+1)$的值\\bracket{20}.\n\\fourch{是正数}{是负数}{是非负数}{与$m$有关}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -566243,7 +571800,9 @@ "id": "040174", "content": "设函数$f(x)=\\dfrac{1}{x}$, $g(x)=-x^2+b x$. 若$y=f(x)$的图像与$y=g(x)$的图像有且仅有两个不同的公共点$A(x_1$, $y_1), B(x_2, y_2)$, 则下列判断正确的是\\bracket{20}.\n\\twoch{$x_1+x_2>0$, $y_1+y_2>0$}{$x_1+x_2>0$, $y_1+y_2<0$}{$x_1+x_2<0$, $y_1+y_2>0$}{$x_1+x_2<0$, $y_1+y_2<0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -566275,7 +571834,9 @@ "id": "040175", "content": "函数$f(x)=a x^2+b x+c$($a \\neq 0$)的图像关于直线$x=-\\dfrac{b}{2 a}$对称, 据此可推测, 对任意的非零实数$a, b, c, m, n, p$, 关于$x$的方程$m(f(x))^2+n f(x)+p=0$的解集不可能是\\bracket{20}.\n\\fourch{$\\{1,2\\}$}{$\\{1,4\\}$}{$\\{1,2,3,4\\}$}{$\\{1,4,16,64\\}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -566307,7 +571868,9 @@ "id": "040176", "content": "求曲线$y=\\cos x$在$x=\\dfrac{\\pi}{2}$处的切线方程.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -566339,7 +571902,9 @@ "id": "040177", "content": "直线$y=-x+b$是下列函数的切线吗? 如果是, 请求出$b$的值; 如果不是, 请说明理由.\\\\\n(1) $f(x)=\\ln x$;\\\\\n(2) $f(x)=\\dfrac{1}{x}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -566371,7 +571936,9 @@ "id": "040178", "content": "已知某产品的总成本函数为$C=Q^2+2Q$, 总成本函数在$Q_0$处的导数$f'(Q_0)$称为在$Q_0$处的边际成本, 用$MC(Q_0)$表示.求边际成本$MC(500)$并说明它的实际意义.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -566403,7 +571970,9 @@ "id": "040179", "content": "如图, 在函数$f(x)=x^3$的图像$\\Gamma$上任取一点$A$(非原点), 过点$A$作$\\Gamma$的切线$l_1$, 交$\\Gamma$于另外一点$B$, 再过点$B$作$\\Gamma$的切线$l_2$, 交$\\Gamma$于另外一点$C$, 过点$C$作$\\Gamma$的切线$l_3$, 交$l_1$于点$D$, 试问: $\\dfrac{|AD|}{|AB|}$是否为定值? 若是, 求出该定值; 若不是, 请说明理由.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.7]\n\\draw [->] (-1,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = -0.79:1.25] plot (\\x,{pow(\\x,3)});\n\\draw (0.3,{pow(0.3,3)}) node [above] {$A$} coordinate (A);\n\\draw (-0.6,{pow(-0.6,3)}) node [below] {$B$} coordinate (B);\n\\draw (1.2,{pow(1.2,3)}) node [right] {$C$} coordinate (C);\n\\draw ($(A)!-0.6!(B)$) node [above right] {$D$} coordinate (D);\n\\draw ($(B)!-0.1!(D)$) -- ($(B)!1.3!(D)$) node [right] {$l_1$};\n\\draw ($(B)!-0.1!(C)$) -- ($(B)!1.1!(C)$) node [right] {$l_2$};\n\\draw ($(C)!-0.1!(D)$) -- ($(C)!1.2!(D)$) node [below] {$l_3$};\n\\foreach \\i in {A,B,C,D}\n{\\filldraw (\\i) circle (0.015);};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -566435,7 +572004,9 @@ "id": "040180", "content": "设函数$g(x)=\\sqrt{x}+1$, 函数$h(x)=\\dfrac{1}{x+3}$, $x \\in(-3, a]$, 其中$a$为常数, 且$a>0$. 令$f(x)$为函数$g(x)$和$h(x)$的积函数.\\\\\n(1) 求函数$f(x)$的表达式, 并求出其定义域;\\\\\n(2) 当$a=\\dfrac{1}{4}$时, 求函数$f(x)$的值域;\\\\\n(3) 是否存在自然数$a$, 使得函数$f(x)$的值域恰为$[\\dfrac{1}{3}, \\dfrac{1}{2}]$? 若存在, 试写出所有满足条件的自然数$a$所构成的集合; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -566467,7 +572038,9 @@ "id": "040181", "content": "已知$\\sin x=\\dfrac{3}{5}, \\cos y=\\dfrac{4}{5}$, 其中$x, y$是第一象限角, 则$\\cos (x+y)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 7{25}$", "solution": "", @@ -566489,7 +572062,9 @@ "id": "040182", "content": "若角$\\alpha$满足$\\dfrac{1}{2} \\cos \\alpha-\\dfrac{\\sqrt{3}}{2} \\sin \\alpha=1$, 则$\\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{\\pi}3+2k\\pi,k \\in \\mathbf{Z}$", "solution": "", @@ -566520,7 +572095,9 @@ "id": "040183", "content": "已知$\\cos (\\alpha+\\dfrac{\\pi}{6})=\\dfrac{3}{5}$, $\\alpha \\in[0, \\pi)$, 则$\\sin \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{4\\sqrt{3}-3}{10}$", "solution": "", @@ -566542,7 +572119,9 @@ "id": "040184", "content": "若$3 \\sin \\alpha=4 \\cos \\alpha$, 则$\\tan (\\alpha-\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 17$", "solution": "", @@ -566562,7 +572141,9 @@ "id": "040185", "content": "将$4 \\sin \\alpha-4 \\cos \\alpha$化成$A \\sin (\\alpha+\\varphi)$(其中$A>0$, $\\varphi \\in[0,2 \\pi)$)的形式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$4\\sqrt{2} \\sin(\\alpha+\\dfrac {7}{4}\\pi))$", "solution": "", @@ -566582,7 +572163,9 @@ "id": "040186", "content": "已知$\\tan \\alpha=-\\dfrac{1}{7}$, $\\tan (2 \\alpha-\\beta)=2$, 则$\\tan (\\alpha-\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -566604,7 +572187,9 @@ "id": "040187", "content": "化简: $\\cos ^2 \\alpha+\\cos ^2(\\dfrac{\\pi}{3}+\\alpha)+\\cos ^2(\\dfrac{\\pi}{3}-\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 32$", "solution": "", @@ -566626,7 +572211,9 @@ "id": "040188", "content": "求值: $\\tan 36^{\\circ}+\\sqrt{3} \\tan 24^{\\circ} \\tan 36^{\\circ}+\\tan 24^{\\circ}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\sqrt{3}$", "solution": "", @@ -566651,7 +572238,9 @@ "id": "040189", "content": "若$A+B=-\\dfrac{\\pi}{4}$, 则$(\\tan A-1)(\\tan B-1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -566671,7 +572260,9 @@ "id": "040190", "content": "若$\\cos (\\dfrac{\\pi}{4}-\\theta) \\cdot \\cos (\\dfrac{\\pi}{4}+\\theta)=\\dfrac{1}{3}$, 则$\\sin ^4 \\theta+\\cos ^4 \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac {13}{18}$", "solution": "", @@ -566694,7 +572285,9 @@ "id": "040191", "content": "已知$\\dfrac{\\pi}{2}<\\boldsymbol{\\alpha}<\\pi$, $-\\pi<\\beta<0$, $\\tan \\alpha=-\\dfrac{1}{3}$, $\\tan \\beta=-\\dfrac{1}{7}$, 则$2 \\alpha+\\beta$的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{7}{4}\\pi$", "solution": "", @@ -566714,7 +572307,9 @@ "id": "040192", "content": "已知$\\dfrac{\\pi}{2}<\\alpha<\\pi$, 且$\\sin \\alpha=\\dfrac{3}{5}$, $\\tan (\\alpha-\\beta)=-1$, 则$2 \\cos ^2 \\beta-\\dfrac{4}{5} \\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{64}{25}$", "solution": "", @@ -566734,7 +572329,9 @@ "id": "040193", "content": "若$\\alpha$、$\\beta$均为锐角, 且$\\sin \\alpha=\\dfrac{m}{\\sqrt{1+m^2}}, \\tan \\beta=\\dfrac{1}{m}$, 则$\\alpha+\\beta$的值为\\bracket{20}.\n\\fourch{$150^{\\circ}$}{$120^{\\circ}$}{$90^{\\circ}$}{$60^{\\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -566754,7 +572351,9 @@ "id": "040194", "content": "若$\\sin 2 x \\sin 3 x=\\cos 2 x \\cos 3 x$, 则$x$的一个值是\\bracket{20}.\n\\fourch{$18^{\\circ}$}{$36^{\\circ}$}{$30^{\\circ}$}{$45^{\\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -566774,7 +572373,9 @@ "id": "040195", "content": "下列关系中, 角$\\alpha$存在的是\\bracket{20}.\n\\twoch{$\\sin \\alpha+\\cos \\alpha=\\dfrac{3}{2}$}{$\\sin \\alpha+\\cos \\alpha=\\dfrac{4}{3}$}{$\\sin \\alpha=\\dfrac{1}{3}$且$\\cos \\alpha=\\dfrac{2}{3}$}{$\\cos \\alpha-\\sin \\alpha=-\\sqrt{3}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -566805,7 +572406,10 @@ "id": "040196", "content": "若$f(x)=2^{\\sin x}, g(x)=2^{\\cos x}, x \\in \\mathbf{R}$, 则积函数$f(x) \\cdot g(x)$必有\\bracket{20}.\n\\fourch{最大值$4$}{最小值$4$}{最大值$2^{\\sqrt{2}}$}{最小值$2^{\\sqrt{2}}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第一单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -566825,7 +572429,9 @@ "id": "040197", "content": "设$\\alpha, \\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, $\\tan \\alpha$、$\\tan \\beta$是一元二次方程$x^2+3 \\sqrt{3} x-8=0$的两个根, 求$\\alpha+\\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{\\pi}6$", "solution": "", @@ -566847,7 +572453,9 @@ "id": "040198", "content": "已知$\\alpha$是$\\triangle ABC$的一个内角, 且满足$\\sin \\alpha-\\sqrt{3} \\cos \\alpha=\\sqrt{3}$, 求$\\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac 23 \\pi$", "solution": "", @@ -566867,7 +572475,9 @@ "id": "040199", "content": "已知$5 \\sin \\beta=\\sin (2 \\alpha+\\beta)$, 求$\\tan (\\alpha+\\beta) \\cot \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac 32$", "solution": "", @@ -566887,7 +572497,9 @@ "id": "040200", "content": "已知$\\dfrac{\\pi}{4}<\\alpha<\\dfrac{\\pi}{2}$, 且$\\dfrac{2 \\sin ^2 \\alpha+\\sin 2 \\alpha}{1+\\tan \\alpha}=k$, 试用$k$表示$\\sin \\alpha-\\cos \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\sqrt{1-k}$", "solution": "", @@ -566909,7 +572521,9 @@ "id": "040201", "content": "已知: $\\cos (\\alpha-\\dfrac{\\beta}{2})=-\\dfrac{1}{9}$, $\\sin (\\dfrac{\\alpha}{2}-\\beta)=\\dfrac{2}{3}$, 且$\\dfrac{\\pi}{2}<\\alpha<\\pi$, $0<\\beta<\\dfrac{\\pi}{2}$. 求$\\cos (\\alpha+\\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-\\dfrac{484}{729}$", "solution": "", @@ -566931,7 +572545,9 @@ "id": "040202", "content": "设函数$y=f_i(x)$($i=1,2,3$)的导数分别为$y'=f_i'(x)$($i=1,2,3$).\\\\\n(1) 若$f_1(x)=\\sqrt{x}$, $f_2(x)=\\sqrt[3]{x}$, $f_3(x)=\\dfrac{1}{\\sqrt{x}}$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(2) 若$f_1(x)=x^2-\\mathrm{e}^x$, $f_2(x)=\\ln x+\\cos x$, $f_3(x)=\\sqrt{x}+\\sin x$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(3) 若$f_1(x)=x^3 \\cdot \\mathrm{e}^x$, $f_2(x)=x \\ln x$, $f_3(x)=\\dfrac{x}{\\sin x}$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(4) 若$f_1(x)=\\mathrm{e}^{2 x-1}$, $f_2(x)=2^x$, $f_3(x)=(\\dfrac{1}{3})^{x+1}$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(5) 若$f_1(x)=\\lg x$, $f_2(x)=\\ln (2 x-1)$, $f_3(x)=\\log _3(3-2 x)$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.\\\\\n(6) 若$f_1(x)=\\sin (x-\\dfrac{\\pi}{3})$, $f_2(x)=\\cos (\\dfrac{x}{3}+\\dfrac{\\pi}{4})$, $f_3(x)=\\tan (2 x)$, 则\\\\$f_1'(x)=$\\blank{50}; $f_2'(x)=$\\blank{50}; $f_3'(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "(1) $\\dfrac 12 x^{-\\frac 12}$; (2) $\\dfrac 13 x^{-\\frac 23}$; (3) $-\\dfrac 12 x^{-\\frac 32}$; (4) $2x-\\mathrm{e}^x$; (5) $\\dfrac 1x-\\sin x$; (6) $\\dfrac 12 x^{-\\frac 12}+\\cos x$; (7) $\\mathrm{e}^x(x^3+3x^2)$; (8) $1+\\ln x$; (9) $\\dfrac{\\sin x-x\\cos x}{\\sin^2 x}$; (10) $2\\mathrm{e}^{2x-1}$; (11) $2^x\\ln 2$; (12) $-(\\dfrac 13)^{x+1}\\ln 3$; (13) $\\dfrac{1}{x\\ln 10}$; (14) $\\dfrac{2}{2x-1}$; (15) $\\dfrac{2}{(2x-3)\\ln 3}$; (16) $\\cos (x-\\dfrac\\pi 3)$; (17) $-\\dfrac 13 \\sin(\\dfrac x 3+\\dfrac \\pi 4)$; (18) $\\dfrac{2}{\\cos^2(2x)}$", "solution": "", @@ -566968,7 +572584,9 @@ "id": "040203", "content": "对于函数$y=f(x)$, 先求导数$y'=f'(x)$, 再写$y=f(x)$的单调区间.\\\\\n(1) 若$f(x)=x^3-x^2-x$, 导数$f'(x)=$\\blank{100};\\\\$y=f(x)$的递增区间为\\blank{100}; 递减区间为\\blank{100}.\\\\\n(2) 若$f(x)=x-\\ln x$, 导数$f'(x)=$\\blank{100};\\\\$y=f(x)$的递增区间为\\blank{100}; 递减区间为\\blank{100}.\\\\\n(3) 若$f(x)=\\sin (\\dfrac{\\pi}{3}-2 x)$, 导数$f'(x)=$\\blank{100};\\\\$y=f(x)$的递增区间为\\blank{100}; 递减区间为\\blank{100}.\\\\\n(4) 若$f(x)=x \\cdot 2^x$, 导数$f'(x)=$\\blank{100};\\\\$y=f(x)$的递增区间为\\blank{100}; 递减区间为\\blank{100}.\\\\\n(5) 若$f(x)=2 x^2-\\sqrt{x}$, 导数$f'(x)=$\\blank{100};\\\\$y=f(x)$的递增区间为\\blank{100}; 递减区间为\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -566994,7 +572612,9 @@ "id": "040204", "content": "设$f(x)=\\cos x$. \\\\\n(1) 函数$y=f(x)$的驻点的集合为\\blank{50};\\\\\n(2) 函数$y=x+2 f(x)$的驻点的集合为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567020,7 +572640,9 @@ "id": "040205", "content": "设函数$y=f(x)$的导数为$y'=f'(x)$. 若$f'(1)=-3$.\\\\\n(1) $\\displaystyle\\lim _{h \\to 0} \\dfrac{f(1+3 h)-f(1)}{h}=$\\blank{50};\\\\\n(2) $\\displaystyle\\lim _{h \\to 0} \\dfrac{f(1+h)-f(1-h)}{h}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "(1) $-9$; (2) $-6$", "solution": "", @@ -567057,7 +572679,9 @@ "id": "040206", "content": "若直线$l$是曲线$f(x)=x^3+x^2$在$x=1$处的切线.\\\\\n(1) $f(1)=$\\blank{50};\n(2) 直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567083,7 +572707,9 @@ "id": "040207", "content": "设$a \\in \\mathbf{R}$, 函数$f(x)=a x^3+3 x^2+2$的导数为$y=f'(x)$.\\\\\n(1) 若$a=-1$, 则$f'(x)=$\\blank{50};\\\\\n(2) 若$f'(-1)=6$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567109,7 +572735,9 @@ "id": "040208", "content": "设$a, b, k \\in \\mathbf{R}$, 已知直线$y=k x+1$与曲线$y=x^3+a x+b$相切于点$(1,3)$.\\\\ \n(1) $k=$\\blank{50};\\\\\n(2) 数对$(a, b)$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567135,7 +572763,9 @@ "id": "040209", "content": "曲线$y=f(x)$在$x=1$处的切线为直线$y=x-2$.\\\\\n(1) $f(1)=$\\blank{50};\\\\ \n(2) $f'(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567161,7 +572791,9 @@ "id": "040210", "content": "点$P$在曲线$y=x^3-3 x^2$上移动, 设$P$处的该曲线的切线的斜率为$k$, 倾斜角为$\\alpha$.\\\\\n(1) $k$的取值范围是\\blank{50};\\\\\n(2) $\\alpha$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567187,7 +572819,9 @@ "id": "040211", "content": "设$a, b, c \\in \\mathbf{R}$, $a \\neq 0$, 函数$f(x)=a x^2+b x+c$的导数为$y=f'(x)$.\\\\\n(1) 设$\\lambda, \\mu \\in \\mathbf{R}$, 且$a: b: c=1: \\lambda: \\mu$, $f'(x)=b x+c$, 则$\\lambda+2 \\mu=$\\blank{50};\\\\\n(2) 若关于$x$的的不等式$f'(x)>0$的解集为$(-\\infty, 1)$, 则关于$x$的的不等式$f(x)>c$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567213,7 +572847,9 @@ "id": "040212", "content": "设$m \\in \\mathbf{R}$, 若点$P$为曲线$y=x^3(x>0)$上一动点, 过$P$作直线$b$平行于直线$a: y=3 x+m$.\\\\\n(1) 若直线$b$是该曲线的切线, 则$P$的坐标为\\blank{50};\\\\\n(2) 若点$P$总不在直线$a$上, 则$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567239,7 +572875,9 @@ "id": "040213", "content": "设$n \\in \\mathbf{R}$, 若点$Q$为曲线$y=\\ln x$上一动点, 过$P$作直线$d$平行于直线$c: y=3 x+n$.\\\\\n(1) 若直线$d$是该曲线的切线, 则$Q$的坐标为\\blank{50};\\\\\n(2) 若点$Q$总不在直线$c$上, 则$n$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567265,7 +572903,9 @@ "id": "040214", "content": "某种动物的体温$T$(单位: 摄氏度) 与太阳落山后的时间$t$(单位: 分钟) 满足函数关系$T(t)=\\dfrac{120}{t+5}+15$. 若在$t=t_0$时刻该动物体温的瞬时变化率是$-2$摄氏度/分钟, 则$t_0=$\\blank{50}分钟.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567291,7 +572931,9 @@ "id": "040215", "content": "吹一个球形气球时, 气球半径将随着空气容量$V$的增加而变化. 当$V=8$时, 气球的瞬时膨胀率(即气球半径关于气球内空气容量的瞬时变化率)为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567317,7 +572959,9 @@ "id": "040216", "content": "设函数$y=f(x)$的导数为$y'=f'(x)$. 若$f(x)=x^2+2 x \\cdot f'(1)$, 则$f'(1)+f'(0)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567343,7 +572987,9 @@ "id": "040217", "content": "设函数$f(x)=\\dfrac{\\mathrm{e}^x-\\mathrm{e}^{-x}}{\\mathrm{e}^{2 x}}$的导数$f'(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567369,7 +573015,9 @@ "id": "040218", "content": "设函数$f(x)=x^2 \\cos 3 x+\\lg (-x)$的导数$f'(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567395,7 +573043,9 @@ "id": "040219", "content": "设函数$f(x)=\\sqrt{x} \\tan x$的导数$f'(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567421,7 +573071,9 @@ "id": "040220", "content": "设函数$f(x)=x \\cdot \\mathrm{e}^x \\cdot \\ln x$的导数$f'(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -567447,7 +573099,9 @@ "id": "040221", "content": "完成``求函数$y=f(x)$的导数$y'=f'(x)$''的过程与结论.\\\\\n(1) $y=\\dfrac{1-\\sin x}{1+\\cos x}$;\\\\\n(2) $y=\\dfrac{x^5+\\sqrt{x}+\\sin x}{x^2}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -567473,7 +573127,9 @@ "id": "040222", "content": "已知函数$y=f(x)$的导数$y=f'(x)$的图像如下图所示, 则$y=f(x)$的图像可能是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:3] plot (\\x,{\\x*(\\x-2)});\n\\draw (2,0.2) -- (2,0) node [below] {$2$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.4:2.2] plot (\\x,{-\\x*(\\x-1)*(\\x-2)});\n\\draw [dashed] ({(3-sqrt(3))/3},{-2/3/sqrt(3)}) -- ({(3-sqrt(3))/3},0) node [above] {$1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.2:2.4] plot (\\x,{1.1*\\x*(\\x-1)*(\\x-2)});\n\\draw (2,0.2) -- (2,0) node [below] {$2$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = -1:2.85] plot (\\x,{\\x*\\x-\\x*\\x*\\x/3-0.5});\n\\draw [dashed] (2,{4-8/3-0.5}) -- (2,0) node [below] {$2$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:2.85] plot (\\x,{-\\x*\\x+\\x*\\x*\\x/3+1});\n\\draw [dashed] (2,{-4+8/3+1}) -- (2,0) node [above] {$2$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -567511,7 +573167,9 @@ "id": "040223", "content": "已知函数$y=f(x)$的图像如下图所示, 则导数$y=f'(x)$的图像可能是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3, samples = 100] plot (\\x,{(2*pow(pow(\\x,2)-5,2)/25-2)*0.75});\n\\draw [dashed] (-3,{-18*0.75/25}) -- (-3,0) node [above] {$-3$};\n\\draw [dashed] (3,{-18*0.75/25}) -- (3,0) node [above] {$3$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-3,3) -- (3,-3);\n\\draw [dashed] (-3,3) -- (-3,0) node [below] {$-3$};\n\\draw [dashed] (3,-3) -- (3,0) node [above] {$3$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot (\\x,{pow(\\x,2)/1.5-3});\n\\draw [dashed] (-3,3) -- (-3,0) node [below] {$-3$};\n\\draw [dashed] (3,3) -- (3,0) node [below] {$3$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot (\\x,{-pow(\\x,2)/1.5+3});\n\\draw [dashed] (-3,-3) -- (-3,0) node [above] {$-3$};\n\\draw [dashed] (3,-3) -- (3,0) node [above] {$3$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot (\\x,{(pow(\\x,2)-5)*\\x*8/25*0.75});\n\\draw [dashed] (-3,-3) -- (-3,0) node [above] {$-3$};\n\\draw [dashed] (3,3) -- (3,0) node [below] {$3$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -567537,7 +573195,9 @@ "id": "040224", "content": "已知函数$y=f(x)$的图像如下图所示, 则下列数值排序正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 1:5.5] plot ({pow(\\x,3)/pow(5.5,3)*6+0.2},\\x);\n\\foreach \\i in {1,2,...,5}\n{\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$};\n\\draw (0,\\i) -- (-0.2,\\i) node [left] {$\\i$};};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$0=latex,scale = 0.6]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0) node [right] {$B$} coordinate (B);\n\\filldraw (3,0) node [below] {$D$} coordinate (D) circle (0.03);\n\\draw (B) ++ (120:{(1+sqrt(33))/2}) node [above] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--cycle(C)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若线段$AD$的长为$3$, 求$\\sin \\angle BCD$的值;\\\\\n(2) 若$\\triangle BCD$的面积为$\\sqrt{3}$, 求点$A$到直线$BC$的距离.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{3}}6$; (2) $\\dfrac{\\sqrt{39}+\\sqrt{3}}2$", "solution": "", @@ -568410,7 +574152,9 @@ "id": "040266", "content": "函数$y=\\sqrt{\\sin x-\\dfrac{1}{2}}$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|\\dfrac{\\pi}6+2k\\pi \\le x \\le \\dfrac 56 \\pi+2k\\pi, k \\in \\mathbb{Z} \\}$", "solution": "", @@ -568430,7 +574174,9 @@ "id": "040267", "content": "函数$y=\\sqrt{\\sin x}+\\dfrac{1}{\\sqrt{9-x^2}}$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[0,3)$", "solution": "", @@ -568450,7 +574196,9 @@ "id": "040268", "content": "函数$y=\\cos (\\dfrac{\\pi x}{2}-\\dfrac{1}{4})$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -568470,7 +574218,9 @@ "id": "040269", "content": "函数$y=|\\sin x|$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi$", "solution": "", @@ -568490,7 +574240,9 @@ "id": "040270", "content": "函数$y=|1+2 \\sin 2 x|$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi$", "solution": "", @@ -568510,7 +574262,9 @@ "id": "040271", "content": "函数$y=|\\sin x|+|\\cos x|$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{2}$", "solution": "", @@ -568530,7 +574284,9 @@ "id": "040272", "content": "若函数$y=f(x)$满足$f(x)+f(x+1)=0$, 且当$x \\in[0,1)$时, $f(x)=\\sin x+1$, 则$f(7.5)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\sin{\\dfrac 12 -1}$", "solution": "", @@ -568550,7 +574306,9 @@ "id": "040273", "content": "下列函数中是周期函数的有\\blank{50}.\\\\\n\\textcircled{1} $y=\\sin |x|$; \\textcircled{2} $y=\\cos |x|$; \\textcircled{3} $y=\\sin x+2 \\cos x$; \\textcircled{4} $y=x+\\sin x$; \\textcircled{5} $y=\\sin ^2 x$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{3}\\textcircled{5}", "solution": "", @@ -568570,7 +574328,9 @@ "id": "040274", "content": "若$\\dfrac{a}{\\sin A}=\\dfrac{b}{\\cos B}=\\dfrac{c}{\\cos C}$, 则$\\triangle ABC$的形状是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "等腰直角三角形", "solution": "", @@ -568590,7 +574350,9 @@ "id": "040275", "content": "函数$y=\\sqrt{\\sin x-\\cos x}$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|\\dfrac{\\pi}4+2k\\pi \\le x \\le \\dfrac 45 \\pi+2k\\pi, k \\in \\mathbb{Z} \\}$", "solution": "", @@ -568612,7 +574374,9 @@ "id": "040276", "content": "函数$y=\\cos (\\dfrac{x}{2}-\\dfrac{\\pi}{4})$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$4\\pi$", "solution": "", @@ -568632,7 +574396,9 @@ "id": "040277", "content": "函数$y=|\\sin 2 x|$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}{2}$", "solution": "", @@ -568652,7 +574418,9 @@ "id": "040278", "content": "在$\\triangle ABC$中, 角$A, B, C$所对的边分别是$a, b, c$. $\\tan A=1$, $\\tan B=2$, $c=3$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\sqrt{5}$", "solution": "", @@ -568672,7 +574440,9 @@ "id": "040279", "content": "在$\\triangle ABC$中, $a=\\sqrt{5}$, $b=\\sqrt{7}$, $c=2 \\sqrt{3}$, 则$b c \\cos A+c a \\cos B+a b \\cos C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$12$", "solution": "", @@ -568692,7 +574462,9 @@ "id": "040280", "content": "在$\\triangle ABC$中, $A=60^\\circ$, 且最长边长和最短边长分别是方程$x^2-6 x+7=0$的两个根, 则此三角形的周长是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$6+\\sqrt{15}$", "solution": "", @@ -568712,7 +574484,9 @@ "id": "040281", "content": "对于$\\triangle ABC$, 下列四个命题中, 正确命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 若$\\sin A=\\cos B$, 则$\\triangle ABC$总是直角三角形; \\textcircled{2} 若$\\sin 2A=\\sin 2B$, 则$\\triangle ABC$总是等腰三角形;\n\\textcircled{3} 若$\\tan A \\tan B<1$, 则$\\triangle ABC$总是钝角三角形; \\textcircled{4} 若$\\cos (A-B) \\cos (A-C)=1$, 则$\\triangle ABC$总是等边三角形.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{3} \\textcircled{4}", "solution": "", @@ -568732,7 +574506,9 @@ "id": "040282", "content": "在$\\triangle ABC$中, 内角$A, B, C$所对的边长分别是$a, b, c$.\\\\\n(1) 若$a=4, C=\\dfrac{\\pi}{3}$, 且$\\triangle ABC$的面积$S=\\sqrt{3}$, 求$b, c$的值;\\\\\n(2) 若$\\sin (B+A)+\\sin (B-A)=\\sin 2A$, 试判断$\\triangle ABC$的形状", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$(1)b=1,c=\\sqrt{13}$;\\\\\n$(2)$等腰三角形或直角三角形", "solution": "", @@ -568752,7 +574528,9 @@ "id": "040283", "content": "已知集合$S=\\{x | x=2 n+1, n \\in \\mathbf{Z}\\}$, $T=\\{x | x=4 n \\pm 1, n \\in \\mathbf{Z}\\}$, 问$S$、$T$之间的关系如何? 并证明.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "$S=T$", "solution": "", @@ -568772,7 +574550,9 @@ "id": "040284", "content": "已知命题$p$: ``对任意$x \\in[-1,1]$, 不等式$x^2-x-m<0$''是真命题.\\\\\n(1) 求实数$m$的范围;\\\\\n(2) 若命题$q$: ``$-40$, 设$p$: 函数$y=c^x$在$\\mathbf{R}$上严格单调递减; $q$: 不等式$x+|x-2 c|>1$的解集是$\\mathbf{R}$, 如果$p$和$q$中有且只有一个正确, 求实数$c$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -568812,7 +574594,9 @@ "id": "040286", "content": "已知``存在$x \\in\\{x |-10$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -568852,7 +574638,9 @@ "id": "040288", "content": "解不等式组$\\begin{cases}x<0, \\\\ \\dfrac{3-x}{3+x}>|\\dfrac{2-x}{2+x}|.\\end{cases}$", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -568872,7 +574660,9 @@ "id": "040289", "content": "设不等式$x^2-2 a x+a+2 \\leq 0$的解集为$M$, 若$M \\subseteq[1,4]$, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -568892,7 +574682,9 @@ "id": "040290", "content": "求使$\\sqrt{x}+\\sqrt{y} \\leq a \\sqrt{x+y}$, ($x>0$, $y>0$)恒成立的$a$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -568914,7 +574706,9 @@ "id": "040291", "content": "若集合$A=\\{a, b, 2\\}$, 集合$B=\\{2, b^2, 2 a\\}$, $a$、$b \\in \\mathbf{R}$, 且$A=B$, 则实数$b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -568934,7 +574728,9 @@ "id": "040292", "content": "王昌龄《从军行》中有``黄沙百战穿金甲, 不破楼兰终不还''之句, 其中``攻破楼兰''是``返回家乡''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -568954,7 +574750,9 @@ "id": "040293", "content": "设集合$P_1=\\{x | x^2+a x+1>0\\}$, $P_2=\\{x | x^2+a x+2>0\\}$, $Q_1=\\{x | x^2+x+b>0\\}$, $Q_2=\\{x | x^2+2 x+b>0\\}$, 其中$a$、$b \\in \\mathbf{R}$, 下列说法正确的是\\bracket{20}.