From 66f63d6e6255ca3d77602ed26b2a79ac9347abf2 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Tue, 6 Jun 2023 22:40:07 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A5=E9=AB=98=E4=B8=AD=E6=95=B0?= =?UTF-8?q?=E5=AD=A6=E8=B4=A8=E9=87=8F=E6=B5=8B=E8=AF=95=E7=BB=BC=E5=90=88?= =?UTF-8?q?=E6=B5=8B=E8=AF=958=E5=BC=A0=E8=AF=95=E5=8D=B7=E8=AF=95?= =?UTF-8?q?=E9=A2=98?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具/批量收录题目.py | 6 +- 题库0.3/Problems.json | 3480 +++++++++++++++++++++++++++++++++++++++++ 2 files changed, 3483 insertions(+), 3 deletions(-) diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index f7246108..85fb0476 100644 --- a/工具/批量收录题目.py +++ b/工具/批量收录题目.py @@ -1,7 +1,7 @@ #修改起始id,出处,文件名 -starting_id = 17551 -raworigin = "高中数学质量测试与监控单元知识测试" -filename = r"C:\Users\weiye\Documents\wwy sync\待整理word题目\质量测试与监控.tex" +starting_id = 17865 +raworigin = "高中数学质量测试综合测试" +filename = r"C:\Users\weiye\Documents\wwy sync\待整理word题目\质量测试与监控第二部分.tex" editor = "20230606\t王伟叶" indexed = True IndexDescription = "试题" diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index da08e8b5..13128000 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -459298,6 +459298,3486 @@ "space": "4em", "unrelated": [] }, + "017865": { + "id": "017865", + "content": "函数$y=\\lg (x-3)+\\dfrac{(x-2)^0}{x+1}$的定义域是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017866": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017867": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017868": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017869": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017870": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017871": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017875": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017876": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017877": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017878": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017879": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017880": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017881": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题18", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017883": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题19", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017884": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题20", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017885": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题21", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017886": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题22", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017887": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试01集合函数不等式试题23", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017888": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017889": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017890": { + "id": "017890", + "content": "若$\\tan \\dfrac{\\alpha}{2}=2$, 则$\\sin (\\alpha+\\dfrac{\\pi}{4})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017891": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017892": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017893": { + "id": "017893", + "content": "函数$y=\\dfrac{1}{2} \\sin (2 x-\\dfrac{\\pi}{3})$的减区间是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017894": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017895": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017896": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017897": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017898": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017899": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017900": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017901": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017902": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017903": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017904": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017905": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题18", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017906": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题19", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017907": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题20", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017908": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题21", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017909": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试02三角与平面向量试题22", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017910": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017911": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017912": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017913": { + "id": "017913", + "content": "若等差数列$\\{a_n\\}$中, $S_{10}-S_5=40$, 则$a_8=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017914": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017915": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017916": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017917": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017918": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017919": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017920": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017921": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017922": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017923": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017924": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017925": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017926": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017927": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题18", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017928": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题19", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017929": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题20", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017930": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题21", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017931": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试03数列试题22", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017932": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017933": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017934": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017935": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017936": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017937": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017938": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017939": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017940": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017941": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017942": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017943": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017944": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017945": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017946": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017947": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017948": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017949": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题18", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017950": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题19", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017951": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题20", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017952": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试04解析几何直线和圆试题21", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017953": { + "id": "017953", + "content": "准线方程为$y=1$的抛物线标准方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017954": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017955": { + "id": "017955", + "content": "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017956": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017957": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017958": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017959": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017960": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017961": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017962": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017963": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017964": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017965": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017966": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017967": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017968": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017969": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": 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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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试05解析几何圆锥曲线试题19", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017972": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017973": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017974": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017975": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017976": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017977": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017978": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017979": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017980": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017981": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017982": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017983": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017984": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017985": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017986": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017987": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017988": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017989": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题18", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017990": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题19", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017991": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题20", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017992": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题21", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017993": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题22", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017994": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题23", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017995": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题24", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017996": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题25", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017997": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题26", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017998": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题27", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017999": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题28", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018000": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题29", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018001": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试06空间直线与平面多面体与旋转体及空间向量试题30", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018002": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018003": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018004": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018005": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018006": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018007": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018008": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018009": { + "id": "018009", + "content": "方程: $\\mathrm{C}_{13}^{x+1}=\\mathrm{C}_{13}^{2 x-3}$的解为$x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018010": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018011": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018012": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018013": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018014": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018015": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018016": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018017": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018018": { + "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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试07排列组合二项式定理概率与复数试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018019": { + "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$S_1 \\mathrm{C}_n^0+S_2 \\mathrm{C}_n^1+S_3 \\mathrm{C}_n^2+S_4 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+ "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试08导数及其应用试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018025": { + "id": "018025", + "content": "函数$y=\\dfrac{x^2}{x+3}$的导数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试08导数及其应用试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018026": { + "id": "018026", + "content": "若函数$f(x)=x(x-a)^2$在$x=2$处取得极小值, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试08导数及其应用试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018027": { + "id": "018027", + "content": "已知函数$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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试08导数及其应用试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018028": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试08导数及其应用试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018029": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试08导数及其应用试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018030": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试08导数及其应用试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018031": { + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试08导数及其应用试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018032": { + "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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试综合测试08导数及其应用试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018033": { + "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{$0