\n\\onech{对任意$a$, $P_1$是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{对任意$a$, $P_1$是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{存在$a$, $P_1$不是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -568974,7 +574772,9 @@ "id": "040294", "content": "已知$A=\\{x | \\dfrac{x^2+3 x+2}{x-1}>0\\}$, $B=\\{x | x^2+a x+b \\leq 0\\}$, 且有$A \\cup B=\\{x | x+2>0\\}$, $A \\cap B=\\{x | 11$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -569034,7 +574839,9 @@ "id": "040297", "content": "若$x>0, y>0, x+y=1$, 求证: $(1+\\dfrac{1}{x})(1+\\dfrac{1}{y}) \\geq 9$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -569054,7 +574861,9 @@ "id": "040298", "content": "已知三数$a$、$b$、$c$满足$a+b+c=1$, $a^2+b^2+c^2=1$, 且$a>b>c$.\\\\\n(1) 求$a+b$的取值范围;\\\\\n(2) $a^2+b^2$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -569074,7 +574883,9 @@ "id": "040299", "content": "已知函数$f(x)$的导函数为$f'(x)$, 且满足$f(x)=2 x f'(\\mathrm{e})+\\ln x$, 则$f'(\\mathrm{e})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569104,7 +574915,9 @@ "id": "040300", "content": "设函数$f(x)$在$\\mathbf{R}$上存在导函数$f'(x)$, $f(x)$的图像在点$M(1, f(1))$处的切线方程为$y=\\dfrac{1}{2} x+2$, 那么$f(1)+f'(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569134,7 +574947,9 @@ "id": "040301", "content": "曲线$y=x \\mathrm{e}^x+2 x+1$在点$(0,1)$处的切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569164,7 +574979,9 @@ "id": "040302", "content": "曲线$y=(x-1) \\mathrm{e}^x$在$(1,0)$处的切线与坐标轴围成的三角形面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569194,7 +575011,9 @@ "id": "040303", "content": "若$f(x)=-\\dfrac{1}{2} x^2+a \\ln (x+2)$在$[-1,+\\infty)$上是减函数, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569224,7 +575043,10 @@ "id": "040304", "content": "如图, 水以恒速(即单位时间内注入水的体积相同)注入下面四种底面积相同的容器中, 试分别找出各容器对应的水面高度$h$与时间$t$的函数图像\\blank{50}.\n\\begin{center}\n\\textcircled{1}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,2) ellipse (1 and 0.25);\n\\draw (-1,0) arc (180:360:1 and 0.25);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.25);\n\\draw (-1,0) --++ (0,2) (1,0) --++ (0,2);\n\\end{tikzpicture}\n\\textcircled{2}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,2) ellipse (2 and 0.5);\n\\draw (-1,0) arc (180:360:1 and 0.25);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.25);\n\\draw (-1,0) --++ (-1,2) (1,0) --++ (1,2);\n\\end{tikzpicture}\n\\textcircled{3}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,2) ellipse (1 and 0.25);\n\\draw (-2,0) arc (180:360:2 and 0.5);\n\\draw [dashed] (-2,0) arc (180:0:2 and 0.5);\n\\draw (-2,0) --++ (1,2) (2,0) --++ (-1,2);\n\\end{tikzpicture}\n\\textcircled{4}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,2) ellipse (1 and 0.25);\n\\draw (-1.5,1) arc (180:360:1.5 and 0.375);\n\\draw [dashed] (-1.5,1) arc (180:0:1.5 and 0.375);\n\\draw (-1,0) arc (180:360:1 and 0.25);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.25);\n\\draw (-1,0) --++ (-0.5,1) --++ (0.5,1) (1,0) --++ (0.5,1) --++ (-0.5,1);\n\\end{tikzpicture}\n\\end{center}\n\\begin{center}\n(I)\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1.5,0) node [below] {$t$};\n\\draw [->] (0,0) -- (0,2.5) node [left] {$h$};\n\\draw [domain = 0:2] plot ({1/3*(3-1.5*\\x+0.5*0.5*\\x*\\x)*\\x},\\x);\n\\end{tikzpicture}\n(II)\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (2.5,0) node [below] {$t$};\n\\draw [->] (0,0) -- (0,2.5) node [left] {$h$};\n\\draw [domain = 0:2] plot (\\x,\\x);\n\\end{tikzpicture}\n(III)\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3.5,0) node [below] {$t$};\n\\draw [->] (0,0) -- (0,2.5) node [left] {$h$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:1] plot ({1/3*(3+1.5*\\x+0.5*0.5*\\x*\\x)*\\x},\\x);\n\\draw [domain = 1:2] plot ({9.5/3-1/3*(3+1.5*(2-\\x)+0.5*0.5*(2-\\x)*(2-\\x))*(2-\\x)},\\x);\n\\end{tikzpicture}\\\\\n(IV)\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (5,0) node [below] {$t$};\n\\draw [->] (0,0) -- (0,2.5) node [left] {$h$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:2] plot ({1/3*(3+1.5*\\x+0.5*0.5*\\x*\\x)*\\x},\\x);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569254,7 +575076,9 @@ "id": "040305", "content": "如图为函数$f(x)$的导函数的图像, 则下列判断正确的有\\blank{50}.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3.5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:1, samples = 100] plot (\\x,{2*sin((\\x+1)/2*90)});\n\\draw [domain = 1:5, samples = 100] plot (\\x,{2*sin((\\x-4)*90)});\n\\foreach \\i/\\j in {-3/below,-2/below,-1/above,1/below,2/below,3/above,4/below} \n{\\draw (\\i,0.2) -- (\\i,0) node [\\j] {$\\i$};};\n\\draw [dashed] (1,2) -- (1,0) (3,-2) -- (3,0);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $f(x)$在$(-3,1)$上单调递增; \\textcircled{2} $x=-1$是$f(x)$的极小值点; \\textcircled{3} $f(x)$在$(2,4)$上单调递减, 在$(-1,2)$上单调递增; \\textcircled{4} $x=2$是$f(x)$的极小值点.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569284,7 +575108,9 @@ "id": "040306", "content": "已知函数$f(x)=\\dfrac{1}{2} x-\\sin x$在$[0, \\dfrac{\\pi}{2}]$上的极小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569314,7 +575140,9 @@ "id": "040307", "content": "已知函数$f(x)=a \\sqrt{x}+\\ln x$在$x=1$处取得极值, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569344,7 +575172,9 @@ "id": "040308", "content": "函数$f(x)=x^3-a x^2+b x$在$x=1$处有极值为$4$, 则$a-b$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569374,7 +575204,9 @@ "id": "040309", "content": "已知函数$f(x)=x^3+3 a x^2+b x+a^2$, 若$x=-1$时, $f(x)$取得极值$0$, 则$a b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569404,7 +575236,9 @@ "id": "040310", "content": "已知函数$f(x)=x^2(x-a)$, 若$x=2$为$f(x)$的极值点, 则此时$f(x)$的极小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -569434,7 +575268,9 @@ "id": "040311", "content": "已知函数$f(x)=\\dfrac{x^2-a}{\\mathrm{e}^x}$, 是否存在实数$a$使得$x=1$是函数$f(x)$的一个极值点, 若存在求出$a$的值, 否则说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -569464,7 +575300,9 @@ "id": "040312", "content": "设函数$f(x)=x^3+x^2-x-2$.\\\\\n(1) 求$f(x)$在$x=-2$处的切线方程;\\\\\n(2) 求$f(x)$的极小值点和极大值点.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -569494,7 +575332,9 @@ "id": "040313", "content": "已知函数$f(x)=(-x^2+a x) \\mathrm{e}^x$, $a \\in \\mathbf{R}$.\\\\\n(1) 若$f'(0)=1$时, 求实数$a$的值;\\\\\n(2) 若函数$f(x)$在$(-1,1)$上单调递增, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -569524,7 +575364,10 @@ "id": "040314", "content": "如图, 有一块半椭圆形钢板, 其长半轴长为$2 r$, 短半轴长为$r$, 计划将此钢板切割成等腰梯形的形状, 下底$AB$是半椭圆的短轴, 上底$CD$的端点在椭圆上, 记$CD=2 x$, 梯形面积为$S$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (1.5,0) arc (0:180:1.5 and 3) -- (1.5,0);\n\\draw (1.5,0) node [below right] {$B$} coordinate (B);\n\\draw (-1.5,0) node [below left] {$A$} coordinate (A);\n\\draw (50:1.5 and 3) node [above right] {$C$} coordinate (C);\n\\draw (130:1.5 and 3) node [above left] {$D$} coordinate (D);\n\\draw (B)--(C)--(D)--(A);\n\\draw (1.5,0) --++ (0.6,0) (0,3) --++ (2.1,0);\n\\draw [<->] (1.8,0) -- (1.8,3) node [midway, fill = white] {$2r$};\n\\draw (1.5,0) --++ (0,-0.6) (-1.5,0) --++ (0,-0.6);\n\\draw [<->] (-1.5,-0.3) -- (1.5,-0.3) node [midway, fill = white] {$2r$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求面积$S$以$x$为自变量的函数式, 并写出其定义域;\\\\\n(2) 求面积$S$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -569554,7 +575397,9 @@ "id": "040315", "content": "已知函数$f(x)=\\dfrac{\\ln x+a x}{x}$, $a \\in \\mathbf{R}$.\\\\\n(1) 若$a=0$, 求$f(x)$的最大值;\\\\\n(2) 若$00$, $\\varphi \\in(0,2 \\pi)$)的形式\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2\\sin(\\alpha+\\dfrac 23 \\pi)$", "solution": "", @@ -570264,7 +576157,9 @@ "id": "040340", "content": "已知$\\tan \\alpha, \\tan \\beta$是方程$3 x^2+5 x-7=0$的两根, 则$\\tan (\\alpha+\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac 12$", "solution": "", @@ -570284,7 +576179,9 @@ "id": "040341", "content": "已知$\\tan (\\pi+\\alpha)=-2$, 则$\\dfrac{2 \\sin (\\dfrac{3}{2} \\pi-\\alpha)-3 \\sin (\\pi+\\alpha)}{4 \\cos (-\\alpha)+\\cos (\\dfrac{\\pi}{2}-\\alpha)}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-4$", "solution": "", @@ -570304,7 +576201,9 @@ "id": "040342", "content": "方程$\\sin (x+\\dfrac{\\pi}{4})=\\dfrac{1}{2}$在$[0,2 \\pi]$内的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{\\dfrac {23}{12}\\pi,\\dfrac{7}{12}\\pi\\}$", "solution": "", @@ -570324,7 +576223,9 @@ "id": "040343", "content": "已知$\\triangle ABC$中的三边分别为$a$、$b$、$c$, 三边所对的角分别为$A$、$B$、$C$, 且满足$\\dfrac{1}{a+b}+\\dfrac{1}{b+c}=\\dfrac{3}{a+b+c}$, $\\triangle ABC$的外接圆的面积为$3 \\pi$, 则$b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -570344,7 +576245,9 @@ "id": "040344", "content": "在角$\\theta_1, \\theta_2, \\theta_3, \\cdots, \\theta_{60}$的终边上分别有一点$P_1, P_2, P_3, \\cdots, P_{60}$. 如果点$P_k$的坐标为$(\\sin (30^{\\circ}-k^{\\circ})$, $\\sin (60^{\\circ}+k^{\\circ}))$, $1 \\leq k \\leq 60$, $k \\in \\mathbf{N}$, 则$\\cos \\theta_1+\\cos \\theta_2+\\cos \\theta_3+\\cdots+\\cos \\theta_{60}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac 12$", "solution": "", @@ -570364,7 +576267,9 @@ "id": "040345", "content": "若$\\sin 2 \\alpha=\\dfrac{1}{4}$, 且$\\alpha \\in(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$, 则$\\cos \\alpha-\\sin \\alpha$的值为().\n\\fourch{$\\dfrac{\\sqrt{3}}{2}$}{$-\\dfrac{\\sqrt{3}}{2}$}{$\\pm \\dfrac{\\sqrt{3}}{2}$}{$\\dfrac{3}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "B", "solution": "", @@ -570384,7 +576289,9 @@ "id": "040346", "content": "在$\\triangle ABC$中, ``$\\cos A<\\cos B$''是``$\\sin A>\\sin B$''的\\bracket{20} 条件.\n\\fourch{充分非必要}{必要非充分}{充要}{非充分非必要}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -570404,7 +576311,9 @@ "id": "040347", "content": "已知$\\triangle ABC$的三边长分别为$\\sqrt{a}$、$\\sqrt{b}$、$\\sqrt{c}$, 若存在角$\\theta \\in(0, \\pi)$使得$a^2=b^2+c^2-2 b c \\cos \\theta$, 则$\\triangle ABC$的形状为\\bracket{20}.\n\\fourch{锐角三角形}{直角三角形}{钝角三角形}{以上都不对}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -570424,7 +576333,9 @@ "id": "040348", "content": "已知$\\cos (\\alpha+\\beta)=\\dfrac{2 \\sqrt{5}}{5}$, $\\tan \\beta=\\dfrac{1}{7}$, 且$\\alpha$、$\\beta \\in(0, \\dfrac{\\pi}{2})$.\\\\\n(1) 求$\\tan \\alpha$的值;\\\\\n(2) 求$2 \\alpha+\\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$\\dfrac 13$\\\\\n(2)$\\dfrac{\\pi}4$", "solution": "", @@ -570444,7 +576355,9 @@ "id": "040349", "content": "如图所示, 我国黄海某处的一个圆形海域上有四个小岛, 小岛$B$与小岛$A$、小岛$C$相距都为$5 k$公里, 与小岛$D$相距为$3 \\sqrt{5} k$公里(其中$k$为常数). 已知角$A$为钝角, 且$\\sin A=\\dfrac{3}{5}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw (-2.5,0) node [below left] {$A$} coordinate (A);\n\\draw (2.5,0) node [below right] {$B$} coordinate (B);\n\\draw (0,5) coordinate (O);\n\\draw (O) ++ ({-atan(2)-acos(0.28)}:{2.5*sqrt(5)}) node [left] {$D$} coordinate (D);\n\\draw (O) ++ ({-atan(2)+2*atan(1/2)}:{2.5*sqrt(5)}) node [right] {$C$} coordinate (C);\n\\draw [dashed] (O) circle ({2.5*sqrt(5)});\n\\draw (B)--(A)--(D)--(C)--cycle;\n\\draw [dashed] (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求小岛$A$与小岛$D$之间的距离; (用$k$表示)\\\\\n(2) 求四个小岛所形成的四边形$ABCD$的面积;(用$k$表示)(提示: 角$A$与角$C$互补)\\\\\n(3) 记$\\angle CDB$为$\\alpha$, $\\angle CBD$为$\\beta$, 求$\\sin (2 \\alpha+\\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$2k$;\\\\\n(2)$10k$;\\\\\n(3)$\\dfrac2{25}\\sqrt{5}$", "solution": "", @@ -570464,7 +576377,9 @@ "id": "040350", "content": "若$\\sin \\alpha=\\dfrac{1}{3}$, 则$\\cos (\\dfrac{\\pi}{2}+\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac 13$", "solution": "", @@ -570484,7 +576399,9 @@ "id": "040351", "content": "函数$y=1-\\sin x$取得最小值时所有$x$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$x=2k\\pi+\\dfrac{\\pi}2,k \\in \\mathbb{Z}$", "solution": "", @@ -570504,7 +576421,9 @@ "id": "040352", "content": "函数$y=\\cos 2 x$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi$", "solution": "", @@ -570524,7 +576443,9 @@ "id": "040353", "content": "已知$\\tan \\theta=\\dfrac{1}{2}$, 则$\\sin 2 \\theta-2 \\cos ^2 \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac 45$", "solution": "", @@ -570544,7 +576465,9 @@ "id": "040354", "content": "函数$y=\\sqrt{\\tan x}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|k\\pi \\leq x < k\\pi+\\dfrac{\\pi}2,k \\in \\mathbb{Z}\\}$", "solution": "", @@ -570564,7 +576487,9 @@ "id": "040355", "content": "已知$\\cos \\theta=-\\dfrac{3}{5}$, 并且$180^{\\circ}<\\theta<270^{\\circ}$, 则$\\tan \\dfrac{\\theta}{2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-2$", "solution": "", @@ -570584,7 +576509,9 @@ "id": "040356", "content": "若$x \\in(-\\pi, 2 \\pi)$, 则方程$\\sin 2 x=\\sin x$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{0,\\pi,-\\dfrac{\\pi}3,\\dfrac{\\pi}3,\\dfrac 53 \\pi\\}$", "solution": "", @@ -570604,7 +576531,9 @@ "id": "040357", "content": "设$y=x^{\\frac{1}{2}}-x^3$, 则满足$y<0$的$x$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$(1,+\\infty)$", "solution": "", @@ -570635,7 +576564,9 @@ "id": "040358", "content": "函数$y=2^{\\cos ^2 x-\\cos x}$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[2^{-\\dfrac 14},4]$", "solution": "", @@ -570655,7 +576586,9 @@ "id": "040359", "content": "函数$y=\\sin (\\dfrac{\\pi}{6}-x)$, $x \\in[0, \\dfrac{3 \\pi}{2}]$的单调递减区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[0,\\dfrac 23\\pi]$", "solution": "", @@ -570697,7 +576630,9 @@ "id": "040360", "content": "在三角形$ABC$中, $3 \\sin A+4 \\cos B=6$, $4 \\sin B+3 \\cos A=1$, 则$\\angle C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac {\\pi}6$", "solution": "", @@ -570717,7 +576652,9 @@ "id": "040361", "content": "设函数$f(x)=\\sin ^6 \\dfrac{k x}{10}+\\cos ^6 \\dfrac{k x}{10}$, 其中$k$是一个正整数, 若对任意实数$a$, 均有$\\{f(x) | a0$, 则$\\alpha$为第一或第二象限角; \\textcircled{3}$a, b, c>0$, 则$\\sqrt{a^2+b^2}, \\sqrt{b^2+c^2}, \\sqrt{c^2+a^2}$必是某一个锐角三角形的三边长. 上述命题中, 正确的命题有\\bracket{20}个.\n\\fourch{$0$}{$1$}{$2$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -570777,7 +576718,9 @@ "id": "040364", "content": "已知函数$f(x)=\\tan (\\omega x+\\dfrac{\\pi}{4})$($\\omega>0$)的最小正周期为$\\dfrac{\\pi}{2}$.\\\\\n(1) 求$\\omega$的值及函数$f(x)$的定义域;\\\\\n(2) 若$f(\\dfrac{\\alpha}{2})=3$, 求$\\tan 2 \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$\\omega =2$，定义域为$\\{x|x\\neq \\dfrac{k\\pi}2+\\dfrac{\\pi}8,k \\in \\mathbb{Z}\\}$\\\\\n(2)$\\dfrac 43$", "solution": "", @@ -570797,7 +576740,9 @@ "id": "040365", "content": "某水产养殖户承包一片靠岸水域. 如图, $AO$、$OB$为直线岸线, $OA=1000$\n米, $OB=1500$米, $\\angle AOB=\\dfrac{\\pi}{3}$, 该承包水域的水面边界是某\n圆的一段弧$\\overset\\frown{AB}$, 过弧$\\overset\\frown{AB}$上一点$P$按线段$PA$和$PB$修建养殖网箱, 已知$\\angle APB=\\dfrac{2 \\pi}{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,{sqrt(3)}) node [right] {$A$} coordinate (A);\n\\draw (-1.5,{1.5*sqrt(3)}) node [left] {$B$} coordinate (B);\n\\draw (A) arc ({atan((sqrt(3)-5/sqrt(12))/1.5)}:{180+atan((1.5*sqrt(3)-5/sqrt(12))/(-1))}:{sqrt(7/3)});\n\\draw (-0.5,{5/sqrt(12)}) ++ (85:{sqrt(7/3)}) node [above] {$P$} coordinate (P);\n\\draw (A) -- (P) -- (B) (O) -- (A) (O) -- (B);\n\\draw [dashed] (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求岸线上点$A$与点$B$之间的直线距离;\\\\\n(2) 如果线段$PA$上的网箱每米可获得$40$元的经济收益, 线段$PB$上的网箱每米可获得$30$元的经济收益. 记$\\angle PAB=\\theta$, 则这两段网箱获得的经济总收益最高为多少? (精确到元)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$500\\sqrt{7}$\\\\\n(2)55706元", "solution": "", @@ -570817,7 +576762,9 @@ "id": "040366", "content": "已知函数$f(x)=2^x$($x \\in \\mathbf{R}$), 记$g(x)=f(x)-f(-x)$.\\\\\n(1) 解不等式: $f(2 x)-f(x) \\leq 6$;\\\\\n(2) 设$k$为实数, 若存在实数$x_0 \\in(1,2]$, 使得$g(2 x_0)=k \\cdot g^2(x_0)-1$成立, 求$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1)$x \\leq \\log_2 3$\\\\\n(2)$k \\in [\\dfrac{225}{271},\\dfrac{19}9)$", "solution": "", @@ -570837,7 +576784,9 @@ "id": "040367", "content": "已知函数$f(x)=\\log _3 \\dfrac{m-x}{x+2}$为奇函数.\\\\\n(1) 求实数$m$的值;\\\\\n(2) 判定函数$f(x)$在定义域内的单调性, 并证明;\\\\\n(3) 若不等式$f(\\sin ^2 x)+f(t-2 \\cos x-3) \\geq 0$对任意$x \\in[-\\dfrac{\\pi}{3}, \\dfrac{\\pi}{6}]$恒成立, 求实数$t$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1)$m=2$\\\\\n(2)在$(-2,2)$上严格减\\\\\n(3)$\\dfrac {13}4$", "solution": "", @@ -570857,7 +576806,9 @@ "id": "040368", "content": "函数$f(x)=(m^2-m-1) x^{m^2+m-3}$是幂函数, 且函数$f(x)$在区间$(0,+\\infty)$上是严格增函数, 则$f(x)$的解析式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -570877,7 +576828,9 @@ "id": "040369", "content": "函数$y=\\log _2(3-2 x-x^2)$的严格减区间\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -570897,7 +576850,9 @@ "id": "040370", "content": "若函数$y=(m x^2+m x+2)^{-\\frac{3}{4}}$的定义域为$\\mathbf{R}$, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -570917,7 +576872,9 @@ "id": "040371", "content": "设$f(x)=\\log _a(1+x)+\\log _a(3-x)$($a>0$, 且$a \\neq 1)$且$f(1)=2$.\\\\\n(1) 求实数$a$的值及$f(x)$的定义域;\\\\\n(2) 求$f(x)$在区间$[0, \\dfrac{3}{2}]$上的最大值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -570937,7 +576894,9 @@ "id": "040372", "content": "已知$f(x)=a x^2+b x+c$($a$、$b$、$c \\in \\mathbf{R}$).\\\\\n(1) 当$f(1)=-1$, 且$f(x)<0$的解集为$(0,2)$, 求函数$f(x)$的解析式;\\\\\n(2) $b=-2 a$, $c=0$, 若关于$x$的不等式$2^{f(x)}-\\dfrac{1}{4}>0$对一切实数$x$恒成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -570957,7 +576916,9 @@ "id": "040373", "content": "已知$f(x)=x|x-a|+b$, $x \\in \\mathbf{R}$.\\\\\n(1) 当$a=1$, $b=0$时, 判断$f(x)$的奇偶性, 并说明理由;\\\\\n(2) 当$a=1$, $b=1$时, 若$f(\\log _2 x)=3$, 求$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -570977,7 +576938,9 @@ "id": "040374", "content": "求证: 函数$f(x)=\\log _{0.5}(\\dfrac{x-1}{x-2})$在区间$(2,+\\infty)$上是严格增函数;", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -570997,7 +576960,9 @@ "id": "040375", "content": "已知$a>0$且$a \\neq 1$, 若$\\log _a(4 x^2-1)<\\log _a(-2 x^2+x+1)$, 求实数$x$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -571017,7 +576982,9 @@ "id": "040376", "content": "某创业投资公司拟投资开发某种新能源产品, 估计能获得$25$万元至$1600$万元的投资收益, 现准备制定一个对科研课题组的奖励方案: 奖金$y$(单位: 万元) 随投资收益$x$(单位: 万元)的增加而增加, 奖金不超过$75$万元, 同时奖金不超过投资收益的$20 \\%$.\\\\\n(1) 请用数学语言列出公司对函数模型的基本要求;\\\\\n(2) 判断函数$f(x)=\\dfrac{x}{40}+10$是否符合公司奖励方案函数模型的要求, 并说明理由;\\\\\n(3) 已知函数$g(x)=a \\sqrt{x}-5$($a \\geq 1$)符合公司奖励方案函数模型要求, 求实数$a$取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -571037,7 +577004,9 @@ "id": "040377", "content": "设$a \\in\\{-2,-\\dfrac{3}{5},-\\dfrac{1}{2},-\\dfrac{1}{3}, \\dfrac{1}{2}, 1,2,3\\}$, 已知幂函数$y=x^a$图像关于原点中心对称, 且在区间$(0,+\\infty)$上是严格减函数, 则满足条件的$a$值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571057,7 +577026,9 @@ "id": "040378", "content": "函数$y=\\sqrt{(\\dfrac{1}{16})^x-64}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571079,7 +577050,9 @@ "id": "040379", "content": "若$\\sqrt[4]{a}+(a-2)^0$有意义, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571099,7 +577072,9 @@ "id": "040380", "content": "函数$y=a^{x+2021}+2021$($a>0$, $a \\neq 1$)的图像恒过定点\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571119,7 +577094,9 @@ "id": "040381", "content": "对任意的实数$a$, 下列各式一定正确的是\\bracket{20}.\n\\fourch{$(a^{\\frac{2}{3}})^{\\frac{1}{2}}=a^{\\frac{1}{3}}$}{$(a^{\\frac{1}{2}})^{\\frac{2}{3}}=a^{\\frac{1}{3}}$}{$(a^{-\\frac{3}{5}})^{-\\frac{1}{3}}=a^{\\frac{1}{5}}$}{$(a^{\\frac{1}{3}})^{\\frac{3}{5}}=a^{\\frac{1}{5}}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -571139,7 +577116,9 @@ "id": "040382", "content": "已知$f(x)=\\begin{cases}x+1,& x \\in[-1,0), \\\\ x^2+1,& x \\in[0,1],\\end{cases}$ 则下列函数的图像错误的是\\bracket{20}的图像.\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) -- (1,1);\n\\draw [domain = 1:2] plot (\\x,{(\\x-1)*(\\x-1)+1});\n\\draw [dashed] (1,0) node [below] {$1$} -- (1,1) -- (0,1) node [left] {$1$} (2,0) node [below] {$2$} -- (2,2) -- (0,2) node [left] {$2$};\n\\draw (1,-1) node {$y=f(x-1)$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) -- (0,1);\n\\draw [domain = -1:0] plot (\\x,{\\x*\\x+1});\n\\draw [dashed] (0,2) node [right] {$2$} coordinate (2) -- (-1,2) -- (-1,0) node [below] {$-1$};\n\\draw (0,1) node [right] {$1$} (1,0) node [below] {$1$};\n\\draw (0,-1) node {$y=f(-x)$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1] plot (\\x,{\\x*\\x+1});\n\\draw (0,2) node [right] {$2$};\n\\draw (-1,0) node [below] {$-1$};\n\\draw (0,1) node [right] {$1$} (1,0) node [below] {$1$};\n\\draw [dashed] (1,0) -- (1,2) -- (-1,2) -- (-1,0);\n\\draw (0,-1) node {$y=f(|x|)$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:1] plot (\\x,{\\x*\\x+1});\n\\draw (-1,2) -- (0,1);\n\\draw (0,2) node [right] {$2$};\n\\draw (-1,0) node [below] {$-1$};\n\\draw (0,1) node [right] {$1$} (1,0) node [below] {$1$};\n\\draw [dashed] (1,0) -- (1,2) -- (-1,2) -- (-1,0);\n\\draw (0,-1) node {$|y=f(x)|$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -571159,7 +577138,9 @@ "id": "040383", "content": "已知幂函数$f(x)=x^{m^2-2 m-3}$($m \\in \\mathbf{Z}$)为偶函数, 且在区间$(0,+\\infty)$上是严格减函数, 求$f(x)$的解析式, 并讨论函数$g(x)=a \\sqrt{f(x)}-\\dfrac{b}{x f(x)}$的奇偶性.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -571179,7 +577160,9 @@ "id": "040384", "content": "设$0 \\leq x \\leq 2$, 求函数$y=4^{x-\\dfrac{1}{2}}-a \\cdot 2^x+\\dfrac{a^2}{2}+1$的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -571199,7 +577182,9 @@ "id": "040385", "content": "设函数$f(x)=\\dfrac{10^x-10^{-x}}{10^x+10^{-x}}$.\\\\\n(1) 证明$f(x)$在$(-\\infty,+\\infty)$上是严格增函数;\\\\\n(2) 求函数$f(x)$的值域.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -571219,7 +577204,9 @@ "id": "040386", "content": "定义: 对函数$y=f(x)$, 对给定的正整数$k$, 若在其定义域内存在实数$x_0$, 使得$f(x_0+k)=f(x_0)+f(k)$, 则称函数$f(x)$为``$k$性质函数''.\\\\\n(1) 若函数$f(x)=2^x$为``$1$性质函数'', 求$x_0$;\\\\\n(2) 证明: 函数$f(x)=\\dfrac{1}{x}$不是``$k$性质函数'';\\\\\n(3) 若函数$f(x)=\\lg \\dfrac{a}{x^2+1}$为``$2$性质函数'', 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -571239,7 +577226,9 @@ "id": "040387", "content": "设$\\alpha \\in(0,2 \\pi)$. 若$\\alpha$的终边与$-\\dfrac{2 \\pi}{3}$的终边关于$y$轴对称, 则$\\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571259,7 +577248,9 @@ "id": "040388", "content": "求值: $\\sin (\\dfrac{19 \\pi}{3})+\\cos (-\\dfrac{17 \\pi}{6})+\\tan (\\dfrac{15}{4} \\pi)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571279,7 +577270,9 @@ "id": "040389", "content": "设$\\alpha \\in(-\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2})$. 若$\\sin \\alpha=-\\dfrac{\\sqrt{3}}{2}$, 则$\\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571299,7 +577292,9 @@ "id": "040390", "content": "若$\\cot \\alpha=3$, 则$\\tan (\\dfrac{\\pi}{2}+\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571319,7 +577314,9 @@ "id": "040391", "content": "若$\\tan (\\dfrac{3 \\pi}{4}-\\alpha)=2$, 则$\\tan (\\alpha+\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571339,7 +577336,9 @@ "id": "040392", "content": "若$\\sin \\alpha+\\cos \\alpha=\\dfrac{3}{4}$, $-\\dfrac{\\pi}{2}<\\alpha<\\dfrac{\\pi}{2}$, 则$\\sin \\alpha-\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571359,7 +577358,9 @@ "id": "040393", "content": "若$\\alpha$是钝角, 且$\\cos \\alpha=-\\dfrac{3}{5}$, 则$\\sin \\alpha+\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571379,7 +577380,9 @@ "id": "040394", "content": "数列$\\{a_n\\}$满足$a_{n+1}=\\begin{cases}2 a_n, & 0 \\leq a_n<\\dfrac{1}{2}, \\\\ 2 a_n-1, & \\dfrac{1}{2} \\leq a_n<1,\\end{cases}$ 若$a_1=\\dfrac{6}{7}$, 则$a_{2023}=$\\blank{50}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -571399,7 +577402,9 @@ "id": "040395", "content": "已知无穷数列$\\{a_n\\}$的前$n$项和为$S_n$, 若对于任意的正整数$n$, 均有$S_{2 n-1} \\geq 0$, $S_{2 n} \\leq 0$, 则称数列$\\{a_n\\}$具有性质$P$.\\\\\n(1) 判断首项为$1$, 公比为$-2$的无穷等比数列$\\{a_n\\}$是否具有性质$P$, 并说明理由;\\\\\n(2) 已知无穷数列$\\{a_n\\}$具有性质$P$, 且任意相邻四项之和都相等, 求证: $S_4=0$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -571419,7 +577424,9 @@ "id": "040396", "content": "函数$y=\\lg (\\sin x-\\dfrac{\\sqrt{2}}{2})$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|2k\\pi+\\dfrac{\\pi}40)$在$[0,1]$内恰取到$50$次最大值, 则$\\omega$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[\\dfrac{197}2 \\pi,\\dfrac{201}2 \\pi)$", "solution": "", @@ -571662,7 +577691,9 @@ "id": "040408", "content": "设$\\omega>0$, 函数$f(x)=2 \\sin \\omega x$在$[-\\dfrac{\\pi}{3}, \\dfrac{\\pi}{4}]$上为严格增函数, 则$\\omega$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$(0,\\dfrac 32]$", "solution": "", @@ -571682,7 +577713,9 @@ "id": "040409", "content": "求函数$y=(\\sin x+1)(\\cos x+1)$的值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$[0,\\sqrt{2}+\\dfrac 32]$", "solution": "", @@ -571702,7 +577735,9 @@ "id": "040410", "content": "已知函数$f(x)=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$, $|\\varphi|<\\dfrac{\\pi}{2}$, $x \\in \\mathbf{R}$)的图像的一部分如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2,0) -- (8,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-1,1,2,3,4,5,6,7}\n{\\draw (\\i,0.3)--(\\i,0) node [below] {$\\i$};};\n\\foreach \\j in {-2,-1,1,2}\n{\\draw (0.3,\\j)--(0,\\j) node [left] {$\\j$};};\n\\draw [dashed] (0,2) -- (1,2)--(1,0.3);\n\\draw [domain = -1.2:7.2, samples = 100] plot (\\x,{2*sin(\\x*45+45)});\n\\end{tikzpicture}\n\\end{center}\n(1) 求函数$f(x)$的解析式;\\\\\n(2) 当$x \\in[-6,-\\dfrac{2}{3}]$时, 求函数$y=f(x)+f(x+2)$的最大值与最小值及相应的$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$f(x)=2\\sin(\\dfrac{\\pi}4x+\\dfrac{\\pi}4)$\\\\\n(2)$x=-\\dfrac 23$时，取最大值为$\\sqrt{6}$;$x=-4$时，取最小值为$-2\\sqrt{2}$", "solution": "", @@ -571722,7 +577757,9 @@ "id": "040411", "content": "试判断方程$\\sin x=\\dfrac{x}{100 \\pi}$实数解的个数.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$199$个", "solution": "", @@ -571742,7 +577779,9 @@ "id": "040412", "content": "已知函数$y=\\sin ^2 x+\\sqrt{3} \\sin x \\cos x+2 \\cos ^2 x$, $x \\in \\mathbf{R}$.\\\\\n(1) 求最小正周期;\\\\\n(2) 判断奇偶性;\\\\\n(3) 求出单调区间;\\\\\n(4) 求出$x \\in[0, \\dfrac{\\pi}{2}]$时, 该函数的最大值与最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$T=\\pi$\\\\\n(2)非奇非偶函数\\\\\n(3)增区间为$[k\\pi-\\dfrac{\\pi}3,k\\pi+\\dfrac{\\pi}6],k\\in \\mathbb{Z}$,减区间为$[k\\pi+\\dfrac{\\pi}{6},k\\pi+\\dfrac 23\\pi],k\\in \\mathbb{Z}$\\\\\n(4)$y_{min}=1,x=\\dfrac{\\pi}2;y_{max}=\\dfrac 52,x=\\dfrac{\\pi}6$", "solution": "", @@ -571762,7 +577801,9 @@ "id": "040413", "content": "已知函数$f(x)=a \\sin x+a \\cos x+1-a$($a \\in \\mathbf{R}$), $x \\in[0, \\dfrac{\\pi}{2}]$, 若定义在非零实数集上的奇函数$g(x)$在$(0,+\\infty)$上是严格增函数, 且$g(2)=0$, 求当$g[f(x)]<0$恒成立时实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-1-\\sqrt{2}0$, $0<\\varphi<\\dfrac{\\pi}{2}$). $y=f(x)$的部分图像如图所示. $P, Q$分别为该图像的最高点和最低点, 点$P$的坐标为$(1, A)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.5:5.5, samples = 100] plot (\\x,{sqrt(3)*sin(60*\\x+30)});\n\\draw [dashed] (1,{sqrt(3)}) node [above] {$P$} coordinate (P)-- (1,0) node [above left] {$R$} coordinate (R) -- (4,{-sqrt(3)}) node [below] {$Q$} coordinate (Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\varphi$的值;\\\\\n(2) 若点$R$的坐标为$(1,0)$, $\\angle PRQ=\\dfrac{2 \\pi}{3}$, 求$A$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$\\dfrac{\\pi}6$\\\\\n(2)$\\sqrt{3}$", "solution": "", @@ -571937,7 +577992,9 @@ "id": "040421", "content": "已知函数$y=2 \\cos x \\sin (x+\\dfrac{\\pi}{3})-\\sqrt{3} \\sin ^2 x+\\dfrac{1}{2} \\sin 2 x$.\\\\\n(1) 求函数的最小正周期和单调递减区间;\\\\\n(2) 求函数在区间$[\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3}]$上的最大值与最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)最小正周期为$\\pi$，单调减区间为$[k\\pi+\\dfrac{\\pi}{12},k\\pi+\\dfrac 7{12}\\pi],k \\in \\mathbb{Z}$\\\\\n(2)$y_{\\max}=\\sqrt{3},x=\\dfrac{\\pi}6$时取; $y_{\\min}=-2,x=\\dfrac 7{12}\\pi$时取", "solution": "", @@ -571957,7 +578014,9 @@ "id": "040422", "content": "不等式$|x-1|<2$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$(-1,3)$", "solution": "", @@ -571977,7 +578036,9 @@ "id": "040423", "content": "函数$y=\\lg (-x)+\\dfrac{2}{\\sqrt{x^2-1}}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-\\infty,-1)$", "solution": "", @@ -571997,7 +578058,9 @@ "id": "040424", "content": "已知复数$z$满足$(2+\\mathrm{i}) z=3+4 \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\sqrt{5}$", "solution": "", @@ -572017,7 +578080,9 @@ "id": "040425", "content": "对于正实数$x$, 代数式$x+\\dfrac{9}{x+1}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -572037,7 +578102,9 @@ "id": "040426", "content": "己知角$x$在第二象限, 且$\\cos (x+\\dfrac{\\pi}{2})=-\\dfrac{4}{5}$, 则$\\tan 2 x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{24}{7}$", "solution": "", @@ -572057,7 +578124,9 @@ "id": "040427", "content": "已知随机变量$X$服从正态分布$N(1.5, \\sigma^2)$, 且$P(1.50$, $|\\varphi|<\\dfrac{\\pi}{2}$), 若点$A(\\dfrac{1}{3}, 0)$为函数$y=f(x)$图像的对称中心, $B$、$C$是图像上相邻的最高点与最低点, 且$|BC|=4$, 则下列结论正确的是\\bracket{20}.\n\\onech{函数$y=f(x)$的对称轴方程为$x=4 k+\\dfrac{4}{3}, k \\in \\mathbf{Z}$}{函数$y=f(x-\\dfrac{\\pi}{3})$的图像关于坐标原点对称}{函数$y=f(x)$在区间$(0,2)$上是严格增函数}{若函数$y=f(x)$在区间$(0, m)$内有$5$个零点, 则它在此区间内有且有$2$个极小值点}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -572277,7 +578366,9 @@ "id": "040438", "content": "已知四棱锥$P-ABCD$的底面$ABCD$为矩形, $PA \\perp$底面$ABCD$, 且$PA=AD=2AB=2$, 设$E$、$F$、$G$分别为$PC$、$BC$、$CD$的中点, $H$为$EG$的中点, 如图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw ($(B)+(D)-(A)$) node [right] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F);\n\\draw ($(P)!0.5!(C)$) node [left] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D)$) node [right] {$G$} coordinate (G);\n\\draw ($(E)!0.5!(G)$) node [above] {$H$} coordinate (H);\n\\draw (F)--(E)--(G);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (P)--(A)--(B)(A)--(D)(B)--(D)(F)--(G)(F)--(H);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $FH\\parallel$平面$PBD$;\\\\\n(2) 求直线$FH$与平面$PBC$所成角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 略; (2) $\\dfrac{\\sqrt{15}}{15}$", "solution": "", @@ -572297,7 +578388,9 @@ "id": "040439", "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 已知$S_n=\\dfrac{1}{2} a_n+n^2+1$($n$为正整数).\\\\\n(1) 求$a_1+a_2$的值, 并证明数列$\\{a_n+a_{n+1}\\}$是等差数列;\\\\\n(2) 求$S_n$的表达式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) $a_1+a_2=6$, 公差为$4$; (2) $S_n=\\begin{cases}n^2+n, & n=2 k, \\\\ n^2+n+2, & n=2 k-1\\end{cases}$, $k$为正整数", "solution": "", @@ -572317,7 +578410,10 @@ "id": "040440", "content": "社会实践是大学生课外教育的一个重要方面, 在校大学生利用要期参加社会实践活动, 是认识社会、了解社会、提高自我能力的重要机会. 某省统计了该省其中的$4$所大学$2023$年毕业生的人数及参加过暑期社会实践活动的人数 (单位: 千人), 得到如下表格:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 大学 &A大学 &B大学 &C大学 &D大学 \\\\\n\\hline 2023 年毕业生人数$x$(千人) & 7 & 6 & 5 & 4 \\\\\n\\hline 2023 年毕业生中参加过社会实践人数$y$(千人) & 0.5 & 0.4 & 0.3 & 0.2 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 己知$y$与$x$具有较强的线性相关性, 求$y$关于$x$的线性回归方程$y=\\hat{a} x+\\hat{b}$;\\\\\n(2) 假设该省对参加过暑期社会实践活动的大学生每人发放$0.5$万元的补贴.\\\\\n(i) 若该省大学$2023$年毕业生人数为$12$万人, 估计该省要发放补贴的总金额;\\\\\n(ii) 若$2023$年毕业生中的小李、小王参加过暑期社会实践活动的概率分别为$p$、$3 p-1$, 该省对小李、小王两人补贴总金额的期望不超过$0.75$万元, 求$p$的取值范围.\\\\\n参考公式: $\\hat{a}=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})(y_i-\\overline {y})}{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})^2}=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i y_i-n \\overline {x} \\cdot \\overline {y})}{\\displaystyle\\sum_{i=1}^n x_i^2-n \\overline {x}^2}$, $\\hat{b}=\\overline {y}-\\hat{a}\\overline{x}$.", "objs": [], - "tags": [], + "tags": [ + "第九单元", + "第八单元" + ], "genre": "解答题", "ans": "(1) $y=0.1 x-0.2$; (2) (i) $5900$万元; (ii) $[\\dfrac{1}{3}, \\dfrac{5}{8}]$", "solution": "", @@ -572337,7 +578433,9 @@ "id": "040441", "content": "己知椭圆$C_1: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的离心率为$\\dfrac{\\sqrt{2}}{2}$, 且点$(-2, \\sqrt{2})$在椭圆$C_1$上.\\\\\n(1) 求椭圆$C_1$的方程;\\\\\n(2) 过点$Q(0,1)$的直线$l$与椭圆$C_1$交于$D$、$E$两点, 已知$\\overrightarrow{DQ}=2 \\overrightarrow{QE}$, 求直线$l$的方程;\\\\\n(3) 点$P$为椭圆$C_1$上任意一点, 过点$P$作$C_1$的切线与圆$C_2: x^2+y^2=12$交于$A$、$B$两点, 设直线$OA$、$OB$的斜率分别为$k_1$、$k_2$, 证明: $k_1 k_2$为定值, 并求该定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{8}+\\dfrac{y^2}{4}=1$; (2) $y= \\pm \\dfrac{\\sqrt{30}}{10} x+1$; (3) $-\\dfrac{1}{2}$", "solution": "", @@ -572357,7 +578455,9 @@ "id": "040442", "content": "设$f(x)=\\mathrm{e}^x+a \\sin x-1$.\\\\\n(1) 求曲线$y=f(x)$在点$(0, f(0))$处的切线方程;\\\\\n(2) 若函数$y=f(x)$在$x=0$处取得极小值, 求$a$的值;\\\\\n(3) 若存在正实数$m$, 使得对任意的$x \\in(0, m)$, 都有$f(x)<0$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $y=(1+a) x$; (2) $-1$; (3) $(-\\infty,-1)$", "solution": "", @@ -572377,7 +578477,9 @@ "id": "040443", "content": "已知集合$A=\\{1,2\\}$, $B=\\{2,3\\}$, 则$A \\cup B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$\\{1,2,3\\}$", "solution": "", @@ -572397,7 +578499,9 @@ "id": "040444", "content": "若复数$z$满足$z=\\dfrac{3+\\mathrm{i}}{\\mathrm{i}}$(其中$\\mathrm{i}$是虚数单位), 则$|\\overline {z}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\sqrt{10}$", "solution": "", @@ -572417,7 +578521,9 @@ "id": "040445", "content": "轴截面是边长为$2$的正三角形的圆锥的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$2 \\pi$", "solution": "", @@ -572437,7 +578543,9 @@ "id": "040446", "content": "若$\\sin \\alpha=\\dfrac{1}{3}$, 则$\\cos (\\alpha+\\dfrac{\\pi}{2})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac{1}{3}$", "solution": "", @@ -572457,7 +578565,9 @@ "id": "040447", "content": "在$(x+2)^4$的展开式中, 含有$x^2$项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$24$", "solution": "", @@ -572477,7 +578587,9 @@ "id": "040448", "content": "$f(x)=x^3+a x^2+3 x-9$, 已知$f(x)$在$x=3$时取得极值, 则$a$等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$-5$", "solution": "", @@ -572497,7 +578609,9 @@ "id": "040449", "content": "在空间直角坐标系中, 点$A(1,0,0)$, 点$B(5,-4,3)$, 点$C(2,0,1)$, 则$\\overrightarrow{AB}$在$\\overrightarrow{CA}$方向上的投影向量的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(\\dfrac{7}{2}, 0, \\dfrac{7}{2})$", "solution": "", @@ -572540,7 +578654,9 @@ "id": "040450", "content": "已知函数$f(x)=x^{\\frac{1}{3}}$, 则关于$t$的表达式$f(t^2-2 t)+f(2 t^2-1)<0$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-\\dfrac{1}{3}, 1)$", "solution": "", @@ -572560,7 +578676,9 @@ "id": "040451", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=(20-n) \\cdot(\\dfrac{3}{2})^n$, 则$a_n$取最大值时, $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$17$或$18$", "solution": "", @@ -572580,7 +578698,9 @@ "id": "040452", "content": "一项研究同年龄段的男、女生的注意力差别的脑功能实验, 实验数据如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 注意力稳定 & 注意力不稳定 \\\\\n\\hline 男生 & 29 & 7 \\\\\n\\hline 女生 & 33 & 5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n依据$P(\\chi^2 \\geq 3.841) \\approx 0.05$, 该实验\\blank{50}该年龄段的学生在注意力的稳定性上对于性别没有显著差异(填拒绝或支持).\n参考公式: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中$n=a+b+c+d$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "支持", "solution": "", @@ -572600,7 +578720,9 @@ "id": "040453", "content": "已知$F_1$、$F_2$是椭圆$\\Gamma$的两个焦点, 点$A$在$\\Gamma$上, 且$\\angle F_1AF_2=90^{\\circ}$, 延长$AF_1$交$\\Gamma$于点$B$, 若$|AB|=|AF_2|$, 则椭圆$\\Gamma$的离心率$e=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$\\sqrt{6}-\\sqrt{3}$", "solution": "", @@ -572620,7 +578742,9 @@ "id": "040454", "content": "已知等差数列共有$n$($n \\geq 4$)项, 各项与公差$d$均不为零, 若将此数列删去某一项后, 得到的数列 (按原来顺序) 是等比数列, 则所有数列$(n, \\dfrac{a_1}{d})$组成的集合为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$\\{(4,-4),(4,1)\\}$", "solution": "", @@ -572640,7 +578764,9 @@ "id": "040455", "content": "已知$P(B | A)=\\dfrac{1}{2}$, $P(AB)=\\dfrac{3}{8}$, 则$P(A)=$\\bracket{20}.\n\\fourch{$\\dfrac{3}{16}$}{$\\dfrac{13}{16}$}{$\\dfrac{3}{4}$}{$\\dfrac{1}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -572660,7 +578786,9 @@ "id": "040456", "content": "``$x=2 k \\pi+\\dfrac{\\pi}{2}$($k \\in \\mathbf{Z}$)''是``$|\\sin x|=1$''的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充要}{既不充分也不必要}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -572680,7 +578808,9 @@ "id": "040457", "content": "对于某一集合$A$, 若满足$a$、$b$、$c \\in A$, 任取$a$、$b$、$c \\in A$都有``$a$、$b$、$c$为某一三角形的三边长'', 则称集合$A$为``三角集'', 下列集合中为三角集的是\\bracket{20}.\n\\twoch{$\\{x | x$是$\\triangle ABC$的高的长度$\\}$}{$\\{x | \\dfrac{x-1}{x-2} \\leq 0\\}$}{$\\{x|| x-1|+| x-3 |=2\\}$}{$\\{x | y=\\log _2(3 x-2)\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -572700,7 +578830,9 @@ "id": "040458", "content": "若存在实常数$k$和$b$, 使得函数$F(x)$和$G(x)$对其公共定义域上的任意实数$x$都满足: $F(x) \\geq k x+b$和$G(x) \\leq k x+b$恒成立, 则称此直线$y=k x+b$为$F(x)$和$G(x)$的``隔离直线'', 已知函数$f(x)=x^2$($x \\in \\mathbf{R}$), $g(x)=\\dfrac{1}{x}$($x<0$), $h(x)=2 \\mathrm{e} \\ln x$($x>0$), 有下列两个命题:\\\\\n命题$\\alpha: f(x)$和$h(x)$之间存在唯一的``隔离直线''$y=2 \\sqrt{\\mathrm{e}} x-\\mathrm{e}$;\\\\\n命题$\\beta: f(x)$和$g(x)$之间存在``隔离直线'', 且$b$的最小值为$-1$. 则下列说法正确的是\\bracket{20}.\n\\twoch{命题$\\alpha$、命题$\\beta$都是真命题}{命题$\\alpha$为真命题, 命题$\\beta$为假命题}{命题$\\alpha$为假命题, 命题$\\beta$为真命题}{命题$\\alpha$、命题$\\beta$都是假命题}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -572720,7 +578852,9 @@ "id": "040459", "content": "已知函数$f(x)=\\cos ^2 x-\\sin ^2 x+\\dfrac{1}{2}$.\\\\\n(1) 求$f(x)$的单调增区间;\\\\\n(2) 设$\\triangle ABC$为锐角三角形, 角$A$、$B$、$C$所对的边分别是$a$、$b$、$c$, $a=\\sqrt{19}$, $b=5$, 若$f(A)=0$, 求$\\triangle ABC$的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $[-\\dfrac{\\pi}{2}+k \\pi, k \\pi]$, $k \\in \\mathbf{Z}$; (2) $\\dfrac{15 \\sqrt{3}}{4}$", "solution": "", @@ -572740,7 +578874,9 @@ "id": "040460", "content": "如图所示, 在四棱锥$P-ABCD$中, $AB \\perp$平面$PAD$, $AB\\parallel CD$且$2AB=CD$, $PD=PA$, 点$H$为线段$AD$的中点, 若$PH=1$, $AD=\\sqrt{2}$, $PB$与平面$ABCD$所成角的大小为$45^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2.5]\n\\draw (0,0,0) node [right] {$H$} coordinate (H);\n\\draw (0,0,{sqrt(2)/2}) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(2)/2},0,{sqrt(2)/2}) node [below] {$B$} coordinate (B);\n\\draw (0,0,{-sqrt(2)/2}) node [above right] {$D$} coordinate (D);\n\\draw (D) ++ ({sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (P)--(H)(P)--(D)--(C)(A)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $PH \\perp$平面$ABCD$;\\\\\n(2) 求四棱锥$P-ABCD$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 略; (2) $\\dfrac{1}{2}$", "solution": "", @@ -572760,7 +578896,9 @@ "id": "040461", "content": "某校举行知识竞赛, 最后一个名额要在$A$、$B$两名同学中产生, 测试方案如下: $A$、$B$两名学生各自从给定的$4$个问题中随机抽取$3$个问题作答, 在这$4$个问题中, 已知$A$能正确作答其中的$3$个, $B$能正确作答每个问题的概率是$\\dfrac{3}{4}$, $A$、$B$两名同学作答问题相互独立.\\\\\n(1) 求$A$、$B$恰好答对$2$个问题的概率;\\\\\n(2) 设$A$答对的题数为$X, B$答对的题数为$Y$, 若让你投票决定参赛选手, 你会选择哪名学生, 说明理由?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{3}{256}$; (2) 选择A同学", "solution": "", @@ -572780,7 +578918,9 @@ "id": "040462", "content": "已知点$F_1$、$F_2$分别为双曲线$\\Gamma: \\dfrac{x^2}{2}-y^2=1$的左、右焦点, 直线$l: y=k x+1$与$\\Gamma$有两个不同的交点$A$、$B$.\\\\\n(1) 当$F_1 \\in l$时, 求$F_2$到$l$的距离;\\\\\n(2) 若$O$为原点, 直线$l$与$\\Gamma$的两条渐近线在一、二象限的交点分别为$C$、$D$, 证明: 当$\\triangle COD$的面积最小时, 直线$CD$平行于$x$轴;\\\\\n(3) 设$P$为$x$轴上一点, 是否存在实数$k$($k>0$), 使得$\\triangle PAB$是点$P$为直角顶点的等腰直角三角形? 若存在, 求出$k$的值及点$P$的坐标; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\sqrt{3}$; (2) 略; (3) 存在, $k=\\dfrac{\\sqrt{3}}{2}$, $P(-3 \\sqrt{3}, 0)$", "solution": "", @@ -572800,7 +578940,9 @@ "id": "040463", "content": "已知函数$g(x)=a \\mathrm{e}^x-2 x-a \\mathrm{e}^{-x}$.\\\\\n(1) 若$a=2$, 求曲线$y=g(x)$在点$(0, g(0))$处的切线方程;\\\\\n(2) 若函数$y=g(x)$在$\\mathbf{R}$上是单调函数, 求实数$a$的取值范围;\\\\\n(3) 设函数$h(x)=a \\mathrm{e}^x$, 若在$\\mathbf{R}$上至少存在一点$x_1$, 使得$g(x_1)>h(x_1)$成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $y=2 x$; (2) $(-\\infty, 0] \\cup[1,+\\infty)$; (3) $(-\\infty,\\dfrac{2}{\\mathrm{e}})$", "solution": "", @@ -572820,7 +578962,9 @@ "id": "040464", "content": "设集合$A=\\{x \\| x |<2, x \\in \\mathbf{R}\\}$, $B=\\{x | x^2-4 x+3 \\geq 0,\\ x \\in \\mathbf{R}\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$(-2,1]$", "solution": "", @@ -572853,7 +578997,9 @@ "id": "040465", "content": "已知$\\mathrm{i}$为虚数单位, 复数$z$满足$\\dfrac{1-z}{1+z}=\\mathrm{i}$, 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -572900,7 +579046,9 @@ "id": "040466", "content": "在平面直角坐标系内, 直线$l: 2 x+y-2=0$, 将$l$与两条坐标轴围成的封闭图形绕$x$轴旋转一周, 所得几何体的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{4 \\pi}{3}$", "solution": "", @@ -572933,7 +579081,9 @@ "id": "040467", "content": "已知$\\sin 2 \\theta+\\sin \\theta=0$, $\\theta \\in(\\dfrac{\\pi}{2}, \\pi)$, 则$\\tan 2 \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\sqrt{3}$", "solution": "", @@ -572969,7 +579119,9 @@ "id": "040468", "content": "设定义在$\\mathbf{R}$上的奇函数$y=f(x)$, 当$x>0$时, $f(x)=2^x-4$, 则不等式$f(x) \\leq 0$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-\\infty,-2] \\cup[0,2]$", "solution": "", @@ -573006,7 +579158,9 @@ "id": "040469", "content": "在平面直角坐标系$xOy$中, 有一定点$A(1,1)$, 若线段$OA$的垂直平分线过抛物线$C: y^2=2 p x(p>0)$的焦点, 则抛物线$C$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$y^2=4 x$", "solution": "", @@ -573042,7 +579196,9 @@ "id": "040470", "content": "设某产品的一个部件来自三个供应商, 这三个供应商的良品率分别是$0.92$、$0.95$、$0.94$, 若这三个供应商的供货比例为$3: 2: 1$, 那么这个部件的总体良品率是\\blank{50}(用最简分数作答).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{14}{15}$", "solution": "", @@ -573075,7 +579231,9 @@ "id": "040471", "content": "记$(2 x+\\dfrac{1}{x})^n$($n \\in \\mathbf{N}$, $n\\ge 1$)的展开式中第$m$项的系数为$b_m$, 若$b_3=2 b_4$, 则$n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -573108,7 +579266,9 @@ "id": "040472", "content": "已知一个正四棱锥的每条棱长均为$2$, 从该正四棱锥的$5$个顶点中任取$3$个点, 设随机变量$X$表示这三个点所构成的三角形的面积, 则其数学期望$E[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{6+2 \\sqrt{3}}{5}$", "solution": "", @@ -573141,7 +579301,9 @@ "id": "040473", "content": "已知函数$f(x)=x^2+p x+q$有两个零点$1$、$2$, 数列$\\{x_n\\}$满足$x_{n+1}=x_n-\\dfrac{f(x_n)}{f'(x_n)}$, 若$a_n=\\ln \\dfrac{x_n-2}{x_n-1}$, 且$a_1=-2$, 则数列$\\{a_n\\}$的前$2023$项的和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$2-2^{2024}$", "solution": "", @@ -573174,7 +579336,9 @@ "id": "040474", "content": "平面直角坐标系$xOy$中, 抛物线$y^2=2 x$的焦点为$F$, 设$M$是抛物线上的动点, 则$\\dfrac{|MO|}{|MF|}$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$\\dfrac{2 \\sqrt{3}}{3}$", "solution": "", @@ -573207,7 +579371,9 @@ "id": "040475", "content": "已知$a>0$, 函数$f(x)=x-\\dfrac{a}{x}$($x \\in[1,2]$)的图像的两个端点分别为$A$、$B$, 设$M$是函数$f(x)$图像上任意一点, 过$M$作垂直于$x$轴的直线$l$, 且$l$与线段$AB$交于点$N$, 若$|MN| \\leq 1$恒成立, 则$a$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$6+4 \\sqrt{2}$", "solution": "", @@ -573240,7 +579406,10 @@ "id": "040476", "content": "``$\\sin \\alpha=0$''是``$\\cos \\alpha=1$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分也非必要}", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第一单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -573273,7 +579442,9 @@ "id": "040477", "content": "设$l_1$、$l_2$为两条不同的直线, $\\alpha$为一个平面, 则下列命题正确的是\\bracket{20}.\n\\onech{若直线$l_1\\parallel$平面$\\alpha$, 直线$l_2\\parallel$平面$\\alpha$, 则$l_1\\parallel l_2$}{若直线$l_1$上有两个点到平面$\\alpha$的距离相等, 则$l_1\\parallel \\alpha$}{直线$l_2$与平面$\\alpha$所成角的取值范围是$(0, \\dfrac{\\pi}{2})$}{若直线$l_1 \\perp$平面$\\alpha$, 直线$l_2 \\perp$平面$\\alpha$, 则$l_1\\parallel l_2$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -573306,7 +579477,9 @@ "id": "040478", "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$是平面内两个互相垂直的单位向量, 若向量$\\overrightarrow {c}$满足$(\\overrightarrow {c}-\\overrightarrow {a}) \\cdot(\\overrightarrow {c}-\\overrightarrow {b})=0$, 则$|\\overrightarrow {c}|$的最大值是\\bracket{20}\n\\fourch{$1$}{$2$}{$\\sqrt{2}$}{$\\dfrac{\\sqrt{2}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -573342,7 +579515,9 @@ "id": "040479", "content": "已知$f(x)=\\begin{cases}|\\log _3 x|, & 0=latex]\n\\draw (0,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (2,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (0,2,2) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$D$} coordinate (D);\n\\draw (A)--(B)--(B_1)--(C_1)--(A_1)--cycle(A_1)--(B_1);\n\\draw [dashed] (A)--(C)--(B)(C)--(C_1)(C_1)--(D)--(C)(D)--(B_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp$平面$ACC_1A_1$;\\\\\n(2) 求二面角$B_1-CD-C_1$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 略; (2) $\\arccos \\dfrac{2}{3}$", "solution": "", @@ -573408,7 +579585,9 @@ "id": "040481", "content": "已知$f(x)=2 \\sin x \\cos x+2 \\cos ^2 x$.\\\\\n(1) 求函数$f(x)$的单调增区间;\\\\\n(2) 将函数$y=f(x)$图像向右平移$\\dfrac{\\pi}{4}$个单位后, 得到函数$y=g(x)$的图像, 求方程$g(x)=1$的解集.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $[k \\pi-\\dfrac{3 \\pi}{8}, k \\pi+\\dfrac{\\pi}{8}]$, $k \\in \\mathbf{Z}$; (2) $\\{x | x=\\dfrac{k \\pi}{2}+\\dfrac{\\pi}{8},\\ k \\in \\mathbf{Z}\\}$", "solution": "", @@ -573441,7 +579620,9 @@ "id": "040482", "content": "如图, 一智能扫地机器人在$A$处发现位于它正西方向的$B$处和北偏东$30^{\\circ}$方向上的$C$处分别有需要清扫的垃圾, 红外线感应测量发现机器人到$B$的距离比到$C$的距离少$0.4 \\text{m}$, 于是选择沿$A \\to B \\to C$路线清扫. 已知智能扫地机器人的直线行走速度为$0.2 \\text{m} / \\text{s}$, 忽略机器人吸入垃圾及在$B$处旋转所用时间, $10$秒钟完成了清扫任务.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0) node [right] {$A$} coordinate (A);\n\\draw (-0.6,0) node [left] {$B$} coordinate (B);\n\\draw (60:1) node [right] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (-0.6,0.6) coordinate (T);\n\\draw [->] (T) --++ (0.3,0) node [right] {东};\n\\draw [->] (T) --++ (0,0.3) node [above] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) $B$、$C$两处垃圾的距离是多少?\\\\\n(2) 智能扫地机器人此次清扫行走路线的夹角$\\angle ABC$是多少?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $1.4$米; (2) $\\arcsin \\dfrac{5 \\sqrt{3}}{14}$", "solution": "", @@ -573474,7 +579655,9 @@ "id": "040483", "content": "如图, 设$F$是椭圆$\\dfrac{x^2}{3}+\\dfrac{y^2}{4}=1$的下焦点, 直线$y=k x-4$($k>0$)与椭圆相交于$A$、$B$两点, 与$y$轴交于$P$点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (O) ellipse ({sqrt(3)} and 2);\n\\draw (0,-1) node [left] {$F$} coordinate (F);\n\\draw (0,-4) node [left] {$P$} coordinate (P);\n\\draw ({3*sqrt(5)/8},{-7/4}) node [below right] {$A$} coordinate (A);\n\\draw ($(P)!2!(A)$) node [right] {$B$} coordinate (B);\n\\draw ($(P)!-0.2!(B)$) -- ($(B)!-0.2!(P)$);\n\\draw (A)--(F)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\overrightarrow{PA}=\\overrightarrow{AB}$, 求$k$的值;\\\\\n(2) 求证: $\\angle AFP=\\angle BFO$;\\\\\n(3) 求$\\triangle ABF$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{6 \\sqrt{5}}{5}$; (2) 略; (3) $\\dfrac{3 \\sqrt{3}}{4}$", "solution": "", @@ -573507,7 +579690,9 @@ "id": "040484", "content": "已知正项数列$\\{a_n\\}$、$\\{b_n\\}$满足: 对任意$n \\in \\mathbf{N}$, $n\\ge 1$都有$a_n$、$b_n$、$a_{n+1}$成等差数列, $b_n$、$a_{n+1}$、$b_{n+1}$成等比数列, 且$a_1=10$, $a_2=15$.\\\\\n(1) 求证: 数列$\\{\\sqrt{b_n}\\}$是等差数列;\\\\\n(2) 求数列$\\{a_n\\}$、$\\{b_n\\}$的通项公式;\\\\\n(3) 设$S_n=\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\cdots+\\dfrac{1}{a_n}$, 如果对任意正整数$n$, 不等式$2 a S_n<2-\\dfrac{b_n}{a_n}$恒成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) 略; (2) $a_n=\\dfrac{(n+3)(n+4)}{2}$, $b_n=\\dfrac{(n+4)^2}{2}$; (3) $(-\\infty,1]$", "solution": "", @@ -573540,7 +579725,9 @@ "id": "040485", "content": "设$a$、$b$、$c$是互不相等的实数, 则满足条件$\\{a, b\\} \\cup A=\\{a, b, c\\}$的所有集合$A$有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -573573,7 +579760,9 @@ "id": "040486", "content": "已知复数$z=(1+2 \\mathrm{i})(3-\\mathrm{i})$, 则$\\dfrac{1}{z}$对应的点在第\\blank{50}象限.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "四", "solution": "", @@ -573606,7 +579795,9 @@ "id": "040487", "content": "若扇形的弧长和面积都是$4$, 那么这个扇形的圆心角的弧度数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -573639,7 +579830,9 @@ "id": "040488", "content": "首项为$1$, 公比为$-\\dfrac{1}{2}$的无穷等比数列$\\{a_n\\}$的各项和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$\\dfrac 23$", "solution": "", @@ -573674,7 +579867,9 @@ "id": "040489", "content": "已知数据$x_1, x_2, x_3, \\cdots, x_8$的方差为$16$, 则数据$3 x_1+1$、$3 x_2+1$、$\\cdots$、$3 x_8+1$的标准差为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$12$", "solution": "", @@ -573707,7 +579902,9 @@ "id": "040490", "content": "若$(x^2-\\dfrac{2}{x^3})^5$展开式中的常数项为\\blank{50}(用数字作答).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$40$", "solution": "", @@ -573740,7 +579937,10 @@ "id": "040491", "content": "已知函数$f(x)=\\log _a(x+2)-2$($a>0$且$a \\neq 1)$的图像恒过定点$A$, 若点$A$在一次函数$y=m x-n$的图像上, 其中$m, n>0$, 则$\\dfrac{1}{m}+\\dfrac{1}{n}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第一单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -573773,7 +579973,9 @@ "id": "040492", "content": "某校数学兴趣小组给一个底面边长互不相等的直四棱柱容器的侧面和下底面染色, 提出如下的``四色问题'': 要求相邻两个面不得使用同一种颜色, 现有$4$种颜色可以选择, 则不同的染色方案有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$72$", "solution": "", @@ -573806,7 +580008,9 @@ "id": "040493", "content": "有一种卫星接收天线, 其曲面与轴截面的交线为抛物线的一部分, 已知该卫星接收天线的口径$AB=6$, 深度$MO=2$, 信号处理中心$F$位于焦点处, 以顶点$O$为坐标原点, 建立如图所示的平面直角坐标系$xOy$, 若$P$是该抛物线上一点, 点$Q(\\dfrac{15}{8}, 2)$, 则$|PF|+|PQ|$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3, samples = 100] plot ({\\x*\\x/4.5},\\x);\n\\draw [dashed, domain = -4:-3, samples = 100] plot ({\\x*\\x/4.5},\\x);\n\\draw [dashed, domain = 3:4, samples = 100] plot ({\\x*\\x/4.5},\\x);\n\\draw (2,3) node [above] {$A$} coordinate (A);\n\\draw (2,-3) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below right] {$M$} coordinate (M);\n\\draw (1.25,0) node [below] {$F$} coordinate (F);\n\\filldraw (F) circle (0.1);\n\\draw [dashed] (A)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -573839,7 +580043,9 @@ "id": "040494", "content": "$y=\\sin x+\\sin 2 x$在$(-a, a)$上恰有$5$个零点, 则实数$a$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{4\\pi}3$", "solution": "", @@ -573872,7 +580078,9 @@ "id": "040495", "content": "设向量$\\overrightarrow{OA}, \\overrightarrow{OB}$满足$|\\overrightarrow{OA}|=|\\overrightarrow{OB}|=2$, $\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=2$, 若$m, n \\in \\mathbf{R}$, $m+n=1$, 则$|m \\overrightarrow{AB}|+|\\dfrac{1}{2} \\overrightarrow{BO}-n \\overrightarrow{BA}|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\sqrt{3}$", "solution": "", @@ -573905,7 +580113,9 @@ "id": "040496", "content": "函数$f(x)=\\begin{cases}-3 x,& x<0, \\\\ x^2-1,& x \\geq 0.\\end{cases}$ 若方程$f(x)+3 \\sqrt{1-x^2}+|f(x)-3 \\sqrt{1-x^2}|-2 a x-6=0$有三个根, 且$x_1\\sqrt{b}$''是``$\\ln a>\\ln b$''的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充要}{既不充分也不必要}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第一单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -573971,7 +580184,9 @@ "id": "040498", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 且$b^2+c^2=a^2+b c$. 若$\\sin B \\sin C=\\sin ^2A$, 则$\\triangle ABC$的形状是\\bracket{20}.\n\\fourch{等腰且非等边三角形}{直角三角形}{等边三角形}{等腰直角三角形}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -574004,7 +580219,9 @@ "id": "040499", "content": "在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, 动点$P$在棱$A_1B_1$上, 动点$Q$在线段$BC_1$上, 若$A_1P=\\lambda$, $BQ=\\mu$, 则三棱锥$D_1-APQ$的体积\\bracket{20}.\n\\fourch{与$\\lambda$无关, 与$\\mu$有关}{与$\\lambda$有关, 与$\\mu$无关}{与$\\lambda, \\mu$都有关}{与$\\lambda, \\mu$都无关}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -574037,7 +580254,9 @@ "id": "040500", "content": "已知数列$\\{a_n\\}$是各项为正数的等比数列, 公比为$q$, 在$a_1, a_2$之间插入$1$个数, 使这$3$个数成等差数列, 记公差为$d_1$; 在$a_2, a_3$之间插入$2$个数, 使这$4$个数成等差数列, 公差为$d_2$, $\\cdots$, 在$a_n, a_{n+1}$之间插入$n$个数, 使这$n+2$个数成等差数列, 公差为$d_n$, 则\\bracket{20}.\n\\twoch{当$01$时, 数列$\\{d_n\\}$单调递增}{当$d_1>d_2$时, 数列$\\{d_n\\}$单调递减}{当$d_1=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(6)},0,0) node [right] {$B$} coordinate (B);\n\\draw (B) ++ (0,{sqrt(2)},0) node [above] {$P$} coordinate (P);\n\\draw (A) ++ ({sqrt(6)/2},0,{sqrt(6)/2}) node [below] {$C$} coordinate (C);\n\\draw ($(C)!0.5!(P)$) node [above] {$D$} coordinate (D);\n\\draw (A)--(C)--(B)--(P)--cycle(P)--(C);\n\\draw (A)--(D)--(B);\n\\draw [dashed] (A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp AC$;\\\\\n(2) 若$AC=\\sqrt{3}$, 求直线$BC$与平面$ADB$所成角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{\\sqrt{14}}7$", "solution": "", @@ -574090,7 +580311,9 @@ "id": "040502", "content": "为了促进地方经济的快速发展, 国家鼓励地方政府实行积极灵活的人才引进政策, 被引进的人才, 可享受地方的福利待遇, 发放高标准的安家补贴费和生活津贴. 某市政府从本年度的$1$月份开始进行人才招聘工作, 参加报名的人员通过笔试和面试两个环节的审查后, 符合一定标准的人员才能被录用. 现对该市$1\\sim 4$月份的报名人员数和录用人才数(单位: 千人)进行统计, 得到如下表格.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 月份 & 1 月份 & 2 月份 & 3 月份 & 4 月份 \\\\\n\\hline 报名人员数$x /$千人 & 3.5 & 5 & 6.5 & 7 \\\\\n\\hline 录用人才数$y /$千人 & 0.2 & 0.33 & 0.4 & 0.47 \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n(1) 求出$y$关于$x$的经验回归方程;\\\\\n(2) 假设该市对被录用的人才每人发放$2$万元的生活津贴\\\\\n(i) 若该市$5$月份报名人员数为$8000$人, 试估计该市对$5$月份招聘的人才需要发放的生活津贴的总金额;\\\\\n(ii) 假设在参加报名的人员中, 小王和小李两人被录用的概率分别为$p, 3p-1$. 若两人的生活津贴之和的均值不超过$3$万元, 求$p$的取值范围.\\\\\n附: 经验回归方程$\\hat{y}=\\hat{a}+\\hat{b} x$中, 斜率和截距的最小二乘法估计公式分别为$\\hat{b}=\\dfrac{\\displaystyle\\sum_{i=1}^n x_i y_i-n \\overline{x}\\cdot \\overline{y}}{\\displaystyle\\sum_{i=1}^n x_i^2-n \\overline {x}^2}$, $\\hat{a}=\\overline {y}-\\hat{b} \\overline {x}$; $\\displaystyle\\sum_{i=1}^4 x_i^2=128.5$, $\\displaystyle\\sum_{i=1}^4 x_i y_i=8.24$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) $y=0.072x-0.046$; (2) (i) $1060$万; (ii) $(\\dfrac 13,\\dfrac 58]$", "solution": "", @@ -574124,7 +580347,9 @@ "id": "040503", "content": "已知函数$f(x)=\\cos ^2 x+\\sin x \\cos x-\\dfrac{1}{2}$, 其中$x \\in \\mathbf{R}$.\\\\\n(1) 求不等式$f(x) \\geq \\dfrac{1}{2}$的解集;\\\\\n(2) 若函数$g(x)=\\dfrac{\\sqrt{2}}{2} \\sin (2 x+\\dfrac{3 \\pi}{4})$, 且对任意的$0 \\leq x_10$, $b>0$)的一条渐近线, 且双曲线$C$经过点$(2 \\sqrt{2}, 1)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-10,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (10,5) -- (-10,-5) (10,-5) -- (-10,5);\n\\draw [domain = -10:{-2*sqrt(2)}, samples = 100] plot (\\x,{sqrt(\\x*\\x/4-2)});\n\\draw [domain = -10:{-2*sqrt(2)}, samples = 100] plot (\\x,{-sqrt(\\x*\\x/4-2)});\n\\draw [domain = {2*sqrt(2)}:10, samples = 100] plot (\\x,{sqrt(\\x*\\x/4-2)});\n\\draw [domain = {2*sqrt(2)}:10, samples = 100] plot (\\x,{-sqrt(\\x*\\x/4-2)}); \n\\end{tikzpicture}\n\\end{center}\n(1) 求双曲线$C$的方程;\\\\\n(2) 设直线$l': x=t y+4$与$C$交于$M, N$, 三角形$OMN$面积为$S$, 判断: 是否存在$t$使得$S=8 \\sqrt{15}$成立? 若存在, 求出$t$的值, 否则说明理由;\\\\\n(3) 设$A, B$是双曲线右支上两点, 若直线$l$上存在点$P$, 使得$\\triangle ABP$为正三角形, 求直线$AB$的斜率的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{4}-y^2=1$; (2) 存在, $t=\\pm \\sqrt{3}$或$t=\\pm \\sqrt{\\dfrac{76}{15}}$; (3) $(-8-5\\sqrt{3},-\\dfrac 12)\\cup (\\dfrac 12,-8+5\\sqrt{3})$", "solution": "", @@ -574191,7 +580418,9 @@ "id": "040505", "content": "已知函数$f(x)=a x$($a \\in \\mathbf{R}$), $g(x)=\\cos x$.\\\\\n(1) 分别写出函数$y=f(g(x))$与$y=g(f(x))$的导函数;\\\\\n(2) 当$x \\in[\\pi,+\\infty)$时, 若不等式$f(x-\\pi) \\leq g(\\dfrac{\\pi}{2}-x)$恒成立, 求实数$a$的取值范围;\\\\\n(3) 令函数$h(x)=f(x)+g(x)$, $x \\in[0, \\pi]$, 若$y=h(x)$恰有两个极值点, 记极大值和极小值分别为$m, n$, 求$2 m-n$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $f(g(x))$的导函数为$y=-a\\sin x$, $g(f(x))$的导函数为$y=-a\\sin (ax)$; (2) $(-\\infty,-1]$; (3) $[\\dfrac 32,3)$", "solution": "", @@ -574224,7 +580453,9 @@ "id": "040506", "content": "已知$\\overrightarrow {a}=(1,2)$, $\\overrightarrow {b}=(\\lambda, 3)$, $(2 \\overrightarrow {a}-\\overrightarrow {b}) \\perp \\overrightarrow {a}$, 则$\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -574244,7 +580475,9 @@ "id": "040507", "content": "设$\\mathrm{i}$为虚数单位, 若复数$(1+\\mathrm{i})(1+a \\mathrm{i})$是纯虚数, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -574264,7 +580497,9 @@ "id": "040508", "content": "$(1-2 x)^4$的展开式中含$x^2$项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$24$", "solution": "", @@ -574284,7 +580519,9 @@ "id": "040509", "content": "已知$A(-\\sqrt{3}, 0)$, $B(\\sqrt{3}, 0)$, $C(0,3)$, 则$\\triangle ABC$外接圆的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$x^2+(y-1)^2=4$", "solution": "", @@ -574304,7 +580541,9 @@ "id": "040510", "content": "已知集合$M=\\{y | y=\\sin x,\\ x \\in \\mathbf{R}\\}$, $N=\\{x | x^2-x-2<0\\}$, 则$M \\cap N=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$(-1,1]$", "solution": "", @@ -574324,7 +580563,9 @@ "id": "040511", "content": "一个正六棱柱的茶叶盒, 底面边长为$10 \\text{cm}$, 高为$20 \\text{cm}$, 则这个茶叶盒的表面积为\\blank{50}$\\text{cm}^2$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$300(4+\\sqrt{3})$", "solution": "", @@ -574344,7 +580585,9 @@ "id": "040512", "content": "已知一个半径为$4$的扇形圆心角为$\\theta$($0<\\theta<2 \\pi$), 面积为$2 \\pi$, 若$\\tan (\\theta+\\varphi)=3$, 则$\\tan \\varphi=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{1}{2}$", "solution": "", @@ -574364,7 +580607,9 @@ "id": "040513", "content": "某校为了了解高三年级学生的身体素质状况, 在开学初举行了一场身体素质体能测试, 以便对体能不达标的学生进行有针对性的训练, 促进他们体能的提升, 现从整个年级测试成绩中抽取$100$名学生的测试成绩, 并把测试成绩分成$[40,50)$、$[50,60)$、$[60,70)$、$[70,80)$、$[80,90)$、$[90,100]$六组, 绘制成频率分布直方图 (如图所示). 其中分数在$[90,100]$这一组中的纵坐标为$a$, 则该次体能测试成绩的第$80$百分位数约为分\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.06, yscale = 110]\n\\draw [->] (0,0) -- (120,0) node [below] {分数};\n\\draw [->] (0,0) -- (0,0.045) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {40/0.002,50/0.004,60/0.014,70/0.02,80/0.035,90/0.025}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {40/0.002,50/0.004,60/0.014,70/0.02,80/0.035,90/0.025/a}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$92$", "solution": "", @@ -574384,7 +580629,9 @@ "id": "040514", "content": "莫高窟坐落在甘肃的敦煌, 它是世界上现存规模最大、内容最丰富的佛教艺术胜地, 每年都会吸引来自世界各地的游客参观旅游. 已知购买莫高窟正常参观套票可以参观$8$个开放洞窟, 在这$8$个洞中莫高窟九层楼$96$号、莫高三层楼$16$号窟、藏经洞$17$号被誉为最值得参观的洞窟. 根据疫情防控的需要, 莫高窟改为极速参观模式, 游客需从套票包含的开放洞、 窟中随机选择$4$个进行参观, 所有选择中至少包含$2$个最值得参观洞窟的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac{1}{2}$", "solution": "", @@ -574404,7 +580651,9 @@ "id": "040515", "content": "已知$F$是椭圆$E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左焦点, 经过原点$O$的直线$l$与椭圆$E$交于$P$、$Q$两点, 若$|PF|=5|QF|$, 且$\\angle PFQ=120^{\\circ}$, 则椭圆$E$的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{21}}{6}$", "solution": "", @@ -574424,7 +580673,9 @@ "id": "040516", "content": "在$\\triangle ABC$中, 内角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 已知$b \\cos C+c \\cos B=3 a \\cos A$, 若$S$为$\\triangle ABC$的面积, 则$\\dfrac{a^2}{S}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2 \\sqrt{2}$", "solution": "", @@ -574444,7 +580695,10 @@ "id": "040517", "content": "对于二元函数$f(x, y)$, $\\min _x\\{\\max _y\\{f(x, y)\\}\\}$表示$f(x, y)$先关于$y$求最大值, 再关于$x$求最小值, 已知平面内非零向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$, 满足$\\overrightarrow {a} \\perp \\overrightarrow {b}$, $\\dfrac{\\overrightarrow {a} \\cdot \\overrightarrow {c}}{|\\overrightarrow {a}|}=2 \\dfrac{\\overrightarrow {b} \\cdot \\overrightarrow {c}}{|\\overrightarrow {b}|}$, 记$f(m, n)=\\dfrac{|m \\overrightarrow {c}-\\overrightarrow {b}|}{|m \\overrightarrow {c}-n \\overrightarrow {a}|}$($m$、$n \\in \\mathbf{R}$, 且$m \\neq 0$, $n \\neq 0$), 则$\\min _m\\{\\max _n\\{f(m, n)\\}\\}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第二单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -574464,7 +580718,9 @@ "id": "040518", "content": "设$l$、$m$、$n$表示直线, $\\alpha$、$\\beta$表示平面, 使``$l \\perp \\alpha$''成立的一个充分条件是\\bracket{20}.\n\\twoch{$\\alpha \\perp \\beta$, $l\\parallel \\beta$}{$\\alpha \\perp \\beta$, $l \\subset \\beta$}{$l\\parallel n$, $n \\perp \\alpha$}{$m \\subset \\alpha$, $n \\subset \\alpha$, $l \\perp m$, $l \\perp n$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -574484,7 +580740,9 @@ "id": "040519", "content": "2020 年初, 新型冠状病毒引起的肺炎疫情暴发以来, 各地医疗机构采取了各种针对性的治疗方法, 取得了不错的成效, 某医疗机构开始使用中西医结合方法后, 每周治愈的患者人数如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 第$x$周 & 1 & 2 & 3 & 4 & 5 \\\\\n\\hline 治愈人数$y$(单位: 十人) & 3 & 8 & 10 & 14 & 15 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n由上表可得$y$关于$x$的线性回归方程为$y=\\hat{a}x+1$, 则此回归模型第$5$周的离差(实际值减去估计值)为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -574504,7 +580762,9 @@ "id": "040520", "content": "设函数$f(x)$定义域为$\\mathbf{R}$, $f(x-1)$为奇函数, $f(x+1)$为偶函数, 当$x \\in(-1,1)$时, $f(x)=-x^2+1$, 则下列四个结论: \\textcircled{1} $f(\\dfrac{7}{2})=-\\dfrac{3}{4}$; \\textcircled{2} $f(x+7)$为奇函数; \\textcircled{3} $f(x)$在$(6,8)$上为减函数; \\textcircled{4} $f(x)$的一个周期为$8$, 其中错误的个数是\\bracket{20}.\n\\fourch{1}{2}{3}{4}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -574524,7 +580784,9 @@ "id": "040521", "content": "已知共有$k$($k$为正整数) 项的数列$\\{a_n\\}$, $a_1=2$, 定义向量$\\overrightarrow{c_n}=(a_n, a_{n+1})$, $\\overrightarrow{d_n}=(n, n+1)$、($n=1,2,\\cdots, k-1$), 若$|\\overrightarrow{c_n}|=|\\overrightarrow{d_n}|$, 则满足条件的数列$\\{a_n\\}$的个数有\\bracket{20}个.\n\\fourch{$2$}{$k$}{$2^{k-1}$}{$2^{\\frac{k(k-1)}{2}}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -574544,7 +580806,9 @@ "id": "040522", "content": "已知公差$d$不为$0$的等差数列$\\{a_n\\}$的前$n$项和为$S_n$, $a_3=6$, $\\dfrac{S_5}{S_9}=\\dfrac{1}{3}$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 若数列$b_n=2^{a_n}$, $c_n=a_n+b_n$, 求数列$\\{c_n\\}$的前$n$项和$T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) $2n$; (2) $n^2+n+\\dfrac{4^{n+1}}{3}-\\dfrac{4}{3}$", "solution": "", @@ -574564,7 +580828,10 @@ "id": "040523", "content": "为了了解员工长假的出游意愿, 某单位从``70 后''至``00 后''的人群中按年龄段分层抽取了$100$名员工进行调查. 调查结果如图所示, 已知每个员工仅有``有出游意愿''和``无出游意愿''两种回答, 且样本中`` 00 后''与``90 后''员工占比分别为$10 \\%$和$30 \\%$. 图中表示了各年龄段员工有出游意愿的人数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [gray!40] (90:2.5) arc (90:126:2.5) -- (126:0.5) arc (126:90:0.5) -- cycle;\n\\fill [gray!10] (126:2.5) arc (126:216:2.5) -- (216:0.5) arc (216:126:0.5) -- cycle;\n\\fill [gray!50] (216:2.5) arc (216:324:2.5) -- (324:0.5) arc (324:216:0.5) -- cycle;\n\\draw (0,0) circle (0.5) circle (1.5) circle (2.5);\n\\draw (90:0.5) -- (90:2.5) (126:0.5) -- (126:2.5) (216:0.5) -- (216:2.5) (324:0.5)--(324:2.5);\n\\draw (97.2:1.5) -- (97.2:2.5) (0:1.5) -- (0:2.5) (162:1.5) -- (162:2.5) (244.8:1.5) -- (244.8:2.5);\n\\draw (108:1) node {\\small{00后}};\n\\draw (27:1) node {\\small{80后}};\n\\draw (171:1) node {\\small{70后}};\n\\draw (270:1) node {\\small{90后}};\n\\draw (111.6:2) node {\\small{8人}};\n\\draw (189:2) node {\\small{15人}};\n\\draw (284.4:2) node {\\small{22人}};\n\\draw (45:2) node {\\small{25人}};\n\\end{tikzpicture}\n\\end{center}\n(1) 现从`` 00 后''样本中随机抽取$3$人, 记$3$人中``无出游意愿''的人数为随机变量$X$, 求$X$的分布及数学期望;\\\\\n(2)若把``00 后''和``90 后''定义为青年, ``80 后''和``70 后''定义为中年, 结合样本数据完成$2 \\times 2$列联表, 并回答能否在犯错误的概率不超过$0.05$的前提下认为该单位员工长假的出游意愿与年龄段有关?\\\\\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 有出游意愿 & 无出游意愿 & 合计 \\\\\n\\hline 青年 & & & \\\\\n\\hline 中年 & & & \\\\\n\\hline 合计 & & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)} \\text {, 其中 } n=a+b+c+d$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$P(\\chi^2 \\geq k_0)$& 0.050 & 0.010 & 0.005 & 0.001 \\\\\n\\hline$k_0$& 3.841 & 6.635 & 7.879 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "(1) $\\begin{pmatrix}0 & 1 & 2 \\\\ \\dfrac{7}{15} & \\dfrac{7}{15} & \\dfrac{1}{15}\\end{pmatrix}$, 期望为$\\dfrac{3}{5}$; (2) $\\chi^2 \\approx 0.794<3.841$, 不能认为有关", "solution": "", @@ -574584,7 +580851,9 @@ "id": "040524", "content": "如图所示, 在四棱锥$S-ABCD$中, $AD \\perp$平面$SCD$, $BC \\perp$平面$SCD$, $AD=CD=2$, $BC=1$, 又$SD=2$, $\\angle SDC=120^{\\circ}$, $F$为$SD$中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$S$} coordinate (S);\n\\draw (0,2,0) node [above] {$A$} coordinate (A);\n\\draw (-0.5,0,{sqrt(3)/2}) node [below] {$C$} coordinate (C);\n\\draw (C) ++ (0,1,0) node [left] {$B$} coordinate (B);\n\\draw ($(D)!0.5!(S)$) node [above left] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(S)--cycle(B)--(S);\n\\draw [dashed] (C)--(D)--(S)(C)--(F)(A)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $CF\\parallel$平面$SAB$;\\\\\n(2) 求平面$SAD$与平面$SAB$所成的锐二面角的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 略; (2) $\\dfrac{\\sqrt{10}}{5}$", "solution": "", @@ -574604,7 +580873,9 @@ "id": "040525", "content": "已知椭圆$E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的长轴长是短轴长的两倍, 且$E$过点$(-\\sqrt{3}, \\dfrac{1}{2})$.\n(1) 求椭圆$E$的方程;\\\\\n(2) 若点$S(1,0)$, 点$T$为圆$E$上的任意一点, 求$|\\overrightarrow{TS}|$的最大值与最小值;\\\\\n(3) 设椭圆$E$的下顶点为点$A$, 若不过点$A$且不垂直于坐标轴的直线$l$交椭圆$E$于$P$、$Q$两点, 直线$AP$、$AQ$分别与$x$轴交于$M$、$N$两点, 若$M$、$N$的横坐标之积是$2$, 证明: 直线$l$过定点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{4}+y^2=1$; (2) 最大值为$3$; 最小值为$\\dfrac{\\sqrt{6}}{3}$; (3) $(0,3)$", "solution": "", @@ -574624,7 +580895,9 @@ "id": "040526", "content": "已知函数$f(x)=\\mathrm{e}^x$, $g(x)=\\sin x+\\cos x$.\\\\\n(1) 求函数$y=g(x)$在点$(0,1)$处的切线方程;\\\\\n(2) 已知$\\mathrm{e}^x \\geq x+1$对于$x \\in \\mathbf{R}$恒成立, 证明: 当$x>-\\dfrac{\\pi}{4}$时, $f(x) \\geq g(x)$;\\\\\n(3) 已知$a \\in \\mathbf{R}$. 若当$x>-\\dfrac{\\pi}{4}$时, 不等式$f(x)+g(x)-2-a x \\geq 0$恒成立, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $y=x+1$; (2) 略; (3) $2$", "solution": "", @@ -574644,7 +580917,9 @@ "id": "040527", "content": "$-2023^{\\circ}$是第\\blank{50}象限角.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "二", "solution": "", @@ -574664,7 +580939,9 @@ "id": "040528", "content": "已知$\\alpha$满足$\\tan \\alpha=\\sqrt{2}$, 那么$2 \\sin ^2 \\alpha-\\cos ^2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -574684,7 +580961,9 @@ "id": "040529", "content": "若函数$y=\\tan (\\omega x+\\dfrac{\\pi}{3})$(其中常数$\\omega \\in \\mathbf{R}$)的最小正周期为$\\pi$, 则常数$\\omega$取值集合的元素个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -574704,7 +580983,9 @@ "id": "040530", "content": "函数$y=\\sin (x+\\dfrac{\\pi}{6})$, $x \\in[-\\dfrac{\\pi}{3}, \\dfrac{\\pi}{2}]$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -574724,7 +581005,9 @@ "id": "040531", "content": "函数$y=\\sin (\\dfrac{\\pi}{2}(x+\\varphi))$($0<\\varphi<2 \\pi$)是奇函数, 那么常数$\\varphi$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$6$", "solution": "", @@ -574744,7 +581027,9 @@ "id": "040532", "content": "以$y=\\sin x$和$y=\\cos x$的图像的连续三个交点$A$、$B$、$C$作为$\\triangle ABC$的三个顶点, 则$\\triangle ABC$的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\sqrt{2}\\pi$", "solution": "", @@ -574764,7 +581049,9 @@ "id": "040533", "content": "设$\\omega>0$, 若函数$y=\\sin \\omega x$在区间$[0,2 \\pi]$上恰有两个零点, 则$\\omega$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[\\dfrac 12,1)$", "solution": "", @@ -574784,7 +581071,9 @@ "id": "040534", "content": "函数$y=\\sin ^2 x+2 \\cos x$的定义域为$[-\\dfrac{2 \\pi}{3}, \\alpha]$, 值域为$[-\\dfrac{1}{4}, 2]$, 则$\\alpha$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[0,\\dfrac 23\\pi]$", "solution": "", @@ -574804,7 +581093,9 @@ "id": "040535", "content": "为了得到函数$y=4 \\sin (2 x+\\dfrac{\\pi}{3})$的图像, 只需将函数$y=4\\sin 2x$的图像向\\blank{20}平移\\blank{50}个单位.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "左，$\\dfrac{\\pi}6$", "solution": "", @@ -574824,7 +581115,9 @@ "id": "040536", "content": "函数$y=\\sin (\\dfrac{\\pi}{6}-x)$的单调递减区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[2k\\pi-\\dfrac{\\pi}3,2k\\pi+\\dfrac 23 \\pi],k \\in \\mathbb{Z}$", "solution": "", @@ -574844,7 +581137,9 @@ "id": "040537", "content": "如果函数$y=\\sin 2 x+a \\cdot \\cos 2 x$的图象关于直线$x=-\\dfrac{\\pi}{8}$对称, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-1$", "solution": "", @@ -574864,7 +581159,9 @@ "id": "040538", "content": "下列说法正确的是\\bracket{20}.\n\\twoch{函数$y=\\sin x$在第一象限内是严格增函数}{函数$y=\\cos x$的图像是中心对称图像}{函数$y=\\tan x$在其定义域内是严格增函数}{函数$y=\\dfrac{\\sin x}{x}$是周期函数}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -574884,7 +581181,9 @@ "id": "040539", "content": "下列函数中, 以$\\dfrac{\\pi}{2}$为周期且单调增区间为$[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$的函数是\\bracket{20}.\n\\fourch{$|\\cos 2 x|$}{$|\\sin 2 x|$}{$\\sin 4 x$}{$\\cos 2 x$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -574904,7 +581203,9 @@ "id": "040540", "content": "$C$、$S$分别表示一个扇形的周长和面积, 下列能作为有序数对$(C,S)$的取值的是\\bracket{20}.\n\\fourch{$(3,1)$}{$(5,1)$}{$(4,2)$}{$(4,3)$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -574924,7 +581225,9 @@ "id": "040541", "content": "对于函数$f(x)=\\max \\{\\sin x, \\cos x\\}$, 下列说法中不正确的是\\blank{50}. (填上符合要求的全部序号)\\\\\n\\textcircled{1} $f(x)$的定义域是$\\mathbf{R}$; \\textcircled{2} $f(x)$的值域是$[-1,1]$; \\textcircled{3} $f(x)$是一个奇函数; \\textcircled{4} $x=2 k \\pi$或$2 k \\pi+\\dfrac{\\pi}{2}$, $k \\in \\mathbf{Z}$时$f(x)$的最大值是$1$; \\textcircled{5} $f(x)$的最小正周期是$2 \\pi$; \\textcircled{6} $f(x)$的递增区间是$[2 k \\pi+\\dfrac{\\pi}{4}, 2 k \\pi+\\dfrac{\\pi}{2}] \\cup[2 k \\pi+\\dfrac{5 \\pi}{4}, 2 k \\pi+2 \\pi]$, $k \\in \\mathbf{Z}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{2},\\textcircled{3},\\textcircled{6}", "solution": "", @@ -574944,7 +581247,9 @@ "id": "040542", "content": "在$\\triangle ABC$中, 下列结论正确的是\\blank{50}.\\\\\n\\textcircled{1} $\\sin (A+B)+\\sin C=0$;\n\\textcircled{2} $\\cos (A+B)+\\cos C=0$;\n\\textcircled{3} $\\sin (2A+2B)+\\sin 2C=0$;\n\\textcircled{4} $\\cos (2A+2B)+\\cos 2C=0$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{2},\\textcircled{3}", "solution": "", @@ -574964,7 +581269,9 @@ "id": "040543", "content": "在$\\triangle ABC$中, 有$A>B$, 下列结论正确的是\\blank{50}.\\\\\n\\textcircled{1} $\\sin A>\\sin B$;\n\\textcircled{2} $\\cos A<\\cos B$;\n\\textcircled{3} $\\sin A>\\cos B$;\n\\textcircled{4} $\\cot A<\\cot B$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{1},\\textcircled{2},\\textcircled{4}", "solution": "", @@ -574984,7 +581291,9 @@ "id": "040544", "content": "已知函数$f(x)=\\sin (2 x+\\pi)$, 则有\\blank{50}.\\\\\n\\textcircled{1} 函数$f(x)$的图像关于直线$x=\\dfrac{3 \\pi}{4}$对称;\n\\textcircled{2} 函数$f(x)$的图像关于点$(\\pi, 0)$对称;\n\\textcircled{3} 函数$f(x)$在区间$[\\dfrac{\\pi}{4}, \\dfrac{3 \\pi}{4}]$上为严格增函数;\n\\textcircled{4} 函数$f(x)$是偶函数.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{1},\\textcircled{2}", "solution": "", @@ -575004,7 +581313,9 @@ "id": "040545", "content": "已知函数$f(x)=|\\cos x|+\\cos |2 x|$, 下列结论正确的是\\blank{50}.\\\\\n\\textcircled{1} 函数$f(x)$是偶函数;\n\\textcircled{2} 函数$f(x)$的最小正周期是$\\pi$;\n\\textcircled{3} 函数$f(x)$在区间$[\\dfrac{3 \\pi}{4}, \\dfrac{5 \\pi}{4}]$上为严格增函数;\n\\textcircled{4} 当$x \\in[\\dfrac{3 \\pi}{4}, \\dfrac{5 \\pi}{4}]$时, 函数$f(x)$的最大值是$2$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "\\textcircled{1},\\textcircled{2},\\textcircled{4}", "solution": "", @@ -575024,7 +581335,9 @@ "id": "040546", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$的对边分别是$a$、$b$、$c$, $a^2+c^2-b^2=a c$, $b=\\sqrt{3}$, 求$2 a+c$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$(\\sqrt{3},2\\sqrt{7}]$", "solution": "", @@ -575044,7 +581357,9 @@ "id": "040547", "content": "判断下列函数的奇偶性:\\\\\n(1) $\\sin (2 x+\\varphi)$;\\\\ \n(2) $\\dfrac{1+\\sin x-\\cos x}{1+\\sin x+\\cos x}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) \\textcircled{1} $\\varphi=k\\pi,k \\in \\mathbb{Z}$时为奇函数；\\\\\n\\textcircled{2} $\\varphi=k\\pi+\\dfrac{\\pi}2,k \\in \\mathbb{Z}$时为偶函数；\\\\\n\\textcircled{3} $\\varphi \\neq \\dfrac{k\\pi}2,k \\in \\mathbb{Z}$时为非奇非偶函数.\\\\\n(2)非奇非偶函数", "solution": "", @@ -575064,7 +581379,9 @@ "id": "040548", "content": "讨论关于实数$x$的方程解的个数:\\\\\n(1) $\\sin x+\\cos x=k$, $x \\in[0, \\pi]$;\\\\\n(2) $\\sin ^2 x+\\cos x=k$, $x \\in[0, \\pi]$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$k=\\sqrt{2}$或$k \\in [-1,1)$时，一解；$k \\in [1,\\sqrt{2})$时，两解；$k \\in (-\\infty，-1)\\cup (\\sqrt{2},+\\infty)$时，无解.\n\\\\\n(2)$k=\\dfrac 54$或$k \\in [-1,1)$时，一解；$k \\in [1,\\dfrac 54)$时，两解；$k \\in (-\\infty，-1)\\cup (\\dfrac 54,+\\infty)$时，无解.", "solution": "", @@ -575084,7 +581401,9 @@ "id": "040549", "content": "如图, 某城市有一矩形街心广场$ABCD$, 其中$AB=4$百米, $BC=3$百米. 现将挖掘一个三角形水池$DMN$种植荷花, 其中$M$点在$BC$边上, $N$点在$AB$边上, 且必须满足要求: $\\angle MDN=\\dfrac{\\pi}{4}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0) node [below] {$B$} coordinate (B);\n\\draw (4,3) node [above] {$C$} coordinate (C);\n\\draw (0,3) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [below] {$N$} coordinate (N);\n\\draw (C) ++ (0,{-4/3}) node [right] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(D)--cycle(D)--(N)--(M)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 若$AN=CM=2$百米, 判断$\\triangle DMN$是否符合要求, 并说明理由;\\\\\n(2) 设$\\angle CDM=\\theta$, 求$\\triangle DMN$的面积$S$关于$\\theta$的函数关系式, 并求出$S$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)不符合；\\\\\n(2)$\\theta=\\dfrac{\\pi}8$时，$S$取最小值，最小值为$12\\sqrt{2}-12$", "solution": "", @@ -575104,7 +581423,10 @@ "id": "040550", "content": "已知函数$y=f(x)$, $x \\in \\mathbf{R}$是周期为$\\pi$的周期函数, 当$x \\in[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$时, $f(x)=\\sin \\dfrac{x}{2}$.\\\\\n(1) 求$f(\\dfrac{365 \\pi}{3})$的值;\\\\\n(2) 当$x \\in[-\\dfrac{5 \\pi}{2},-\\dfrac{\\pi}{2})$时, 求$f(x)$的表达式;\\\\\n(3) 设$g(x)=f(x)+2|f(x)|$, 求方程$g(x)=\\dfrac{1}{2}$的解集.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "解答题", "ans": "(1)$-\\dfrac 12$\\\\\n(2)\\\\\n(3)$\\{x|x=k\\pi+2\\arcsin{\\dfrac 16}$或$x=k\\pi-\\dfrac{\\pi}3,k \\in \\mathbb{Z}\\}$", "solution": "", @@ -575124,7 +581446,9 @@ "id": "040551", "content": "已知函数$f(x)=4 \\sin x \\cos (x+\\dfrac{\\pi}{3})+\\sqrt{3}$, $x \\in \\mathbf{R}$.\\\\\n(1) 将函数$f(x)$化简为$y=A \\sin (\\omega x+\\varphi)+B$的形式, 写出其振幅、初相与最小正周期;\\\\\n(2) 求函数$f(x)$的单调增区间、最小值及此时所有$x$的取值;\\\\\n(3) 将函数$f(x)$的图像向右平移$\\dfrac{\\pi}{6}$个单位, 再将所得图像上各点的横坐标缩短为原来的$a$($01,\\end{cases}$若不等式$f(x) \\leq|x-k|$对任意的$x \\in \\mathbf{R}$成立, 则$k$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$[0,2-\\ln 2]$", "solution": "", @@ -575301,7 +581639,9 @@ "id": "040559", "content": "已知向量$\\overrightarrow{AC}, \\overrightarrow{AD}$和$\\overrightarrow{AB}$在单位正方形拼成的矩形中的位置如图所示, 若$\\overrightarrow{AD}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AC}$, 其中$\\lambda$、$\\mu \\in \\mathbf{R}$, 则$\\lambda \\mu=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale= 0.6]\n\\foreach \\i in {0,1,2,3,4}\n{\\draw [dashed] (\\i,-1) -- (\\i,5) (-1,\\i) -- (5,\\i);};\n\\draw [->] (1,2) node [above left] {$A$} -- (2,2) node [above right] {$D$};\n\\draw [->] (1,2) -- (3,0) node [above right] {$C$};\n\\draw [->] (1,2) -- (2,4) node [below right] {$B$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575321,7 +581661,9 @@ "id": "040560", "content": "设$a \\in \\mathbf{R}$, 若不等式$|x-2 a| \\geq \\dfrac{1}{2} x+a-1$对任意的$x \\in \\mathbf{R}$成立, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-\\infty,\\dfrac 12]$", "solution": "", @@ -575348,7 +581690,9 @@ "id": "040561", "content": "如图, 在等腰直角$\\triangle ABC$中, $AB=AC=2$, 点$P$是边$AB$上异于$A$、$B$的一点, 光线从点$P$出发, 经$BC$、$CA$反射后又回到点$P$. 若光线$QR$经过$\\triangle ABC$的内心, 则$AP=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$C$} coordinate (C);\n\\draw ({2*sqrt(2)-2},0) node [below] {$P$} coordinate (P);\n\\draw ({4-2*sqrt(2)},{2*sqrt(2)-2}) node [above right] {$Q$} coordinate (Q);\n\\draw (0,{6-4*sqrt(2)}) node [left] {$R$} coordinate (R);\n\\draw (A)--(B)--(C)--cycle(P)--(Q)--(R)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575368,7 +581712,9 @@ "id": "040562", "content": "若$y=f(x)$是$\\mathbf{R}$上的偶函数, 在区间$[0,+\\infty)$上是严格减函数, 且$f(a)=0$($a>0$), 则不等式$x \\cdot f(x)<0$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575388,7 +581734,9 @@ "id": "040563", "content": "设$m \\in \\mathbf{R}$, 若方程$x^2-4|x|+5=m$有$4$个不相等的实根, 则$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$(1,5)$", "solution": "", @@ -575415,7 +581763,10 @@ "id": "040564", "content": "函数$y=\\dfrac{-1}{x-1}$的图像与函数$y=2 \\sin \\pi x(-2 \\leq x \\leq 4)$的图像所有交点的横坐标之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "$8$", "solution": "", @@ -575442,7 +581793,9 @@ "id": "040565", "content": "在$\\triangle ABC$中, $AC=4$, 若$\\overrightarrow{AC}$在$\\overrightarrow{AB}$方向上的数量投影是$-2$, 则$|\\overrightarrow{BC}-\\lambda \\overrightarrow{BA}|$($\\lambda \\in \\mathbf{R}$)的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575466,7 +581819,9 @@ "id": "040566", "content": "设$\\{a_n\\}$是等差数列, 其前$n$项和为$S_n$, 若$S_5S_8$, 则下列结论错误的是\\bracket{20}.\n\\twoch{$\\{a_n\\}$的公差小于$0$}{$a_7=0$}{$S_9>S_5$}{$S_6$与$S_7$均为数列$\\{S_n\\}$的最大项}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -575486,7 +581841,9 @@ "id": "040567", "content": "若复数$z$满足$|z|=1$, 则$|z+2-\\mathrm{i}|$($\\mathrm{i}$为虚数单位)的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575506,7 +581863,9 @@ "id": "040568", "content": "若正实数$a$、$b$满足$3 a+2 b=6$, 则$b+\\sqrt{a^2+b^2-2 b+1}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$\\dfrac{29}{13}$", "solution": "", @@ -575533,7 +581892,9 @@ "id": "040569", "content": "设$a>0, b$、$c \\in \\mathbf{R}, f(x)=a x^3+b x^2+x$, 若$y=f'(x)$在区间$(-\\infty, 1]$上是严格减函数, 且$b-a^2+2 a+2 \\geq 0$, 则$\\dfrac{b-3}{a-2}$的取值范围是\\bracket{20}.\n\\fourch{$[\\dfrac{3}{2}, 6]$}{$[\\dfrac{1}{2}, \\dfrac{3}{2}]$}{$(\\dfrac{3}{2}, 6]$}{$[\\dfrac{1}{2}, \\dfrac{3}{2})$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -575555,7 +581916,9 @@ "id": "040570", "content": "$(x^3-2 x)^7$的展开式的第$4$项的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575587,7 +581950,9 @@ "id": "040571", "content": "$(x+\\dfrac{2}{\\sqrt{x}})^6$展开式中常数项是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575619,7 +581984,9 @@ "id": "040572", "content": "$(2 x-5 y)^{20}$的展开式中各项系数之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575651,7 +582018,9 @@ "id": "040573", "content": "$(x-1)^{11}$展开式中$x$的偶次项系数之和是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575683,7 +582052,9 @@ "id": "040574", "content": "若$\\mathrm{C}_n^0+2\\mathrm{C}_n^1+4\\mathrm{C}_n^2+\\cdots+2^n \\mathrm{C}_n^n=729$, 则$\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\mathrm{C}_n^3+\\cdots+\\mathrm{C}_n^n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575715,7 +582086,9 @@ "id": "040575", "content": "若今天是星期五, 再过$3^{100}$天是星期\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575747,7 +582120,9 @@ "id": "040576", "content": "$(1+x+x^2)(1-x)^{10}$展开式中, 含$x^6$的项为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575779,7 +582154,9 @@ "id": "040577", "content": "求$(1+x)+(1+x)^2+\\cdots+(1+x)^{10}$展开式中$x^3$的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575811,7 +582188,9 @@ "id": "040578", "content": "三棱柱$ABC-A_1B_1C_1$中, 若$\\overrightarrow{CA}=\\overrightarrow{a}$, $\\overrightarrow{CB}=\\overrightarrow{b}$, $\\overrightarrow{CC_1}=\\overrightarrow{c}$, 则用$\\overrightarrow{a}$、$\\overrightarrow{b}$、$\\overrightarrow{c}$表示$\\overrightarrow{A_1B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575843,7 +582222,9 @@ "id": "040579", "content": "若$\\overrightarrow {a}=2 \\overrightarrow{i}+2 \\overrightarrow {j}+x \\overrightarrow {k}$, $\\overrightarrow {b}=x \\overrightarrow{i}-3 \\overrightarrow {j}-5 \\overrightarrow {k}$, 且$\\overrightarrow {a}$与$\\overrightarrow {b}$垂直, 则实数$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575875,7 +582256,9 @@ "id": "040580", "content": "已知$\\overrightarrow {a}, \\overrightarrow {b}$为平面$\\alpha$内的两个不相等的向量, $\\overrightarrow {c}$在直线$l$上, 则``$\\overrightarrow {c} \\perp \\overrightarrow {a}$且$\\overrightarrow {c} \\perp \\overrightarrow {b}$''是``直线$l$垂直平面$\\alpha$''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575907,7 +582290,9 @@ "id": "040581", "content": "已知$S$是$\\triangle ABC$所在平面外一点, $D$是$SC$的中点, 若$\\overrightarrow{BD}=x \\overrightarrow{AB}+y \\overrightarrow{AC}+z \\overrightarrow{AS}$, 则$x+y+z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575939,7 +582324,9 @@ "id": "040582", "content": "已知向量$\\overrightarrow {a} \\perp \\overrightarrow{b}$, 向量$\\overrightarrow{c}$与$\\overrightarrow{a}$、$\\overrightarrow{b}$的夹角都是$60^\\circ$, 且$|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, $| \\overrightarrow{c}|=3$, 则$\\overrightarrow{a}$与$2\\overrightarrow {b}-\\overrightarrow{c}$的夹角的余弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -575971,7 +582358,9 @@ "id": "040583", "content": "已知空间四边形$ABCD$的每条边和对角线的长都等于$1$, 点$E$、$F$分别是$AB$、$AD$的中点, 则$\\overrightarrow{EF} \\cdot \\overrightarrow{DC}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576003,7 +582392,9 @@ "id": "040584", "content": "如图, 一个结晶体的形状为平行六面体, 其中, 以顶点$A$为端点的三条棱长都等于$1$, 且它们彼此的夹角都是$60^{\\circ}$, 那么以这个顶点为端点的晶体的对角线的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$B$} coordinate (B);\n\\draw ({1/2},0,{-sqrt(3)/2}) node [below] {$D$} coordinate (D);\n\\draw ($(B)+(D)-(A)$) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0.5,{sqrt(2/3)},{-1/2/sqrt(3)}) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(B)+(A_1)-(A)$) node [above] {$B_1$} coordinate (B_1);\n\\draw ($(C)+(A_1)-(A)$) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(D)+(A_1)-(A)$) node [above] {$D_1$} coordinate (D_1);\n\\draw (A)--(B)--(C)--(C_1)--(D_1)--(A_1)--cycle;\n\\draw (A_1)--(B_1)--(C_1)(B)--(B_1);\n\\draw [dashed] (A)--(C_1)(A)--(D)--(C)(D)--(D_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576035,7 +582426,9 @@ "id": "040585", "content": "已知$(1-2 x)^7=a_0+a_1 x+a_2 x^2+\\cdots+a_7 x^7$, 求:\\\\\n(1) $a_1+a_2+\\cdots+a_7$;\\\\\n(2) $a_1+a_3+a_5+a_7$;\\\\\n(3) $|a_0|+|a_1|+\\cdots+|a_7|$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576067,7 +582460,9 @@ "id": "040586", "content": "已知$(\\sqrt{x}-\\dfrac{2}{x^2})^n$的展开式中, 第五项与第三项的系数之比为$56: 3$, 求展开式中所有的有理项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576099,7 +582494,9 @@ "id": "040587", "content": "求$(5 x-2 y)^{20}$展开式中, 第几项的系数最小.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576131,7 +582528,9 @@ "id": "040588", "content": "直三棱柱$ABC-A_1 B_1 C_1$中, 若$\\overrightarrow{CA}=\\overrightarrow {a}, \\overrightarrow{CB}=\\overrightarrow {b}, \\overrightarrow{CC_1}=\\overrightarrow {c}$, 则$\\overrightarrow{A_1B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576151,7 +582550,9 @@ "id": "040589", "content": "设点$B$是点$A(2,-3,5)$关于$xOy$平面的对称点, 则$|\\overrightarrow{AB}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576171,7 +582572,9 @@ "id": "040590", "content": "已知点$A$、$B$、$C$、$D$的坐标分别为$(-1,0,-1)$、$(-1,0,0)$、$(-2,-2,-2)$、$(-3,0,0)$, 则$\\overrightarrow{AB}$与$\\overrightarrow{CD}$的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576191,7 +582594,9 @@ "id": "040591", "content": "已知$A$、$B$、$C$三点的坐标分别为$A(4,1,3)$, $B(2,-5,1)$, $C(3,7, \\lambda)$, $\\overrightarrow{AB} \\perp \\overrightarrow{AC}$, 则$\\lambda$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576211,7 +582616,9 @@ "id": "040592", "content": "在长方体$ABCD-A' B' C' D'$中, $AB=2$, $AD=4$, $AA'=3$, $E$、$F$在棱$AA'$, 且$AE=EF=FA'$, $G$是$B' C'$的中点, 则直线$FG$与$EC'$所成角的大小是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576231,7 +582638,9 @@ "id": "040593", "content": "已知$\\overrightarrow {a}=(2,-1,3)$, $\\overrightarrow {b}=(-1,4,-2)$, $\\overrightarrow {c}=(7, \\lambda, 5)$, 若$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$三向量共面, 则实数$\\lambda$等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576251,7 +582660,9 @@ "id": "040594", "content": "在长方体$ABCD-A'B'C'D'$中, $AB=1$, $AD=2$, $AA'=3$, 则直线$A'C'$到平面$ACB'$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576271,7 +582682,9 @@ "id": "040595", "content": "设$\\overrightarrow {a}=(a_1, a_2, a_3), \\overrightarrow {b}=(b_1, b_2, b_3)$, 且$\\overrightarrow {a} \\neq \\overrightarrow {b}$, 记$|\\overrightarrow {a}-\\overrightarrow {b}|=m$, 则$\\overrightarrow {a}-\\overrightarrow {b}$与$x$轴正方向的夹角的余弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576291,7 +582704,9 @@ "id": "040596", "content": "已知$\\overrightarrow{OA}=(1,2,3), \\overrightarrow{OB}=(2,1,2), \\overrightarrow{OP}=(1,1,2)$, 点$Q$在直线$OP$上运动, 则当$\\overrightarrow{QA} \\cdot \\overrightarrow{QB}$取得最小值时, 点$Q$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576311,7 +582726,9 @@ "id": "040597", "content": "设$A_1, A_2, A_3, A_4, A_5$是空间中给定的$5$个不同点, 则使$\\overrightarrow{MA_1}+\\overrightarrow{MA_2}+\\overrightarrow{MA_3}+\\overrightarrow{MA_4}+\\overrightarrow{MA_5}=\\overrightarrow{0}$成立的点$M$的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -576331,7 +582748,9 @@ "id": "040598", "content": "已知平行六面体$ABCD-A'B'C'D'$中, $AB=4$, $AD=3$, $AA'=5$, $\\angle BAD=90^{\\circ}$, $\\angle BAA'=\\angle DAA'=60^{\\circ}$, 求$AC'$的长.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576351,7 +582770,9 @@ "id": "040599", "content": "如图, 已知四棱锥$P-ABCD$, 底面$ABCD$为矩形, $PA=AB=2$, $AD=2AB$, $PA \\perp$平面$ABCD$, $E, F$分别是$BC, PC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (4,0,2) node [right] {$C$} coordinate (C);\n\\draw ($(P)!0.5!(C)$) node [above] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw (P)--(B)--(C)--(D)--cycle(E)--(F)(P)--(C);\n\\draw [dashed] (P)--(A)--(B)(A)--(D)(A)--(E)(A)--(C)(A)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$PD$与平面$AEF$所成的角的正弦值;\\\\\n(2) 求二面角$F-AE-D$的大小 (用反三角函数表示).", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576371,7 +582792,9 @@ "id": "040600", "content": "如图, 在直三棱柱$ABO-A_1B_1O_1$中, $OO_1=4$, $OA=4$, $OB=3$, $\\angle AOB=90^{\\circ}$, $D$是线段$A_1B_1$的中点, $P$是侧棱$BB_1$上的一点. 若$OP \\perp BD$, 求$OP$与底面$AOB$所成角的大小 (结果用反三角函数值表示). \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [above right] {$O$} coordinate (O);\n\\draw (4,0,0) node [above right] {$A$} coordinate (A);\n\\draw (0,0,3) node [left] {$B$} coordinate (B);\n\\draw (0,4,0) node [above right] {$O_1$} coordinate (O_1);\n\\draw ($(A)+(O_1)-(O)$) node [right] {$A_1$} coordinate (A_1);\n\\draw ($(B)+(O_1)-(O)$) node [left] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)!0.5!(B_1)$) node [below] {$D$} coordinate (D);\n\\draw (B)++(0,1.25) node [left] {$P$} coordinate (P);\n\\draw (B)--(D);\n\\draw (B)--(A)--(A_1)--(O_1)--(B_1)--cycle(A_1)--(B_1);\n\\draw [dashed] (B)--(O)--(A)(O)--(O_1)(O)--(P);\n\\draw [->] (B)--($(O)!1.6!(B)$) node [left] {$x$} coordinate (x);\n\\draw [->] (A)--($(O)!1.3!(A)$) node [right] {$y$} coordinate (y);\n\\draw [->] (O_1)--($(O)!1.3!(O_1)$) node [left] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576391,7 +582814,9 @@ "id": "040601", "content": "正方体$AC_1$的棱长为$1$, $E$为$BB_1$的中点, $G$、$F$分别为对角线$AD_1$、$BD$的中点, $M$在$CD_1$上, 且$CM=\\dfrac{1}{4} CD_1$.\\\\\n(1) 求$EM$与$GF$所成的角;\\\\\n(2) 是否存在过$G$、$F$两点的截面, 使得$EF$垂直于此截面? 若存在, 求出与棱$D_1C_1$的交点的位置; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576411,7 +582836,9 @@ "id": "040602", "content": "已知平行六面体$ABCD-A_1B_1C_1D_1$的底面$ABCD$是菱形, 且$\\angle C_1CB=\\angle C_1CD=\\angle BCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [below] {$C$} coordinate (C);\n\\draw (-1,0,0) node [below] {$B$} coordinate (B);\n\\draw ({cos(80)},0,{-sin(80)}) node [right] {$D$} coordinate (D);\n\\draw ($(B)+(D)-(C)$) node [below] {$A$} coordinate (A);\n\\draw (C) ++ ({1.3*sin(10)},{1.3*sqrt(1-sin(10)*sin(10)-tan(10)*tan(10)-2*sin(10)*tan(10)*tan(10)-sin(10)*sin(10)*tan(10)*tan(10))},{1.3*tan(10)*(1+sin(10))}) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(C_1)-(C)+(D)$) node [right] {$D_1$} coordinate (D_1);\n\\draw ($(C_1)-(C)+(B)$) node [left] {$B_1$} coordinate (B_1);\n\\draw ($(C_1)-(C)+(A)$) node [above] {$A_1$} coordinate (A_1);\n\\draw (B)--(C)--(D)--(D_1)--(A_1)--(B_1)--cycle(B_1)--(C_1)--(D)(C)--(C_1)(D)--(C_1)--(D_1);\n\\draw [dashed] (B)--(D)(A_1)--(C)(B)--(C_1)(B)--(A)--(D)(A)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $CC_1 \\perp BD$;\\\\\n(2) 找出$|\\dfrac{CD}{CC_1}|$的一个比值, 使得$A_1 C \\perp$平面$C_1 BD$, 并予以证明.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576431,7 +582858,9 @@ "id": "040603", "content": "正三棱柱$ABC-A_1B_1C_1$中, $AB=2$, $AA_1=\\dfrac{\\sqrt{6}}{2}$, $O$为$AB$中点.\\\\\n(1) $P$点在棱$A_2B_1$上什么位置时, 异面直线$AP$与$A_1C$互相垂直;\\\\\n(2) 求平面$A_1CO$与侧面$B_1BCC_1$所成的锐二面角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576451,7 +582880,9 @@ "id": "040604", "content": "在直三棱柱$ABC-A_1B_1C_1$中, 底面是等腰直角三角形, $\\angle ACB=90^{\\circ}$, 侧棱$AA_1=2$, $D$、$E$分别是$CC_1$与$A_1B$的中点, 点$E$在平面$ABD$上的射影是$\\triangle ABD$的重心$G$.\\\\\n(1) 求$A_1B$与平面$ABD$所成角的大小 (结果用反三角函数值表示);\\\\\n(2) 求点$A_1$到平面$AED$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -576471,7 +582902,9 @@ "id": "040605", "content": "四人互相传球, 由甲开始发球, 并作为第一次传球, 经过$3$次传球后, 球仍回到甲手中, 则不同的传球方式共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$6$", "solution": "", @@ -576491,7 +582924,9 @@ "id": "040606", "content": "书架上某层有$8$本书, 新买$2$本插进去, 要保持原有$8$本书的顺序, 则有\\blank{50}种不同的插法. (具体数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$90$", "solution": "", @@ -576511,7 +582946,9 @@ "id": "040607", "content": "若$(x+1)^n$的展开式中第$3$项与第$9$项的二项式系数相等, 则$n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$10$", "solution": "", @@ -576531,7 +582968,9 @@ "id": "040608", "content": "$7$ 个志愿者的名额分给$3$个班, 每班至少一个名额, 则有\\blank{50}种不同的分配方法. (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$15$", "solution": "", @@ -576551,7 +582990,9 @@ "id": "040609", "content": "$A$、$B$、$C$、$D$、$E$五名同学站成一排合影, 若$A$不站在两端, $B$和$C$相邻, 则不同的站队方式共有\\blank{50}种. (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$24$", "solution": "", @@ -576571,7 +583012,9 @@ "id": "040610", "content": "设函数$f(x)=\\dfrac{1}{3} x^2-27 \\ln x$在区间$[a, 2 a+1]$上严格减, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(0,\\dfrac{9\\sqrt{2}-2}4]$", "solution": "", @@ -576591,7 +583034,9 @@ "id": "040611", "content": "$6$ 位大学毕业生分配到$3$家单位, 每家单位至少录用$1$人, 则不同的分配方法共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$540$", "solution": "", @@ -576611,7 +583056,9 @@ "id": "040612", "content": "已知在四面体$V-ABC$中, $VA=VB=VC=2$, $AB=1$, $\\angle ACB=\\dfrac{\\pi}{6}$, 则该四面体外接球的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{16\\pi}3$", "solution": "", @@ -576631,7 +583078,9 @@ "id": "040613", "content": "用 1、2、3、4、5 组成没有重复数字的五位数$\\overline{a b c d e}$, 其中满足$a>b>c$, 且$c=latex]\n\\draw [->] (-3,0) -- (3.2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [above right] {$O$};\n\\foreach \\i in {-2,-1,1,2,3}\n{\\draw (\\i,0) -- (\\i,0.1) node [above] {$\\i$};};\n\\draw (-3,0.5) -- (-1,-0.5) -- (1,0) -- (3.2,-0.8);\n\\draw [dashed] (-1,0) -- (-1,-0.5) (3,0) -- (3,-0.75);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $f(x)$在区间$(-1,1)$上严格增;\\\\\n\\textcircled{2} $f(x)$的图像在$x=-2$处的切线斜率等于$0$;\\\\\n\\textcircled{3} $f(x)$在$x=1$处取得极大值;\\\\\n\\textcircled{4} $f'(x)$在$x=-1$处取得极小值.\\\\\n正确的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "\\textcircled{2}\\textcircled{4}", "solution": "", @@ -576671,7 +583122,9 @@ "id": "040615", "content": "平面直角坐标系$xOy$中, 已知点$M(2,-1)$, 若直线$l: 3 x-4 y+5=0$上总存在$P$、$Q$两点, 使得$\\angle PMQ \\geq \\dfrac{\\pi}{2}$恒成立, 则线段$PQ$长度的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$[6,+\\infty)$", "solution": "", @@ -576691,7 +583144,9 @@ "id": "040616", "content": "设$x_1$、$x_2$是函数$f(x)=a x^2-\\mathrm{e}^x$ ($a \\in \\mathbf{R}$)的两个极值点, 若$\\dfrac{x_2}{x_1} \\geq 2$, 则$a$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$\\log_2 \\mathrm{e}$", "solution": "", @@ -576711,7 +583166,9 @@ "id": "040617", "content": "下列求导运算正确的是\\bracket{20}.\n\\twoch{$(\\ln x+\\dfrac{3}{x})'=\\dfrac{1}{x}+\\dfrac{3}{x^2}$}{$(x^2 \\mathrm{e}^x)'=2 x \\mathrm{e}^x$}{$(3^x \\cos 2 x)'=3^x(\\ln 3 \\cdot \\cos 2 x-2 \\sin 2 x)$}{$(\\ln \\dfrac{1}{2}+\\log _2 x)'=2+\\dfrac{1}{x \\ln 2}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -576731,7 +583188,9 @@ "id": "040618", "content": "函数$f(x)=\\dfrac{x-\\sin x}{\\mathrm{e}^x+\\mathrm{e}^{-x}}$在$[-\\pi, \\pi]$上的图像大致为\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, xscale = 0.4, yscale = 5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,0.2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain =-pi:pi, samples = 100] plot (\\x,{(\\x-sin(\\x/pi*180))/(exp(\\x)+exp(-\\x))});\n\\draw (0.3,0.1) -- (0,0.1) node [left] {$0.1$};\n\\draw (pi,0.02) -- (pi,0) node [below] {$\\pi$};\n\\draw (-pi,0.02) -- (-pi,0) node [below] {$-\\pi$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.4, yscale = 5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,0.2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain =-pi:pi, samples = 100] plot (\\x,{(\\x-1.5*sin(\\x/pi*180))/(exp(\\x)+exp(-\\x))});\n\\draw (0,0.1) -- (0.3,0.1) node [right] {$0.1$};\n\\draw (pi,0.02) -- (pi,0) node [below] {$\\pi$};\n\\draw (-pi,0.02) -- (-pi,0) node [below] {$-\\pi$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.4, yscale = 5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,0.2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain =-pi:pi, samples = 100] plot (\\x,{-(\\x-sin(\\x/pi*180))/(exp(\\x)+exp(-\\x))});\n\\draw (0.3,0.1) -- (0,0.1) node [left] {$0.1$};\n\\draw (pi,0.02) -- (pi,0) node [below] {$\\pi$};\n\\draw (-pi,0.02) -- (-pi,0) node [below] {$-\\pi$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.4, yscale = 5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,0.2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain =-pi:pi, samples = 100] plot (\\x,{sqrt(abs(\\x))/10-0.1});\n\\draw (0.3,0.1) -- (0,0.1) node [left] {$0.1$};\n\\draw (pi,0.02) -- (pi,0) node [below] {$\\pi$};\n\\draw (-pi,0.02) -- (-pi,0) node [below] {$-\\pi$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -576751,7 +583210,9 @@ "id": "040619", "content": "设$(2 x-1)^5=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4+a_5 x^5$, 则$|a_1|+2|a_2|+3|a_3|+4|a_4|+5|a_5|=$\\bracket{20}.\n\\fourch{$80$}{$242$}{$405$}{$810$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -576771,7 +583232,9 @@ "id": "040620", "content": "点$P$为抛物线$C: y^2=4 x$准线上的点, 若存在过$P$的直线交抛物线$C$于$A$、$B$两点, 且$|PA|=|AB|$, 则称点$P$为``$\\Omega$点'', 那么下列结论中正确的是\\bracket{20}.\n\\onech{准线上的所有点都不是``$\\Omega$点''}{准线上的所有点都是``$\\Omega$点''}{准线上仅有有限个点是``$\\Omega$点''}{准线上有无穷多个点(不是所有的点)是``$\\Omega$点''}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -576791,7 +583254,9 @@ "id": "040621", "content": "如图, 已知四棱锥$P-ABCD$的底面是菱形, 对角线$AC$、$BD$交于点$O$, $OA=3$, $OB=4$, $OP=3$, $OP \\perp$底面$ABCD$, 设点$M$满足$\\overrightarrow{PM}=\\dfrac{1}{2} \\overrightarrow{MC}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw ({-3/sqrt(2)},0,{3/sqrt(2)}) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,{2*sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw ($(A)!2!(O)$) node [right] {$C$} coordinate (C);\n\\draw ($(B)!2!(O)$) node [left] {$D$} coordinate (D);\n\\draw (O) ++ (0,3,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!{1/3}!(C)$) node [right] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B)(M)--(B)(P)--(D)--(A);\n\\draw [dashed] (A)--(C)(B)--(D)(P)--(O)(D)--(M)(D)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$PA$与平面$BDM$所成角的正弦值;\\\\\n(2) 求点$P$到平面$BDM$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{10}}{10}$; (2) $\\dfrac{3\\sqrt{5}}{5}$", "solution": "", @@ -576811,7 +583276,9 @@ "id": "040622", "content": "对于代数式$(2 x-\\dfrac{1}{\\sqrt{x}})^5$,\\\\\n(1) 求其展开式中含$x^2$的项的系数;\\\\\n(2) 设该代数式的展开式中前三项的二项式系数的和为$M$, $(1+a x)^4$的展开式中各项系数的和为$N$, 若$M=N$, 求实数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "(1) $80$; (2) $1$或$-3$", "solution": "", @@ -576831,7 +583298,9 @@ "id": "040623", "content": "已知直线$l: y=k x$($k \\neq 0$)与圆$C: x^2+y^2-2 x-3=0$相交于$A$、$B$两点.\\\\\n(1) 若$|AB|=\\sqrt{13}$, 求$k$;\\\\\n(2) 在$x$轴上是否存在点$M$, 使得当$k$变化时, 总有直线$MA$、$MB$的斜率之和为 $0$ , 若存在, 求出点$M$的坐标; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\pm \\sqrt{3}$; (2) 存在, 点$M$的坐标为$(-3,0)$", "solution": "", @@ -576851,7 +583320,9 @@ "id": "040624", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+y^2=1$($a>1$)的离心率为$\\dfrac{\\sqrt{3}}{2}$.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 若直线$l: y=k x-2$与椭圆$C$交于两个不同点$D$、$E$, 以线段$DE$为直径的圆经过原点, 求实数$k$的值;\\\\\n(3) 设$A$、$B$为椭圆$C$的左、右顶点, $H$为椭圆$C$上除$A$、$B$外任意一点, 线段$BH$的垂直平分线分别交直线$BH$和直线$AH$于点$P$和点$Q$, 分别过点$P$和$Q$作$x$轴的垂线, 垂足分别为$M$和$N$, 求证: 线段$MN$的长为定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}4+y^2=1$; (2) $k=\\pm 2$; (3) 定值为$\\dfrac 23$, 证明略", "solution": "", @@ -576871,7 +583342,9 @@ "id": "040625", "content": "已知函数$f(x)=x-\\ln x-3$.\\\\\n(1) 求曲线$y=f(x)$在$x=1$处的切线方程;\\\\\n(2) 函数$f(x)$在区间$(k, k+1)$ ($k \\in \\mathbf{N}$)上有零点, 求$k$的值;\\\\\n(3) 记函数$g(x)=x^2-b x-3-f(x)$, 设$x_1$、$x_2$($x_10$对任意$x \\in(0,+\\infty)$恒成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-\\infty,1)$", "solution": "", @@ -577091,7 +583584,9 @@ "id": "040636", "content": "若$(1+x)^8+(2+x)^8=a_0+a_1(1-x)^1+a_2(1-x)^2+\\cdots+a_8(1-x)^8$对任意$x \\in \\mathbf{R}$恒成立, 则$a_4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$6790$", "solution": "", @@ -577111,7 +583606,10 @@ "id": "040637", "content": "已知$A(a, 1-a^2)$, $B(b, 1-b^2)$, 其中$a b<0$, 过$A$、$B$分别作二次函数$y=1-x^2$的切线, 则两条切线与$x$轴围成的三角形面积的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第七单元" + ], "genre": "填空题", "ans": "$\\dfrac{8\\sqrt{3}}9$", "solution": "", @@ -577131,7 +583629,9 @@ "id": "040638", "content": "在古典概率模型中, $\\Omega$是样本空间, $x$是样本点, $A$是随机事件, 则下列表述正确的\\bracket{20}.\n\\fourch{$x \\in \\Omega$}{$x \\subseteq \\Omega$}{$A \\in \\Omega$}{$\\Omega \\subseteq A$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -577151,7 +583651,9 @@ "id": "040639", "content": "已知$A$、$B$为两个随机事件, 则``$A$、$B$为互斥事件''是``$A$、$B$为对立事件''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{非充分非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -577171,7 +583673,9 @@ "id": "040640", "content": "下列关于排列数$\\mathrm{P}_n^{m-1}$和组合数$\\mathrm{C}_n^{m-1}$的计算中正确的是\\bracket{20}.\n\\twoch{$\\mathrm{P}_n^{m-1}=\\dfrac{n !}{(m-1) !}$}{$\\mathrm{P}_n^{m-1}=\\dfrac{n !}{(n-m-1) !}$}{$\\mathrm{C}_n^{m-1}=\\dfrac{n !}{(m-1) !(n-m+1) !}$}{$\\mathrm{C}_n^{m-1}=\\dfrac{n !}{(m-1) !(n-m-1) !}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -577191,7 +583695,9 @@ "id": "040641", "content": "已知$x \\in \\mathbf{N}$, $y \\in \\mathbf{N}$, $x=latex]\n\\def\\t{50}\n\\draw (3,0) node [below] {$N$} coordinate (N);\n\\draw (-3,0) node [below] {$M$} coordinate (M);\n\\filldraw (0,0) circle (0.03) node [below] {$O$} coordinate (O);\n\\draw (N) arc (0:180:3) -- cycle;\n\\draw (30:3) node [above right] {$F$} coordinate (F);\n\\draw (150:3) node [above left] {$I$} coordinate (I);\n\\draw (F) -- ($(M)!(F)!(N)$) node [below] {$E$} coordinate (E);\n\\draw (I) -- ($(M)!(I)!(N)$) node [below] {$H$} coordinate (H);\n\\draw (\\t:3) node [above right] {$C$} coordinate (C);\n\\draw ({180-\\t}:3) node [above left] {$D$} coordinate (D);\n\\draw (C) -- ($(M)!(C)!(N)$) node [below] {$B$} coordinate (B);\n\\draw (D) -- ($(M)!(D)!(N)$) node [below] {$A$} coordinate (A);\n\\draw (C)--(D);\n\\draw (I) -- ($(A)!(I)!(D)$) node [right] {$J$} coordinate (J);\n\\draw (F) -- ($(B)!(F)!(C)$) node [left] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(C)$) node {海洋球池};\n\\draw ($(A)!0.5!(I)$) node {息};\n\\draw ($(A)!0.5!(I)$) ++ (0,0.4) node {休};\n\\draw ($(A)!0.5!(I)$) ++ (0,-0.4) node {区};\n\\draw ($(B)!0.5!(F)$) node {息};\n\\draw ($(B)!0.5!(F)$) ++ (0,0.4) node {休};\n\\draw ($(B)!0.5!(F)$) ++ (0,-0.4) node {区};\n\\end{tikzpicture}\n\\end{center}\n(1) 求当$\\theta=\\dfrac{\\pi}{4}$时该亲子乐园可供人活动的区域面积$S$, 并求出此时的``得地率''(结果精确到$1 \\%)$;\\\\\n(2) 求当$\\theta$为多大时, 该亲子乐园的``得地率''最大?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $\\theta=\\dfrac\\pi 4$时, $S=\\dfrac{2-\\sqrt{2}+\\sqrt{3}}2R^2$, ``得地率''约为$74\\%$; (2) $\\theta = \\arcsin\\dfrac{1+\\sqrt{33}}8$时, ``得地率''最大", "solution": "", @@ -577271,7 +583783,9 @@ "id": "040645", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点分别为$F_1$、$F_2$. 椭圆$\\Gamma$上有互异的且不在$x$轴上的三点$A$、$B$、$C$满足直线$AC$经过$F_1$, 直线$BC$经过$F_2$.\\\\\n(1) 若椭圆$\\Gamma$的长轴长为 $4$ , 离心率为$\\dfrac{1}{2}$, 求$b$的值;\\\\\n(2) 若点$C$的坐标为$(0,1)$, $\\triangle ABC$的面积$S=\\dfrac{64}{49} \\sqrt{3}$, 求$a$的值;\\\\\n(3) 若$a=\\sqrt{2}$, $b=1$, 直线$AB$经过点$(\\dfrac{3}{2}, 0)$, 求$C$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $b=\\sqrt{3}$; (2) $a=2$; (3) $C$的坐标为$(-\\dfrac 43,-\\dfrac 13)$或$(-\\dfrac 43,\\dfrac 13)$", "solution": "", @@ -577291,7 +583805,9 @@ "id": "040646", "content": "已知定义在$\\mathbf{R}$上的函数$f(x)$的导函数为$f'(x)$, 若$|f'(x)| \\leq 1$对任意$x \\in \\mathbf{R}$恒成立, 则称函数$f(x)$为``线性控制函数''.\\\\\n(1) 判断函数$f(x)=\\sin x$和$g(x)=\\mathrm{e}^x$是否为``线性控制函数'', 并说明理由;\\\\\n(2) 若函数$f(x)$为``线性控制函数'', 且$f(x)$在$\\mathbf{R}$上严格增, 设$A$、$B$为函数$f(x)$图像上互异的两点, 设直线$AB$的斜率为$k$, 判断命题``$00$)为周期的周期函数, 证明: 对任意$x_1$、$x_2$都有$|f(x_1)-f(x_2)| \\leq T$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $f(x)$是``线性控制函数'', $g(x)$不是``线性控制函数'', 理由略; (2) 是真命题, 理由略; (3) 证明略", "solution": "", @@ -577311,7 +583827,9 @@ "id": "040647", "content": "$10$件产品中混有$2$件次品, 如果从中任意抽出$3$件进行检验, 其中恰有一件次品的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577343,7 +583861,9 @@ "id": "040648", "content": "在所有由$1$到$9$的$5$个不同数字所组成的没有重复数字的$5$位数中, 任取一个数能被$5$整除的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577375,7 +583895,9 @@ "id": "040649", "content": "$6$个人排成一排拍照, 其中甲、乙、丙三人正好相邻而坐的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577407,7 +583929,9 @@ "id": "040650", "content": "一副扑克牌 ($52$ 张) 中任取$3$张, 正好是同一花色的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577439,7 +583963,9 @@ "id": "040651", "content": "七人站成一排, 如果甲乙两人必须不相邻, 那么有\\blank{50}种不同的排法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577471,7 +583997,9 @@ "id": "040652", "content": "已知$P(A | B)=\\dfrac{1}{2}$, $P(B)=\\dfrac{1}{3}$, 则$P(A \\cap B)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577503,7 +584031,9 @@ "id": "040653", "content": "三张奖券中只有一张能中奖, 现分别由三名同学无放回地抽取.\\\\ \n(1) 最后一名同学抽到中奖奖券的概率是\\blank{50};\\\\\n(2) 若已经知道第一名同学没有抽到中奖奖券, 则最后一名同学抽到奖券的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577535,7 +584065,9 @@ "id": "040654", "content": "某射击选手射击一次击中$10$环的概率是$\\dfrac{4}{5}$, 连续两次均击中$10$环的概率是$\\dfrac{1}{2}$, 已知该选手某次击中$10$环, 则随后一次击中$10$环的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577567,7 +584099,9 @@ "id": "040655", "content": "从编号为$1,2, \\cdots, 10$的$10$个大小与质地相同的球中任取$4$个, 已知取出$4$号球的条件下, 取出球的最大号码为$6$的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577599,7 +584133,9 @@ "id": "040656", "content": "某保险公司把被保险人分为$3$类: ``谨慎的''``一般的''``冒失的''. 统计资料表明, 这$3$类人在一年内发生事故的概率依次为$0.05$, $0.15$和 $0.30$. 如果``谨慎的''被保险人占$20 \\%$, ``一般的''被保险人占$50 \\%$, ``冒失的''被保险人占$30 \\%$, 则一个被保险人在一年内出事故的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577631,7 +584167,9 @@ "id": "040657", "content": "两批相同的产品分别有$12$件和$10$件, 每批产品中各有$1$件废品, 现在先从第$1$批产品中任取$1$件放入第$2$批中, 然后从第$2$批中任取$1$件, 则取到废品的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577663,7 +584201,9 @@ "id": "040658", "content": "设袋中共有$10$个大小与质地相同的球, 其中$2$个红球, 其余为白球, 两人分别从袋中任取一球, 则第二个人取得红球的概率为\\blank{50}.(第一人取出的球不放回)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577695,7 +584235,9 @@ "id": "040659", "content": "某小组有$20$名射手, 其中$1$、$2$、$3$、$4$级射手分别为$2$、$6$、$9$、$3$名. 若选$1$、$2$、$3$、$4$ 级射手参加比赛, 则在比赛中射中目标的概率分别为$0.85$、$0.64$、\n$0.45$、$0.32$. 今随机选一人参加比赛, 则该小组比赛中射中目标的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577727,7 +584269,9 @@ "id": "040660", "content": "播种用的一等小麦种子中混有$2 \\%$的二等种子、$1.5 \\%$的三等种子、$1 \\%$的四等种子. 用一、二、三、四等种子结出的穗含有$50$颗以上麦粒的概率分别为 $0.5$ 、\n$0.15$、$0.1$、$0.05$, 这批种子所结的穗含有$50$颗以上麦粒的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -577759,7 +584303,9 @@ "id": "040661", "content": "抛掷一枚质地均匀的硬币两次.\\\\\n(1) 两次都是正面向上的概率是多少?\\\\\n(2) 在已知有一次出现正面向上的条件下, 两次都是正面向上的概率是多少?\\\\\n(3) 在第一次出现正面向上的条件下, 第二次出现正面向上的概率是多少?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -577791,7 +584337,9 @@ "id": "040662", "content": "某地区气象台统计, 该地区下雨的概率为$\\dfrac{4}{15}$, 刮风的概率为$\\dfrac{2}{15}$, 既刮风又下雨的概率是$\\dfrac{1}{10}$, 设下雨为事件$A$, 刮风为事件$B$. 求:\\\\\n(1) $P(A|B)$;\\\\\n(2) $P(B|A)$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -577823,7 +584371,9 @@ "id": "040663", "content": "从一副不含大小王的$52$张扑克牌中随机取出一张, 用$A$表示取出的牌是Q, 用$B$表示取出的牌是红桃, 试计算$P( A| B)$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -577855,7 +584405,9 @@ "id": "040664", "content": "甲、乙两个口袋中各有大小与质地相同的$3$只白球、$2$ 只黑球. 从甲口袋中任取一球放入乙口袋中, 求再从乙口袋中取出一球为白球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -577887,7 +584439,9 @@ "id": "040665", "content": "设有两箱同一种商品: 第一箱内装$50$件, 其中$10$件优质品; 第二箱内装$30$件, 其中$18$件优质品. 现在随机地打开一箱, 然后从箱中随意取出一件, 求取到优质品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -577919,7 +584473,9 @@ "id": "040666", "content": "两台机床加工同样的零件, 第一台的废品率为$0.04$, 第二台的废品率为$0.07$, 加工出来的零件混放, 并设第一台加工的零件是第二台加工零件的$2$倍, 现任取一零件, 求它是合格品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -577951,7 +584507,9 @@ "id": "040667", "content": "$\\overrightarrow{AC}+\\overrightarrow{CD}+\\overrightarrow{DB}+\\overrightarrow{BA}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\overrightarrow{0}$", "solution": "", @@ -577971,7 +584529,9 @@ "id": "040668", "content": "若$\\sin (\\alpha+\\dfrac{\\pi}{4})=\\dfrac{\\sqrt{3}}{2}$, 则$\\tan \\alpha+\\dfrac{1}{\\tan \\alpha}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -577991,7 +584551,9 @@ "id": "040669", "content": "若扇形的弧长和半径都是$3$, 则扇形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 92$", "solution": "", @@ -578011,7 +584573,9 @@ "id": "040670", "content": "函数$f(x)=\\sin x \\cdot \\cos x$的最大值是\\blank{50}, 此时$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 12,\\dfrac{\\pi}4+k\\pi,k \\in \\mathbb{Z}$", "solution": "", @@ -578031,7 +584595,9 @@ "id": "040671", "content": "已知$\\triangle ABC$内角$A, B, C$的对边分别为$a, b, c$, 那么当$a=$\\blank{50}时, 满足条件``$b=2$, $A=30^{\\circ}$''的$\\triangle ABC$有两个. (仅写出一个$a$的具体数值即可)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "填$(1,2)$内的任意数均可", "solution": "", @@ -578051,7 +584617,9 @@ "id": "040672", "content": "已知向量$\\overrightarrow {a}$, $\\overrightarrow {b}$满足$|\\overrightarrow {a}|=3$, $|\\overrightarrow {b}|=4$, $\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$\\dfrac{5 \\pi}{6}$, 则$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-\\dfrac 23 \\sqrt{3}\\overrightarrow{a}$", "solution": "", @@ -578071,7 +584639,9 @@ "id": "040673", "content": "函数$f(x)=\\cos (\\dfrac{\\pi}{6}-x)$的严格增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[2k\\pi-\\dfrac{5\\pi}6,2k\\pi+\\dfrac{\\pi}6], k \\in \\mathbb{Z}$", "solution": "", @@ -578091,7 +584661,9 @@ "id": "040674", "content": "通过研究正五边形和正十边形的作图, 古希腊数学家毕达哥拉斯发现了黄金分割率, 黄金分割率的值也可以用$2 \\sin 18^{\\circ}$表示, 即$\\dfrac{\\sqrt{5}-1}{2}=2 \\sin 18^{\\circ}$. 记$m=2 \\sin 18^{\\circ}$, 则$\\dfrac{\\sqrt{1+\\cos 36^{\\circ}}}{(m^2-2) \\cdot \\sin 144^{\\circ}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\sqrt{2}$", "solution": "", @@ -578111,7 +584683,10 @@ "id": "040675", "content": "函数$y=\\lg (2 \\cos x-\\sqrt{3})$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|-\\dfrac{\\pi}6+2k\\pi-3$)上至少有$7$个零点, 则实数$m$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -578151,7 +584728,9 @@ "id": "040677", "content": "若存在区间$[a, b]$($a, b \\in \\mathbf{R}$)使得函数$f(x)=\\sin x-\\dfrac{\\sqrt{3}}{2}$在此区间上仅有两个零点, 则$b-a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[\\dfrac{\\pi}3,\\dfrac{11\\pi}3)$", "solution": "", @@ -578171,7 +584750,9 @@ "id": "040678", "content": "已知非零向量$\\overrightarrow {a}$, $\\overrightarrow {b}$, $\\overrightarrow {c}$, 则``$\\overrightarrow {a} \\cdot \\overrightarrow {c}=\\overrightarrow {b} \\cdot \\overrightarrow {c}$''是``$\\overrightarrow {a}=\\overrightarrow {b}$''的 \\bracket{20} 条件.\n\\fourch{充分不必要}{必要不充分}{充分必要}{既不充分也不必要}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -578191,7 +584772,9 @@ "id": "040679", "content": "已知函数$f(x)=A \\cos \\omega x-\\sqrt{3} \\sin \\omega x$($\\omega>0$)的部分图像如图, $y=f(x)$的对称轴方程为$x=\\dfrac{5 \\pi}{12}+\\dfrac{k \\pi}{2}$($k \\in \\mathbf{Z}$), 则$f(0)=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.25]\n\\draw [->] ({-pi/6},0) -- ({pi/6*5},0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-pi/6}:{5*pi/6}, samples = 100] plot (\\x,{3*cos(2*\\x/pi*180)-sqrt(3)*sin(2*\\x/pi*180)});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$3$}{$2$}{$\\dfrac{3}{2}$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -578211,7 +584794,9 @@ "id": "040680", "content": "阻尼器是一种以提供阻力达到减震效果的专业工程装置. 我国第一高楼上海中心大厦的阻尼器减震装置, 被称为``镇楼神器''. 由物理学知识可知, 某阻尼器的运动过程可近似为单摆运动, 单摆运动离开平衡位置的位移$y(\\text{m})$和时间$t(\\text{s})$的函数关系为$y=\\sin (\\omega t+\\varphi)$($\\omega>0$, $|\\varphi|<\\pi$), 如图. 若该阻尼器在摆动过程中连续三次到达同一位置的时间分别为$t_1$, $t_2$, $t_3$($0=latex]\n\\draw [->] (-0.5,0) -- (4,0) node [below] {$t$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.5:4, samples = 100] plot (\\x,{sin(\\x/pi*540)});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{1}{3} \\text{s}$}{$\\dfrac{2}{3} \\text{s}$}{$1 \\text{s}$}{$\\dfrac{4}{3} \\text{s}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -578231,7 +584816,9 @@ "id": "040681", "content": "已知函数$f(x)=\\sin \\omega x+\\cos \\omega x(\\omega>0)$图象的相邻两条对称轴之间的距离为$\\dfrac{\\pi}{2}$, 则\\blank{50}.(写出所有正确结论的序号)\\\\\n\\textcircled{1} $f(x)$的图象关于点$(\\dfrac{3 \\pi}{8}, 0)$对称;\\\\\n\\textcircled{2} 将$f(x)$的图象向左平移$\\dfrac{\\pi}{8}$个单位长度, 得到的函数图象关于$y$轴对称;\\\\\n\\textcircled{3} $f(x)$在$[0, \\dfrac{\\pi}{2}]$上的值域为$[-1,1]$;\\\\\n\\textcircled{4} $f(x)$在$[-\\dfrac{\\pi}{4}, 0]$上单调递增.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "ABD", "solution": "", @@ -578251,7 +584838,9 @@ "id": "040682", "content": "在$\\triangle ABC$中, 已知$\\tan _2 \\dfrac{C}{2} \\sin (A+B)$, 则以下四个结论正确的是\\blank{50}.(写出所有正确结论的序号)\\\\\n\\textcircled{1} $\\cos A \\cos B$的最大值为$\\dfrac{1}{2}$;\\\\\n\\textcircled{2} $\\sin A+\\sin B$的最小值为$1$;\\\\\n\\textcircled{3} $\\tan A+\\tan B$的取值范围为$[2,+\\infty)$;\\\\\n\\textcircled{4} $\\sin ^2A+\\sin ^2B+\\sin ^2C$为定值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "ACD", "solution": "", @@ -578271,7 +584860,9 @@ "id": "040683", "content": "已知$|\\overrightarrow {a}|=6$, $|\\overrightarrow {b}|=4$,$(\\overrightarrow {a}-2 \\overrightarrow {b}) \\cdot(\\overrightarrow {a}+3 \\overrightarrow {b})=-72$.\\\\\n(1) 求向量$\\overrightarrow {a}, \\overrightarrow {b}$的夹角$\\theta$;\\\\\n(2) 求$|\\overrightarrow {a}+3 \\overrightarrow {b}|$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1)$\\dfrac 23 \\pi$;(2)$6\\sqrt{3}$", "solution": "", @@ -578291,7 +584882,9 @@ "id": "040684", "content": "在$\\triangle ABC$中, $a^2+c^2=b^2+\\sqrt{2} a c$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 求$\\sqrt{2} \\cos A+\\cos C$的最大值;\\\\\n(3) 若$b=4$, 求$\\triangle ABC$面积的最大值和$\\triangle ABC$周长的范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$\\dfrac{\\pi}4$;(2)$1$;\\\\\n(3)面积最大值为$4+4\\sqrt{2}$,周长的取值范围是$(8,4+4\\sqrt{4+2\\sqrt{2}})$", "solution": "", @@ -578311,7 +584904,9 @@ "id": "040685", "content": "已知函数$f(x)=2 \\cos x(\\cos x+\\sqrt{3} \\sin x)-1$.\\\\\n(1) 求函数$f(x)$的最大值, 并写出此时$x$的取值;\\\\\n(2) 求函数$f(x)$严格增区间;\\\\\n(3) 设点$P_1(x_1, y_1)$、$P_2(x_2, y_2)$、$\\cdots$、$P_n(x_n, y_n)$($n \\geq 1$, $n \\in \\mathbf{N}$)都在函数$y=f(x)$的图像上, 且满足$x_1=\\dfrac{\\pi}{3}$, $x_2-x_1=\\dfrac{\\pi}{2}$, $x_3-x_2=\\dfrac{\\pi}{2}$, $\\cdots$, $x_{n+1}-x_n=\\dfrac{\\pi}{2}$($n \\geq 1$, $n \\in \\mathbf{N}$). 求$y_1, y_2$和$y_1+y_2+y_3+\\cdots+y_{2025}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)最大值为$2$，此时$x=\\dfrac{\\pi}6+k\\pi,k\\in\\mathbb{Z}$;\\\\\n(2)$[k\\pi-\\dfrac{\\pi}3,k\\pi+\\dfrac{\\pi}6],,k\\in\\mathbb{Z}$;\\\\\n(3)$y_1=1,y_2=-1,y_1+y_2+y_3+\\cdots+y_{2025}=1$", "solution": "", @@ -578331,7 +584926,9 @@ "id": "040686", "content": "如图, 一块直角梯形区域$ABCD$, $AB=AD=1$, $BC=2$, 在$D$处有一个可以转动的探照灯, 其照射角$\\angle EDF$始终为$45^{\\circ}$, 设$\\angle ADE=\\alpha$, $\\alpha \\in[0, \\dfrac{\\pi}{2}]$, 探照灯照射在该梯形$ABCD$内部区域的面积为$S$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (0,1) node [left] {$A$} coordinate (A);\n\\draw (1,1) node [above] {$D$} coordinate (D);\n\\draw (D) ++ (205:{1/cos(25)}) node [left] {$E$} coordinate (E);\n\\draw (D) ++ (250:{1/cos(20)}) node [below] {$F$} coordinate (F);\n\\draw (E)--(D)--(F)(A)--(B)--(C)--(D)--cycle;\n\\draw (D) pic [scale = 0.5, draw] {angle = E--D--F};\n\\end{tikzpicture}\n\\end{center}\n(1) 求$S$关于$\\alpha$的函数关系式;\\\\\n(2) 求$S$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$S=$;\\\\\n(2)$[\\sqrt{2}-1,2-\\sqrt{2}]$", "solution": "", @@ -578351,7 +584948,9 @@ "id": "040687", "content": "函数$y=\\sqrt{\\dfrac{3}{1-2 \\sin x}}$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|\\dfrac{5\\pi}{6}+2k\\pi=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0) node [right] {$C$} coordinate (C);\n\\draw (2,2) node [above] {$A$} coordinate (A);\n\\draw ($(A)!{2/3}!(B)$) node [above left] {$D$} coordinate (D);\n\\draw (D)--(C)(A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "$\\dfrac 23$", "solution": "", @@ -578711,7 +585344,9 @@ "id": "040705", "content": "已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$.\\\\\n(1) 求证: $\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\dfrac{|\\overrightarrow {a}+\\overrightarrow {b}|^2-|\\overrightarrow {a}|^2-|\\overrightarrow {b}|^2}{2}$;\\\\\n(2) 求证: $|\\overrightarrow {a}+\\overrightarrow {b}|^2+|\\overrightarrow {a}-\\overrightarrow {b}|^2=2(|\\overrightarrow {a}|^2+|\\overrightarrow {b}|^2)$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "略", "solution": "", @@ -578731,7 +585366,9 @@ "id": "040706", "content": "设$f(x)$是定义$\\mathbf{R}$上的偶函数, 且对任意$x_1, x_2 \\in[0, \\dfrac{1}{2}]$, 都有$f(x_1+x_2)=f(x_1) \\cdot f(x_2)$, 又$f(1)=2$.\\\\\n(1) 分别求$f(\\dfrac{1}{2}), f(\\dfrac{1}{4}), f(0)$的值;\\\\\n(2) 请写出一个满足条件的函数解析式 (不必写求解过程);\\\\\n(3) 若$f(x)$还满足: 对任意$x\\in \\mathbf{R}$, 都成立$f(1+x)=f(1-x)$, 问: $f(x)$是否是周期函数? 为什么?\\\\\n(4) 请写出一个满足满足题干与(3)中条件的函数 (不必写求解过程).", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1)$f(\\dfrac 12)=\\sqrt{2},f(\\dfrac 14)=2^{\\dfrac 14},f(0)=1$\\\\\n(2)$f(x)=2^{|x|}$\\\\\n(3)周期为2\\\\\n(4)$f(x)=2^{|x-2k|},x \\in [2k-1,2k+1],k \\in \\mathbb{Z}$", "solution": "", @@ -578751,7 +585388,9 @@ "id": "040707", "content": "$\\overrightarrow {a}=\\overrightarrow{0}$或$\\overrightarrow {b}=\\overrightarrow{0}$是$\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow{0}$的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "充分非必要", "solution": "", @@ -578771,7 +585410,9 @@ "id": "040708", "content": "若向量$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=\\sqrt{3}$, 且$\\overrightarrow {a}$和$\\overrightarrow {b}$的夹角为$30^{\\circ}$, 则$|\\overrightarrow {a}-\\overrightarrow {b}|=$\\blank{50}, $|\\overrightarrow {a}+\\overrightarrow {b}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$1,\\sqrt{7}$", "solution": "", @@ -578792,7 +585433,9 @@ "id": "040709", "content": "已知向量$\\overrightarrow{e_1}$与$\\overrightarrow{e_2}$满足$|\\overrightarrow{e_1}|=4$, $|\\overrightarrow{e_2}|=3$, 且$(2 \\overrightarrow{e_1}-3 \\overrightarrow{e_2}) \\cdot(2 \\overrightarrow{e_1}+\\overrightarrow{e_2})=61$, 则向量$\\overrightarrow{e_1}$与$\\overrightarrow{e_2}$的夹角$\\theta$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\dfrac{2\\pi}3$", "solution": "", @@ -578812,7 +585455,9 @@ "id": "040710", "content": "函数$y=\\sin x \\cdot \\cos x$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi$", "solution": "", @@ -578832,7 +585477,9 @@ "id": "040711", "content": "函数$y=\\sqrt{\\sin x}$的严格增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[2k\\pi,2k\\pi+\\dfrac{\\pi}2],k \\in \\mathbb{Z}$", "solution": "", @@ -578852,7 +585499,9 @@ "id": "040712", "content": "函数$y=\\dfrac{1}{\\sin x-\\cos x}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|x \\neq \\dfrac{\\pi}4+k\\pi,k \\in \\mathbb{Z}\\}$", "solution": "", @@ -578872,7 +585521,9 @@ "id": "040713", "content": "已知$\\omega \\in \\mathbf{R}$, $\\omega>0$, 函数$y=\\sqrt{3} \\sin \\omega x-\\cos \\omega x$在区间$[0,2]$上有唯一的最小值$-2$, 则$\\omega$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[\\dfrac{5\\pi}6,\\dfrac{11\\pi}6)$", "solution": "", @@ -578892,7 +585543,9 @@ "id": "040714", "content": "在$\\triangle ABC$中, 三边长分别为$AB=7$, $BC=5$, $AC=6$, 求$\\overrightarrow{AB} \\cdot \\overrightarrow{BC}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$-19$", "solution": "", @@ -578912,7 +585565,9 @@ "id": "040715", "content": "已知函数$f(x)=\\cos (\\omega x)$($\\omega>0$)的最小正周期为$\\pi$.\\\\\n(1) 求$\\omega$的值及函数$g(x)=f(\\dfrac{\\pi}{4}-x)-\\sqrt{3} f(x)$的严格减区间;\\\\\n(2) 在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$. 若$A \\in(0, \\dfrac{\\pi}{2})$, $f(A)=-\\dfrac{1}{2}$, $a=2$, 求$b+2 c$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1)$\\omega=2$,严格减区间为$[k\\pi+\\dfrac{5\\pi}{12},k\\pi+\\dfrac{11\\pi}{12}],k \\in \\mathbb{Z}$\\\\\n(2)$(2,\\dfrac{4\\sqrt{21}}{3}]$", "solution": "", @@ -578932,7 +585587,9 @@ "id": "040716", "content": "若$\\overrightarrow{AB}=(2,4)$, $\\overrightarrow{AC}=(1,3)$, 则$\\overrightarrow{BC}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(-1,-1)$", "solution": "", @@ -578952,7 +585609,9 @@ "id": "040717", "content": "已知$P_1(2,-1)$, $P_2(0,5)$若点$P$在直线$P_1P_2$上, 且满足$|\\overrightarrow{P_1P_2}|=2|\\overrightarrow{PP_2}|$, 则$P$点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(-1,8),(1,2)$", "solution": "", @@ -578972,7 +585631,9 @@ "id": "040718", "content": "若三点$A(2,2)$、$B(a,0)$、$C(0,4)$共线, 则$a$的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -578992,7 +585653,9 @@ "id": "040719", "content": "点$P$在平面上作匀速直线运动, 速度向量$\\overrightarrow {v}=(4,-3)$. 设开始时点$P$的坐标为$(-10,10)$, 则$5$秒后点$P$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(10,-5)$", "solution": "", @@ -579012,7 +585675,9 @@ "id": "040720", "content": "在直角坐标平面上, $O$是原点, $\\overrightarrow{OA}=(2,-4)$, $\\overrightarrow{OB}=(-2,-2)$, 若$x \\overrightarrow{OA}+y \\overrightarrow{OB}=3 \\overrightarrow{AB}$, 则实数$x=$\\blank{50}, $y=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-3,3$", "solution": "", @@ -579032,7 +585697,9 @@ "id": "040721", "content": "若$\\overrightarrow {a}=(k^2+k-3) \\overrightarrow {i}+(1-k) \\overrightarrow {j}$, $\\overrightarrow {b}=-3 \\overrightarrow {i}+(k-1) \\overrightarrow {j}$, $\\overrightarrow {a}$和$\\overrightarrow {b}$平行, 则$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$1,2,-3$", "solution": "", @@ -579052,7 +585719,9 @@ "id": "040722", "content": "已知点$A(1,0)$、$B(0,2)$、$C(-1,-2)$, 则以$A$、$B$、$C$为顶点的平行四边形的第四个顶点$D$的坐标\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(2,4),(0,-4),(-2,0)$", "solution": "", @@ -579072,7 +585741,9 @@ "id": "040723", "content": "已知$\\overrightarrow {e_1}, \\overrightarrow {e_2}$是不平行的向量, 设$\\overrightarrow {a}=\\overrightarrow {e_1}+k \\overrightarrow {e_2}$, $\\overrightarrow {b}=k \\overrightarrow {e_1}+\\overrightarrow {e_2}$, 则$\\overrightarrow {a}$与$\\overrightarrow {b}$共线的一个充要条件是实数$k$等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\pm 1$", "solution": "", @@ -579092,7 +585763,9 @@ "id": "040724", "content": "在$\\triangle ABC$中, $O$为中线$AM$上一个动点, 若$AM=2$, 则$\\overrightarrow{OA} \\cdot(\\overrightarrow{OB}+\\overrightarrow{OC})$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-2$", "solution": "", @@ -579112,7 +585785,9 @@ "id": "040725", "content": "直角坐标平面上三点$A(1,2)$、$B(3,-2)$、$C(9,7)$, 若$E$、$F$为线段$BC$的三等分点, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{AF}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "22", "solution": "", @@ -579132,7 +585807,9 @@ "id": "040726", "content": "设$P$是$\\triangle ABC$所在平面内的一点, $\\overrightarrow{BC}+\\overrightarrow{BA}=2 \\overrightarrow{BP}$, 则\\bracket{20}.\n\\fourch{$\\overrightarrow{PA}+\\overrightarrow{PB}=\\overrightarrow{0}$}{$\\overrightarrow{PC}+\\overrightarrow{PA}=\\overrightarrow{0}$}{$\\overrightarrow{PB}+\\overrightarrow{PC}=\\overrightarrow{0}$}{$\\overrightarrow{PA}+\\overrightarrow{PB}+\\overrightarrow{PC}=\\overrightarrow{0}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -579152,7 +585829,9 @@ "id": "040727", "content": "如图, 在$\\triangle ABC$中, $\\angle ACB=90^{\\circ}$, $AB=5$, $BC=4$, $CA=3$, 点$D$是边$BC$的中点, 点$H$在$AB$上, $CH \\perp AB$, 则$\\overrightarrow{AD} \\cdot \\overrightarrow{CH}=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (5,0) node [below] {$B$} coordinate (B);\n\\draw ({9/5},{12/5}) node [above] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [right] {$D$} coordinate (D);\n\\draw ({9/5},0) node [below] {$H$} coordinate (H);\n\\draw (A)--(B)--(C)--cycle(C)--(H)(A)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-\\dfrac{72}{25}$", "solution": "", @@ -579172,7 +585851,9 @@ "id": "040728", "content": "$\\triangle ABC$中, $\\overrightarrow{AB}=(2,4)$, $\\overrightarrow{BC}=(-6,2)$, 则$\\triangle ABC$的面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$14$", "solution": "", @@ -579192,7 +585873,9 @@ "id": "040729", "content": "已知向量$\\overrightarrow{a}=(1,2), \\overrightarrow{b}=(2,-3)$. 若向量$\\overrightarrow{c}$满足$(\\overrightarrow{c}+\\overrightarrow{a})\\parallel \\overrightarrow{b}, \\overrightarrow{c} \\perp(\\overrightarrow{a}+\\overrightarrow{b})$, 则$\\overrightarrow {c}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(-\\dfrac 79,\\dfrac 73)$", "solution": "", @@ -579212,7 +585895,9 @@ "id": "040730", "content": "设$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$为同一平面内具有相同起点的任意三个非零向量, 且满足$\\overrightarrow {a}$与$\\overrightarrow {b}$不共线, $\\overrightarrow {a} \\perp \\overrightarrow {c}$, $|\\overrightarrow {a}|=|\\overrightarrow {c}|$, 则$|\\overrightarrow {b} \\cdot \\overrightarrow {c}|$的值一定等于\\bracket{20}.\n\\twoch{以$\\overrightarrow {a}, \\overrightarrow {b}$为邻边的平行四边形的面积}{以$\\overrightarrow {b}, \\overrightarrow {c}$为两边的三角形面积}{$\\overrightarrow {a}, \\overrightarrow {b}$为两边的三角形面积}{以$\\overrightarrow {b}, \\overrightarrow {c}$为邻边的平行四边形的面积}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -579232,7 +585917,9 @@ "id": "040731", "content": "$O$是平面上一定点, $A,B,C$是平面上不共线的三点, 动点$P$满足$\\overrightarrow{OP}=\\overrightarrow{OA}+\\lambda(\\dfrac{\\overrightarrow{AB}}{|\\overrightarrow{AB}|}+\\dfrac{\\overrightarrow{AC}}{|\\overrightarrow{AC}|})$, $\\lambda \\in(0,+\\infty)$, \n则$P$的轨迹一定通过$\\triangle ABC$的\\bracket{20}.\n\\fourch{外心}{内心}{重心}{垂心}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -579252,7 +585939,9 @@ "id": "040732", "content": "向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$以及$\\overrightarrow {b}-\\overrightarrow {c}$都是非零向量, 则``$\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow {a} \\cdot \\overrightarrow {c} ''$是``$\\overrightarrow {a} \\perp(\\overrightarrow {b}-\\overrightarrow {c})$''的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充分必要}{既不充分又不必要}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -579272,7 +585961,9 @@ "id": "040733", "content": "已知$|\\overrightarrow {a}|=3,|\\overrightarrow {b}|=4$. 若$\\overrightarrow {c}$使得$\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}+\\overrightarrow {b} \\cdot \\overrightarrow {c}+\\overrightarrow {c} \\cdot \\overrightarrow {a}$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$[-37,-13]$", "solution": "", @@ -579292,7 +585983,9 @@ "id": "040734", "content": "设$\\lambda$是正实数, $\\triangle ABC$所在平面上的三点$A_1, B_1, C_1$(与$A,B,C$共六点两两不重合)满足: $\\overrightarrow{AA_1}=\\lambda(\\overrightarrow{AB}+\\overrightarrow{AC})$, $\\overrightarrow{BB_1}=\\lambda(\\overrightarrow{BC}+\\overrightarrow{BA})$, $\\overrightarrow{CC_1}=\\lambda(\\overrightarrow{CA}+\\overrightarrow{CB})$. 若$\\triangle ABC$与$\\triangle A_1B_1C_1$的面积相等, 则$\\lambda$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\dfrac 23$", "solution": "", @@ -579312,7 +586005,9 @@ "id": "040735", "content": "已知$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=1$, 且向量$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$\\dfrac{\\pi}{3}$.\\\\\n(1) 设$\\overrightarrow {x}=2 \\overrightarrow {a}-\\overrightarrow {b}$, $\\overrightarrow {y}=3 \\overrightarrow {b}-\\overrightarrow {a}$, 求$\\overrightarrow {x}$与$\\overrightarrow {y}$的夹角;\\\\\n(2) 若向量$\\overrightarrow {a}$与$\\overrightarrow {a}+k \\overrightarrow {b}$的夹角为锐角, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1)$\\pi-\\arccos{\\dfrac{\\sqrt{21}}{14}}$\\\\\n(2)$k \\in (-2,0) \\cup (0,+\\infty)$", "solution": "", @@ -579332,7 +586027,9 @@ "id": "040736", "content": "如图, $\\triangle AOE$和$\\triangle BOE$都是边长为$1$的等边三角形, 延长$OB$到$C$使$|BC|=t$($t>0$), 连$AC$交$BE$于$D$点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-0.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$E$} coordinate (E);\n\\draw (60:2) node [above] {$A$} coordinate (A);\n\\draw (-60:2) node [below] {$B$} coordinate (B);\n\\draw ($(O)!1.3!(B)$) node [below] {$C$} coordinate (C);\n\\path [name path = AC] (A)--(C);\n\\path [name path = BE] (B)--(E);\n\\path [name intersections = {of = AC and BE, by = D}];\n\\draw (O)--(D) node [right] {$D$};\n\\draw (O)--(A)--(E)--(B)--cycle(A)--(C)(B)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 用$t$表示向量$\\overrightarrow{OC}$和$\\overrightarrow{OD}$的坐标\\\\\n(2) 求向量$\\overrightarrow{OD}$和$\\overrightarrow{EC}$的夹角的大小.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1)$\\overrightarrow{OC}=(\\dfrac{1+t}2,-\\dfrac{\\sqrt{3}}{2} \\cdot (1+t)),\\overrightarrow{OD}=(\\dfrac{2t+1}{2t+2},-\\dfrac{\\sqrt{3}}{2} \\cdot \\dfrac{1}{1+t})$\\\\\n(2)$\\dfrac{\\pi}3$", "solution": "", @@ -579352,7 +586049,9 @@ "id": "040737", "content": "在同一个平面内, 有向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {p}$、$\\overrightarrow {q}$、$\\overrightarrow {r}$, 其中$\\overrightarrow {b} \\neq \\overrightarrow{0}$, 且$\\overrightarrow {a} \\perp \\overrightarrow {b}$, $|\\overrightarrow {a}|=2|\\overrightarrow {b}|$.\\\\\n(1) 若$|\\overrightarrow {b}|=1$, 求$|\\overrightarrow {a}-\\overrightarrow {b}|$的值;\\\\\n(2) 设常数$k \\in \\mathbf{R}$. 若向量$\\overrightarrow {p}=k \\overrightarrow {a}+2 \\overrightarrow {b}$, $\\overrightarrow {q}=\\overrightarrow {a}+2 k \\overrightarrow {b}$, $\\overrightarrow {p}$与$\\overrightarrow {q}$的夹角大小为$120^{\\circ}$, 求$k$的值;\\\\\n(3) 设$\\overrightarrow {r} \\neq \\overrightarrow{0}$, 且$\\theta$是$\\overrightarrow {r}$与$\\overrightarrow {a}$的夹角. 若$\\overrightarrow {r} \\cdot \\overrightarrow {a}=\\overrightarrow {r} \\cdot \\overrightarrow {b}$, 求$\\theta$的大小.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1)$\\sqrt{5}$\\\\(2)$k=-2\\pm\\sqrt{3}$\\\\(3)$\\theta=\\arctan{2}$或$\\theta=\\pi-\\arctan{2}$", "solution": "", @@ -579372,7 +586071,9 @@ "id": "040738", "content": "若向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$\\overrightarrow {a}+\\overrightarrow {b}=(2,-2)$, $\\overrightarrow {a}-\\overrightarrow {b}=(-2,4)$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579392,7 +586093,9 @@ "id": "040739", "content": "若非零向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$|\\overrightarrow {a}+\\overrightarrow {b}|=2|\\overrightarrow {a}|$, $|\\overrightarrow {b}|=\\sqrt{3}|\\overrightarrow {a}|$, 则$\\overrightarrow {a}$与$\\overrightarrow {b}$所成角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579412,7 +586115,9 @@ "id": "040740", "content": "若$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=3$, 且$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$\\dfrac{2 \\pi}{3}$, 则$|\\overrightarrow {a}-3 \\overrightarrow {b}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579432,7 +586137,9 @@ "id": "040741", "content": "设向量$\\overrightarrow {a}=(2,3)$, 点$A$的坐标为$(1,4)$. 若向量$\\overrightarrow{AB}$与$\\overrightarrow {a}$方向相反, 且$|\\overrightarrow{AB}|=3|\\overrightarrow {a}|$, 则点$B$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579452,7 +586159,9 @@ "id": "040742", "content": "已知向量$\\overrightarrow {p}=\\dfrac{\\overrightarrow {a}}{|\\overrightarrow {a}|}+\\dfrac{\\overrightarrow {b}}{|\\overrightarrow {b}|}$, 其中$\\overrightarrow {a}, \\overrightarrow {b}$均为非零向量, 则$|\\overrightarrow {p}|$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579472,7 +586181,9 @@ "id": "040743", "content": "已知向量$\\overrightarrow {a}=(1,2)$, $\\overrightarrow {b}=(2,-3)$, 若向量$\\overrightarrow {c}$满足$(\\overrightarrow {c}+\\overrightarrow {a})\\parallel \\overrightarrow {b}$, $\\overrightarrow {c} \\perp(\\overrightarrow {a}+\\overrightarrow {b})$, 则$\\overrightarrow {c}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579492,7 +586203,9 @@ "id": "040744", "content": "设$O$是坐标原点, $\\overrightarrow{OA}=(k, 12)$, $\\overrightarrow{OB}=(4,5)$, $\\overrightarrow{OC}=(10, k)$, 则$k=$\\blank{50}时, $A$、$B$、$C$三点共线.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579512,7 +586225,9 @@ "id": "040745", "content": "设$A(a, 1), B(2, b), C(4,5)$为坐标平面上三点, $O$为坐标原点, 若$\\overrightarrow{OA}$与$\\overrightarrow{OB}$在$\\overrightarrow{OC}$方向上的投影相同, 则$a$、$b$满足的关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579532,7 +586247,9 @@ "id": "040746", "content": "在$\\triangle ABC$中, $AD \\perp AB$, $\\overrightarrow{BC}=\\sqrt{3} \\overrightarrow{BD}$, $|\\overrightarrow{AD}|=1$($D$为垂足), 则$\\overrightarrow{AC} \\cdot \\overrightarrow{AD}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579552,7 +586269,9 @@ "id": "040747", "content": "设常数$t \\in \\mathbf{R}$, 向量$\\overrightarrow {a}=(2, t)$, $\\overrightarrow {b}=(-1,2)$. 若$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为钝角, 则$t$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579572,7 +586291,9 @@ "id": "040748", "content": "设$\\overrightarrow {e_1}$、$\\overrightarrow {e_2}$为单位向量, 且$\\overrightarrow {e_1}$、$\\overrightarrow {e_2}$的夹角大小为$60^{\\circ}$. 若$\\overrightarrow {a}=\\overrightarrow {e_1}-\\overrightarrow {e_2}$, $\\overrightarrow {b}=2 \\overrightarrow {e_1}$, 则$\\overrightarrow {a}$在$\\overrightarrow {b}$方向上的数量投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579592,7 +586313,9 @@ "id": "040749", "content": "给定两个不平行的非零向量$\\overrightarrow{OA}$、$\\overrightarrow{OB}$, $M$是$OB$的中点, $N$是$AB$上靠近$B$的三等分点, 点$C$在$MN$上变动, 若$\\overrightarrow{OC}=x \\overrightarrow{OA}+y \\overrightarrow{OB}$, 其中$x$、$y \\in \\mathbf{R}$, 则$x+y$的范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579612,7 +586335,9 @@ "id": "040750", "content": "已知两点$A(-1,2)$, $B(2,8)$, $C$、$D$两点满足$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=1$, $\\overrightarrow{OA} \\cdot(\\dfrac{1}{2} \\overrightarrow{BC})=0$, $\\overrightarrow{DC}+2 \\overrightarrow{AD}=0$, 求$C$、$D$两点的坐标(其中$O$是原点).", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -579632,7 +586357,9 @@ "id": "040751", "content": "已知$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=2$, $\\overrightarrow {a}$、$\\overrightarrow {b}$的夹角为$\\dfrac{\\pi}{3}$, 记$\\overrightarrow {m}=3 \\overrightarrow {a}-\\overrightarrow {b}$, $\\overrightarrow {n}=2 \\overrightarrow {a}+2 \\overrightarrow {b}$, 求$\\overrightarrow {m}$与$\\overrightarrow {n}$的夹角.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -579652,7 +586379,9 @@ "id": "040752", "content": "设$t \\in \\mathbf{R}$, $O(0,0)$, $A(1,2)$, $B(4,5)$, $C$四点满足$\\overrightarrow{OC}=\\overrightarrow{OA}+t \\overrightarrow{AB}$.\\\\\n(1) $t$为何值时, 点$C$在$y$轴上?\\\\\n(2) 四边形$OABC$能否成为平行四边形? 若能, 求出相应的$t$的值; 若不能, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -579672,7 +586401,9 @@ "id": "040753", "content": "计算: $\\dfrac{1-\\mathrm{i}}{1+\\mathrm{i}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579692,7 +586423,9 @@ "id": "040754", "content": "复数$\\dfrac{5}{1+2 \\mathrm{i}}$的共轭复数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579713,7 +586446,9 @@ "id": "040755", "content": "求值: $\\dfrac{(1-4 \\mathrm{i})(1+\\mathrm{i})+2+4 \\mathrm{i}}{3+4 \\mathrm{i}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579733,7 +586468,9 @@ "id": "040756", "content": "已知$x$是实数, $y$是纯虚数且滿足$(2 x-1)+(3-y) \\mathrm{i}=y-\\mathrm{i}$, 则$x=$\\blank{50}, $y=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579753,7 +586490,9 @@ "id": "040757", "content": "已知复数$z_1=1+3 \\mathrm{i}$, $z_2=\\sqrt{3} \\cos \\alpha+\\mathrm{i} \\sin \\alpha$, $z=z_1 \\cdot z_2$, 则$\\text{Re} z$的取大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579773,7 +586512,9 @@ "id": "040758", "content": "复数$z=3+a \\mathrm{i}$满足条件$|z-2|<2$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579793,7 +586534,9 @@ "id": "040759", "content": "若$z=\\dfrac{(3-4 \\mathrm{i})^2 \\cdot(-\\dfrac{\\sqrt{3}}{2}-\\dfrac{1}{2} \\mathrm{i})^{10}}{(\\sqrt{2}-\\sqrt{3} \\mathrm{i})^4}$, 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579813,7 +586556,9 @@ "id": "040760", "content": "已知$z$、$w$为复数, $(1+3 \\mathrm{i}) z$为纯虚数, $w=\\dfrac{z}{2+\\mathrm{i}}$, 且$|w|=5 \\sqrt{2}$, 则$w=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579833,7 +586578,9 @@ "id": "040761", "content": "若 $3-\\mathrm{i}$ 是关于$x$的实系数一元二次方程$2 x^2+p x+q=0$的一个根, 则$p+q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579853,7 +586600,9 @@ "id": "040762", "content": "复数$5-12 \\mathrm{i}$的平方根为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579873,7 +586622,9 @@ "id": "040763", "content": "在复数范围内分解因式: $x^4-y^4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579893,7 +586644,9 @@ "id": "040764", "content": "若关于$x$的方程$x^2-(2 \\mathrm{i}-1) x+3 m-\\mathrm{i}=0$有实根, 则实数$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579913,7 +586666,9 @@ "id": "040765", "content": "$\\triangle ABC$三个顶点所对应的复数分别为$z_1, z_2, z_3$, 复数$z$满足$|z-z_1|=|z-z_2|=|z-z_3|$, 则$z$所对应的点是$\\triangle ABC$的\\blank{50}(填内心、外心、重心、垂心等).", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579933,7 +586688,9 @@ "id": "040766", "content": "设$z, z_1, z_2 \\in \\mathbf{C}$, 有下列命题: \\textcircled{1} $z+\\overline {z}=0$是$z$为纯虚数的充要条件; \\textcircled{2} $z^2 \\geq 0$是$z \\in \\mathbf{R}$的充要条件; \\textcircled{3} 若$z_1^2-z_2^2>0$, 则$z_1^2>z_2{ }^2$; \\textcircled{4} 总有$z_1 \\cdot \\overline{z_2}+\\overline{z_1} \\cdot z_2 \\in \\mathbf{R}$. 其中是真命题的有\\blank{50}. (写出所有正确序号)", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -579953,7 +586710,9 @@ "id": "040767", "content": "设全体虚数组成的集合为$I$, 那么下列关系中不正确的是\\bracket{20}.\n\\fourch{$\\mathbf{R} \\cup I=\\mathbf{C}$}{$\\mathbf{R} \\cap I=\\varnothing$}{$\\mathbf{R} \\cup \\mathbf{C}=\\mathbf{C}$}{$\\mathbf{R} \\cap \\mathbf{C}=I$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -579973,7 +586732,9 @@ "id": "040768", "content": "若$(\\dfrac{1+\\mathrm{i}}{1-\\mathrm{i}})^n=\\mathrm{i}$, 则整数$n$等于\\bracket{20}.\n\\fourch{$4 k$($k \\in \\mathbf{Z}$)}{$4 k+1$($k \\in \\mathbf{Z}$)}{$4 k+2$($k \\in \\mathbf{Z}$)}{$4 k+3$($k \\in \\mathbf{Z}$)}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -579993,7 +586754,9 @@ "id": "040769", "content": "已知复数$z_1=a^2-3+(a+5) \\mathrm{i}$, $z_2=a-1+(a^2+2 a-1) \\mathrm{i}$($a \\in \\mathbf{R}$)分别对应向量$\\overrightarrow{OZ_1}$、$\\overrightarrow{OZ_2}$($O$为原点), 若向量$\\overrightarrow{Z_1Z_2}$对应的复数为纯虚数, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -580013,7 +586776,9 @@ "id": "040770", "content": "已知$z$是虚数, $w=z+\\dfrac{1}{z}$, 且$-1=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,-0.2) -- (2,0.5) node [right] {$l_1$};\n\\draw (-1.5,-2) -- (1,2) node [right] {$l_2$};\n\\draw (-1,-2) -- (-0.5,2) node [above] {$l_3$};\n\\draw (2,-1) -- (-2,2) node [above] {$l_4$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -580974,7 +587833,9 @@ "id": "040818", "content": "已知方程$(a^2+a) x+(a^2-1) y-1=0$表示一条直线, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -580994,7 +587855,9 @@ "id": "040819", "content": "直线$(m^2+1) x+y=1$, $m \\in \\mathbf{R}$的倾斜角的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581014,7 +587877,9 @@ "id": "040820", "content": "已知$P$是直线$l$上的一点, 将直线$l$绕点$P$沿逆时针方向旋转$\\alpha$($0<\\alpha<\\dfrac{\\pi}{2}$), 所得的直线方程是$x-y-2=0$, 若将它继续逆时针旋转$\\dfrac{\\pi}{2}-\\alpha$, 所得的直线方程是$2 x+y-1=0$, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581034,7 +587899,9 @@ "id": "040821", "content": "一直线$l$被两直线$4 x+y+6=0$和$3 x-5 y-6=0$截得的线段中点恰好是坐标原点, 则直线$l$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581054,7 +587921,9 @@ "id": "040822", "content": "已知$m, n$是正的常数, $l$是经过原点的直线, 若直线$l$上的任意点先向左平移$m$个单位再向上平移$n$个单位后仍在直线$l$上, 则直线$l$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581074,7 +587943,9 @@ "id": "040823", "content": "已知$\\triangle ABC$的三个顶点为$A(1,6), B(-1,-2), C(6,3)$.\\\\\n(1) 求$BC$边上的高所在直线的方程;\\\\\n(2) 求与$AC$边平行的中位线所在直线的方程;\\\\\n(3) $AB$边上的垂直平分线所在的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581094,7 +587965,9 @@ "id": "040824", "content": "在正方形$ABCO$中, $O$为原点, $\\overrightarrow{OA}=(3,4)$, $O, A, B, C$按逆时针排列, 求正方形$ABCO$各边所在的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581114,7 +587987,9 @@ "id": "040825", "content": "已知直线$l: (3 a-1) x-(a-2) y-1=0$($a \\in \\mathbf{R}$).\\\\\n(1) 求证: 不论$a$取何值, $l$恒过定点;\\\\\n(2) 若$l$不通过第二象限, 求实数$a$的取值范围;\\\\\n(3) 记 (1) 中的定点为$P$, 若$l \\perp OP$($O$为原点), 求实数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581134,7 +588009,9 @@ "id": "040826", "content": "过点$P(2,1)$作直线$l$分别交$x$轴、$y$轴的正半轴于$A, B$两点, $O$是坐标原点.\\\\\n(1) 当$\\triangle AOB$面积最小时, 求直线$l$的方程;\\\\\n(2) 当$|OA|+|OB|$最小时, 求直线$l$的方程;\\\\\n(3) 当$|PA| \\cdot|PB|$最大时, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581154,7 +588031,9 @@ "id": "040827", "content": "已知$P$是直线$3 x-y=0$上位于第一象限的点, $M(3,2)$为一定点, 直线$PM$交$x$轴正半轴于点$Q$, 当$\\triangle POQ$的面积最小时, 求直线$PQ$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581174,7 +588053,9 @@ "id": "040828", "content": "设$u, v$是非零常数, $a$是正的常数, 在直角坐标平面内, 过点$P(u, v)$作直线$l$, 使其与两坐标轴相交所成的三角形的面积等于$a$, 当$u, v, a$取不同值时, 这样的直线$l$可能有\\blank{50}条.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581194,7 +588075,9 @@ "id": "040829", "content": "经过点$A(2,2)$、$B(3,-2)$的直线$l$的两点式方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581214,7 +588097,9 @@ "id": "040830", "content": "经过点$(3,2)$, 且倾斜角为$\\dfrac{5 \\pi}{6}$的直线$l$的点斜式方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581234,7 +588119,9 @@ "id": "040831", "content": "若直线$l_1: 2 x-3 y+2=0$与直线$l_2: x+t y+1=0$垂直, 则实数$t=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581254,7 +588141,9 @@ "id": "040832", "content": "若直线$(a-2) x+3 a y-4=0$在$x$轴上的截距为$2$, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581274,7 +588163,9 @@ "id": "040833", "content": "若直线的倾斜角$\\theta \\in[0, \\dfrac{\\pi}{3}) \\cup[\\dfrac{3 \\pi}{4}, \\pi)$, 则其斜率$k$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581294,7 +588185,9 @@ "id": "040834", "content": "若$\\overrightarrow {n}=(a, 1-a)$是直线$3 x+4 y-1=0$的一个法向量, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581314,7 +588207,9 @@ "id": "040835", "content": "已知点$A(2,1), B(-1,3)$, 若过$(0,-2)$且斜率为$k$的直线$l$与线段$AB$有公共点, 则实数$k$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581334,7 +588229,9 @@ "id": "040836", "content": "已知直线$(1-a) x+(2 a-3) y+1-2 a=0$不经过第一象限, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581354,7 +588251,9 @@ "id": "040837", "content": "已知点$P$是直线$l: a x+y+2=0$上的一点, 将$l$绕点$P$沿逆时针方向旋转$\\dfrac{\\pi}{4}$后与直线$4 x-3 y+5=0$重合, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581374,7 +588273,9 @@ "id": "040838", "content": "已知$\\triangle ABC$的顶点$B(2,3)$、$C(-2,1)$, 若其垂心$H$为$(-4,4)$, 则其顶点$A$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581394,7 +588295,9 @@ "id": "040839", "content": "已知直线$l: (1-2 a) x+(a-1) y-a+2=0$, 点$A(-1,4), B(3,2)$, 是否存在实数$a$, 使得$l$与直线$AB$平行? 若存在, 求出$a$的值; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581414,7 +588317,9 @@ "id": "040840", "content": "已知直线$l_1: m x-y-m+2=0$, 直线$l_2: (m+1) x+(m-1) y+1-3 m=0$, 其中$m>0$.\\\\\n(1) 求证: 不论$m$取何值时, 直线$l_1$与直线$l_2$均恒过一个定点;\\\\\n(2) 设$l_1$与$y$轴正半轴相交于点$M$, $l_2$与$x$轴正半轴相交于点$N$, $l_1$与$l_2$相交于点$A$, $O$为坐标原点, 求四边形$OMAN$的面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581434,7 +588339,9 @@ "id": "040841", "content": "已知复数$z=1-2 \\mathrm{i}$, $\\mathrm{i}$为虚数单位, 则$\\mathrm{Im} z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581454,7 +588361,9 @@ "id": "040842", "content": "复数$z$满足$(z-3)(2-\\mathrm{i})=5$, $\\mathrm{i}$为虚数单位, 则$\\overline {z}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581474,7 +588383,9 @@ "id": "040843", "content": "若复数$z=-1-\\mathrm{i}$, $\\mathrm{i}$为虚数单位, 则$\\arg z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581494,7 +588405,9 @@ "id": "040844", "content": "在$\\triangle ABC$中, $A$、$B$、$C$所对的边是$a$、$b$、$c$, $A=60^{\\circ}$, $b=1$, $\\triangle ABC$的面积为$\\sqrt{3}$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581514,7 +588427,9 @@ "id": "040845", "content": "已知向量$\\overrightarrow {a}=(\\sqrt{3}, 1)$, $\\overrightarrow {b}=(0,2)$, 则向量$\\overrightarrow {b}$在向量$\\overrightarrow {a}$方向上的数量投影是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581534,7 +588449,9 @@ "id": "040846", "content": "已知向量$\\overrightarrow {a}=(-1,2)$, $\\overrightarrow {b}=(1,2)$, $\\overrightarrow {c}=(k, 6)$, 若$(\\overrightarrow {a}-\\overrightarrow {c})$与$\\overrightarrow {b}$共线, 则$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581554,7 +588471,9 @@ "id": "040847", "content": "已知函数$f(x)=\\sin 2 x$的两条对称轴之间距离的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581574,7 +588493,9 @@ "id": "040848", "content": "设复数$z$满足$z^2-5|z|+6=0$, 则符合条件的$z$的解的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581594,7 +588515,10 @@ "id": "040849", "content": "在集合$\\{1,2,3,4\\}$中, 任取一个偶数$a$和一个奇数$b$, 构成一个位置向量$\\overrightarrow {x}=(a, b)$, 以所有这些向量为邻边作的平行四边形中, 面积不超过$4$的平行四边形的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581614,7 +588538,9 @@ "id": "040850", "content": "已知平面非零向量$\\overrightarrow {\\alpha}$、$\\overrightarrow {\\beta}$($|\\overrightarrow {\\alpha}| \\neq|\\overrightarrow {\\beta}|$), 满足$|\\overrightarrow {\\beta}|=2$, 且$\\overrightarrow {\\alpha}$与$\\overrightarrow {\\beta}-\\overrightarrow {\\alpha}$的夹角为$135^{\\circ}$, 则$|\\overrightarrow {\\alpha}|$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581634,7 +588560,9 @@ "id": "040851", "content": "已知点$P$在单位圆$O$的圆周上运动, 点$A(-3,0)$, 则$\\overrightarrow{AO} \\cdot \\overrightarrow{AP}$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581654,7 +588582,9 @@ "id": "040852", "content": "已知$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$为不共线向量, 且$|\\overrightarrow {a}|=3,|\\overrightarrow {b}|=2|\\overrightarrow {c}|$, 且$(\\overrightarrow {b}-\\overrightarrow{2 a}) \\perp(\\overrightarrow {c}-\\overrightarrow{2 a})$, 则$|\\overrightarrow {b}|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581675,7 +588605,10 @@ "id": "040853", "content": "函数$y=\\sqrt{\\cos (x-\\dfrac{\\pi}{3})}$的一个严格单调增区间为\\bracket{20}.\n\\fourch{$[0, \\dfrac{\\pi}{2}]$}{$[0, \\dfrac{\\pi}{4}]$}{$[-\\dfrac{\\pi}{2}, 0]$}{$[\\dfrac{\\pi}{3}, \\dfrac{\\pi}{2}]$}", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -581695,7 +588628,9 @@ "id": "040854", "content": "设$O$为坐标原点, 复数$z_1$、$z_2$在复平面内对应的点分别为$P$、$Q$, 则下列结论中不一定正确的是\\bracket{20}.\n\\twoch{$|z_1+z_2|=|\\overrightarrow{OP}+\\overrightarrow{OQ}|$}{$|z_1-z_2|=| \\overrightarrow{OP}-\\overrightarrow{OQ}|$}{$|z_1|+|z_2|=|\\overrightarrow{OP}|+|\\overrightarrow{OQ}|$}{$|z_1 \\cdot z_2|=|\\overrightarrow{OP} \\cdot \\overrightarrow{OQ}|$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -581715,7 +588650,9 @@ "id": "040855", "content": "下列所给的四个命题中, 不是真命题的是\\bracket{20}.\n\\twoch{$|z|=|\\overline {z}|$}{$z \\in \\mathbf{R} \\Leftrightarrow z=\\overline {z}$}{$|z|^2=z \\cdot \\overline {z}$}{$z^2=-|z|^2 \\Leftrightarrow z$为纯虚数}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -581735,7 +588672,9 @@ "id": "040856", "content": "若向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$满足$\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$, 且$\\overrightarrow {a}^2<\\overrightarrow {b}^2<\\overrightarrow {c}^2$, 则``$\\overrightarrow {a} \\cdot \\overrightarrow {b}$''、``$\\overrightarrow {b} \\cdot \\overrightarrow {c}$''、``$\\overrightarrow {c} \\cdot \\overrightarrow {a}$''中, 最大的是\\bracket{20}.\n\\fourch{$\\overrightarrow {a} \\cdot \\overrightarrow {b}$}{$\\overrightarrow {b} \\cdot \\overrightarrow {c}$}{$\\overrightarrow {c} \\cdot \\overrightarrow {a}$}{不能确定}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -581755,7 +588694,9 @@ "id": "040857", "content": "已知复数$z=1+m \\mathrm{i}$, 且$\\overline {z} \\cdot(1+\\mathrm{i})$是纯虚数($m \\in \\mathbf{R}$, $\\mathrm{i}$是虚数单位, $\\overline {z}$是$z$的共轭复数).\\\\\n(1) 求实数$m$的值和$|z|$;\\\\\n(2) 设$z_1=\\dfrac{a-\\mathrm{i}^{2023}}{z}$, 且复数$z_1$对应的点在第二象限, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581775,7 +588716,9 @@ "id": "040858", "content": "已知关于$x$的实系数一元二次方程$x^2+m x+9=0$.\\\\\n(1) 若复数$z$是该方程的一个虚根, 且$|z|+\\overline {z}=4-2 \\sqrt{2} \\mathrm{i}$, 求实数$m$的值;\\\\\n(2) 若该方程的两根为$x_1$、$x_2$, 若$|x_1-x_2|=2 \\sqrt{3}$, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581795,7 +588738,9 @@ "id": "040859", "content": "复数$z_1=2 \\sin \\theta-\\sqrt{3} \\cdot \\mathrm{i}$, $z_2=1+2 \\cos \\theta \\cdot \\mathrm{i}$, $i$是虚数单位.\\\\\n(1) 若$\\theta \\in[0, \\dfrac{\\pi}{2}]$, 且$z_1 \\cdot z_2$为实数, 求$\\theta$的值;\\\\\n(2) 若$\\theta \\in[\\dfrac{\\pi}{2}, \\dfrac{5 \\pi}{6}]$, 复数$z_1$、$z_2$对应的向量分别是$\\overrightarrow {a}$、$\\overrightarrow {b}$, 且存在$\\theta$使等式$(\\lambda \\overrightarrow {a}-\\overrightarrow {b}) \\cdot(\\overrightarrow {a}-\\lambda \\overrightarrow {b})=0$, 求实数$\\lambda$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581815,7 +588760,9 @@ "id": "040860", "content": "如图是函数$f(x)=A \\sin (\\omega x+\\varphi),(A>0, \\omega>0,0<\\omega<\\dfrac{\\pi}{2})$图像的一部分, $M$、$N$是它与$x$轴的两个交点, $C$、$D$分别是它的最高点和最低点, $E(0,1)$是线段$MC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {-pi/4}:{7*pi/4}, samples = 100] plot (\\x,{2*sin(\\x/pi*180+45)});\n\\draw ({pi/4},2) node [above] {$C$} coordinate (C);\n\\draw ({-pi/4},0) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(M)$) node [right] {$E$} coordinate (E);\n\\draw ({3*pi/4},0) node [below left] {$N$} coordinate (N);\n\\draw ({5*pi/4},-2) node [below] {$D$} coordinate (D);\n\\draw (M)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 若点$M$的坐标为$(-1,0)$, 求$C$、$N$、$D$的坐标;\\\\\n(2) 若点$M$的坐标为$(-m, 0)$($m>0$), 且$\\overrightarrow{MC} \\cdot \\overrightarrow{MD}=\\dfrac{3 \\pi^2}{4}-4$, 试确定函数$f(x)$的解析式, 并求$f(x)$在$[0, \\dfrac{\\pi}{2}]$上的值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581835,7 +588782,9 @@ "id": "040861", "content": "设$\\triangle ABC$是边长为$2$的正三角形, 点$P_1$、$P_2$、$P_3$依次四等分线段$BC$(如图所示).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (60:2) node [above] {$A$} coordinate (A);\n\\draw (0.5,0) node [below] {$P_1$} coordinate (P_1);\n\\draw (1,0) node [below] {$P_2$} coordinate (P_2);\n\\draw (1.5,0) node [below] {$P_3$} coordinate (P_3);\n\\draw (A)--(B)--(C)--cycle;\n\\foreach \\i in {1,2,3}\n{\\draw [->] (A) -- (P_\\i);};\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot \\overrightarrow{AP_1}+\\overrightarrow{AP_1} \\cdot \\overrightarrow{AP_2}+\\overrightarrow{AP_2} \\cdot \\overrightarrow{AP_3}+\\overrightarrow{AP_3} \\cdot \\overrightarrow{AC}$的值;\\\\\n(2) 若$Q$为线段$AP_1$上的一点, 且$\\overrightarrow{AQ}=m \\overrightarrow{AB}+\\dfrac{1}{16} \\overrightarrow{AC}$, 求实数$m$的值;\\\\\n(3) $P$在边$BC$的何处时, $\\overrightarrow{PA} \\cdot \\overrightarrow{PC}$取得最小值, 并求出此最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -581855,7 +588804,9 @@ "id": "040862", "content": "直线$(x+2)+2(y-3)=0$的斜率为\\blank{50}, 倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581875,7 +588826,9 @@ "id": "040863", "content": "若直线$l$过点$(2,-3)$, 且平行于向量$\\overrightarrow {d}=(3,4)$, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581895,7 +588848,9 @@ "id": "040864", "content": "若直线$l$经过两点$A(1,-2)$、$B(3,1)$, 且与向量$\\overrightarrow {d}=(2, m)$平行, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581915,7 +588870,9 @@ "id": "040865", "content": "直线$x+2=0$与直线$\\sqrt{3} x+y+3=0$的夹角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581935,7 +588892,9 @@ "id": "040866", "content": "若实数$x$、$y$满足关系式$4 x+3 y-12=0$, 则$\\sqrt{x^2+y^2}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581955,7 +588914,9 @@ "id": "040867", "content": "若直线$l_1: (2-m) x+m x+3=0$的法向量恰为直线$l_2: x-m y-3=0$的方向向量, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581975,7 +588936,9 @@ "id": "040868", "content": "已知直线$l$过点$A(-2,5)$、$B(3,-1)$, 若将直线$l$绕点$A$旋转$90^{\\circ}$得到直线$m$, 则直线$m$的点法式方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -581995,7 +588958,9 @@ "id": "040869", "content": "直线$x \\cos a-y+1=0$的倾斜角的范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582015,7 +588980,9 @@ "id": "040870", "content": "在$\\triangle ABC$中, 已知顶点$A$、$B$、$C$的坐标分别为$(2,1)$、$(-3,-4)$、$(3,6)$, 则过点$A$且平分$\\triangle ABC$面积的直线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582035,7 +589002,9 @@ "id": "040871", "content": "若直线$l$经过点$(3,4)$, 且与直线$x+2 y-1=0$的夹角为$\\arctan \\dfrac{1}{2}$, 则直线$l$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582055,7 +589024,9 @@ "id": "040872", "content": "已知三角形$ABC$的各顶点分别为$A(1,3)$、$B(-2,-1)$、$C(4,-3)$, 则$AC$边上中线$BD$所在直线的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582075,7 +589046,9 @@ "id": "040873", "content": "若直线$l$过点$A(-1,1)$、且与点$B(2,5)$的距离最大的直线$l$的点法式方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582095,7 +589068,9 @@ "id": "040874", "content": "过点$(-1,1)$与点$(3,9)$的直线在$x$轴上的截距是\\bracket{20}.\n\\fourch{$-\\dfrac{3}{2}$}{$\\dfrac{3}{2}$}{$-3$}{$3$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -582115,7 +589090,9 @@ "id": "040875", "content": "经过点$A(1,0)$、$B(m, 2)$的直线方程是\\bracket{20}.\n\\twoch{$x-1=0$}{$\\dfrac{x-1}{m-1}=\\dfrac{y}{2}$}{$x-1=0$或$\\dfrac{x-1}{m-1}=\\dfrac{y}{2}$}{$x-1=0$或$(m-1) x+2 y=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -582135,7 +589112,9 @@ "id": "040876", "content": "下列结论中, 正确的是\\bracket{20}.\n\\onech{一条直线的倾斜角的正切叫做直线的斜率}{若一条直线经过$(x_1, y_1) 、(x_2, y_2)$两点, 则这条直线的斜率$k=\\dfrac{y_2-y_1}{x_2-x_1}$}{直线的倾斜角的变化范围是$[0, \\dfrac{\\pi}{2}) \\cup(\\dfrac{\\pi}{2}, \\pi)$}{若直线$l$平行于$x$轴, 则$l$的倾斜角为$0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -582155,7 +589134,9 @@ "id": "040877", "content": "已知三条直线$l_1: 2 x+1=0$, 直线$l_2: m x+y=0$, 直线$l_3: x+m y-1=0$, 若这三条直线中有两条直线平行, 则实数$m$所有可能的值的个数为\\bracket{20}.\n\\fourch{$2$}{$3$}{$4$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -582175,7 +589156,9 @@ "id": "040878", "content": "已知直线$l$过点$(1,2)$, 且$M(2,3)$、$N(0,5)$两点到直线$l$的距离相等, 求直线$l$的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582195,7 +589178,9 @@ "id": "040879", "content": "已知直线$l$过点$P(3,2)$, 且与直线$m: a(x-1)+b(y+2)=0$($a$、$b \\in \\mathbf{R}$)垂直.\\\\ \n(1) 求直线$l$的方程;\\\\\n(2) 若直线$m$过点$(2,1)$, 求直线$2 a x+b y=0$所经过的定点坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582215,7 +589200,9 @@ "id": "040880", "content": "已知直线$l_1: (m+2) x+(m^2-3 m) y+4=0$和直线$l_2: 2 m x+2(m-3) y+m+2=0$.\\\\\n(1) 当$m$为何值时, 直线$l_1$与直线$l_2$相交?\\\\\n(2) 当$m$为何值时, 直线$l_1$与直线$l_2$平行?\\\\\n(3) 当$m$为何值时, 直线$l_1$直线$l_2$重合?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582235,7 +589222,9 @@ "id": "040881", "content": "在直线$y=-2 x+4$上选一点$A$, 在抛物线$y=-x^2+1$上选一点$B$.\\\\\n(1) 求使$A$、$B$两点距离最短时点$A$、$B$的坐标;\\\\\n(2) 若点$P$为$x$轴上的一点, 且三角形$PAB$的面积为$4$($A,B$是前一小题中所得的点), 求点$P$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582256,7 +589245,9 @@ "id": "040882", "content": "若不论$a$为何实数, 直线$(a+3) x+(2 a-1) y+7=0$恒过一个定点, 则这个定点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582276,7 +589267,9 @@ "id": "040883", "content": "若直线$l_1$与直线$l_2: 3 x+4 y-7=0$垂直, 且原点到直线$l_1$的距离为 $5$, 则直线$l_1$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582296,7 +589289,9 @@ "id": "040884", "content": "直线$l$经过点$(2,4)$, 且在两坐标轴上的截距相等, 则直线$l$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582316,7 +589311,9 @@ "id": "040885", "content": "若$a, b \\in \\mathbf{R}$, 直线$a x+b y+c=0$($a>0$, $b>0$)的倾斜角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582336,7 +589333,9 @@ "id": "040886", "content": "若直线$l$经过点$(3,2)$, 且与直线$x-y+3=0$的夹角是$45^{\\circ}$, 则直线$l$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582356,7 +589355,9 @@ "id": "040887", "content": "已知$4A+5B+6=0$. 若直线$l: A x+B y+1=0$必经过定点, 则这个定点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582376,7 +589377,9 @@ "id": "040888", "content": "已知$C_1, C_2 \\in \\mathbf{R}$, $C_1 \\neq C_2$, 直线$l$与直线$l_1: A x+B y+C_1=0$和直线$l_2: A x+B y+C_2=0$都平行, 且到它们的距离相等, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582396,7 +589399,9 @@ "id": "040889", "content": "已知集合$A=\\{(x, y) | \\dfrac{y-3}{x-2}=a+1\\}$, 集合$E=\\{(x, y) |(a^2-1) x+(a-1) y-15=0\\}$, 是否存在实数$a$, 使$A \\cap B=\\varnothing$? 若存在, 求出$a$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582416,7 +589421,9 @@ "id": "040890", "content": "圆心为$C(-\\dfrac{3}{2}, 3)$, 半径为$\\sqrt{3}$的圆的标准方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582444,7 +589451,9 @@ "id": "040891", "content": "圆心为$C(\\sqrt{2}, 1)$, 过点$A(-1, \\sqrt{2})$的圆的标准方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582472,7 +589481,9 @@ "id": "040892", "content": "圆$(x+2)^2+(y-1)^2=1$的圆心到直线$3 x+4 y+4=0$的距离$d=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582500,7 +589511,9 @@ "id": "040893", "content": "以点$(-3,2)$为圆心、且与$x$轴相切的圆的标准方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582528,7 +589541,9 @@ "id": "040894", "content": "与$x$轴相交于$A(1,0)$、$B(5,0)$两点, 且半径等于$\\sqrt{5}$的圆的标准方程\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582556,7 +589571,9 @@ "id": "040895", "content": "已知一个圆的直径端点是$A(4,1), B(2,3)$, 则这个圆的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582584,7 +589601,9 @@ "id": "040896", "content": "若圆$C$与圆$(x+2)^2+(y-1)^2=1$关于原点对称, 则圆$C$的方程是\\bracket{20}.\n\\fourch{$(x-2)^2+(y+1)^2=1$}{$(x-2)^2+(y-1)^2=1$}{$(x-1)^2+(y+2)^2=1$}{$(x+1)^2+(y-2)^2=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -582612,7 +589631,9 @@ "id": "040897", "content": "由曲线$y=|x|$和圆$x^2+y^2=4$可围成两个面积不等的封闭图形, 其中较小的一个面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582640,7 +589661,9 @@ "id": "040898", "content": "已知圆$(x-a)^2+(y-b)^2=r^2(r>0)$. 求在下列情况下, 实数$a$、$b$、$r$分别应满足什么条件.\\\\\n(1) 圆过原点;\\\\\n(2) 圆心在$x$轴上;\\\\\n(3) 圆与$x$轴相切;\\\\\n(4) 圆与坐标轴相切.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582668,7 +589691,9 @@ "id": "040899", "content": "若圆$C$经过$A(2,-3)$、$B(-2,-5)$两点, 且圆心$C$在直线$x-2 y-3=0$上, 求圆$C$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582696,7 +589721,9 @@ "id": "040900", "content": "求过点$(2,-1)$, 圆心在直线$2 x+y=0$上, 且与直线$x-y-1=0$相切的圆的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582724,7 +589751,9 @@ "id": "040901", "content": "求圆心在直线$5 x-3 y=8$上, 且与两个坐标轴都相切的圆的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582752,7 +589781,9 @@ "id": "040902", "content": "画出下列方程的曲线, 并分别指出变量$x$和$y$的取值范围.\\\\\n(1) $y=\\sqrt{4-x^2}$;\\\\\n(2) $y=1-\\sqrt{9-x^2}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582780,7 +589811,9 @@ "id": "040903", "content": "判断下列方程是否是圆的方程? 如果是, 请写出该圆的标准方程、圆心坐标以及半径.\\\\\n(1) $9 x^2+16 y^2=25$;\\\\\n(2) $x^2+y^2+2 x y=4$;\\\\\n(3) $x^2+y^2-4 x=0$;\\\\\n(4) $x^2+y^2-x+3 y+2=0$;\\\\\n(5) $2 x^2+2 y^2+x-2 y+1=0$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582810,7 +589843,9 @@ "id": "040904", "content": "``$A=C \\neq 0$且$B=0$''是``$A x^2+B x y+C y^2+D x+E y+F=0$表示圆的方程''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582840,7 +589875,9 @@ "id": "040905", "content": "判断下列点与圆之间的位置关系, 并在空格中选填: ``点在圆内''、``点在圆上''或``点在圆外''.\\\\\n(1) 已知圆$x^2+y^2-4 x-6 y-4=0$及点$(4,-\\sqrt{3})$, 则点与圆之间的位置关系是\\blank{50};\\\\\n(2) 设常数$\\theta \\in \\mathbf{R}$, 已知圆$x^2+y^2=3$及点$(\\sqrt{2} \\cos \\theta, \\sqrt{2} \\sin \\theta)$, 则点与圆的位置关系是\\blank{50};\\\\\n(3) 设常数$\\theta \\in \\mathbf{R}$, 已知圆$x^2+y^2=x \\cos \\theta+y \\sin \\theta+1$及点$(0,0)$, 则点与圆的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582870,7 +589907,9 @@ "id": "040906", "content": "若方程$a^2 x^2+(a+2) y^2+2 a x+a=0$表示圆, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582900,7 +589939,9 @@ "id": "040907", "content": "若方程$x^2+y^2-2(a+3) x+2(1-4 a^2) y+16 a^4+9=0$表示圆, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582930,7 +589971,9 @@ "id": "040908", "content": "若点$(0,2)$在方程$x^2+y^2-2 x+6 y+a=0$所表示的圆外, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -582960,7 +590003,9 @@ "id": "040909", "content": "求圆$C: x^2+y^2+6 x-2 y+6=0$关于直线$y=x$的对称的圆的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -582990,7 +590035,9 @@ "id": "040910", "content": "已知圆过原点, 且与$x$轴、$y$轴的交点的坐标分别为$(a, 0)$、$(0, b)$($a$、$b \\neq 0$), 求这个圆的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583020,7 +590067,9 @@ "id": "040911", "content": "已知$\\triangle ABC$三个顶点的坐标分别为$A(-1,5), B(-2,-2), C(5,5)$, 求其外接圆的圆心坐标与半径.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583050,7 +590099,9 @@ "id": "040912", "content": "已知圆过原点和$A(-1,3)$, 且在$x$轴上截得的线段长为$2$, 求圆的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583080,7 +590131,9 @@ "id": "040913", "content": "求过两点$A(1,4)$、$B(3,2)$, 且圆心在直线$y=2 x$上的圆的方程. 并判断点$M_1(2,3), M_2(2,4)$与圆的位置关系.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583110,7 +590163,9 @@ "id": "040914", "content": "已知$A$、$B$两点相距$10$厘米, 动点$P$到点$A$的距离是它到点$B$的距离的$3$倍, 求点$P$的轨迹.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583140,7 +590195,9 @@ "id": "040915", "content": "已知集合$A=\\{(x, y) | x^2+y^2=1,\\ x, y \\in \\mathbf{R}\\}$, $B=\\{(x, y) | x+y=0,\\ x, y \\in \\mathbf{R}\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583167,7 +590224,9 @@ "id": "040916", "content": "若直线$l$经过点$A(1,-1)$且与圆$x^2+y^2+2 x+6 y+2=0$相切, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583194,7 +590253,9 @@ "id": "040917", "content": "直线$2 x-y+1=0$和圆$x^2+y^2-2 m x-4 m y+m^2-1=0$($m \\in \\mathbf{R}$)的位置关系是\\bracket{20}.\n\\fourch{相交但直线不过圆心}{相交且直线过圆心}{相切}{相离}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -583221,7 +590282,9 @@ "id": "040918", "content": "``$D^2=4F$''是``方程$x^2+y^2+D x+E y+F=0$表示与$x$轴相切的圆''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分又非必要}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -583248,7 +590311,9 @@ "id": "040919", "content": "若直线$4 x-3 y-2=0$与圆$x^2+y^2-2 a x+4 y+a^2-12=0$总有两个不同的交点, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583275,7 +590340,9 @@ "id": "040920", "content": "设常数$A, B \\in \\mathbf{R}$, 则直线$A x+B y=0$与圆$x^2+y^2+A x+B y=0$的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583302,7 +590369,9 @@ "id": "040921", "content": "若直线$m x+n y-3=0$($m, n \\in \\mathbf{R}$)与圆$x^2+y^2=3$没有公共点, 则$m^2+n^2$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583329,7 +590398,9 @@ "id": "040922", "content": "(1) 若直线$y=x+m$和曲线$x^2+y^2=2$有两个公共点, 则实数$m$的取值范围是\\blank{50};\\\\\n(2) 若直线$y=x+m$和曲线$y=\\sqrt{2-x^2}$有两个公共点, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583356,7 +590427,9 @@ "id": "040923", "content": "已知曲线$\\Gamma: y=\\sqrt{4-x^2}$.\\\\\n(1) 设直线$l: y=k(x-1)+2$, 当实数$k$为何值时, 直线$l$与曲线$\\Gamma$分别有两个公共点? 一个公共点?\\\\\n(2) 求$x+y$的取值范围;\\\\\n(3) 求$\\dfrac{y}{x-3}$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583383,7 +590456,9 @@ "id": "040924", "content": "已知直线$x \\cdot \\sin \\alpha+y \\cdot \\cos \\alpha+m=0$(常数$\\alpha \\in(0, \\dfrac{\\pi}{2})$)被圆$x^2+y^2=2$所截得的线段的长为$\\dfrac{4}{3} \\sqrt{3}$, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583410,7 +590485,9 @@ "id": "040925", "content": "已知直线$y=k x+3$与圆$(x-3)^2+(y-2)^2=4$相交于$M$、$N$两点, 若$|MN| \\geq 2 \\sqrt{3}$, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583437,7 +590514,9 @@ "id": "040926", "content": "已知定点$A(3,1)$, 动点$B$在圆$x^2+y^2=4$上, $P$在线段$AB$上, 且$BP: PA=1: 2$, 求点$P$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583464,7 +590543,9 @@ "id": "040927", "content": "已知圆$C$与$y$轴相切, 圆心$C$在直线$x-3 y=0$上, 且直线$y=x$被圆$C$所截得的线段的长为$2 \\sqrt{7}$, 求圆$C$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583491,7 +590572,9 @@ "id": "040928", "content": "已知圆$C: x^2+y^2+2 x-4 y-4=0$, 是否存在斜率为$1$的直线$l$, 使以$l$被圆$C$截得的弦$AB$为直径的圆经过坐标原点? 若存在, 写出直线$l$的方程; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583518,7 +590601,9 @@ "id": "040929", "content": "圆$x^2+y^2-4 y=0$与圆$x^2+y^2-16=0$的位置关系是\\bracket{20}.\n\\fourch{相交}{内切}{外切}{内含}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -583541,7 +590626,9 @@ "id": "040930", "content": "圆$C: x^2+y^2-2 x-6 y+9=0$关于直线$x-y-1=0$对称的曲线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583564,7 +590651,9 @@ "id": "040931", "content": "圆$x^2+y^2=8$内一点$P(-1,2)$, 过点$P$的直线$l$交该圆于$A$、$B$两点, 若弦$AB$恰好被点$P$平分, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583587,7 +590676,9 @@ "id": "040932", "content": "与圆$x^2+y^2=25$外切于点$P(4,-3)$且半径为$1$的圆的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583610,7 +590701,9 @@ "id": "040933", "content": "已知圆$x^2+y^2-2 x+2 y-3=0$和圆$x^2+y^2+4 x-1=0$关于直线$l$对称, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -583633,7 +590726,9 @@ "id": "040934", "content": "已知圆$x^2+y^2=4$, $P$为该圆上任意一点, 定点$A(3,0)$, 若点$Q$在线段$PA$的延长线上且$\\overrightarrow{PQ}=-2 \\overrightarrow{QA}$, 求动点$Q$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583656,7 +590751,9 @@ "id": "040935", "content": "已知两圆$x^2+y^2-10 x-10 y=0$和$x^2+y^2+6 x-2 y-40=0$.\\\\\n(1) 求这两个圆的公共弦所在直线的方程;\\\\\n(2) 求这两个圆的公共弦长.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583679,7 +590776,9 @@ "id": "040936", "content": "若动直线$k x-y+1=0$和圆$x^2+y^2=1$相交于$A$、$B$两点, 求弦$AB$的中点的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583702,7 +590801,9 @@ "id": "040937", "content": "已知圆$C: x^2+y^2+x-6 y+m=0$与直线$x+2 y-3=0$交于$P$、$Q$两点, 且满足$CP \\perp CQ$(注: 这里的点$C$为圆心), 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -583725,7 +590826,9 @@ "id": "040938", "content": "已知实数$x$、$y$满足方程$x^2+y^2-4 x+1=0$.\\\\\n(1) 求$\\dfrac{y}{x}$的最大值和最小值;\\\\\n(2) 求$x^2+y^2$的最大值和最小值;\\\\\n(3) 求$x-y$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "",