diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 0928665c..987eea87 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -643095,7 +643095,9 @@ "id": "023601", "content": "作出函数 $y=\\sin |x|$, $x \\in[-2 \\pi, 2 \\pi]$ 的图像.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643140,7 +643142,9 @@ "id": "023603", "content": "函数 $y=3 \\sin (\\dfrac{x}{3}+\\dfrac{\\pi}{4})$ 的严格减区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -643211,7 +643215,9 @@ "id": "023606", "content": "函数 $y=\\cos \\dfrac{x}{3}$ 的最小正周期是\\blank{50}; 函数 $y=2 \\cos (-2 x+\\dfrac{\\pi}{6})$ 的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -643231,7 +643237,9 @@ "id": "023607", "content": "求函数 $y=\\cos (2 x+\\dfrac{\\pi}{3})$, $x \\in[-\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3}]$ 的单调区间和值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643251,7 +643259,9 @@ "id": "023608", "content": "样本数据 $16,24,14,10,20,30,12,14,40$ 的中位数为\\bracket{20}.\n\\fourch{$14$}{$16$}{$18$}{$20$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -643271,7 +643281,9 @@ "id": "023609", "content": "椭圆 $\\dfrac{x^2}{a^2}+y^2=1$($a>1$) 的离心率为 $\\dfrac{1}{2}$, 则 $a=$\\bracket{20}.\n\\fourch{$\\dfrac{2 \\sqrt{3}}{3}$}{$\\sqrt{2}$}{$\\sqrt{3}$}{2}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -643291,7 +643303,9 @@ "id": "023610", "content": "记等差数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, $a_3+a_7=6$, $a_{12}=17$, 则 $S_{16}=$\\bracket{20}.\n\\fourch{$120$}{$140$}{$160$}{$180$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -643311,7 +643325,9 @@ "id": "023611", "content": "设 $\\alpha, \\beta$ 是两个平面, $m, l$ 是两条直线, 则下列命题为真命题的是\\bracket{20}.\n\\twoch{若 $\\alpha \\perp \\beta$, $m \\parallel \\alpha$, $l \\parallel \\beta$, 则 $m \\perp l$}{若 $m \\subset \\alpha, l \\subset \\beta, m \\parallel l$, 则 $\\alpha \\parallel \\beta$}{若 $\\alpha \\cap \\beta=m$, $l \\parallel \\alpha$, $l \\parallel \\beta$, 则 $m \\parallel l$}{若 $m \\perp \\alpha$, $l \\perp \\beta$, $m \\parallel l$, 则 $\\alpha \\perp \\beta$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -643331,7 +643347,9 @@ "id": "023612", "content": "甲、乙、丙等 $5$ 人站成一排, 且甲不在两端, 乙和丙之间恰有 $2$ 人, 则不同排法共有\\bracket{20}.\n\\fourch{$20$ 种}{$16$ 种}{$12$ 种}{$8$ 种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -643351,7 +643369,9 @@ "id": "023613", "content": "已知 $Q$ 为直线 $l: x+2 y+1=0$ 上的动点, 点 $P$ 满足 $\\overrightarrow{QP}=(1,-3)$, 记 $P$ 的轨迹为 $E$, 则\\bracket{20}.\n\\twoch{$E$ 是一个半径为 $\\sqrt{5}$ 的圆}{$E$ 是一条与 $l$ 相交的直线}{$E$ 上的点到 $l$ 的距离均为 $\\sqrt{5}$}{$E$ 是两条平行直线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -643371,7 +643391,9 @@ "id": "023614", "content": "已知 $\\theta \\in(\\dfrac{3 \\pi}{4}, \\pi)$, $\\tan 2 \\theta=-4 \\tan (\\theta+\\dfrac{\\pi}{4})$, 则 $\\dfrac{1+\\sin 2 \\theta}{2 \\cos ^2 \\theta+\\sin 2 \\theta}=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{4}$}{$\\dfrac{3}{4}$}{1}{$\\dfrac{3}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -643391,7 +643413,9 @@ "id": "023615", "content": "设双曲线 $C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的左、右焦点分别为 $F_1, F_2$, 过坐标原点的直线与 $C$ 交于 $A, B$ 两点, $|F_1B|=2|F_1A|$, $\\overrightarrow{F_2A}\\cdot \\overrightarrow{F_2B}=4 a^2$, 则 $C$ 的离心率为\\bracket{20}.\n\\fourch{$\\sqrt{2}$}{$2$}{$\\sqrt{5}$}{$\\sqrt{7}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -643411,7 +643435,9 @@ "id": "023616", "content": "已知函数 $f(x)=\\sin (2 x+\\dfrac{3 \\pi}{4})+\\cos (2 x+\\dfrac{3 \\pi}{4})$, 则\\blank{50}.\\\\\n\\textcircled{1} 函数 $f(x-\\dfrac{\\pi}{4})$ 为偶函数; \\textcircled{2} 曲线 $y=f(x)$ 的对称轴为 $x=k \\pi$, $k \\in \\mathbf{Z}$; \\textcircled{3} $f(x)$ 在区间 $(\\dfrac{\\pi}{3}, \\dfrac{\\pi}{2})$ 单调递增; \\textcircled{4} $f(x)$ 的最小值为 $-2$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -643431,7 +643457,9 @@ "id": "023617", "content": "已知复数 $z, w$ 均不为 $0$ , 则\\blank{50}.\\\\\n\\textcircled{1} $z^2=|z|^2$; \\textcircled{2} $\\dfrac{z}{\\overline{z}}=\\dfrac{z^2}{|z|^2}$; \\textcircled{3} $\\overline{z-w}=\\overline{z}-\\overline{w}$; \\textcircled{4} $|\\dfrac{z}{w}|=\\dfrac{|z|}{|w|}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -643451,7 +643479,9 @@ "id": "023618", "content": "已知函数 $f(x)$ 的定义域为 $\\mathbf{R}$, 且 $f(\\dfrac{1}{2}) \\neq 0$, 若 $f(x+y)+f(x) f(y)=4 x y$, 则\\blank{50}.\\\\\n\\textcircled{1} $f(-\\dfrac{1}{2})=0$; \\textcircled{2} $f(\\dfrac{1}{2})=-2$; \\textcircled{3} 函数 $f(x-\\dfrac{1}{2})$ 是偶函数; \\textcircled{4} 函数 $f(x+\\dfrac{1}{2})$ 是减函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -643471,7 +643501,9 @@ "id": "023619", "content": "已知集合 $A=\\{-2,0,2,4\\}$, $B=\\{x|| x-3 | \\leq m\\}$, 若 $A \\cap B=A$, 则 $m$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -643491,7 +643523,9 @@ "id": "023620", "content": "已知轴截面为正三角形的圆锥 $MM'$ 的高与球 $O$ 的直径相等, 则圆锥 $MM'$ 的体积与球 $O$ 的体积的比值是\\blank{50}, 圆锥 $MM'$ 的表面积与球 $O$ 的表面积的比值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -643511,7 +643545,9 @@ "id": "023621", "content": "以 $\\max M$ 表示数集 $M$ 中最大的数. 设 $0=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (-1,{sqrt(2)},1) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (-1,{sqrt(2)},1) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (-1,{sqrt(2)},1) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (-1,{sqrt(2)},1) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (A)--(C)(B)--(D);\n\\draw ($(A)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw [dashed] (C1)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $C_1O \\perp$ 平面 $ABCD$;\\\\\n(2) 求二面角 $B-AA_1-D$ 的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643591,7 +643633,9 @@ "id": "023625", "content": "已知抛物线 $C: y^2=4 x$ 的焦点为 $F$, 过 $F$ 的直线 $l$ 交 $C$ 于 $A, B$ 两点, 过 $F$ 与 $l$ 垂直的直线交 $C$ 于 $D, E$ 两点, 其中 $B, D$ 在 $x$ 轴上方, $M, N$ 分别为 $AB, DE$ 的中点.\\\\\n(1) 证明: 直线 $MN$ 过定点;\\\\\n(2) 设 $G$ 为直线 $AE$ 与直线 $BD$ 的交点, 求 $\\triangle GMN$ 面积的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643611,7 +643655,9 @@ "id": "023626", "content": "离散对数在密码学中有重要的应用. 设 $p$ 是素数, 集合 $X=\\{1,2, \\cdots, p-1\\}$, 若 $u, v \\in X$, $m \\in \\mathbf{N}$, 记 $u \\otimes v$ 为 $u v$ 除以 $p$ 的余数, $u^{m, \\otimes}$ 为 $u^m$ 除以 $p$ 的余数; 设 $a \\in X$ , $1, a, a^{2, \\otimes}, \\cdots, a^{p-2, \\otimes}$ 两两不同, 若 $a^{n, \\otimes}=b$($n \\in\\{0,1, \\cdots, p-2\\}$), 则称 $n$ 是以 $a$ 为底 $b$ 的离散对数, 记为 $n=\\log (p)_a b$.\\\\\n(1) 若 $p=11$, $a=2$, 求 $a^{p-1, \\otimes}$;\\\\\n(2) 对 $m_1, m_2 \\in\\{0,1, \\cdots, p-2\\}$, 记 $m_1 \\oplus m_2$ 为 $m_1+m_2$ 除以 $p-1$ 的余数 (当 $m_1+m_2$ 能被 $p-1$ 整除时, $m_1 \\oplus m_2=0$). 证明: $\\log (p)_a(b \\otimes c)=\\log (p)_a b \\oplus \\log (p)_a c$, 其中 $b, c \\in X$;\\\\\n(3) 已知 $n=\\log (p)_a b$. 对 $x \\in X$, $k \\in\\{1,2, \\cdots, p-2\\}$, 令 $y_1=a^{k, \\otimes}$, $y_2=x \\otimes b^{k, \\otimes}$. 证明: $x=y_2 \\otimes y_1^{n(p-2), \\otimes}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643631,7 +643677,9 @@ "id": "023627", "content": "已知 $\\overrightarrow{a}$、$\\overrightarrow{b}$ 和 $\\overrightarrow{c}$ 是平面上任意给定的向量, 求证: $(\\overrightarrow{a}+\\overrightarrow{b})+\\overrightarrow{c}=\\overrightarrow{a}+(\\overrightarrow{b}+\\overrightarrow{c})$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643651,7 +643699,9 @@ "id": "023628", "content": "已知 $\\overrightarrow{a}$、$\\overrightarrow{b}$ 为非零向量. 若 $\\overrightarrow{a}\\cdot \\overrightarrow{b}=-\\dfrac{\\sqrt{3}}{3}$, 向量 $\\overrightarrow{b}$ 在向量 $\\overrightarrow{a}$ 上的投影向量是 $\\overrightarrow{c}$, 向量 $\\overrightarrow{a}$ 在向量 $\\overrightarrow{b}$ 上的投影向量是 $\\overrightarrow{d}$, 则下列结论正确的是 (填上所有你认为正确的选项).\n\\fourch{$|\\overrightarrow{c}|=|\\overrightarrow{d}|$}{$\\overrightarrow{a}\\cdot \\overrightarrow{b}=\\overrightarrow{a}\\cdot \\overrightarrow{c}$}{$\\overrightarrow{d}=\\dfrac{\\sqrt{3}}{3}\\overrightarrow{b}$}{$\\overrightarrow{c}\\cdot \\overrightarrow{d}=-\\dfrac{\\sqrt{3}}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643671,7 +643721,9 @@ "id": "023629", "content": "已知向量 $\\overrightarrow{a}$ 在向量 $\\overrightarrow{b}$ 上的投影向量的模是 $1$, 向量 $\\overrightarrow{b}$ 在向量 $\\overrightarrow{a}$ 上的投影向量的模是 $\\dfrac{1}{2}$, 且 $|\\overrightarrow{b}|=1$, 则 $|\\overrightarrow{a}-\\overrightarrow{b}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -643691,7 +643743,9 @@ "id": "023630", "content": "已知在平行四边形 $ABCD$ 中, $\\overrightarrow{DE}=\\dfrac{1}{2}\\overrightarrow{EC}$, $\\overrightarrow{BF}=\\dfrac{1}{2}\\overrightarrow{FC}$, $|\\overrightarrow{AE}|=2$, $|\\overrightarrow{AF}|=\\sqrt{6}$, 求 $\\overrightarrow{AC}\\cdot \\overrightarrow{DB}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643711,7 +643765,9 @@ "id": "023631", "content": "已知 $|\\overrightarrow{a}|=4$, $\\overrightarrow{e}$ 为单位向量, $\\langle\\overrightarrow{a}, \\overrightarrow{e}\\rangle=\\dfrac{2 \\pi}{3}$, 则向量 $\\overrightarrow{a}$ 在向量 $\\overrightarrow{e}$ 上的投影向量是\\blank{50}, 向量 $\\overrightarrow{e}$ 在向量 $\\overrightarrow{a}$ 上的投影向量是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -643731,7 +643787,9 @@ "id": "023632", "content": "证明: 设 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$、$\\overrightarrow{c}$ 是向量, $\\lambda$ 是实数, 则\\\\\n(1) 向量数量积的交换律: $\\overrightarrow{a}\\cdot \\overrightarrow{b}=\\overrightarrow{b}\\cdot \\overrightarrow{a}$;\\\\\n(2) 向量数量积对数乘的结合律: $(\\lambda \\overrightarrow{a}) \\cdot \\overrightarrow{b}=\\overrightarrow{a}\\cdot(\\lambda \\overrightarrow{b})$;\\\\\n(3) 向量数量积对加法的分配律: $\\overrightarrow{a}\\cdot(\\overrightarrow{b}+\\overrightarrow{c})=\\overrightarrow{a}\\cdot \\overrightarrow{b}+\\overrightarrow{a}\\cdot \\overrightarrow{c}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643751,7 +643809,9 @@ "id": "023633", "content": "如图, 给定边长为 $6$ 的正三角形 $ABC$. 求 $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}$ 和 $\\overrightarrow{AB}\\cdot \\overrightarrow{BC}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$B$} coordinate (B);\n\\draw (2,0) node [below right] {$C$} coordinate (C);\n\\draw (60:2) node [above] {$A$} coordinate (A);\n\\draw [->] (A)--(B);\n\\draw [->] (A)--(C);\n\\draw [->] (B)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643771,7 +643831,9 @@ "id": "023634", "content": "已知向量 $\\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ 满足 $\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, $(\\overrightarrow{a}-\\overrightarrow{b}) \\cdot(\\overrightarrow{a}-\\overrightarrow{c})=0$, $|\\overrightarrow{b}-\\overrightarrow{c}|=9$, 求 $|\\overrightarrow{a}|$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643791,7 +643853,9 @@ "id": "023635", "content": "如果 $\\overrightarrow{e_1}$ 与 $\\overrightarrow{e_2}$ 是平面上两个不平行的向量, 那么该平面上的任意向量 $\\overrightarrow{a}$, 都可唯一地表示为 $\\overrightarrow{e_1}$ 与 $\\overrightarrow{e_2}$ 的线性组合, 即存在唯一的一对实数 $\\lambda$ 与 $\\mu$, 使得\n$\\overrightarrow{a}=\\lambda \\overrightarrow{e_1}+\\mu \\overrightarrow{e_2}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643811,7 +643875,9 @@ "id": "023636", "content": "给定平面上不共线的三点 $O$、$A$、$B$, 根据平面向量基本定理, 对平面上任定一点 $P$, 都有唯一的一对实数 $\\lambda, \\mu$, 使得 $\\overrightarrow{OP}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$. 求证: $A$、$B$、$P$ 三点共线的一个充要条件是 $\\lambda+\\mu=1$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643831,7 +643897,9 @@ "id": "023637", "content": "矩形 $ABCD$ 中, $AB=1$, $AD=2$, $CE \\perp BD$ 于 $E$, 点 $P$ 在以 $C$ 为圆心, $CE$为半径的圆周上运动, 若 $\\overrightarrow{AP}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AD}$ ($\\lambda$、$\\mu$ 为实数), 求 $\\lambda+\\mu$ 的范围.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,1) node [above] {$C$} coordinate (C);\n\\draw (0,1) node [above] {$D$} coordinate (D);\n\\draw ($(B)!0.2!(D)$) node [left] {$E$} coordinate (E);\n\\draw (C) circle ({2/sqrt(5)});\n\\draw (C) ++ (30:{2/sqrt(5)}) node [above right] {$P$} coordinate (P);\n\\draw (A) rectangle (C) (B)--(D)(C)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643851,7 +643919,9 @@ "id": "023638", "content": "自由落体运动, 物体下落的距离 $S$ (单位: $\\mathrm{m}$) 与时间 $t$ (单位: $\\mathrm{s}$) 满足函数关系 $S(t)=5 t^2$. 试求物体在 $t=2$ 时的瞬时速度.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643875,7 +643945,9 @@ "id": "023639", "content": "如图, 已知曲线 $y=\\sqrt{2-x^2}$($-\\sqrt{2}\\leq x \\leq \\sqrt{2}$) 上两点 $P(1,1)$、$Q(0, \\sqrt{2})$.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw ({sqrt(2)},0) arc (0:180:{sqrt(2)});\n\\draw (45:{sqrt(2)}) node [below] {$P$} coordinate (P);\n\\draw (0,{sqrt(2)}) node [below left] {$Q$} coordinate (Q);\n\\draw ($(P)!-0.5!(Q)$) -- ($(Q)!-0.5!(P)$);\n\\filldraw (P) circle (0.03) (Q) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 求割线 $PQ$ 的斜率;\\\\\n(2) 对正整数 $n$ , 令 $x_n=1-\\dfrac{1}{n}$, $y_n=\\sqrt{2-x_n^2}$, 在该曲线上取一系列点 $Q_n(x_n, y_n)$ , 借助现代信息技术, 适当地计算一些割线 $PQ_n$ 的斜率, 观察并总结当 $n$ 逐渐增大时, 割线 $PQ_n$ 的斜率的变化趋势.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643895,7 +643967,9 @@ "id": "023640", "content": "求常数函数 $y=C$ 的导数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643915,7 +643989,9 @@ "id": "023641", "content": "是否存在实数$b$, 使得直线 $y=-x+b$ 是下列函数图像的切线? 如果存在, 请求出 $b$ 的值; 如果不存在, 请说明理由.\\\\\n(1) $f(x)=\\ln x$;\\\\\n(2) $f(x)=\\dfrac{1}{x}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643938,7 +644014,9 @@ "id": "023642", "content": "设 $f(x)=\\ln x$, 已知 $f(x)$ 的图像上有且只有三个点到直线 $y=x+a$ 的距离为 $\\sqrt{2}$, 求实数 $a$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643958,7 +644036,9 @@ "id": "023643", "content": "已知曲线 $y=\\mathrm{e}^x$ 在点 $(x_1, \\mathrm{e}^{x_1})$ 处的切线与曲线 $y=\\ln x$ 在点 $(x_2, \\ln x_2)$ 处的切线相同, 求 $(x_1+1)(x_2-1)$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643978,7 +644058,9 @@ "id": "023644", "content": "证明: 对函数 $y=f(x)$ 与任何常数 $C\\in \\mathbf{R}$, 都有 $(C f(x))'=C f'(x)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -643998,7 +644080,9 @@ "id": "023645", "content": "利用导数求函数 $f(x)=-3 x^2+6 x-1$ 在 $[0,3]$ 上的最大值与最小值. 从而一般化, 此处所得的结果与之前的认识是否一致? 哪种方法更简便?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -644018,7 +644102,9 @@ "id": "023646", "content": "某种型号的汽车在匀速行驶中每小时的耗油量 $y$ (单位: $\\mathrm{L}$) 关于行驶速度 $x$ (单位: $\\mathrm{km}$/$\\mathrm{h}$) 满足函数关系 $y=\\dfrac{1}{128000}x^3-\\dfrac{3}{80}x+8$($0=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,1) node [left] {$D$} coordinate (D);\n\\draw (2,1) node [above right] {$C$} coordinate (C);\n\\draw (D) ++ (0,{2*tan(35)}) node [above] {$Q$} coordinate (Q);\n\\draw (B) ++ ({1/tan(35)},0) node [right] {$P$} coordinate (P);\n\\fill [pattern = north east lines] (A)--(B)--(C)--(D)--cycle;\n\\draw (A) rectangle (C) (D)--(Q)--(P)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644062,7 +644150,9 @@ "id": "023648", "content": "在微积分中``以直代曲''是最基本、最朴素的思想方法, 中国古代科学家刘徽创立的``割圆术'', 用圆的外切正 $n$ 边形和内接正 $n$ 边形``内外夹逼''的办法求出了圆周率 $\\pi$ 的精度较高的近似值, 事实上就是用``以直代曲''的思想进行近似计算的, 它是我国最优秀的传统科学文化之一. 借用``以直代曲''的方法, 在切点附近、可以用函数图像的切线代替在切点附近的曲线来``近似计算''. 请用函数 $f(x)=\\mathrm{e}^x$``近似计算''$\\sqrt[2024]{\\mathrm{e}}$ 的值为\\blank{50}(结果用分数表示).", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644082,7 +644172,9 @@ "id": "023649", "content": "某厂生产产品 $x$ 件的总成本为 $C(x)=1200+\\dfrac{2}{75}x^3$ (单位: 万元) . 已知产品单价 $P$ (单位: 万元) 和产品件数 $x$ 满足函数关系 $P^2=\\dfrac{k}{x}$, 且生产 $100$ 件这样的产品时, 单价定为 $50$ 万元. 问产量为多少件时, 总利润最大?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -644154,7 +644246,9 @@ "id": "023652", "content": "下列求导运算中正确的是\\bracket{20}.\n\\fourch{$(\\ln 2)'=0$}{$(\\cos x)'=\\sin x$}{$(\\mathrm{e}^{-x})'=\\mathrm{e}^{-x}$}{$(\\dfrac{1}{x^5})'=-\\dfrac{1}{5}x^{-6}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -644174,7 +644268,9 @@ "id": "023653", "content": "若函数 $f(x)=\\cos x+\\sin x$, 则 $f'(\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644194,7 +644290,9 @@ "id": "023654", "content": "已知函数 $f(x)=a x^3+3 x^2+2$, 若 $f'(-1)=4$, 则 $a$ 的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644214,7 +644312,9 @@ "id": "023655", "content": "若函数 $f(x)=\\mathrm{e}^x \\ln x$, $f'(x)$ 为 $f(x)$ 的导函数, 则 $f'(1)$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644234,7 +644334,9 @@ "id": "023656", "content": "若函数 $f(x)=x^4-2 x^3$, 则曲线 $y=f(x)$ 在点 $(1, f(1))$ 处的切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644254,7 +644356,9 @@ "id": "023657", "content": "若函数 $f(x)=x^3-f'(1) x^2+2$, 则 $f(2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644274,7 +644378,9 @@ "id": "023658", "content": "已知函数 $f(x)=\\mathrm{e}^{-x}(\\sin x+\\cos x)$, 记 $f'(x)$ 是 $f(x)$ 的导函数, 若满足 $f'(x)=0$ 的所有正数 $x$ 从小到大排成数列 $\\{x_n\\}, n \\in \\mathbf{N}$, $n \\geq 1$, 则数列 $\\{f(x_n)\\}$ 的通项公式是\\bracket{20}.\n\\fourch{$(-1)^n \\mathrm{e}^{-(n+1) \\pi}$}{$(-1)^{n+1}\\mathrm{e}^{-n \\pi}$}{$(-1)^n \\mathrm{e}^{-n \\pi}$}{$(-1)^{n+1}\\mathrm{e}^{-(n+1) \\pi}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -644294,7 +644400,9 @@ "id": "023659", "content": "已知 $f(x)=x \\ln x+x^2-x+2$, 曲线 $y=f(x)$ 在点 $(x_0, f(x_0))$($x_0>0$) 处的切线恰好经过坐标原点, 求 $x_0$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -644314,7 +644422,9 @@ "id": "023660", "content": "若函数 $f(x)=x^3+a x^2-a x$ 在 $\\mathbf{R}$ 上是严格增函数, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644334,7 +644444,9 @@ "id": "023661", "content": "若函数 $f(x)=-\\dfrac{1}{3}x^3+a x$ 有三个单调区间, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644354,7 +644466,9 @@ "id": "023662", "content": "若函数 $f(x)=\\dfrac{1}{2}x^2-16 \\ln(2x)$ 在区间 $[a-\\dfrac{1}{2}, a+\\dfrac{1}{2}]$ 上严格减, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644374,7 +644488,9 @@ "id": "023663", "content": "若函数 $f(x)=\\dfrac{x-1}{x}-\\ln x$, 则 $f(x)$ 的单调增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644394,7 +644510,9 @@ "id": "023664", "content": "已知函数 $f(x)=\\mathrm{e}^{2 x}-a(x+2)$. 当 $a=2$ 时, $f(x)$ 的单调增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644414,7 +644532,9 @@ "id": "023665", "content": "已知函数 $f(x)=k x^3+3(k-1) x^2-k^2+1$($k>0$).\\\\\n(1) 若 $f(x)$ 的单调减区间是 $(0,4)$, 求实数 $k$ 的值;\\\\\n(2) 若 $f(x)$ 在 $(0,4)$ 上为严格减函数, 求实数 $k$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -644434,7 +644554,9 @@ "id": "023666", "content": "若 $a>0$, 函数 $f(x)=a^2 x-\\dfrac{1}{3}x^3$ 的极小值为 $-\\dfrac{4}{3}$, 则实数 $a$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644454,7 +644576,9 @@ "id": "023667", "content": "函数 $f(x)=6+12 x-x^3$ 在 $[-1,3]$ 上的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644474,7 +644598,9 @@ "id": "023668", "content": "若函数 $f(x)=x^3+a x^2+3 x-9$ 在 $x=-3$ 时取得极值, 则实数 $a$ 等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644497,7 +644623,9 @@ "id": "023669", "content": "函数 $f(x)=(1-x) \\mathrm{e}^x$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644517,7 +644645,9 @@ "id": "023670", "content": "若函数 $f(x)=\\mathrm{e}^x(-x^2+2 x+a)$ 在区间 $(a, a+1)$ 上存在最大值, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644537,7 +644667,9 @@ "id": "023671", "content": "若函数 $f(x)=\\dfrac{1}{3}x^3+x^2-\\dfrac{2}{3}$ 在区间 $(a, a+3)$ 内既存在最大值也存在最小值, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644557,7 +644689,9 @@ "id": "023672", "content": "若函数 $f(x)$ 的导函数 $f'(x)$ 的图像如图所示, 则下列结论中正确的序号是\\blank{50}.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-4,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-3,-2,1,3,5,6}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\draw (-1,0.1) -- (-1,0) node [above] {$-1$};\n\\draw (-3,-3) sin (-2,1) cos (-1,0) sin (0,-1) cos (1,0) sin (3,2) cos (4.5,0) sin (5,-0.5) cos (6,1.5);\n\\draw [dashed] (-2,0) -- (-2,1) (3,0) -- (3,2) (6,0) -- (6,1.5);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 当 $x=1$ 时, 函数 $f(x)$ 取得极小值;\\\\\n\\textcircled{2} 函数 $f(x)$ 在区间 $(-1,1)$ 上是严格增函数;\\\\\n\\textcircled{3} 当 $x=3$ 时, 函数 $f(x)$ 取得极大值;\\\\\n\\textcircled{4} 函数 $f(x)$ 在区间 $(5,6)$ 上是严格增函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644581,7 +644715,9 @@ "id": "023673", "content": "已知函数 $f(x)=-x^3+a x^2+b x+c$($a$、$b$、$c \\in \\mathbf{R}$), 且 $f'(-1)=f'(3)=0$.\\\\\n(1) 求 $a-b$ 的值;\\\\\n(2) 若函数 $f(x)$ 在 $[-2,2]$ 上的最大值为 $20$, 求函数 $f(x)$ 在 $[-1,4]$ 上的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -644601,7 +644737,9 @@ "id": "023674", "content": "已知某生产厂家的年利润 $y$ (单位: 万元) 与年产量 $x$ (单位: 万件) 的函数关系式为 $y=-\\dfrac{1}{3}x^3+81 x-234$, 则使该生产厂家获取最大年利润的年产量为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644621,7 +644759,9 @@ "id": "023675", "content": "统计表明: 某种型号的汽车在匀速行驶中每小时的耗油量 $y(L)$ 关于行驶速度 $x(\\mathrm{km}/ \\mathrm{h})$的函数解析式可以表示为 $y=\\dfrac{1}{128000}x^3-\\dfrac{3}{80}x+8$, $x \\in(0,120]$, 且甲、乙两地相距 $100 \\mathrm{km}$, 则当汽车以\\blank{50}$\\mathrm{km}/ \\mathrm{h}$ 的速度匀速行驶时, 从甲地到乙地的耗油量最少.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644646,7 +644786,9 @@ "id": "023676", "content": "某市一特色酒店由一些完全相同的帐篷构成. 每顶帐篷的体积为 $54 \\pi \\mathrm{m}^3$, 且分上、下两层,其中上层是半径为 $r(r \\geq 1) m$ 的半球体, 下层是底面半径为 $r m$, 高为 $h m$ 的圆柱体 (如图). 经测算, 上层半球体部分每平方米的建造费用为 $2$ 千元, 下层圆柱体的侧面、隔层和地面三个部分每平方米的建造费用均为 $3$ 千元, 设每顶帐篷的建造费用为 $y$ 千元.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0) arc (180:360:1 and 0.25) --++ (0,1) arc (0:180:1) --cycle;\n\\draw (-1,1) arc (180:360:1 and 0.25);\n\\draw [dashed] (-1,1) arc (180:0:1 and 0.25) (-1,0) arc (180:0:1 and 0.25);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $y$ 关于 $r$ 的函数解析式, 并指出该函数的定义域;\\\\\n(2) 当半径 $r$ 为何值时, 每顶帐篷的建造费用最小?并求出最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -644666,7 +644808,9 @@ "id": "023677", "content": "若直线 $l$ 过点 $A(2,-1)$、$B(1,2)$, 则直线 $l$ 的两点式方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644688,7 +644832,9 @@ "id": "023678", "content": "若直线 $3 x+2 y+5=0$ 的一个法向量为 $(a, a-2)$, 则 $a$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644710,7 +644856,9 @@ "id": "023679", "content": "有一直线与 $y$ 轴交于 $(0,-2)$, 若其倾斜角的正弦满足方程 $6 x^2+x-1=0$, 则此直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644730,7 +644878,9 @@ "id": "023680", "content": "(1) 经过点 $M(2,-1)$, 倾斜角为直线 $4 x+3 y-1=0$ 的倾斜角的一半的直线方程是\\blank{50}.\\\\\n(2) 经过点 $P(3,-4)$, 且在坐标轴上的横截距和纵截距相等的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644750,7 +644900,9 @@ "id": "023681", "content": "(1) 若过点 $M(-2,3)$ 引一直线, 使它夹在两坐标轴间的线段被 $M$ 平分, 则该直线方程为\\blank{50}.\\\\\n(2) 若将直线 $x+2 y-4=0$ 绕着它与 $x$ 轴的交点按逆时针方向旋转 $\\dfrac{\\pi}{4}$, 旋转后所得直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644770,7 +644922,9 @@ "id": "023682", "content": "已知三角形的一个顶点 $A(-4,2)$, 两条中线所在直线方程分别为 $3 x-2 y+2=0$ 和 $3 x+5 y-12=0$, 求三角形各边所在直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -644790,7 +644944,9 @@ "id": "023683", "content": "若直线 $l_1: x+m y+6=0$ 和 $l_2:(m-2) x+3 y+2 m=0$, 则当且仅当 $m \\in$\\blank{50}时, $l_1$ 与 $l_2$ 相交; 当且仅当 $m \\in$\\blank{50}时, $l_1$ 与 $l_2$ 平行.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644814,7 +644970,9 @@ "id": "023684", "content": "(1) 若直线 $l_1: 2 x+y-3=0$, $l_2: 3 x-2 y-4=0$, 则 $l_1$ 与 $l_2$ 的夹角为\\blank{50}.\\\\\n(2) 若平行直线 $l_1: 2 x+y-1=0$, $l_2: 2 x+y+1=0$, 则 $l_1$、$l_2$ 的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644834,7 +644992,9 @@ "id": "023685", "content": "(1) 过点 $P(1,1)$ 且与直线 $\\sqrt{3}x+y+1=0$ 的夹角为 $60^{\\circ}$ 的直线方程是\\blank{50}.\\\\\n(2) 若三条直线 $4 x+y+4=0$、$m x+y+1=0$、$x-y+1=0$ 不能围成三角形, 则 $m$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644854,7 +645014,9 @@ "id": "023686", "content": "若 $0 \\leq \\theta \\leq \\dfrac{\\pi}{2}$, 当点 $(1, \\cos \\theta)$ 到直线 $x \\sin \\theta+y \\cos \\theta-1=0$ 的距离是 $\\dfrac{1}{4}$, 则直线的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644876,7 +645038,9 @@ "id": "023687", "content": "设直线 $l$ 过点 $A(2,4)$, 它被两平行线 $x-y+1=0$ 和 $x-y-1=0$ 所截得的线段的中点在直线 $x+2 y-3=0$ 上, 则直线 $l$ 的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644898,7 +645062,9 @@ "id": "023688", "content": "求经过点 $A(2,3)$, 且被两平行线 $3 x+4 y-7=0$ 和 $3 x+4 y+8=0$ 截得长为 $3 \\sqrt{2}$ 的线段所在直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -644921,7 +645087,9 @@ "id": "023689", "content": "若圆 $C: x^2+y^2+k x+2 y=-k^2$($k \\in \\mathbf{R}$), 则当圆 $C$ 面积最大时, 圆心 $C$ 的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644943,7 +645111,9 @@ "id": "023690", "content": "若实数 $x$、$y$ 满足 $x^2+y^2-4 x+1=0$, 则 $x-y$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -644965,7 +645135,9 @@ "id": "023691", "content": "在平面直角坐标系 $x O y$ 中, 曲线 $\\Gamma: y=x^2-m x+2 m$($m \\in \\mathbf{R}$) 与 $x$ 轴交于不同的两点\n$A$、$B$, 曲线 $\\Gamma$ 与 $y$ 轴交于点 $C$.\\\\\n(1) 是否存在以 $AB$ 为直径的圆过点 $C$ ? 若存在, 求出该圆的方程; 若不存在, 请说明理由;\\\\\n(2) 求证: 过 $A$、$B$、$C$ 三点的圆过定点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -644985,7 +645157,9 @@ "id": "023692", "content": "若圆心为 $(2,-1)$ 的圆在直线 $x-y-1=0$ 上截得的弦长为 $2 \\sqrt{2}$, 则此圆方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645005,7 +645179,9 @@ "id": "023693", "content": "若过点 $(3,1)$ 作圆 $(x-1)^2+y^2=r^2$ 的切线有且只有一条, 则该切线的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645025,7 +645201,9 @@ "id": "023694", "content": "已知以点 $C(t, \\dfrac{2}{t})$($t \\in \\mathbf{R}$, $t \\neq 0$) 为圆心的圆与 $x$ 轴交于点 $O$、$A$, 与 $y$ 轴交于点 $O$、$B$, 其中 $O$ 为坐标原点.\\\\\n(1) 求证: $\\triangle OAB$ 的面积为定值;\\\\\n(2) 设直线 $y=-2 x+4$ 与圆 $C$ 交于点 $M$、$N$, 若 $|OM|=|ON|$, 求圆 $C$ 的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -645045,7 +645223,9 @@ "id": "023695", "content": "若中心在原点, 两焦点坐标分别为 $(-2,0)$、$(2,0)$ 的椭圆过点 $(\\dfrac{5}{2},-\\dfrac{3}{2})$, 则该椭圆的标准方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645067,7 +645247,9 @@ "id": "023696", "content": "若椭圆的长轴长为 $12$, 一个焦点是 $(0,2)$, 则该椭圆的标准方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645087,7 +645269,9 @@ "id": "023697", "content": "椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$ 上的点 $M$ 到焦点 $F_1$ 的距离为 $2$, 若 $N$ 是 $MF_1$ 的中点, 则 $|ON|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645109,7 +645293,9 @@ "id": "023698", "content": "若椭圆 $\\dfrac{y^2}{9}+\\dfrac{x^2}{4}=1$ 焦点为 $F_1$、$F_2$, 点 $P$ 在椭圆上, 则 $\\Delta F_1PF_2$ 的面积的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645129,7 +645315,9 @@ "id": "023699", "content": "若方程 $\\dfrac{x^2}{a+2}-\\dfrac{y^2}{a-1}=1$ 表示的曲线为椭圆, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645151,7 +645339,9 @@ "id": "023700", "content": "已知 $A$、$B$ 是椭圆 $E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 上的两点, 且 $A$、$B$ 关于坐标原点对称, $F$是椭圆的一个焦点, 若 $\\triangle ABF$ 面积的最大值恰为 $2$, 则椭圆 $E$ 的长轴长的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645171,7 +645361,9 @@ "id": "023701", "content": "椭圆 $\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$ 与椭圆 $\\dfrac{x^2}{9-m}+\\dfrac{y^2}{4-m}=1$($m<4$) 的 \\bracket{20}.\n\\twoch{长轴长相等}{焦距相等}{短轴长相等}{长轴长、短轴长、焦距都不相等}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -645193,7 +645385,9 @@ "id": "023702", "content": "已知椭圆 $C$ 的焦点为 $F_1(-1,0)$、$F_2(1,0)$, 过 $F_2$ 的直线与椭圆 $C$ 交于 $A$、$B$ 两点. 若 $|AF_2|=2|F_2B|$, $|AB|=|BF_1|$, 则椭圆 $C$ 的标准方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{2}+y^2=1$}{$\\dfrac{x^2}{3}+\\dfrac{y^2}{2}=1$}{$\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$ }{$\\dfrac{x^2}{5}+\\dfrac{y^2}{4}=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -645215,7 +645409,9 @@ "id": "023703", "content": "在平面直角坐标系 $x O y$ 中, 点 $P$ 到两点 $(0,-\\sqrt{3})$、$(0, \\sqrt{3})$ 的距离之和等于 $4$, 设点 $P$ 的轨迹为 $C$.\\\\\n(1) 求曲线 $C$ 的方程;\\\\\n(2) 设直线 $y=k x+1$ 与 $C$ 交于 $A$、$B$ 两点. $k$ 为何值时 $\\overrightarrow{OA}\\perp \\overrightarrow{OB}$ ? 此时 $|\\overrightarrow{AB}|$ 的值是多少?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -645235,7 +645431,9 @@ "id": "023704", "content": "已知椭圆 $C$ 的中心为平面直角坐标系 $x O y$ 的原点, 焦点在 $x$ 轴上, 它的一个顶点到两个焦点的距离分别为 $7$ 和 $1$.\\\\\n(1) 求椭圆 $C$ 的方程;\\\\\n(2) 若 $P$ 为椭圆 $C$ 上的动点, $M$ 为过 $P$ 且垂直于 $x$ 轴的直线上的点, $\\dfrac{|OP|}{|OM|}=e$($e$ 为椭圆离心率), 求点 $M$ 的轨迹方程, 并说明轨迹是什么曲线.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -645255,7 +645453,9 @@ "id": "023705", "content": "若平面上到两定点距离差的绝对值为非零常数的点的轨迹存在, 则轨迹可以是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645275,7 +645475,9 @@ "id": "023706", "content": "方程 $m x^2+n y^2+m n=0$($m<-n<0$)所表示的曲线的焦点坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645295,7 +645497,9 @@ "id": "023707", "content": "以椭圆 $\\dfrac{x^2}{3}+\\dfrac{y^2}{4}=1$ 的顶点为焦点, 焦点为顶点的双曲线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645319,7 +645523,9 @@ "id": "023708", "content": "实、虚轴长之和为 $28$, 焦距为 $20$, 且焦点在 $x$ 轴上的双曲线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645339,7 +645545,9 @@ "id": "023709", "content": "以椭圆 $3 x^2+13 y^2=39$ 的焦点为焦点, 以 $y= \\pm \\dfrac{1}{2}x$ 为渐近线的双曲线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645361,7 +645569,9 @@ "id": "023710", "content": "渐近线方程为 $3 x \\pm 4 y=0$, 焦点为椭圆 $\\dfrac{x^2}{10}+\\dfrac{y^2}{5}=1$ 的一对顶点的双曲线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645384,7 +645594,9 @@ "id": "023711", "content": "双曲线 $C: \\dfrac{x^2}{4}-\\dfrac{y^2}{2}=1$ 的右焦点为 $F$, 若点 $P$ 在 $C$ 的一条渐近线上, $O$ 为坐标原点, 若 $|PO|=|PF|$, 则 $\\triangle PFO$ 的面积为\\bracket{20}.\n\\fourch{$\\dfrac{3 \\sqrt{2}}{4}$}{$\\dfrac{3 \\sqrt{2}}{2}$}{$2 \\sqrt{2}$}{$3 \\sqrt{2}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -645404,7 +645616,9 @@ "id": "023712", "content": "设 $F$ 为双曲线 $C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的右焦点, $O$ 为坐标原点, 以 $OF$ 为直径的圆与圆 $x^2+y^2=a^2$ 交于 $P$、$Q$ 两点. 若 $|PQ|=|OF|$, 则双曲线 $C$ 的离心率为 \\bracket{20}.\n\\fourch{$\\sqrt{2}$}{$\\sqrt{3}$}{$2$}{$\\sqrt{5}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -645424,7 +645638,9 @@ "id": "023713", "content": "已知双曲线的中心在原点, 焦点 $F_1$、$F_2$ 在坐标轴上, 离心率为 $\\sqrt{2}$, 且过点 $(4,-\\sqrt{10})$, 点 $M(3, m)$ 在双曲线上.\\\\\n(1) 求双曲线的方程;\\\\\n(2) 求证:点 $M$ 在以 $F_1F_2$ 为直径的圆上;\\\\\n(3) 求 $\\Delta F_1MF_2$ 的面积.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -645444,7 +645660,9 @@ "id": "023714", "content": "设 $A$、$B$ 分别为双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的左、右顶点, 双曲线的实轴长为 $4 \\sqrt{3}$,焦点到渐近线的距离为 $\\sqrt{3}$.\\\\\n(1) 求双曲线的方程;\\\\\n(2) 已知直线 $y=\\dfrac{\\sqrt{3}}{3}x-2$ 与双曲线的右支交于 $M$、$N$ 两点, 且在双曲线的右支上存在点 $D$, 使 $\\overrightarrow{OM}+\\overrightarrow{ON}=t \\overrightarrow{OD}$, 求 $t$ 的值及点 $D$ 的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -645464,7 +645682,9 @@ "id": "023715", "content": "若将 $1$、$2$、$3$ 填人 $3 \\times 3$ 的方格中, 要求每行、每列都没有重复数字, 如图是其中一种填法, 则不同的填写方法共有\\blank{50}种.\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline 1 & 2 & 3 \\\\\n\\hline 3 & 1 & 2 \\\\\n\\hline 2 & 3 & 1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645484,7 +645704,9 @@ "id": "023716", "content": "学校组织春游活动,每个学生可以选择去四个地方:崇明、朱家角、南汇和嘉定, 有四位同学恰好分别来自这四个地方, 若他们不去家乡, 且分别去了不同地方,则四位同学去向的所有可能结果数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645504,7 +645726,9 @@ "id": "023717", "content": "甲、乙、丙三个人玩``剪刀、石头、布''游戏,一次游戏中可以出现的不同结果数为\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645524,7 +645748,9 @@ "id": "023718", "content": "我们把各位数字之和为 6 的四位数称为``六合数''(如 2013), 则``六合数''中首位为 2 的六合数共有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645544,7 +645770,10 @@ "id": "023719", "content": "从$-1,0,1,2$ 这四个数中选三个不同的数作为函数 $f(x)=a x^2+b x+c$ 的系数, 则可组成个不同的二次函数,其中偶函数有\\blank{50}个(用数字作答).", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645564,7 +645793,9 @@ "id": "023720", "content": "如图所示,用 $4$ 种不同的颜色涂入图中的矩形 $A$、$B$、$C$、$D$ 中,要求相邻的矩形涂色不同,则不同的涂法有\\blank{50}种.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) rectangle (1.5,2);\n\\draw (0,{2/3}) --++ (1.5,0);\n\\draw (0,{4/3}) --++ (1.5,0);\n\\draw (0.75,{1/3}) node {D};\n\\draw (0.75,1) node {C};\n\\draw (0.375,{5/3}) node {A} ({9/8},{5/3}) node {B};\n\\draw (0.75,{4/3}) -- (0.75,2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645586,7 +645817,10 @@ "id": "023721", "content": "从正方体六个面的对角线中,任取两条作为一对,其中所成角为 $60^{\\circ}$ 的共有\\blank{50}对.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -645606,7 +645840,9 @@ "id": "023722", "content": "定义数列 $\\{a_n\\}$ 如下: 存在 $k \\in \\mathbf{N}$ 且 $k \\geq 1$, 满足 $a_ka_{s+1}$,已知数列 $\\{a_n\\}$ 共 $4$ 项, 若 $a_i \\in\\{t, x, y, z\\}$($i=1$、$2$、$3$、$4$) 且 $t=latex, xscale = 0.05, yscale = 50]\n\\draw [->] (40,0) -- (42,0) -- (43,-0.002) -- (45,0.002) -- (46,0) -- (120,0) node [below] {分数};\n\\draw [->] (40,0) -- (40,0.06) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.008,60/0.028,70/0.04,80/0.016,90/0.008}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {60/0.028,70/0.04,80/0.016,90/0.008}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\filldraw [gray!50] (78,0.025) circle (20 and 0.02);\n\\end{tikzpicture}\n\\end{minipage}\n\\end{center}\n(1) 求高三(1)班全体女生的人数;\\\\\n(2) 求分数在 $[80,90)$ 之间的女生人数, 并计算频率分布直方图中 $[80,90)$ 之间的矩形的高;\\\\\n(3) 根据频率分布直方图, 估计高三(1)班全体女生的数学平均成绩(同一组中的数据用该组区间的中点值代表).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -646651,7 +646986,9 @@ "id": "023772", "content": "判断正误(正确的画``\\checkmark'', 错误的画``$\\times$'')\\\\\n(1) 相关关系与函数关系都是一种确定性的关系, 也是一种因果关系;\\blank{20}\\\\\n(2) 利用散点图可以直观判断两个变量的关系是否可以用线性关系表示;\\blank{20}\\\\\n(3) 只有两个变量有相关关系, 所得到的回归模型才有预测价值;\\blank{20}\\\\\n(4) 事件 $X$、$Y$ 的关系越密切, 由观测数据计算得到的 $\\chi^2$ 的观测值越大;\\blank{20}\\\\\n(5) 通过回归方程 $y=\\hat{a}x+\\hat{b}$ 可以估计和观测变量的取值和变化趋势.\\blank{20}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646671,7 +647008,9 @@ "id": "023773", "content": "观察下列各图形, 其中两个变量 $x$、$y$ 具有相关关系的图是\\blank{50}.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1.6,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,1.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.31/1,0.56/0.84,0.71/0.55,1.09/0.24,1.48/0.31,1.26/0.41,1.37/0.64,1.07/0.53,0.94/0.71,1.15/0.76,1.02/0.92,1.28/0.85,1.4/1.02,1.28/1.16}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (0.8,-0.3) node {\\textcircled{1}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1.6,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,1.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.28/0.68,0.46/1.01,0.49/0.6,0.41/0.31,0.7/0.46,0.65/0.82,0.93/1.01,0.9/0.66,0.81/0.35,1/0.4,1.11/0.57}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (0.8,-0.3) node {\\textcircled{2}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1.6,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,1.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.2/0.4,0.45/0.51,0.65/0.67,0.78/0.9,0.91/0.86,0.96/1.08,1.09/1.13,1.21/1.27}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (0.8,-0.3) node {\\textcircled{3}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1.6,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,1.6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.17/1.44,0.24/1.3,0.36/1.2,0.5/1.09,0.64/0.96,0.82/0.84,0.88/0.68,0.99/0.55,1.14/0.45}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (0.8,-0.3) node {\\textcircled{4}};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646693,7 +647032,9 @@ "id": "023774", "content": "某公司在 $2020$ 年上半年的月收入 $x$ (单位:万元)与月支出 $y$ (单位:万元)的统计资料如表所示, 根据统计资料, 则\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 月份 & 1 月 & 2 月 & 3月 & 4 月 & 5 月 & 6 月 \\\\\n\\hline 月收入 $x$ & 12.3 & 14.5 & 15.0 & 17.0 & 19.8 & 20.6 \\\\\n\\hline 月支出 $y$ & 5.63 & 5.75 & 5.82 & 5.89 & 6.11 & 6.18 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\onech{月收入的中位数是 $15, x$ 与 $y$ 有正线性相关关系}{月收入的中位数是 $17, x$ 与 $y$ 有负线性相关关系}{月收入的中位数是 $16, x$ 与 $y$ 有正线性相关关系}{月收入的中位数是 $16, x$ 与 $y$ 有负线性相关关系}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -646713,7 +647054,9 @@ "id": "023775", "content": "某研究机构对高三学生的记忆力 $x$ 和判断力 $y$ 进行统计分析, 所得数据如表, 则 $y$ 对 $x$ 的线性回归直线方程为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$x$ & 6 & 8 & 10 & 12 \\\\\n\\hline$y$ & 2 & 3 & 5 & 6 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646735,7 +647078,9 @@ "id": "023776", "content": "某沙漠地区经过治理, 生态系统得到很大改善, 野生动物数量有所增加. 为调查该地区某种野生动物的数量, 将其分成面积相近的 $200$ 个地块, 从这些地块中用简单随机抽样的方法抽取 $20$ 个作为样区, 调查得到样本数据 $(x_i, y_i)$($i=1$、$2$、$\\cdots, 20$), 其中 $x_i$ 和 $y_i$ 分别表示第 $i$ 个样区的植物覆盖面积(单位:公顷)和这种野生动物的数量, 并计算得\n$\\displaystyle\\sum_{i=1}^{20}x_i=60$, $\\displaystyle\\sum_{i=1}^{20}y_i=1200$, $\\displaystyle\\sum_{i=1}^{20}(x_i-\\overline{x})^2=80$, $\\displaystyle\\sum_{i=1}^{20}(y_i-\\overline{y})^2=9000$, $\\displaystyle\\sum_{i=1}^{20}(x_i-\\overline{x}) \\cdot(y_i-\\overline{y})=800$.\\\\\n(1) 求该地区这种野生动物数量的估计值(这种野生动物数量的估计值等于样区这种野生动物数量的平均数乘以地块数);\\\\\n(2) 求样本 $(x_i, y_i)$($i=1$、$2$、$\\cdots$、$20$) 的相关系数 (精确到 $0.01$).\\\\\n附: 相关系数\n$r=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline{x})(y_i-\\overline{y})}{\\sqrt{\\displaystyle\\sum_{i=1}^n(x_i-\\overline{x})^2 \\displaystyle\\sum_{i=1}^n(y_i-\\overline{y})^2}}$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -646758,7 +647103,9 @@ "id": "023777", "content": "有人收集了某 $10$ 年中某城市居民年收人(即该城市所有居民在一年内收人的总和)与某种商品的销售额的相关数据如表, \n且已知 $\\displaystyle\\sum_{i=1}^{10}x_i=380.0$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 第$n$年 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 年收人 $x$ / 亿元 & 32.0 & 31.0 & 33.0 & 36.0 & 37.0 & 38.0 & 39.0 & 43.0 & 45.0 & $x_{10}$ \\\\\n\\hline 商品销售额 $y /$ 万元 & 25.0 & 30.0 & 34.0 & 37.0 & 39.0 & 41.0 & 42.0 & 44.0 & 48.0 & $y_{10}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 求第 $10$ 年的年收人 $x_{10}$;\\\\\n(2) 若该城市居民年收人 $x$ 与该种商品的销售额 $y$ 之间满足线性回归方程 $y=\\dfrac{363}{254}x+\\hat{b}$,\\\\\n\\textcircled{1} 求该种商品第 $10$ 年的销售额 $y_{10}$;\\\\\n\\textcircled{2} 若该城市居民年收人为 $40.0$ 亿元, 估计这种商品的销售额是多少?(精确到 $0.01$ 万元).\\\\\n附: (i) 在线性回归方程 $y=\\hat{a}x+\\hat{b}$ 中, $\\hat{a}=\\dfrac{\\displaystyle\\sum_{i=1}^n x_i y_i-n \\overline{x}\\cdot \\overline{y}}{\\displaystyle\\sum^n x^2}$, $\\hat{b}=\\hat{y}-\\hat{a}\\overline{x}$, $\\displaystyle\\sum_{i=1}^n x_i^2-n \\overline{x}^2$;\\\\\n(ii) $\\displaystyle\\sum_{i=1}^{10}x_i^2-10 x^{-2}=254.0$, $\\displaystyle\\sum_{i=1}^9 x_i y_i=12875.0$, $\\displaystyle\\sum_{i=1}^9 y_i=340.0$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -646778,7 +647125,10 @@ "id": "023778", "content": "某学生兴趣小组随机调查了某市 $100$ 天中每天的空气质量等级和当天到某公园锻炼的人次, 整理数据得到下表(单位:天):\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline 空气质量等级 &{$[0,200]$}& $(200,400]$ & $(400,600]$ \\\\\n\\hline 1 (优) & 2 & 16 & 25 \\\\\n\\hline 2 (良) & 5 & 10 & 12 \\\\\n\\hline 3 (轻度污染) & 6 & 7 & 8 \\\\\n\\hline 4 (中度污染) & 7 & 2 & 0 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 分别估计该市一天的空气质量等级为 $1,2,3,4$ 的概率;\\\\\n(2) 求一天中到该公园锻炼的平均人次的估计值(同一组中的数据用该组区间的中点值为代表);\\\\\n(3) 若某天的空气质量等级为 1 或 2 , 则称这天``空气质量好'';若某天的空气质量等级为 3 或 4 , 则称这天``空气质量不好''. 根据所给数据, 完成下面的 $2 \\times 2$ 列联表, 并根据列联表, 判断是否有 $95 \\%$ 的把握认为一天中到该公园锻炼的人次与该市当天的空气质量有关?\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\\hline & 人次 $\\leq 400$ & 人次 $>400$ \\\\\n\\hline 空气质量好 & & \\\\\n\\hline 空气质量不好 & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$,\n\\begin{tabular}{|l|l|l|l|}\\hline$P$($\\chi^2 \\geq k$) & $0.050$ & $0.010$ & $0.001$ \\\\\n\\hline $k$ & $3.841$ & $6.635$ & $10.828$ \\\\\n\\hline\n\\end{tabular}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -646798,7 +647148,9 @@ "id": "023779", "content": "若 $\\alpha=\\dfrac{2}{3}\\pi$, 则与 $\\alpha$ 终边相同的角的集合是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646818,7 +647170,9 @@ "id": "023780", "content": "在半径为 $10 \\mathrm{m}$ 的圆形弯道中, $120^{\\circ}$ 角所对应的弯道长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646838,7 +647192,9 @@ "id": "023781", "content": "若扇形的面积是 $1$ , 周长是 $4$ , 则扇形的中心角的弧度数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646861,7 +647217,9 @@ "id": "023782", "content": "\\textcircled{1} $-\\dfrac{3 \\pi}{4}$ 是第二象限角;\\textcircled{2} $\\dfrac{4 \\pi}{3}$ 是第三象限角;\\textcircled{3} $-400^{\\circ}$ 是第四象限角; \\textcircled{4} $-315^{\\circ}$ 是第一象限角. 其中正确的命题的个数为\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646881,7 +647239,9 @@ "id": "023783", "content": "已知角 $\\alpha$ 的终边过点 $(3 a-9, a+2)$, 若 $\\cos \\alpha<0$, $\\sin \\alpha>0$, 则 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646901,7 +647261,9 @@ "id": "023784", "content": "若角 $\\alpha$ 的终边落在函数 $y=-3 x$ 的图像上, 则 $\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646921,7 +647283,9 @@ "id": "023785", "content": "函数 $f(x)=\\dfrac{|\\sin x|}{\\sin x}+\\dfrac{|\\cos x|}{\\cos x}+\\dfrac{|\\tan x|}{\\tan x}$ 的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646945,7 +647309,9 @@ "id": "023786", "content": "在半径为 $2$ 的圆中, 一个扇形的周长等于半圆的弧长, 则该扇形圆心角的弧度数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646965,7 +647331,9 @@ "id": "023787", "content": "已知角 $\\alpha$ 的终边上一点 $P(-\\sqrt{3}, m)$, 且 $\\sin \\alpha=\\dfrac{\\sqrt{2}}{4}m$, 则 $\\tan \\alpha$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -646987,7 +647355,9 @@ "id": "023788", "content": "设点 $A$ 是单位圆与 $x$ 正半轴的交点, 点 $A$ 在圆周上依逆时针方向作匀速圆周运动, 已知点 $A$ $1$ 分钟转过 $\\theta$($0<\\theta<\\pi$), $2$ 分钟到达第三象限, $14$ 分钟回到原来位置, 求 $\\theta$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -647007,7 +647377,9 @@ "id": "023789", "content": "若 $\\tan \\alpha=2$, 且 $\\alpha$ 为第三象限的角, 则 $\\sin \\alpha=$\\blank{50}, $\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647027,7 +647399,9 @@ "id": "023790", "content": "若 $1+\\cos ^2 \\theta=3 \\sin \\theta \\cos \\theta$, 则 $\\tan \\theta$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647047,7 +647421,9 @@ "id": "023791", "content": "若 $\\sin \\theta \\cos \\theta=\\dfrac{1}{8}$, $\\theta \\in(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$, 则 $\\cos \\theta-\\sin \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647067,7 +647443,9 @@ "id": "023792", "content": "若 $\\dfrac{2 \\cos x+3 \\sin x}{\\sin x-2 \\cos x}=7$, 则 $\\tan x=$\\blank{50}, $\\cos ^2 x-5 \\sin ^2 x-2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647087,7 +647465,9 @@ "id": "023793", "content": "若 $f(\\alpha)=\\dfrac{2 \\sin (\\pi+\\alpha) \\cos (\\pi-\\alpha)-\\cos (\\pi+\\alpha)}{1+\\sin ^2 \\alpha+\\cos (\\dfrac{3 \\pi}{2}+\\alpha)-\\sin ^2(\\dfrac{\\pi}{2}+\\alpha)}$($1+2 \\sin \\alpha \\neq 0$), 则$f(-\\dfrac{23 \\pi}{6})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647107,7 +647487,9 @@ "id": "023794", "content": "若 $\\dfrac{\\sin \\alpha+3 \\cos \\alpha}{3 \\cos \\alpha-\\sin \\alpha}=5$, 则 $\\cos ^2 \\alpha+\\dfrac{1}{2}\\sin 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647127,7 +647509,9 @@ "id": "023795", "content": "若 $\\sin \\alpha=\\dfrac{k-3}{k+5}$, $\\cos \\alpha=\\dfrac{4-2 k}{k+5}$, $\\dfrac{\\pi}{2}<\\alpha<\\pi$, 则 $\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647147,7 +647531,9 @@ "id": "023796", "content": "已知 $\\cos \\alpha=\\dfrac{4}{5}$, 则 $\\dfrac{\\cos (\\alpha-\\dfrac{\\pi}{2})+2 \\sin (3 \\pi-\\alpha)}{2 \\tan (3 \\pi+\\alpha)+\\cot (\\dfrac{\\pi}{2}+\\alpha)}$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647167,7 +647553,9 @@ "id": "023797", "content": "已知 $\\sin \\theta+\\cos \\theta=-\\dfrac{1}{5}$ 且 $\\theta \\in(0, \\pi)$, 则 $\\tan \\theta$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647187,7 +647575,9 @@ "id": "023798", "content": "已知 $\\tan x-\\dfrac{1}{\\cos x}=\\sqrt{\\dfrac{1-\\sin x}{1+\\sin x}}$, 求 $x$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -647207,7 +647597,9 @@ "id": "023799", "content": "若 $\\cot (\\pi-\\alpha)=2$, 则 $\\cos (\\dfrac{3 \\pi}{2}+2 \\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647227,7 +647619,9 @@ "id": "023800", "content": "若 $\\dfrac{3}{2}\\pi<\\alpha<\\dfrac{5}{2}\\pi$, 则 $\\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2}\\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2}\\cos \\alpha}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647251,7 +647645,9 @@ "id": "023801", "content": "已知 $\\sin \\alpha=\\dfrac{4}{5}$, $\\cos (\\alpha+\\beta)=-\\dfrac{3}{5}$, 若 $\\alpha$、$\\beta$ 都是第一象限的角, 则 $\\sin \\beta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647273,7 +647669,9 @@ "id": "023802", "content": "若 $\\sin \\theta, \\cos \\theta$ 是二次方程 $2 x^2-(\\sqrt{3}+1) x+m=0$ 的两个根, 则 $\\dfrac{\\sin \\theta}{1-\\cot \\theta}+\\dfrac{\\cos \\theta}{1-\\tan \\theta}$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647293,7 +647691,9 @@ "id": "023803", "content": "若 $\\alpha+\\beta=\\dfrac{\\pi}{3}$, 则 $\\tan \\alpha+\\tan \\beta+\\sqrt{3}\\tan \\alpha \\tan \\beta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647315,7 +647715,9 @@ "id": "023804", "content": "化简: $\\cos (20^{\\circ}+x) \\cos (25^{\\circ}-x)-\\cos (70^{\\circ}-x) \\sin (25^{\\circ}-x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647335,7 +647737,9 @@ "id": "023805", "content": "已知 $\\tan (\\dfrac{\\pi}{4}+\\theta)=3$, 则 $\\sin 2 \\theta-2 \\cos ^2 \\theta$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647358,7 +647762,9 @@ "id": "023806", "content": "若表达式 $3 \\sin ^2 \\theta+\\sqrt{3}\\sin \\theta \\cos \\theta+4 \\cos ^2 \\theta+k$ 可化为 $\\sin (2 \\theta+\\varphi)$, $0 \\leq \\varphi<2 \\pi$ 的形式, 则 $k$ 的值为\\blank{50}, $\\varphi$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647378,7 +647784,9 @@ "id": "023807", "content": "已知 $\\tan \\alpha, \\tan \\beta$ 是方程 $x^2+a x+a+1=0$ 的两个实根, 求证: $\\sin (\\alpha+\\beta)=\\cos (\\alpha+\\beta)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -647401,7 +647809,9 @@ "id": "023808", "content": "若 $180^{\\circ}<\\alpha<360^{\\circ}$, 则 $\\cos \\dfrac{\\alpha}{2}$ 的值等于\\bracket{20}.\n\\fourch{$-\\sqrt{\\dfrac{1-\\cos \\alpha}{2}}$}{$\\sqrt{\\dfrac{1-\\cos \\alpha}{2}}$}{$-\\sqrt{\\dfrac{1+\\cos \\alpha}{2}}$}{$\\sqrt{\\dfrac{1+\\cos \\alpha}{2}}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -647421,7 +647831,9 @@ "id": "023809", "content": "若 $\\dfrac{\\sin \\alpha-\\cos \\alpha}{\\sin \\alpha+\\cos \\alpha}=\\dfrac{1}{2}$, 则 $\\tan 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647443,7 +647855,9 @@ "id": "023810", "content": "若 $\\alpha$、$\\beta \\in(0, \\dfrac{\\pi}{2})$, 且 $\\cos \\alpha=\\dfrac{5}{13}$, $\\sin (\\alpha+\\beta)=\\dfrac{3}{5}$, 则 $\\cos \\beta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647466,7 +647880,9 @@ "id": "023811", "content": "若 $\\sin (\\dfrac{\\pi}{6}-\\alpha)=\\dfrac{1}{3}$, 则 $\\cos (\\dfrac{2 \\pi}{3}+2 \\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647488,7 +647904,9 @@ "id": "023812", "content": "若 $\\tan (\\alpha+\\dfrac{\\pi}{6})=\\dfrac{3}{2}$, 则 $\\cos (2 \\alpha-\\dfrac{\\pi}{6})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647508,7 +647926,9 @@ "id": "023813", "content": "已知 $\\alpha+\\beta=\\dfrac{\\pi}{4}$, 则 $(1+\\tan \\alpha)(1+\\tan \\beta)$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647531,7 +647951,9 @@ "id": "023814", "content": "已知 $0<\\alpha<\\dfrac{\\pi}{2}$, $\\cos \\alpha=\\dfrac{4}{5}$. 若 $0<\\beta<\\dfrac{\\pi}{2}$ 且 $\\cos (\\alpha+\\beta)=-\\dfrac{1}{2}$, 则 $\\sin \\beta$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647553,7 +647975,9 @@ "id": "023815", "content": "若 $\\sin \\alpha+\\sin \\beta=\\dfrac{1}{4}$, $\\cos \\alpha+\\cos \\beta=\\dfrac{1}{3}$, 则 $\\tan (\\alpha+\\beta)$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647573,7 +647997,9 @@ "id": "023816", "content": "已知 $\\alpha$ 与 $\\beta$ 都是锐角, 且 $\\sin (\\alpha-\\beta)=\\dfrac{1}{3}$, $\\cos (\\alpha+\\beta)=\\dfrac{\\sqrt{3}}{2}$.\\\\\n(1) 求 $\\sin 2 \\alpha$ 的值;\\\\\n(2) 求证: $\\sin \\alpha \\cos \\beta=5 \\cos \\alpha \\sin \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -647593,7 +648019,9 @@ "id": "023817", "content": "如图, 已知面积为 $\\dfrac{\\pi}{6}$ 的扇形 $AOB$, 半径为 $1$, $C$ 是弧 $AB$ 上任意一点, 作矩形 $CDEF$ 内接于该扇形.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\t{35}\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (60:2) node [above] {$B$} coordinate (B);\n\\draw (\\t:2) node [above right] {$C$} coordinate (C);\n\\draw (C) ++ ({-2/sin(120)*sin(60-\\t)},0) node [above left] {$D$} coordinate (D);\n\\draw ($(O)!(D)!(A)$) node [below] {$E$} coordinate (E);\n\\draw ($(O)!(C)!(A)$) node [below] {$F$} coordinate (F);\n\\draw (E)--(D)--(C)--(F)(A)--(O)--(B) arc (60:0:2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求扇形圆心角 $\\angle AOB$ 的大小;\\\\\n(2) 点 $C$ 在什么位置时, 矩形 $CDEF$ 的面积最大?并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -647613,7 +648041,10 @@ "id": "023818", "content": "函数 $y=\\lg (1-2 \\cos x)$ 的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647633,7 +648064,9 @@ "id": "023819", "content": "函数 $y=\\dfrac{\\sin ^2 x+2}{\\sin x}$ 的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647653,7 +648086,9 @@ "id": "023820", "content": "函数 $y=\\sin ^2 x+2 \\cos x$, $x \\in[\\dfrac{\\pi}{3}, \\dfrac{4 \\pi}{3}]$ 的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647676,7 +648111,9 @@ "id": "023821", "content": "函数 $y=\\tan 3 \\pi x$ 的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647699,7 +648136,9 @@ "id": "023822", "content": "函数 $y=\\tan (x+\\dfrac{\\pi}{4})$ 的严格\\blank{50}(填写``增''或``减'') 区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647721,7 +648160,9 @@ "id": "023823", "content": "函数 $y=\\sin (\\dfrac{\\pi}{3}-2 x)$ 的严格减区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647744,7 +648185,9 @@ "id": "023824", "content": "设 $a, b \\in \\mathbf{R}$. 若函数 $y=2 \\cos x$ 的定义域为 $[\\dfrac{\\pi}{3}, \\pi]$, 值域为 $[a, b]$, 则 $b-a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647764,7 +648207,9 @@ "id": "023825", "content": "若 $y=\\sin (2 x+\\alpha)+\\cos (2 x+\\alpha)$ 为奇函数, 则最小正数 $\\alpha$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647784,7 +648229,9 @@ "id": "023826", "content": "设实数 $t>0$, $f(x)=2 \\sin x+t \\cos x$, \\\\\n(1) 当 $t=1$ 时, $f(x)$ 的最大值为\\blank{50}.\\\\\n(2) 已知 $f(x)=4$ 有实数解, 则 $t$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647804,7 +648251,9 @@ "id": "023827", "content": "设 $a \\in \\mathbf{R}$. 当 $x \\in[0, \\dfrac{\\pi}{2}]$ 时, 求 $y=\\cos ^2 x-2 a \\cos x$ 的最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -647824,7 +648273,9 @@ "id": "023828", "content": "函数 $y=\\sin (2 x+\\dfrac{\\pi}{4})$ 的对称轴方程为\\blank{50}, 对称中心为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647846,7 +648297,9 @@ "id": "023829", "content": "若函数 $f(x)=\\sin (\\omega x+\\dfrac{\\pi}{6})$($\\omega>0$) 的最小正周期为 $\\pi$, 则 $f(\\dfrac{\\pi}{3})$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647866,7 +648319,9 @@ "id": "023830", "content": "把函数 $y=\\cos x-\\sqrt{3}\\sin x$ 的图像向左平移 $m$($m>0$) 个单位所得图像关于 $y$ 轴对称, 则实数 $m$ 的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647889,7 +648344,9 @@ "id": "023831", "content": "电流强度 $I$(A) 随时间 $t$(s) 变化的函数 $I=A \\sin (\\omega t+\\varphi) $($A>0$, $\\omega>0$, $0<\\varphi<\\dfrac{\\pi}{2}$) 的部分图像如图所示, 则当 $t=\\dfrac{1}{100}$s时, 电流强度是\\blank{50}A.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 1.5, yscale = 0.1]\n\\draw [->] (0,0) -- ({7/3},0) node [below] {$x$};\n\\draw [->] (0,-15) -- (0,15) node [left] {$y$};\n\\draw (0,0) node [left] {$O$};\n\\draw [domain = 0:{11/6}, samples = 100] plot (\\x,{10*sin(180*\\x+30)});\n\\foreach \\i in {-10,-5,5,10}\n{\\draw (0.04,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw [dashed] ({4/3},0) --++ (0,-10) -- (0,-10) ({1/3},0)--++ (0,10) -- (0,10);\n\\draw ({1/3},0) node [below] {$\\dfrac{1}{300}$};\n\\draw ({4/3},0) node [above] {$\\dfrac{4}{300}$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647909,7 +648366,9 @@ "id": "023832", "content": "若函数 $f(x)=A \\sin (\\omega x+\\varphi)+k$($A>0$, $\\omega>0$, $|\\varphi|<\\dfrac{\\pi}{2}$)的图像如图所示, 则 $f(x)$ 的表达式是 $f(x)=$\\blank{50}.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = -1.5:2.5, samples = 100] plot (\\x,{1.5*sin(\\x*360/pi+60)+1});\n\\draw [dashed] ({7*pi/12},0) --++ (0,-0.5) -- (0,-0.5);\n\\draw [dashed] ({pi/12},0) --++ (0,2.5) -- (0,2.5);\n\\draw (0,-0.5) node [left] {$-\\frac{1}{2}$} (0,2.5) node [left] {$\\frac{5}{2}$};\n\\draw ({pi/12},0) node [below] {$\\frac{\\pi}{12}$} ({7*pi/12},0) node [above] {$\\frac{7\\pi}{12}$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647929,7 +648388,9 @@ "id": "023833", "content": "若函数 $f(x)=2 \\sin (\\dfrac{\\pi}{3}x+\\varphi)$($|\\varphi|<\\dfrac{\\pi}{2}$) 的图像经过点 $(0,1)$, 则其最小正周期 $T=$\\blank{50}, 初始相位 $\\varphi=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647949,7 +648410,10 @@ "id": "023834", "content": "设 $\\omega \\in \\mathbf{R}$. 已知向量 $\\overrightarrow{m}=(\\cos \\omega x, \\sin \\omega x)$, $\\overrightarrow{n}=(\\cos \\omega x, \\sqrt{3}\\cos \\omega x)$, 函数 $y=f(x)$, 其中 $f(x)=\\overrightarrow{m}\\cdot \\overrightarrow{n}$.\\\\\n(1) 若 $f(x)$ 的最小正周期是 $2 \\pi$, 求 $f(x)$ 的单调增区间;\\\\\n(2) 若 $f(x)$ 的图像的一条对称轴是 $x=\\dfrac{\\pi}{6}$($0<\\omega<2$), 求 $f(x)$ 的最小正周期和值域.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -647969,7 +648433,9 @@ "id": "023835", "content": "在 $\\triangle ABC$ 中, 若 $a=10$, $b=20$, $C=120^{\\circ}$, 则边 $c=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -647989,7 +648455,9 @@ "id": "023836", "content": "在 $\\triangle ABC$ 中, 若 $B=45^{\\circ}$, $c=2 \\sqrt{2}$, $b=\\dfrac{4 \\sqrt{3}}{3}$, 则 $A=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648011,7 +648479,9 @@ "id": "023837", "content": "在 $\\triangle ABC$ 中, 若 $a=2$, $b=3$, $c=4$, 则角 $C=$\\blank{50}, $S_{\\triangle ABC}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648033,7 +648503,9 @@ "id": "023838", "content": "在 $\\triangle ABC$ 中, 若 $2R(\\sin ^2A-\\sin ^2C)=(a-b) \\sin B$, $R$ 为 $\\triangle ABC$ 外接圆的半径, 则角 $C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648053,7 +648525,10 @@ "id": "023839", "content": "在 $\\triangle ABC$ 中, 若 $A(-4,0)$, $C(4,0)$, $B$ 在椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$ 上, 则 $\\dfrac{\\sin A+\\sin C}{\\sin B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648073,7 +648548,9 @@ "id": "023840", "content": "在 $\\triangle ABC$ 中, 角 $A$、$B$、$C$ 所对边的长分别为 $a$、$b$、$c$, $S$ 表示 $\\triangle ABC$ 的面积. 若 $a \\cos B+b \\cos A=c \\sin C$, $S=\\dfrac{1}{4}(b^2+c^2-a^2)$, 则 $\\triangle ABC$ 的形状是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648093,7 +648570,9 @@ "id": "023841", "content": "在 $\\triangle ABC$ 中, 若 $\\cos A \\cos B>\\sin A \\sin B$, 则 $\\triangle ABC$ 的形状为\\blank{50}三角形.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648115,7 +648594,9 @@ "id": "023842", "content": "在 $\\triangle ABC$ 中, 已知 $a^2 \\tan B=b^2 \\tan A$, 试判断三角形的形状.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648135,7 +648616,9 @@ "id": "023843", "content": "在 $\\triangle ABC$ 中, 角 $A$、$B$、$C$ 所对边的长分别为 $a$、$b$、$c$, 且 $(a+b-c)(a+b+c)=a b$.\\\\\n(1) 求角 $C$ 的大小;\\\\\n(2) 若 $c=2 a \\cos B$, $b=2$, 求 $\\triangle ABC$ 的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648157,7 +648640,9 @@ "id": "023844", "content": "选择正确的序号填空: (\\textcircled{1} 充要; \\textcircled{2} 充分非必要; \\textcircled{3} 必要非充分; \\textcircled{4} 既非充分又非必要)\\\\\n(1) 在 $\\triangle ABC$ 中, ``$A>B$''是``$\\sin A>\\sin B$''的\\blank{50}条件;\\\\\n(2) 在 $\\triangle ABC$ 中, ``$A>B$''是``$\\tan A>\\tan B$''的\\blank{50}条件;\\\\\n(3) 在 $\\triangle ABC$ 中, ``$A>B$''是``$\\cos A<\\cos B$''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648177,7 +648662,9 @@ "id": "023845", "content": "甲船在 $A$ 处测得乙船在北偏东 $70^{\\circ}$ 方向, 两船相距 $10$ 海里, 且乙船正沿着南偏东 $40^{\\circ}$ 方向以每小时 $12$ 海里的速度航行, 经过半小时, 甲船追上乙船, 则甲船的航行速度是每小时\\blank{50}海里(精确到 $0.1$).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648197,7 +648684,9 @@ "id": "023846", "content": "$\\triangle ABC$ 的内角 $A$、$B$、$C$ 所对边的长分别为 $a$、$b$、$c$. 已知 $2 \\cos C(a \\cos B+b \\cos A)=c$.\\\\\n(1) 求 $C$;\\\\\n(2) 若 $c=\\sqrt{7}, \\triangle ABC$ 的面积为 $\\dfrac{3 \\sqrt{3}}{2}$, 求 $\\triangle ABC$ 的周长.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648217,7 +648706,9 @@ "id": "023847", "content": "某动物园喜迎虎年的到来, 拟用一块形如直角三角形 $ABC$ 的地块建造小老虎的休息区和活动区. 如图, $\\angle BAC=90^{\\circ}$, $AB=AC=20$ (单位: 米), $E$、$F$ 为 $BC$ 上的两点, 且 $\\angle EAF=45^{\\circ}, \\triangle AEF$ 区域为休息区, $\\triangle ABE$ 和 $\\triangle ACF$ 区域均为活动区. 设 $\\angle EAB=\\alpha$($0<\\alpha<\\dfrac{\\pi}{4}$).\\\\\n\\begin{center}\n\\begin{tikzpicture}\n\\path (0,0) node [below left] {$A$} coordinate (A)-- (3,0) node [below right] {$B$} coordinate (B)-- (0,3) node [above left] {$C$} coordinate (C)-- cycle;\n\\path [name path = lineBC] (3,0) -- (0,3);\n\\path [name path = lineAE] (0,0) -- (15:3);\n\\path [name path = lineAF] (0,0) -- (60:3);\n\\path [name intersections = {of = lineBC and lineAE, by = E}];\n\\path [name intersections = {of = lineBC and lineAF, by = F}];\n\\filldraw [gray!30] (A) -- (B) -- (E) -- cycle;\n\\filldraw [gray!30] (A) -- (F) -- (C) -- cycle;\n\\draw (0,0) -- (E) node [above right] {$E$};\n\\draw (0,0) -- (F) node [above right] {$F$};\n\\draw (A) -- (B) -- (C) -- cycle;\n\\draw (2,0) node [above] {\\tiny{活动区}};\n\\draw (0,2) node [right] {\\tiny{活动区}};\n\\draw (1.2,1) node {\\small{休息区}};\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $AE$ 的长 (用 $\\alpha$ 的代数式表示);\\\\\n(2) 为了使小老虎能健康成长, 要求所建造的活动区面积尽可能大 (即休息区尽可能小), 当 $\\alpha$ 为多少时, 活动区的面积最大? 最大面积为多少?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648239,7 +648730,9 @@ "id": "023848", "content": "在边长为 $1$ 的正六边形 $ABCDEF$ 中, $|\\overrightarrow{BA}+\\overrightarrow{CD}+\\overrightarrow{EF}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648259,7 +648752,9 @@ "id": "023849", "content": "已知平行四边形 $ABCD$ 的对角线 $AC$ 和 $BD$ 相交于 $O$, 且 $\\overrightarrow{OA}=\\overrightarrow{a}$, $\\overrightarrow{OB}=\\overrightarrow{b}$, 则 $\\overrightarrow{BC}$ 用 $\\overrightarrow{a}, \\overrightarrow{b}$ 表示为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648281,7 +648776,9 @@ "id": "023850", "content": "设 $\\overrightarrow{a}, \\overrightarrow{b}$ 是不共线的两个平面向量, 已知 $\\overrightarrow{PQ}=\\overrightarrow{a}+\\sin \\alpha \\cdot \\overrightarrow{b}$, 其中 $\\alpha \\in(0,2 \\pi)$, $\\overrightarrow{QR}=2 \\overrightarrow{a}-\\overrightarrow{b}$. 若 $P$、$Q$、$R$ 三点共线, 则角 $\\alpha$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648301,7 +648798,9 @@ "id": "023851", "content": "设向量 $\\overrightarrow{a}, \\overrightarrow{b}$ 不共线, 且 $\\overrightarrow{c}=\\lambda \\overrightarrow{a}+\\overrightarrow{b}$, $\\overrightarrow{d}=\\overrightarrow{a}+(2 \\lambda-1) \\overrightarrow{b}$, 若 $\\overrightarrow{c}$ 与 $\\overrightarrow{d}$ 同向, 则实数 $\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648321,7 +648820,9 @@ "id": "023852", "content": "如图, 已知等边三角形 $ABC$ 内接于圆 $O$, $D$ 为线段 $OA$ 的中点, 若 $\\overrightarrow{BA}=\\overrightarrow{a}$, $\\overrightarrow{BC}=\\overrightarrow{b}$, 则 $\\overrightarrow{BD}$ 用 $\\overrightarrow{a}, \\overrightarrow{b}$ 表示为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\filldraw (0,0) circle (0.02) node [right] {$O$} coordinate (O);\n\\draw (O) circle (1);\n\\draw (90:1) node [above] {$A$} coordinate (A);\n\\draw (210:1) node [below left] {$B$} coordinate (B);\n\\draw (-30:1) node [below right] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(O)$) node [below right] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--cycle(A)--(E)(B)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648341,7 +648842,9 @@ "id": "023853", "content": "在等腰梯形 $ABCD$ 中, $\\overrightarrow{AB}=2 \\overrightarrow{DC}$, 点 $E$ 是线段 $BC$ 的中点, 若 $\\overrightarrow{AE}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AD}$, 则 $\\lambda=$\\blank{50}, $\\mu=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648361,7 +648864,9 @@ "id": "023854", "content": "在 $\\triangle ABC$ 中, $D$、$E$ 分别为 $BC$、$AC$ 边上的中点, $G$ 为 $BE$ 上一点, 且 $GB=2GE$, 设 $\\overrightarrow{AB}=\\overrightarrow{a}$, $\\overrightarrow{AC}=\\overrightarrow{b}$, 则 $\\overrightarrow{AG}$ 可用 $\\overrightarrow{a}, \\overrightarrow{b}$ 表示为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648381,7 +648886,9 @@ "id": "023855", "content": "已知 $O$、$A$、$B$ 是不共线的三点, 且 $\\overrightarrow{OP}=m \\overrightarrow{OA}+n \\overrightarrow{OB}$($m, n \\in \\mathbf{R}$).\\\\\n(1) 若 $m+n=1$, 求证: $A$、$P$、$B$ 三点共线;\\\\\n(2) 若 $A$、$P$、$B$ 三点共线, 求证: $m+n=1$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648401,7 +648908,9 @@ "id": "023856", "content": "设向量 $\\overrightarrow{a}=(3,2)$, $\\overrightarrow{b}=(1,-4)$, 则 $\\overrightarrow{b}$ 在 $\\overrightarrow{a}$ 方向上的数量投影为\\blank{50}, 投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648423,7 +648932,9 @@ "id": "023857", "content": "已知点 $A(8,-1)$、$B(1,-3)$, 若点 $C(2 m-1, m+2)$ 在直线 $AB$ 上, 则实数 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648446,7 +648957,9 @@ "id": "023858", "content": "设向量 $\\overrightarrow{p}=(2,7)$, $\\overrightarrow{q}=(x,-3)$. 若 $\\overrightarrow{p}$ 与 $\\overrightarrow{q}$ 的夹角为钝角, 则 $x$ 的取值范围是\\blank{50}, 若 $\\overrightarrow{p}$ 与 $\\overrightarrow{q}$ 的夹角为锐角, 则 $x$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648469,7 +648982,9 @@ "id": "023859", "content": "已知 $|\\overrightarrow{a}|=5$, $|\\overrightarrow{b}|=8, \\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为 $60^{\\circ}$, 则 $|\\overrightarrow{a}+\\overrightarrow{b}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648489,7 +649004,9 @@ "id": "023860", "content": "设 $\\overrightarrow{a}=(m+1,-3)$, $\\overrightarrow{b}=(1, m-1)$, 且 $\\overrightarrow{a}+\\overrightarrow{b}$ 与 $\\overrightarrow{a}-\\overrightarrow{b}$ 垂直, 则实数 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648511,7 +649028,9 @@ "id": "023861", "content": "若 $|\\overrightarrow{a}|=3$, $|\\overrightarrow{b}|=4$, $(\\overrightarrow{a}+2 \\overrightarrow{b}) \\cdot(2 \\overrightarrow{a}-\\overrightarrow{b})=-32$, 则 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648531,7 +649050,9 @@ "id": "023862", "content": "若将向量 $\\overrightarrow{a}=(3,4)$ 绕原点按逆时针方向旋转 $45^{\\circ}$ 得到向量 $\\overrightarrow{b}$, 则 $\\overrightarrow{b}$ 的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648553,7 +649074,9 @@ "id": "023863", "content": "已知 $O$ 为坐标原点, $\\overrightarrow{OA}=(2,2)$, $\\overrightarrow{OB}=(4,1)$, 点 $P$ 在 $x$ 轴上, 则 $\\overrightarrow{AP}\\cdot \\overrightarrow{BP}$ 最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648573,7 +649096,9 @@ "id": "023864", "content": "在平面直角坐标系 $x O y$ 中, 点 $A(-1,-2)$、$B(2,3)$、$C(-2,-1)$.\\\\\n(1) 求以线段 $AB$、$AC$ 为邻边的平行四边形两条对角线的长;\\\\\n(2) 设实数 $t$ 满足 $(\\overrightarrow{AB}-t \\overrightarrow{OC}) \\cdot \\overrightarrow{OC}=0$, 求实数 $t$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648593,7 +649118,9 @@ "id": "023865", "content": "如图, 已知 $BC=3BP$, $CA=3CQ$, 设 $\\overrightarrow{AB}=\\overrightarrow{c}$, $\\overrightarrow{BC}=\\overrightarrow{a}$, 用 $\\overrightarrow{a}$、$\\overrightarrow{c}$ 表示: $\\overrightarrow{PA}=$\\blank{50}. $\\overrightarrow{PQ}=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0.5) node [right] {$A$} coordinate (A);\n\\draw (2,2) node [above] {$C$} coordinate (C);\n\\draw ($(B)!{1/3}!(C)$) node [above left] {$P$} coordinate (P);\n\\draw ($(C)!{1/3}!(A)$) node [above right] {$Q$} coordinate (Q);\n\\draw (A)--(B)--(C)--cycle(Q)--(P)--(A);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648613,7 +649140,9 @@ "id": "023866", "content": "设 $\\overrightarrow{a}=(1,2)$, $\\overrightarrow{b}=(x, 1)$, $\\overrightarrow{u}=\\overrightarrow{a}+2 \\overrightarrow{b}$, $\\overrightarrow{v}=2 \\overrightarrow{a}+\\overrightarrow{b}$, 若 $\\overrightarrow{u}$ 与 $\\overrightarrow{v}$ 平行, 则实数 $x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648636,7 +649165,9 @@ "id": "023867", "content": "对于非零向量 $\\overrightarrow{a}$ 和 $\\overrightarrow{b}$, ``$\\overrightarrow{a}$ 和 $\\overrightarrow{b}$ 垂直''是``$|\\overrightarrow{a}+\\overrightarrow{b}|=|\\overrightarrow{a}-\\overrightarrow{b}|$''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648656,7 +649187,9 @@ "id": "023868", "content": "在平面直角坐标系 $x O y$ 中, 已知 $A(1,0)$、$B(0,1), C$ 为坐标平面内第一象限的点, 且 $\\angle AOC=\\dfrac{\\pi}{4}$, $|OC|=2$, 若 $\\overrightarrow{OC}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$, 则 $\\lambda+\\mu=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648676,7 +649209,9 @@ "id": "023869", "content": "已知 $|\\overrightarrow{a}|=|\\overrightarrow{b}|=2, \\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为 $60^{\\circ}$, 若 $\\overrightarrow{OP}=3 \\overrightarrow{a}+2 \\overrightarrow{b}$, $\\overrightarrow{OQ}=-2 \\overrightarrow{a}+3 \\overrightarrow{b}$, 则 $P$、$Q$ 两点之间的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648696,7 +649231,9 @@ "id": "023870", "content": "如图, 经过 $\\triangle OAB$ 的重心 $G$ 的直线与 $OA$、$OB$ 分别交于点 $P$、$Q$, 若 $\\overrightarrow{OP}=m \\overrightarrow{OA}$, $\\overrightarrow{OQ}=n \\overrightarrow{OB}$($m, n \\in \\mathbf{R}$), 则 $\\dfrac{1}{n}+\\dfrac{1}{m}$ 的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0) node [below] {$B$} coordinate (B);\n\\draw (2,2.5) node [above] {$O$} coordinate (O);\n\\filldraw ($1/3*(A)+1/3*(B)+1/3*(O)$) node [below] {$G$} coordinate (G) circle (0.03);\n\\draw (0,0.7) coordinate (S);\n\\draw ($(S)!2!(G)$) coordinate (T);\n\\path [name path = PQ, draw] (S)--(T);\n\\path [name path = AOB, draw] (A)--(O)--(B)--cycle;\n\\path [name intersections = {of = PQ and AOB, by = {P,Q}}];\n\\draw (P) node [above] {$P$};\n\\draw (Q) node [above] {$Q$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648719,7 +649256,9 @@ "id": "023871", "content": "已知向量 $\\overrightarrow{OA}$、$\\overrightarrow{OB}$ 的夹角为 $\\dfrac{\\pi}{3},|\\overrightarrow{OA}|=4$, $|\\overrightarrow{OB}|=1$, 若点 $M$ 在直线 $OB$ 上, 则 $|\\overrightarrow{OA}-\\overrightarrow{OM}|$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648739,7 +649278,9 @@ "id": "023872", "content": "已知向量 $\\overrightarrow{a}=(2,-1)$, $\\overrightarrow{b}=(2 m, 3 n)$ (其中 $m, n$ 是非零实数).\\\\\n(1) 若 $\\overrightarrow{a}\\perp \\overrightarrow{b}$, 求 $\\dfrac{m}{n}$ 的值;\\\\\n(2) 若 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$, 求 $\\dfrac{m}{n}$ 的值;\\\\\n(3) 若 $\\langle\\overrightarrow{a}, \\overrightarrow{b}\\rangle=\\arctan 2$, 求 $\\dfrac{m}{n}$ 的值;\\\\\n(4) 当 $\\dfrac{m}{n}$ 取 (3) 中所求得的值时, 是否总有 $\\langle\\overrightarrow{a}, \\overrightarrow{b}\\rangle=\\arctan 2$? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648759,7 +649300,9 @@ "id": "023873", "content": "若复数 $z$ 满足 $(1+\\mathrm{i}) z=2$, 则复数 $z$ 的虚部为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648779,7 +649322,9 @@ "id": "023874", "content": "若复数 $z=\\dfrac{2 \\mathrm{i}}{1+\\mathrm{i}}$, 则 $z$ 的共轭复数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648801,7 +649346,9 @@ "id": "023875", "content": "若 $x \\in \\mathbf{R}$, 则``$x=2$''是``复数 $z=(x^2-4)+(x+2) \\mathrm{i}$ 为纯虚数''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648821,7 +649368,9 @@ "id": "023876", "content": "若 $(1+\\mathrm{i}) \\cdot z=\\sqrt{3}\\mathrm{i}$, 则复数 $z$ 在复平面内对应的点位于第象限.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648841,7 +649390,9 @@ "id": "023877", "content": "已知 $z=1-\\mathrm{i}$, 计算 $\\dfrac{1+3 i}{\\overline{z}}+\\sqrt{2}|z| \\mathrm{i}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648861,7 +649412,9 @@ "id": "023878", "content": "若复数 $z$ 满足 $z-2 \\mathrm{i}=\\dfrac{1}{1-\\mathrm{i}}$, 则复数 $z$ 在复平面内的点到原点的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648881,7 +649434,9 @@ "id": "023879", "content": "若复数 $z_1=1+2 a \\mathrm{i}$, $z_2=a-\\mathrm{i}$($a \\in \\mathbf{R}$), 集合 $A=\\{z|| z-z_1 | \\leq \\sqrt{2}\\}$, $B=\\{z|| z-z_2 | \\leq 2 \\sqrt{2}\\}$, 且 $A \\cap B=\\varnothing$, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648903,7 +649458,10 @@ "id": "023880", "content": "若 $z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$), 且满足 $|z+2|=-x$, 则复数 $z$ 对应点的轨迹是 \\bracket{20}.\n\\fourch{圆}{抛物线}{椭圆}{双曲线}", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -648923,7 +649481,9 @@ "id": "023881", "content": "已知复数 $z$ 满足 $z \\cdot \\overline{z}+(1-2 \\mathrm{i}) z+(1+2 \\mathrm{i}) \\overline{z}=3$, 求 $|z|$ 的最大值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648943,7 +649503,9 @@ "id": "023882", "content": "已知平行四边形的三个顶点分别对应复数 $2 \\mathrm{i}, 4-4 \\mathrm{i}, 2+6 \\mathrm{i}$, 求第四个顶点对应的复数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -648965,7 +649527,9 @@ "id": "023883", "content": "若方程 $x^2+p x+q=0$($p, q \\in \\mathbf{R}$) 有一个根为 $1+\\mathrm{i}$, 则 $p q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -648989,7 +649553,9 @@ "id": "023884", "content": "在复数范围内因式分解:\\\\\n(1) $2 x^2-4 x+5=$\\blank{50}.\\\\\n(2) $x^4-16=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649011,7 +649577,9 @@ "id": "023885", "content": "若复数 $z$ 满足 $|z|-z-5 \\mathrm{i}-1=0$, 则 $z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649033,7 +649601,9 @@ "id": "023886", "content": "设 $a \\in \\mathbf{R}$, 若关于 $x$ 的方程 $2 x^2-8 x+a+1=0$ 的一个虚根的模是 $\\sqrt{5}$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649053,7 +649623,10 @@ "id": "023887", "content": "设 $a \\in \\mathbf{R}, x_1, x_2$ 是关于 $x$ 的实系数一元二次方程 $x^2+3 a x+a^2+1=0$ 的两个不同虚根, 求 $\\dfrac{1}{x_1}+\\dfrac{1}{x_2}$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -649073,7 +649646,9 @@ "id": "023888", "content": "已知关于 $x$ 的方程 $x^2+x+p=0$ 的两个复数根 $\\alpha, \\beta$ 满足 $|\\alpha-\\beta|=4$, 求实数 $p$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -649095,7 +649670,9 @@ "id": "023889", "content": "平行六面体 $ABCD-A_1B_1C_1D_1$ 中, $M$ 为 $AC$ 与 $BD$ 的交点, 设 $\\overrightarrow{A_1B_1}=\\overrightarrow{a}$, $\\overrightarrow{A_1D_1}=\\overrightarrow{b}$,\n$\\overrightarrow{AA_1}=\\overrightarrow{c}$, 则 $\\overrightarrow{B_1M}=$\\blank{50}. (用 $\\overrightarrow{a}$、$\\overrightarrow{b}$、$\\overrightarrow{c}$ 表示)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649117,7 +649694,9 @@ "id": "023890", "content": "下列所有命题中正确的命题序号是\\blank{50}.\\\\\n\\textcircled{1} 若 $|\\overrightarrow{a}|=|\\overrightarrow{b}|$, 则 $\\overrightarrow{a}, \\overrightarrow{b}$ 的长度相同, 方向相反或相同; \\textcircled{2} 若 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 是互为负向量, 则 $|\\overrightarrow{a}|=|\\overrightarrow{b}|$; \\textcircled{3} 空间向量的加法满足结合律; \\textcircled{4} 在四边形 $ABCD$ 中, 一定有 $\\overrightarrow{AB}+\\overrightarrow{AD}=\\overrightarrow{AC}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649137,7 +649716,9 @@ "id": "023891", "content": "在正四面体 $O-ABC$ 中, $\\overrightarrow{OA}=\\overrightarrow{a}$, $\\overrightarrow{OB}=\\overrightarrow{b}$, $\\overrightarrow{OC}=\\overrightarrow{c}, D$ 为 $BC$ 的中点, $E$ 为 $AD$ 的中点, 则 $\\overrightarrow{OE}=$\\blank{50}. (用 $\\overrightarrow{a}$、$\\overrightarrow{b}$、$\\overrightarrow{c}$ 表示)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649159,7 +649740,9 @@ "id": "023892", "content": "给出下列命题: \\textcircled{1} 空间任意两个向量 $\\overrightarrow{a}$、$\\overrightarrow{b}$ 一定是共面的; \\textcircled{2} $\\overrightarrow{a}$、$\\overrightarrow{b}$ 为空间两个向量, 则 $|\\overrightarrow{a}|=|\\overrightarrow{b}| \\Leftrightarrow \\overrightarrow{a}=\\overrightarrow{b}$; \\textcircled{3} 若 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$, 则 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 所在直线平行; \\textcircled{4} 如果 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$、$\\overrightarrow{b}\\parallel \\overrightarrow{c}$, 那么 $\\overrightarrow{a}\\parallel \\overrightarrow{c}$. 其中假命题的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649179,7 +649762,9 @@ "id": "023893", "content": "已知向量 $\\overrightarrow{AB}$、$\\overrightarrow{AC}$、$\\overrightarrow{BC}$ 满足 $|\\overrightarrow{AB}|=|\\overrightarrow{AC}|+|\\overrightarrow{BC}|$, 则有\\bracket{20}.\n\\fourch{$\\overrightarrow{AB}=\\overrightarrow{AC}+\\overrightarrow{BC}$}{$\\overrightarrow{AB}=-\\overrightarrow{AC}-\\overrightarrow{BC}$}{$\\overrightarrow{AC}$ 与 $\\overrightarrow{BC}$ 同向}{$\\overrightarrow{AC}$ 与 $\\overrightarrow{CB}$ 同向}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -649199,7 +649784,9 @@ "id": "023894", "content": "已知向量 $\\overrightarrow{a}=\\overrightarrow{x}+2 \\overrightarrow{y}-\\overrightarrow{z}$, 且 $|\\overrightarrow{x}|=|\\overrightarrow{y}|=|\\overrightarrow{z}|=1$, 若向量 $\\overrightarrow{x}$、$\\overrightarrow{y}$、$\\overrightarrow{z}$ 两两垂直, 则 $|\\overrightarrow{a}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649219,7 +649806,9 @@ "id": "023895", "content": "已知平行六面体 $ABCD-A' B' C' D'$ 中, $AB=4$, $AD=3$, $AA'=5$, $\\angle BAD=90^{\\circ}$, $\\angle BAA'=\\angle DAA'=60^{\\circ}$, 则$|AC'|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649241,7 +649830,9 @@ "id": "023896", "content": "如图, 已知 $M$、$N$ 分别为四面体 $A-BCD$ 的面 $BCD$ 与面 $ACD$ 的重心, $G$ 为 $AM$ 上一点, 且 $GM: GA=1: 3$. 求证: $B$、$G$、$N$ 三点共线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ($1/3*(D)+1/3*(B)+1/3*(C)$) ++ (0,{2*sqrt(6)/3},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)(A)--(C)(A)--(D)(B)--(C)--(D);\n\\draw [dashed] (B)--(D);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\draw [dashed] (B)--(D);\n\\filldraw ($1/3*(D)+1/3*(B)+1/3*(C)$) node [below] {$M$} coordinate (M) circle (0.03);\n\\filldraw ($1/3*(D)+1/3*(A)+1/3*(C)$) node [below] {$N$} coordinate (N) circle (0.03);\n\\filldraw ($(A)!0.75!(M)$) node [left] {$G$} coordinate (G) circle (0.03);\n\\draw [dashed] (A)--(M);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -649263,7 +649854,9 @@ "id": "023897", "content": "设 $\\theta \\in \\mathbf{R}$. 若 $A(3 \\cos \\theta, 0,1)$、$B(3 \\sin \\theta, 1,1)$, 则 $|\\overrightarrow{AB}|$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649283,7 +649876,9 @@ "id": "023898", "content": "空间三点 $P(-2,0,2)$、$Q(-1,1,2)$、$R(-3,0,4)$, 则 $\\overrightarrow{PQ}$ 与 $\\overrightarrow{PR}$ 的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649305,7 +649900,9 @@ "id": "023899", "content": "已知向量 $\\overrightarrow{a}=(x+3, y+1,3 x+2)$, $\\overrightarrow{b}=(2,0,5)$, 若 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$, 则 $x=$\\blank{50}, $y=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649325,7 +649922,9 @@ "id": "023900", "content": "已知向量 $\\overrightarrow{a}=(1,-1,2)$, 向量 $\\overrightarrow{AB}$ 与 $\\overrightarrow{a}$ 平行且 $|\\overrightarrow{AB}|=2 \\sqrt{6}$, 则 $\\overrightarrow{AB}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649345,7 +649944,9 @@ "id": "023901", "content": "已知向量 $\\overrightarrow{a}=(-1,0,1)$, $\\overrightarrow{b}=(1,2,3)$, 若 $k \\overrightarrow{a}-\\overrightarrow{b}$ 与 $\\overrightarrow{b}$ 垂直, 则实数 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649365,7 +649966,9 @@ "id": "023902", "content": "已知三个力 $\\overrightarrow{OF}_1=(1,2,3)$, $\\overrightarrow{OF}_2=(-2,3,-1)$, $\\overrightarrow{OF}_3=(3,-4,5)$, 若三个力作用于同一物体, 使物体从点 $M_1(1,-2,1)$ 移动到 $M_2(3,1,2)$, 则合力所作的功为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649385,7 +649988,9 @@ "id": "023903", "content": "已知 $A$、$B$、$C$ 三点不共线, $O$ 为平面 $ABC$ 外一点, 下列条件中能确定 $P$、$A$、$B$、$C$ 四点共面的是\\bracket{20}.\n\\twoch{$\\overrightarrow{OP}=\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}$}{$\\overrightarrow{OP}=2 \\overrightarrow{OA}-\\overrightarrow{OB}-\\overrightarrow{OC}$}{$\\overrightarrow{OP}=\\dfrac{1}{5}\\overrightarrow{OA}+\\dfrac{1}{3}\\overrightarrow{OB}+\\dfrac{1}{2}\\overrightarrow{OC}$}{$\\overrightarrow{OP}=\\dfrac{1}{3}\\overrightarrow{OA}+\\dfrac{1}{3}\\overrightarrow{OB}+\\dfrac{1}{3}\\overrightarrow{OC}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -649407,7 +650012,9 @@ "id": "023904", "content": "与向量 $\\overrightarrow{AB}=(-4,6,-1)$, $\\overrightarrow{AC}=(4,3,-2)$ 都垂直的单位向量的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649427,7 +650034,9 @@ "id": "023905", "content": "(1) 设 $m, n \\in \\mathbf{R}$. 若 $A(-1,2,3)$、$B(2,1,4)$、$C(m, n, 1)$ 三点共线, 求 $m+n$ 的值;\\\\\n(2) 已知向量 $\\overrightarrow{a}=(2,-1,3)$, $\\overrightarrow{b}=(-1,4,-2)$, $\\overrightarrow{c}=(7,5, \\lambda)$ 共面, 求实数 $\\lambda$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -649447,7 +650056,9 @@ "id": "023906", "content": "若 $A(1,0,0)$、$B(0,2,0)$、$C(0,0,3)$ 为空间三点, 则平面 $ABC$ 的一个法向量为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649470,7 +650081,9 @@ "id": "023907", "content": "设平面 $\\alpha$ 的一个法向量为 $\\overrightarrow{n}=(-1,2,4)$, 直线 $l$ 的一个方向向量为 $\\overrightarrow{d}=(x, y, z)$.\\\\\n(1) 当 $l \\parallel $ 平面 $\\alpha$ 时, $x, y, z$ 满足\\blank{50};\\\\\n(2) 当 $l \\perp$ 平面 $\\alpha$ 时, $x, y, z$ 满足\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649492,7 +650105,9 @@ "id": "023908", "content": "``直线的方向向量与平面的法向量垂直''是``直线与平面平行''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649512,7 +650127,9 @@ "id": "023909", "content": "异面直线所成的角 $\\theta$ 与它们方向向量的夹角 $\\varphi$ 之间的关系为\\blank{50}; 直线与平面所成的角 $\\theta$ 与直线方向向量和平面法向量的夹角 $\\varphi$ 之间的关系为\\blank{50}; 二面角的大小 $\\theta$ 与两平面法向量的夹角 $\\varphi$ 之间的关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649532,7 +650149,9 @@ "id": "023910", "content": "如图所示是一个直三棱柱 (以 $A_1B_1C_1$ 为底面) 被一平面所截得到的几何体, 截面为 $ABC$. 已知 $A_1B_1=B_1C_1=1$, $\\angle A_1B_1C_1=90^{\\circ}$, $AA_1=4$, $BB_1=2$, $CC_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(2)/2},0,0) node [left] {$A_1$} coordinate (A_1);\n\\draw ({sqrt(2)/2},0,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (0,0,{sqrt(2)/2}) node [below] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (0,4,0) node [left] {$A$} coordinate (A);\n\\draw (B_1) ++ (0,2,0) node [right] {$B$} coordinate (B);\n\\draw (C_1) ++ (0,3,0) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(B)$) node [below right] {$O$} coordinate (O);\n\\draw (A_1)--(B_1)--(C_1)--(C)--(A)--cycle(A)--(B)--(C)(O)--(C)(B)--(B_1);\n\\draw [dashed] (A_1)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 设点 $O$ 是 $AB$ 的中点, 证明: $OC \\parallel $ 平面 $A_1B_1C_1$;\\\\\n(2) 求二面角 $B-AC-A_1$ 的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -649552,7 +650171,9 @@ "id": "023911", "content": "如图, 试用适当的符号表示下列点、直线和平面之间的关系:\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) --++ (3,0) --++ (-1.5,1.5) --++ (-3,0) node [below right = 0 and 0.4] {$\\beta$} --cycle;\n\\draw (3,0) --++ (-1.5,-1.5) --++ (-3,0) node [above right = 0 and 0.4] {$\\alpha$} -- (0,0);\n\\draw (1,0) node [above] {$D$} coordinate (D) --++ (-135:1.5) node [right] {$C$} coordinate (C);\n\\draw (2,0) node [below] {$B$} coordinate (B) --++ (135:1.5) node [right] {$A$} coordinate (A);\n\\end{tikzpicture}\n\\end{center}\n(1) 点 $C$ 与平面 $\\beta$:\\blank{100};\\\\\n(2) 点 $A$ 与平面 $\\alpha$:\\blank{100};\\\\\n(3) 直线 $AB$ 与平面 $\\alpha$:\\blank{100};\\\\\n(4) 直线 $CD$ 与平面 $\\alpha$:\\blank{100};\\\\\n(5) 平面 $\\alpha$ 与平面 $\\beta$:\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649572,7 +650193,9 @@ "id": "023912", "content": "下列所有正确命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 过三点确定一个平面;\\\\\n\\textcircled{2} 四边形是平面图形;\\\\\n\\textcircled{3} 三条直线两两相交则确定一个平面;\\\\\n\\textcircled{4} 两个相交平面把空间分成四个区域.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649592,7 +650215,9 @@ "id": "023913", "content": "下图是正方体或四面体, 若 $P$、$Q$、$R$、$S$ 分别是所在棱的中点, 则这四个点共面的图共有\\blank{50}个.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$R$} coordinate (R);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$S$} coordinate (S);\n\\draw ($(D_1)!0.5!(A_1)$) node [left] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$Q$} coordinate (Q);\n\\foreach \\i in {P,Q,R,S}\n{\\filldraw (\\i) circle (0.03);};\n\\draw (1.2,-0.8) node {\\textcircled{1}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(C)!0.5!(B)$) node [right] {$R$} coordinate (R);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$S$} coordinate (S);\n\\draw ($(D_1)!0.5!(A_1)$) node [left] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(B)$) node [below] {$Q$} coordinate (Q);\n\\foreach \\i in {P,Q,R,S}\n{\\filldraw (\\i) circle (0.03);};\n\\draw (1.2,-0.8) node {\\textcircled{2}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\draw (0,0,0) coordinate (A);\n\\draw (2,0,0) coordinate (B);\n\\draw (1,0,{-sqrt(3)}) coordinate (C);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(C)$) ++ (0,{2*sqrt(6)/3},0) coordinate (D);\n\\draw (A)--(B)--(D)--cycle;\n\\draw [dashed] (A)--(C)--(B)(C)--(D);\n\\draw ($(C)!0.5!(B)$) node [below] {$R$} coordinate (R);\n\\draw ($(A)!0.5!(C)$) node [below] {$S$} coordinate (S);\n\\draw ($(D)!0.5!(A)$) node [left] {$P$} coordinate (P);\n\\draw ($(D)!0.5!(B)$) node [right] {$Q$} coordinate (Q);\n\\foreach \\i in {P,Q,R,S}\n{\\filldraw (\\i) circle (0.03);};\n\\draw (1.2,-0.8) node {\\textcircled{3}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\draw (0,0,0) coordinate (A);\n\\draw (2,0,0) coordinate (B);\n\\draw (1,0,{-sqrt(3)}) coordinate (C);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(C)$) ++ (0,{2*sqrt(6)/3},0) coordinate (D);\n\\draw (A)--(B)--(D)--cycle;\n\\draw [dashed] (A)--(C)--(B)(C)--(D);\n\\draw ($(C)!0.5!(B)$) node [below] {$R$} coordinate (R);\n\\draw ($(A)!0.5!(B)$) node [below] {$S$} coordinate (S);\n\\draw ($(D)!0.5!(A)$) node [left] {$P$} coordinate (P);\n\\draw ($(D)!0.5!(B)$) node [right] {$Q$} coordinate (Q);\n\\foreach \\i in {P,Q,R,S}\n{\\filldraw (\\i) circle (0.03);};\n\\draw (1.2,-0.8) node {\\textcircled{4}};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649612,7 +650237,9 @@ "id": "023914", "content": "若 $A$、$B$、$C$ 表示不同的点, $l$ 表示直线, $\\alpha$、$\\beta$ 表示不同的平面, 则下列所有正确推理的序号为\\blank{50}.\\\\\n\\textcircled{1} $A \\in l$, $A \\in \\alpha$, $B \\in l$, $B \\in \\alpha \\Rightarrow l \\subset \\alpha$;\\\\\n\\textcircled{2} $A \\in \\alpha$, $A \\in \\beta$, $B \\in \\alpha$, $B \\in \\beta \\Rightarrow \\alpha \\cap \\beta=AB$;\\\\\n\\textcircled{3} $l$ 不在平面 $\\alpha$ 上, $A \\in l \\Rightarrow A \\notin \\alpha$;\\\\\n\\textcircled{4} $A \\in \\alpha$, $A \\in l, l$ 不在平面 $\\alpha$ 上 $\\Rightarrow l \\cap \\alpha=A$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649632,7 +650259,9 @@ "id": "023915", "content": "下列命题中正确的是\\bracket{20}.\n\\onech{空间不同的三点确定一个平面}{空间两两相交的三条直线确定一个平面}{空间有三个角为直角的四边形一定是平面图形}{和同一条直线相交的三条平行直线一定在同一平面上}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -649652,7 +650281,9 @@ "id": "023916", "content": "如图, 平面 $ABEF \\perp$ 平面 $ABCD$, 四边形 $ABEF$ 与四边形 $ABCD$ 都是直角梯形, $\\angle BAD=\\angle FAB=90^{\\circ}$, $BC \\parallel AD$ 且 $BC=\\dfrac{1}{2}AD$, $BE \\parallel AF$ 且 $BE=\\dfrac{1}{2}AF$, $G$、$H$ 分别为 $FA$、$FD$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,3,0) node [above] {$F$} coordinate (F);\n\\draw (0,0,1.5) node [below] {$B$} coordinate (B);\n\\draw (B) ++ (1.5,0,0) node [below] {$C$} coordinate (C);\n\\draw (B) ++ (0,1.5,0) node [left] {$E$} coordinate (E);\n\\draw (B)--(C)--(D)--(F)--(E)--cycle(E)--(C);\n\\draw [dashed] (A)--(D)(A)--(F)(A)--(B);\n\\draw ($(A)!0.5!(F)$) node [above right] {$G$} coordinate (G);\n\\draw ($(F)!0.5!(D)$) node [above right] {$H$} coordinate (H);\n\\draw (C)--(H);\n\\draw [dashed] (B)--(G)--(H);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 四边形 $BCHG$ 为平行四边形;\\\\\n(2) 判断 $C$、$D$、$F$、$E$ 四点是否共面? 为什么?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -649672,7 +650303,9 @@ "id": "023917", "content": "如图所示, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, 若 $M$、$N$ 分别为棱 $C_1D_1$、$C_1C$ 的中点, 则以下所有正确结论的序号是\\blank{50}.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [below] {$D$} 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 ($(C)!0.5!(C_1)$) node [right] {$N$} coordinate (N);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$M$} coordinate (M);\n\\draw (B_1)--(M)(B)--(N);\n\\draw [dashed] (A)--(M);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 直线 $AM$ 与 $CC_1$ 是相交直线;\\\\\n\\textcircled{2} 直线 $AM$ 与 $BN$ 是平行直线;\\\\\n\\textcircled{3} 直线 $BN$ 与 $MB_1$ 是异面直线;\\\\\n\\textcircled{4} 直线 $AM$ 与 $DD_1$ 是异面直线.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649692,7 +650325,9 @@ "id": "023918", "content": "如图是表示一个正方体表面的一种平面展开图, 图中的四条线段 $AB$、$CD$、$EF$ 和 $GH$ 在原正方体中相互异面的有\\blank{50}对.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 1]\n\\draw (0,0) -- (3,0) (0,1) -- (4,1) (1,-1) -- (2,-1) (2,2) node [above left] {$C$} -- (4,2);\n\\draw (0,0) -- (0,1) (1,-1) -- (1,1) (2,-1) -- (2,2) (3,0) -- (3,2) (4,1) -- (4,2);\n\\draw (2,-1) node [below right] {$F$} -- (1,0) node [below left] {$E$} coordinate (E) (2,2) -- (3,1) node [below right] {$D$};\n\\draw (0,0) node [below left] {$H$} -- (1,1) node [above] {$G$};\n\\draw (3,2) node [above] {$A$} -- (4,1) node [below right] {$B$};\n\\draw [dashed] (1,0) --++ (45:1/2) --++ (1,0) --++ (225:1/2);\n\\draw [dashed] (1,0) ++ (0,1) --++ (45:1/2) --++ (1,0) --++ (225:1/2);\n\\draw [dashed] (1,0) ++ (45:1/2) --++ (0,1) ++ (1,0) --++ (0,-1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649712,7 +650347,9 @@ "id": "023919", "content": "如图, 棱长为 $1$ 的正方体 $ABCD-A_1B_1C_1D_1$, 若 $E$ 为棱 $BC$ 的中点, $F$ 为棱 $DD_1$ 的中点. 则异面直线 $EF$ 与 $BD_1$ 所成角的余弦值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,\\l,0) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B_1) -- (C_1) -- (D_1) -- (A_1) -- cycle;\n\\draw (B) -- (B_1) (C) -- (C_1) (D) -- (D_1);\n\\draw [dashed] (A) -- (A_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(D_1)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (E)--(F)(B)--(D_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649732,7 +650369,9 @@ "id": "023920", "content": "如图, 直线 $a$、$b$ 相交于点 $O$ 且 $a$、$b$ 成 $60^{\\circ}$ 角, 过点 $O$ 与 $a$、$b$ 都成 $60^{\\circ}$ 的直线有\\blank{50}条.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (30:2) node [right] {$a$} coordinate (a);\n\\draw (-30:2) node [right] {$b$} coordinate (b);\n\\draw (b) -- ($(b)!2!(O)$) (a) -- ($(a)!2!(O)$);\n\\draw pic [draw, \"$60^\\circ$\", scale = 0.5, angle eccentricity = 2.5] {angle = b--O--a};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649752,7 +650391,9 @@ "id": "023921", "content": "在正方体 $ABCD-A_1B_1C_1D_1$ 中, 若 $E$、$F$ 分别是 $AA_1$ 与 $CC_1$ 的中点, 则直线 $ED$ 与 $D_1F$ 所成角的大小是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649772,7 +650413,9 @@ "id": "023922", "content": "若空间中有两条直线, 则``这两条直线为异面直线''是``这两条直线没有公共点''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -649792,7 +650435,9 @@ "id": "023923", "content": "过正方体 $ABCD-A_1B_1C_1D_1$ 的顶点 $A$ 作直线 $l$, 使 $l$ 与棱 $AB$、$AD$、$AA_1$ 所成的角都相等, 这样的直线 $l$ 可以作 \\bracket{20}.\n\\fourch{1 条}{2 条}{3 条}{4 条}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -649812,7 +650457,9 @@ "id": "023924", "content": "如图, 若 $P$ 是 $\\triangle ABC$ 所在平面外一点, $PA \\neq PB$, $PN \\perp AB, N$ 为垂足. $M$ 为 $AB$ 的中点, 求证: $PN$ 与 $MC$ 为异面直线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (1.4,-0.8) node [below] {$B$} coordinate (B);\n\\draw (1.5,1.3) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw ($(A)!0.7!(M)$) node [below left] {$N$} coordinate (N);\n\\draw (A)--(B)--(C)--(P)--cycle(B)--(P)(P)--(N);\n\\draw [dashed] (C)--(M)(A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -649832,7 +650479,9 @@ "id": "023925", "content": "如图, 已知 $\\angle BAC=90^{\\circ}$, $PC \\perp$ 平面 $ABC$, 则在 $\\triangle ABC$、$\\triangle PAC$ 的边所在的直线中, 与 $PC$垂直的直线有\\blank{50}; 与 $AP$ 垂直的直线有\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,-2) node [above right] {$C$} coordinate (C);\n\\draw (C) ++ (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle(A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649852,7 +650501,9 @@ "id": "023926", "content": "给出下列四个命题:\\\\\n\\textcircled{1} 垂直于同一直线的两条直线互相平行;\\\\\n\\textcircled{2} 垂直于同一平面的两个平面互相平行;\\\\\n\\textcircled{3} 若直线 $l_1$、$l_2$ 与同一平面所成的角相等, 则 $l_1, l_2$ 互相平行;\\\\\n\\textcircled{4} 若直线 $l_1$、$l_2$ 是异面直线, 则与 $l_1$、$l_2$ 都相交的两条直线是异面直线.\n其中假命题的个数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649872,7 +650523,9 @@ "id": "023927", "content": "在正三棱柱 $ABC-A_1B_1C_1$ 中, 所有棱长均为 $1$ , 则点 $B_1$ 到平面 $ABC_1$ 的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649892,7 +650545,9 @@ "id": "023928", "content": "如图, 在四棱锥 $P-ABCD$ 中, $AD \\perp$ 平面 $PDC, AD \\parallel BC$, $PD \\perp PB$, $AD=1$, $BC=3$, $CD=4$, $PD=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (1,0,0) node [below] {$A$} coordinate (A);\n\\draw (0,0,-4) node [below] {$C$} coordinate (C);\n\\draw (3,0,-4) node [right] {$B$} coordinate (B);\n\\draw (0,{sqrt(3)},-1) node [above] {$P$} coordinate (P);\n\\draw (D)--(A)--(B)--(P)--cycle(P)--(A);\n\\draw [dashed] (D)--(C)--(B)(C)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线 $AP$ 与 $BC$ 所成角的余弦值;\\\\\n(2) 求证: $PD \\perp$ 平面 $PBC$;\\\\\n(3) 求直线 $AB$ 与平面 $PBC$ 所成角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -649912,7 +650567,9 @@ "id": "023929", "content": "已知直线$l$和平面$\\alpha$、$\\beta$, 若$l\\subset \\alpha$, 则``$l\\perp \\beta$''是``$\\alpha\\perp \\beta$''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649934,7 +650591,9 @@ "id": "023930", "content": "设两个平面 $\\alpha, \\beta$, 直线 $l$, 下列三个条件: \\textcircled{1} $l \\perp \\alpha$; \\textcircled{2} $l \\parallel \\beta$; \\textcircled{3} $\\alpha \\perp \\beta$. 若以其中两个作为前提, 另一个作为结论, 则可构成三个命题, 这三个命题中正确的命题个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649954,7 +650613,9 @@ "id": "023931", "content": "有以下命题:\\\\\n\\textcircled{1} 若直线 $m, n$ 都平行于平面 $\\alpha$, 则 $m \\parallel n$;\\\\\n\\textcircled{2} 设 $a-l-\\beta$ 是直二面角, 若直线 $m \\perp l$, 则 $m \\perp \\beta$;\\\\\n\\textcircled{3} 若直线 $m$、$n$ 在平面 $\\alpha$ 内的射影依次是一个点和一条直线, 且 $m \\perp n$, 则 $n$ 在 $\\alpha$ 内或 $n$ 与 $\\alpha$ 平行;\\\\\n\\textcircled{4} 设 $m$、$n$ 是异面直线, 若 $m$ 与平面 $\\alpha$ 平行, 则 $n$ 与 $\\alpha$ 相交.\n其中所有真命题的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649974,7 +650635,9 @@ "id": "023932", "content": "以等腰直角三角形 $ABC$ 斜边 $AB$ 中线 $CD$ 为棱, 若将 $\\triangle ABC$ 折叠, 使平面 $ACD \\perp$ 平面 $BCD$, 则 $AC$ 与 $BC$ 夹角 $\\angle ACB$ 的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -649996,7 +650659,9 @@ "id": "023933", "content": "正方体 $ABCD-A' B' C' D'$ 中, 若 $E$、$F$ 分别是 $BC$、$CD$ 的中点, 则截面 $B' D' EF$ 与半平面 $B' ECC'$ 所成二面角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650016,7 +650681,9 @@ "id": "023934", "content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, $M$、$N$、$P$ 分别是 $C_1C$、$B_1C_1$ 、 $C_1D_1$ 的中点, 求证:平面 $MNP \\parallel$ 平面 $A_1BD$.\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] {$B$} 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 ($(C)!0.5!(C_1)$) node [right] {$M$} coordinate (M);\n\\draw ($(B_1)!0.5!(C_1)$) node [below] {$N$} coordinate (N);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$P$} coordinate (P);\n\\draw (M)--(N)--(P);\n\\draw [dashed] (M)--(P);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -650038,7 +650705,9 @@ "id": "023935", "content": "如图, 在直四棱柱 $ABCD-A_1B_1C_1D_1$ 中, 四边形 $ABCD$ 为菱形、 $E$ 为棱 $A_1A$ 的中点, 且 $O$ 为 $A_1C_1$ 与 $B_1D_1$ 的交点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below] {$B$} 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 ($(A)!0.5!(A_1)$) node [left] {$E$} coordinate (E);\n\\draw (B)--(C_1)(E)--(B_1)(A_1)--(C_1)(B_1)--(D_1);\n\\draw ($(A_1)!0.5!(C_1)$) node [above] {$O$} coordinate (O);\n\\draw [dashed] (E)--(D_1)(A)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $OE \\parallel $ 平面 $ABC_1$;\\\\\n(2) 求证: 平面 $AA_1C_1\\perp$ 平面 $B_1D_1E$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -650058,7 +650727,9 @@ "id": "023936", "content": "若正三棱柱 $ABC-A_1B_1C_1$ 的侧棱长与底面边长相等, 则 $AB_1$ 与侧面 $ACC_1A_1$ 所成角的正弦值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650080,7 +650751,9 @@ "id": "023937", "content": "如图, 正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $1$ , 将该正方体沿对角面 $BB_1D_1D$ 切成两块, 再将这两块拼接成一个不是正方体的四棱柱, 那么所得四棱柱的全面积为\\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\\fill [pattern = north east lines] (B)--(D)--(D_1)--(B_1)--cycle;\n\\draw (B_1)--(D_1);\n\\draw [dashed] (B)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650100,7 +650773,9 @@ "id": "023938", "content": "若某圆柱底面的半径为 $1$, 高为 $2$, 则该圆柱的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650123,7 +650798,9 @@ "id": "023939", "content": "已知一个正方体与一个圆柱等高, 若侧面积相等, 则这个正方体和圆柱的体积之比为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650145,7 +650822,9 @@ "id": "023940", "content": "正三棱柱 $ABC-A_1B_1C_1$ 的各棱长都 $2$, 若 $E$、$F$ 分别是 $AB$、$A_1C_1$ 的中点, 则 $EF$ 的长是\\bracket{20}.\n\\fourch{$2$}{$\\sqrt{3}$}{$\\sqrt{5}$}{$\\sqrt{7}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -650165,7 +650844,9 @@ "id": "023941", "content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, 点 $P$ 是线段 $BC_1$ 上的一个动点, 有下列三个结论: \\textcircled{1} $A_1P \\parallel $ 平面 $ACD_1$; \\textcircled{2} $B_1D \\perp A_1P$; \\textcircled{3} 平面 $A_1PB \\perp$ 平面 $B_1CD$. 其中所有正确结论的序号是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (B)--(C_1);\n\\draw ($(B)!0.6!(C_1)$) node [right] {$P$} coordinate (P);\n\\draw [dashed] (A_1)--(P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{2}\\textcircled{3}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{2}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -650185,7 +650866,9 @@ "id": "023942", "content": "如图, $AA_1$ 是圆柱的一条母线, $AB$ 是圆柱的底面直径, $C$ 在圆柱下底面圆周上, $M$ 是线段 $A_1C$ 的中点. 已知 $AA_1=AC=4$, $BC=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (-2.5,0) node [left] {$A$} coordinate (A);\n\\draw (2.5,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,4) node [left] {$A_1$} coordinate (A_1);\n\\draw (-70:2.5 and {2.5/4}) node [below] {$C$} coordinate (C);\n\\draw ($(A_1)!0.5!(C)$) node [above right] {$M$} coordinate (M);\n\\draw [dashed] (A)--(C)--(B)--cycle(A_1)--(C)(A)--(M);\n\\draw [dashed] (A) arc (180:0:2.5 and {2.5/4});\n\\draw (A)--++(0,4)(B)--++(0,4)(A_1)arc (180:-180:2.5 and {2.5/4})(A) arc (180:360:2.5 and {2.5/4});\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆柱的侧面积;\\\\\n(2) 求证: $BC \\perp AM$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -650205,7 +650888,9 @@ "id": "023943", "content": "若母线长是 $4$ 的圆锥的轴截面的面积是 $8$, 则该圆锥的高为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650225,7 +650910,9 @@ "id": "023944", "content": "若棱台上、下底面的对应边之比为 $1: 2$, 则上、下底面的面积之比是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650245,7 +650932,9 @@ "id": "023945", "content": "已知一个高为 $1$ 的三棱锥, 若各侧棱长都相等, 底面是边长为 $2$ 的等边三角形, 则三棱锥的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650265,7 +650954,9 @@ "id": "023946", "content": "已知圆锥的顶点为 $S$, 母线 $SA$、$SB$ 所成角的余弦值为 $\\dfrac{7}{8}$, $SA$ 与圆锥底面所成角为 $45^{\\circ}$, 若 $\\triangle SAB$ 的面积为 $5 \\sqrt{15}$, 则该圆锥的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650285,7 +650976,9 @@ "id": "023947", "content": "金字塔的形状可视为一个正四棱锥.若以该四棱锥的高为边长的正方形面积等于该四棱锥一个侧面三角形的面积, 则其侧面三角形底边上的高与底面正方形的边长的比值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650307,7 +651000,9 @@ "id": "023948", "content": "已知体积为 $\\dfrac{\\sqrt{3}}{24}a^3$ 的正三棱锥 $P-ABC$ 中, $AB=a$, 则侧棱与底面所成角的大小为\\blank{50};侧面与底面所成二面角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650327,7 +651022,9 @@ "id": "023949", "content": "如图, $SA$、$SB$ 是圆锥 $SO$ 的两条母线, $O$ 是底面圆的圆心, 底面圆的半径为 $10$, $C$ 是 $SB$ 中点, $\\angle AOB=60^{\\circ}, AC$ 与底面所成角的大小为 $45^{\\circ}$. 求此圆锥的侧面积和体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (0,0) node [above right] {$O$} coordinate (O);\n\\draw (0,{2*sqrt(3)}) node [above] {$S$} coordinate (S);\n\\draw (-120:2 and 0.5) node [below] {$B$} coordinate (B);\n\\draw ($(S)!0.5!(B)$) node [left] {$C$} coordinate (C);\n\\draw (A)--(S)--(2,0)(S)--(B)(A) arc (180:360:2 and 0.5);\n\\draw [dashed] (S)--(O)(A)--++(4,0)(B)--(O)(A)--(C)(A) arc (180:0:2 and 0.5);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -650349,7 +651046,9 @@ "id": "023950", "content": "若一个圆柱和一个圆锥的轴截面分别是边长为 $a$ 的正方形和正三角形, 则它们的表面积之比为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650369,7 +651068,9 @@ "id": "023951", "content": "若在长方体 $ABCD-A_1B_1C_1D_1$ 中, $A_1D$ 和 $CD_1$ 与底面所成的角分别为 $30^{\\circ}$ 和 $45^{\\circ}$, 则异面直线 $A_1D$ 和 $CD_1$ 所成角的余弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650391,7 +651092,9 @@ "id": "023952", "content": "如图, $S$ 是圆锥的顶点, $O$ 是底面圆的圆心, $AB$、$CD$ 是底面圆的两条直径. 若 $AB \\perp CD$, $SO=4$, $OB=2, P$ 为 $SB$ 的中点, 则点 $S$ 到平面 $PCD$ 的距离为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (0,4) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(B)$) node [above right] {$P$} coordinate (P);\n\\draw (120:2 and 0.5) node [above] {$C$} coordinate (C);\n\\draw (-60:2 and 0.5) node [below] {$D$} coordinate (D);\n\\draw (A)--(S)--(B)(A) arc (180:360:2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw [dashed] (A)--(B)(C)--(D)(C)--(P)--(D)(O)--(S);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650413,7 +651116,9 @@ "id": "023953", "content": "若正四棱柱 $ABCD-A_1B_1C_1D_1$ 中, $BE=\\dfrac{1}{4}BB_1=2$, $4AB=3AA_1$,\n则该四棱柱被过点 $A_1, C, E$ 的平面截得的截面面积为\\bracket{20}.\n\\fourch{$24 \\sqrt{2}$}{$36$}{$12 \\sqrt{19}$}{$6 \\sqrt{95}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -650433,7 +651138,9 @@ "id": "023954", "content": "已知球的表面积为 $64 \\pi \\mathrm{cm}^2$, 若用一个平面截球, 使截面圆的半径为 $2 \\mathrm{cm}$, 则截面与球心的距离是\\blank{50} $\\mathrm{cm}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650453,7 +651160,9 @@ "id": "023955", "content": "若棱长为 $2$ 的正方体的八个顶点在同一球面上, 则此球的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650473,7 +651182,9 @@ "id": "023956", "content": "若三球的半径之比是 $1: 2: 3$, 则半径最大的球体积是其余两球体积和的倍.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -650493,7 +651204,9 @@ "id": "023957", "content": "把一个表面积为 $16 \\pi \\mathrm{cm}^2$ 的实心铁球铸成一个底面半径与球的半径一样的圆锥, 若没有任何损耗, 则圆锥的高是\\blank{50}$\\mathrm{cm}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650513,7 +651226,9 @@ "id": "023958", "content": "右图是一个装有水的倒圆锥形杯子, 杯子口径 $6 \\mathrm{cm}$, 高 $8 \\mathrm{cm}$ (不含杯脚), 已知水的高度是 $4 \\mathrm{cm}$, 现往杯子中放入一种直径为 $1 \\mathrm{cm}$ 的珍珠, 该珍珠放入水中后直接沉入杯底, 且体积不变. 若放完珍珠后水不溢出, 则最多可以放入珍珠\\blank{50}颗.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw (0,0) coordinate (O);\n\\draw (-3,8) coordinate (A);\n\\draw (3,8) coordinate (B);\n\\draw (A)--(O)--(B);\n\\draw ($(O)!0.5!(A)$) coordinate (P);\n\\draw (P) arc (180:540:1.5 and {1.5/4});\n\\draw (A) arc (180:540:3 and 0.75);\n\\filldraw (O)--++ (0.1,-0.1) --++ (0,-1.9) --++ (-0.2,0) --++ (0,1.9) -- cycle;\n\\filldraw (0,-2) ellipse (1.5 and {1.5/4});\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650533,7 +651248,9 @@ "id": "023959", "content": "已知球的半径为 $R$, 在球内作一个内接圆柱, 这个圆柱底面半径与高为何值时, 它的侧面积最大? 侧面积的最大值是多少?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -650553,7 +651270,9 @@ "id": "023960", "content": "函数 $f(x)=\\sqrt{x+3}+\\log _2(6-x)$ 的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650573,7 +651292,9 @@ "id": "023961", "content": "已知函数 $f(x)=\\sqrt{-x^2+2 x+3}$, 则函数 $f(3 x-2)$ 的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650593,7 +651314,9 @@ "id": "023962", "content": "已知 $f(\\dfrac{1}{2}x-1)=2 x-5$, 且 $f(a)=6$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650613,7 +651336,9 @@ "id": "023963", "content": "已知 $f(2 x+1)=x^2-2 x$, 则 $f(3)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650633,7 +651358,9 @@ "id": "023964", "content": "高斯是德国著名的数学家, 近代数学奠基者之一, 享有``数学王子''的称号, 用其名字命名的``高斯函数''为设 $x \\in \\mathbf{R}$, 用 $[x]$ 表示不超过 $x$ 的最大整数, 则 $y=[x]$ 称为高斯函数.例如: $[-2.1]=-3$, $[3.1]=3$, 已知函数 $f(x)=\\dfrac{2^x+3}{2^x+1}$, 则函数 $y=[f(x)]$ 的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650653,7 +651380,9 @@ "id": "023965", "content": "行驶中的汽车在刹车时由于惯性作用, 要继续往前滑行一段距离才能停下, 这段距离叫做刹车距离. 在某种路面上, 某种型号汽车的刹车距离 $y(\\mathrm{m})$ 与汽车的车速 $x(\\mathrm{km}/ \\mathrm{h})$ 满足下列关系: $y=\\dfrac{x^2}{200}+m x+n(m$、$n$ 是常数 $)$. 下图是根据多次实验数据绘制的刹车距离 $y(m)$ 与汽车的车速 $x(\\mathrm{km}/ \\mathrm{h})$ 的关系图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 0.05]\n\\draw [->] (-10,0) -- (100,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,50) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:100, samples = 100] plot (\\x,{\\x*\\x/200+\\x/100});\n\\foreach \\i/\\j in {40/8.4,60/18.6,80/32.8}\n{\\draw [dashed] (\\i,0) node [below] {$\\i$} -- (\\i,\\j) -- (0,\\j) node [left] {$\\j$};}; \n\\end{tikzpicture}\n\\end{center}\n(1) 求出 $y$ 关于 $x$ 的函数表达式;\\\\\n(2) 如果要求刹车距离不超过 $25.2 m$, 求行驶的最大速度.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -650673,7 +651402,9 @@ "id": "023966", "content": "函数 $f(x)=x+\\dfrac{1}{x}+1$, $f(a)=3$, 则 $f(-a)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650693,7 +651424,9 @@ "id": "023967", "content": "设函数 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 且 $f(x)=\\begin{cases}\\log _2(x+1),& x \\geq 0,\\\\g(x),& x<0,\\end{cases}$ 则 $f(-7)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650713,7 +651446,9 @@ "id": "023968", "content": "已知函数 $f(x)$ 是周期为 $2$ 的奇函数, 当 $x \\in[0,1)$ 时, $f(x)=\\lg (x+1)$, 则 $f(\\dfrac{2016}{5})+\\lg 18=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650733,7 +651468,10 @@ "id": "023969", "content": "设函数 $f(x)$ 是定义在 $\\mathbf{R}$ 上周期为 $2$ 的偶函数, 若当 $x \\in[0,1]$ 时, $f(x)=x+1$, 则 $f(\\dfrac{3}{2})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650755,7 +651493,9 @@ "id": "023970", "content": "函数 $f(x)$ 的定义域为 $D=\\{x | x \\neq 0\\}$, 且满足对于任意 $x_1$、$x_2\\in D$, 有 $f(x_1\\cdot x_2)=f(x_1)+f(x_2)$.\\\\\n(1) 求 $f(1)$ 的值;\\\\\n(2) 判断 $f(x)$ 的奇偶性并证明你的结论;\\\\\n(3) 如果 $f(4)=1$, $f(x-1)<2$, 且 $f(x)$ 在 $(0,+\\infty)$ 上是严格增函数, 求 $x$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -650778,7 +651518,9 @@ "id": "023971", "content": "函数 $y=x^2+x+1$($x \\in \\mathbf{R}$) 的单调减区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650798,7 +651540,9 @@ "id": "023972", "content": "函数 $f(x)=(\\dfrac{1}{2})^x-\\log _3(x+2)$ 在区间 $[-1,1]$ 上的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650818,7 +651562,9 @@ "id": "023973", "content": "若函数 $f(x)=\\dfrac{1}{x}$ 在区间 $[2, a]$ 上的最大值与最小值的和为 $\\dfrac{3}{4}$, 则实数 $a$ 的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650838,7 +651584,9 @@ "id": "023974", "content": "已知函数 $f(x)=\\begin{cases}x+4,& x0$) 在 $[3,+\\infty)$ 上是严格增函数, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650898,7 +651650,9 @@ "id": "023977", "content": "已知函数 $f(x)=\\begin{cases}\\mathrm{e}^x,& x \\leq 0,\\\\\\ln x,& x>0,\\end{cases}$, $f(x)=f(x)+x+a$. 若 $g(x)$ 存在 $2$ 个零点, 则实数 $a$的取值范围是\\bracket{20}.\n\\fourch{$[-1,0)$}{$[0,+\\infty)$}{$[-1,+\\infty)$}{$[1,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -650918,7 +651672,9 @@ "id": "023978", "content": "已知函数 $f(x)=\\dfrac{x^2+2 x+a}{x}$, $x \\in[1,+\\infty)$.\\\\\n(1) 当 $a=\\dfrac{1}{2}$ 时, 用定义证明函数的单调性并求函数 $f(x)$ 的最小值;\\\\\n(2) 若对任意 $x \\in[1,+\\infty)$, $f(x)>0$ 恒成立, 试求实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -650940,7 +651696,9 @@ "id": "023979", "content": "若函数 $f(x+1)$ 是偶函数, 则 $f(x)$ 的图像的对称轴方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650962,7 +651720,9 @@ "id": "023980", "content": "若 $f(x)=x^2+2 x-3$, 则 $y=|f(x)|$ 的严格增区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -650982,7 +651742,9 @@ "id": "023981", "content": "若奇函数 $f(x)$ 的定义域为 $[-5,5]$, 当 $x \\in[0,5]$ 时, $f(x)$ 的图像如图所示, 则不等式 $f(x)<0$ 的解是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, yscale = 0.5]\n\\draw [->] (-1,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (2,0) node [below] {$2$};\n\\draw (5,0) node [above] {$5$};\n\\draw [domain = 0:2,samples = 100] plot (\\x,-\\x*\\x+2*\\x);\n\\draw [domain = 2:5,samples = 100] plot (\\x, \\x*\\x/2-4*\\x+6);\n\\draw [dashed] (5,0) -- (5,-1.5);\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651004,7 +651766,9 @@ "id": "023982", "content": "若函数 $f(x)=\\dfrac{x-a}{x-a-1}$ 图像的对称中心是 $(4,1)$, 则实数 $a$ 的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651024,7 +651788,9 @@ "id": "023983", "content": "若定义域为 $\\mathbf{R}$ 的函数 $f(x)=\\begin{cases}|\\lg x|,& x>0,\\\\-x^2-2 x,& x \\leq 0,\\end{cases}$ 则函数 $f(x)$ 的零点为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651044,7 +651810,9 @@ "id": "023984", "content": "若直线 $y=1$ 与曲线 $y=x^2-|x|+a$ 有四个交点, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651064,7 +651832,9 @@ "id": "023985", "content": "讨论关于 $x$ 的方程 $|x^2-2 x-3|-a=0$($a \\in \\mathbf{R}$) 的实数解的个数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -651084,7 +651854,9 @@ "id": "023986", "content": "若函数 $h(x)=4 x^2-k x-8$ 在 $[5,20]$ 上是单调函数, 则实数 $k$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651104,7 +651876,9 @@ "id": "023987", "content": "若幂函数 $f(x)=(m^2-3 m+3) x^{m+1}$ 为偶函数, 则实数 $m$ 的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651124,7 +651898,9 @@ "id": "023988", "content": "已知函数 $f(x)=x^2-6 x+8$, $x \\in[1, a]$, 若函数 $f(x)$ 的最小值为 $f(a)$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651144,7 +651920,9 @@ "id": "023989", "content": "若 $f(x)=x^{\\frac{1}{4}}$, 则不等式 $f(x)>f(8 x-16)$ 的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651164,7 +651942,9 @@ "id": "023990", "content": "若对于任意实数 $x$, 函数 $f(x)=(5-a) x^2-6 x+a+5$ 恒为正值, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651184,7 +651964,9 @@ "id": "023991", "content": "下图给出四个幂函数的图像, 则图像与函数表达式对应是\\bracket{20}.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-exp(1/3*ln(2))}:{exp(1/3*ln(2))}, samples = 100] plot (\\x,{\\x*\\x*\\x});\n\\draw (0,-2) node [below] {\\textcircled{1}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-exp(1/2*ln(2))}:{exp(1/2*ln(2))}, samples = 100] plot (\\x,{\\x*\\x});\n\\draw (0,-2) node [below] {\\textcircled{2}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:2, samples = 100] plot (\\x,{sqrt(\\x)});\n\\draw (0,-2) node [below] {\\textcircled{3}};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.5:2, samples = 100] plot (\\x,{1/\\x});\n\\draw [domain = -2:-0.5, samples = 100] plot (\\x,{1/\\x});\n\\draw (0,-2) node [below] {\\textcircled{4}};\n\\end{tikzpicture}\n\\end{center}\n\\onech{\\textcircled{1} $y=x^{\\frac{1}{3}}$, \\textcircled{2} $y=x^2$, \\textcircled{3} $y=x^{\\frac{1}{2}}$, \\textcircled{4} $y=x^{-1}$}{\\textcircled{1} $y=x^3$, \\textcircled{2} $y=x^2$, \\textcircled{3} $y=x^{\\frac{1}{2}}$, \\textcircled{4} $y=x^{-1}$}{\\textcircled{1} $y=x^2$, \\textcircled{2} $y=x^3$, \\textcircled{3} $y=x^{-1}$, \\textcircled{4} $y=x^{\\frac{1}{2}}$}{\\textcircled{1} $y=x^{\\frac{1}{3}}$, \\textcircled{2} $y=x^{\\frac{1}{2}}$, \\textcircled{3} $y=x^2$, \\textcircled{4} $y=x^{-1}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -651204,7 +651986,9 @@ "id": "023992", "content": "已知函数 $y=f(x)$ 是偶函数, 当 $x>0$ 时, $f(x)=(x-1)^2$, 若当 $x \\in[-2,-\\dfrac{1}{2}]$时, $n \\leq f(x) \\leq m$ 恒成立, 则 $m-n$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651224,7 +652008,9 @@ "id": "023993", "content": "已知函数 $f(x)=x^2+(2 a-1) x-3$. 若函数 $f(x)$ 在 $[-1,3]$ 上最大值为 $1$, 求实数 $a$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -651247,7 +652033,9 @@ "id": "023994", "content": "若函数 $f(x)=4+2 a^{x-1}$ 的图像过定点 $P$, 则点 $P$ 的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651269,7 +652057,9 @@ "id": "023995", "content": "若函数 $f(x)=a^x+b$($a>0$, 且 $a \\neq 1$) 的定义域和值域都是 $[-1,0]$, 则 $a+b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651289,7 +652079,9 @@ "id": "023996", "content": "若函数 $y=|2^x-1|$ 的图像与直线 $y=b$ 有两个公共点, 则实数 $b$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651309,7 +652101,9 @@ "id": "023997", "content": "已知实数 $a \\neq 1$, 函数 $f(x)=\\begin{cases}4^x,& x \\geq 0,\\\\2^{a-x},& x<0,\\end{cases}$ 若 $f(1-a)=f(a-1)$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651329,7 +652123,9 @@ "id": "023998", "content": "若偶函数 $f(x)$ 满足 $f(x)=2^x-4$($x \\geq 0$), 则不等式 $f(x-2)>0$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651352,7 +652148,9 @@ "id": "023999", "content": "若函数 $f(x)=a^{x-b}$ 的图像如图所示, 其中 $a$、$b$ 为实常数, 则下列结论中正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:2, samples = 100] plot (\\x,{exp((\\x+0.8)*ln(0.6))});\n\\draw (0,1) -- (0.1,1) node [right] {$1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$a>1$, $b<0$}{$a>1$, $b>0$}{$0K.\\end{cases}$ 给出函数 $f(x)=2^{x+1}-4^x$, 若对于任意$x\\in (-\\infty,1]$, 恒有 $f_K(x)=f(x)$, 则\\bracket{20}.\n\\fourch{$K$ 的最大值为 $0$}{$K$ 的最小值为 $0$}{$K$ 的最大值为 $1$}{$K$ 的最小值为 $1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -651392,7 +652192,9 @@ "id": "024001", "content": "已知函数 $f(x)=\\dfrac{3^x+a}{3^x+1}$($a \\in \\mathbf{R}$) 为奇函数.\\\\\n(1) 求 $a$ 的值;\\\\\n(2) 判断函数 $f(x)$ 的单调性, 并用定义证明.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -651412,7 +652214,9 @@ "id": "024002", "content": "函数 $y=\\sqrt{\\log _3(2 x-1)+1}$ 的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651432,7 +652236,9 @@ "id": "024003", "content": "计算: $\\lg \\dfrac{5}{2}+2 \\lg 2-(\\dfrac{1}{2})^{-1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651452,7 +652258,9 @@ "id": "024004", "content": "若 $2^a=5^b=m$, 且 $\\dfrac{1}{a}+\\dfrac{1}{b}=2$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651476,7 +652284,9 @@ "id": "024005", "content": "若函数 $f(x)=\\begin{cases}-x+6,& x \\geq 2,\\\\3+\\log _ax,& x>2\\end{cases}$($a>0$, $a \\neq 1$) 的值域是 $[4,+\\infty)$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651496,7 +652306,9 @@ "id": "024006", "content": "已知 $a>0$, 且 $a \\neq 1$, 函数 $y=\\log _a(2 x-3)+\\sqrt{2}$ 的图像恒过点 $P$. 若点 $P$ 也在幂函数 $f(x)$ 的图像上, 则 $f(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651516,7 +652328,9 @@ "id": "024007", "content": "若函数 $f(x)=\\log _2(x^2-a x-3 a)$ 在区间 $(-\\infty,-2]$ 上是严格减函数, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651536,7 +652350,9 @@ "id": "024008", "content": "若函数 $f(x)=\\lg (x^2-4 x-5)$ 在 $(a,+\\infty)$ 上是严格增函数, 则 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651556,7 +652372,9 @@ "id": "024009", "content": "已知 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, $f(x-2)$ 为偶函数, 若当 $00$ 时, $f(x)=\\log _{\\frac{1}{2}}x$, 则不等式 $f(x^2-1)>-2$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651596,7 +652416,9 @@ "id": "024011", "content": "已知函数 $f(x)=\\lg (x+\\dfrac{a}{x}-2)$, 其中 $a$ 是大于 $0$ 的常数. 若对任意 $x \\in[2,+\\infty)$ 恒有 $f(x)>0$, 试确定实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -651616,7 +652438,9 @@ "id": "024012", "content": "若函数 $f(x)=a x^2+b x-3 a$($x \\in[a, 2 a+1]$) 是偶函数, 则 $f(1)$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651639,7 +652463,9 @@ "id": "024013", "content": "函数 $f(x)=|x|(2-x)$ 的单调增区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651659,7 +652485,9 @@ "id": "024014", "content": "若函数 $y=\\log _{\\frac{1}{3}}(x^2-a x+a)$ 在区间 $(-\\infty, \\sqrt{2})$ 上是严格增函数, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651681,7 +652509,9 @@ "id": "024015", "content": "若关于 $x$ 的方程 $2^x+3 x=k$ 的解在 $[1,2)$ 内, 则实数 $k$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651701,7 +652531,9 @@ "id": "024016", "content": "若函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的以 $3$ 为周期的奇函数, 且 $f(2)=0$, 则方程 $f(x)=0$ 在区间 $(0,6)$ 内零点的个数的最小值是\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651724,7 +652556,9 @@ "id": "024017", "content": "若定义在 $\\mathbf{R}$ 上的函数 $f(x)$ 满足: \\textcircled{1} $f(x)+f(2-x)=0$; \\textcircled{2} $f(x-2)=f(-x)$; \\textcircled{3} 当 $x \\in[-1,1]$ 时, $f(x)=\\begin{cases}\\sqrt{1-x^2},& x \\in[-1,0],\\\\ \\cos \\dfrac{\\pi}{2}x,& x \\in(0,1].\\end{cases}$ 则函数 $y=f(x)-(\\dfrac{1}{2})^{|x|}$ 在区间 $[-3,3]$ 上的零点个数为\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651747,7 +652581,9 @@ "id": "024018", "content": "设 $f(x)=x^3+a x^2-2 x$($x \\in \\mathbf{R}$), 其中常数 $a \\in \\mathbf{R}$.\\\\\n(1) 判断函数 $y=f(x)$ 的奇偶性, 并说明理由;\\\\\n(2) 若不等式 $f(x)>\\dfrac{3}{2}x^3$ 在区间 $[\\dfrac{1}{2}, 1]$ 上有解, 求实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -651769,7 +652605,9 @@ "id": "024019", "content": "已知集合 $A=\\{1,2,3,4\\}$, 那么 $A$ 的真子集的个数是\\bracket{20}.\n\\fourch{$15$ 个}{$16$ 个}{$3$ 个}{$4$ 个}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -651789,7 +652627,9 @@ "id": "024020", "content": "设集合 $M=\\{1,2,3\\}$, $N=\\{x | x \\subseteq M\\}$, 则 $M$ 与 $N$ 的关系是\\bracket{20}.\n\\fourch{$M \\subset N$}{$N \\subset M$}{$M \\in N$}{$M \\cap N=M$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -651812,7 +652652,9 @@ "id": "024021", "content": "已知集合 $A=\\{1,2, a^2-2 a\\}$, 若 $3 \\in A$, 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651834,7 +652676,10 @@ "id": "024022", "content": "已知集合 $M=\\{-1,0,1\\}$, $N=\\{y | y=\\cos x, x \\in M\\}$, 则 $M \\cap N=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651854,7 +652699,9 @@ "id": "024023", "content": "设 $a$、$b \\in R$, 若集合 $\\{1, a+b, a\\}=\\{0, b, \\dfrac{b}{a}\\}$, 则 $a^{2023}+b^{2023}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651874,7 +652721,9 @@ "id": "024024", "content": "已知函数 $f(x)=a x^2+b x+c(a \\neq 0)$, $a$、$b$、$c \\in \\mathbf{R}$, 集合 $A=\\{x | f(x)=x\\}$, 当 $A=\\{2\\}$ 时, $a: c=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651894,7 +652743,9 @@ "id": "024025", "content": "已知集合 $A=\\{1,2\\}$, $B=\\{x | x^2+m x+1=0, x \\in \\mathbf{R}\\}$, 若 $B \\subseteq A$, 则实数 $m$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651914,7 +652765,9 @@ "id": "024026", "content": "设集合 $A=\\{x|| x |<3\\}$, $B=\\{x | x^2-3 x+2>0\\}$, 集合 $P=\\{x | x \\in A, x \\notin A \\cap B\\}$, 集合 $P$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651934,7 +652787,9 @@ "id": "024027", "content": "同时满足条件: \\textcircled{1} $M \\subseteq\\{1,2,3,4,5\\}$; \\textcircled{2} 若 $a \\in M$, 则 $6-a \\in M$, 所有这样的非空集合 $M$为\\blank{50}(试列举出这些集合).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651957,7 +652812,9 @@ "id": "024028", "content": "已知集合 $A=\\{x | x^2+x-6=0\\}$, $B=\\{x | a x+1=0\\}$, 且 $B \\subseteq A$, 则实数 $a$ 的取值集合为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -651980,7 +652837,9 @@ "id": "024029", "content": "已知集合 $A=\\{1, a, b\\}$, $B=\\{a, a^2, a b\\}$, 若 $A=B$, 则实数 $a$、$b$ 的值分别为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652000,7 +652859,9 @@ "id": "024030", "content": "已知集合 $A=\\{x | \\dfrac{a x-1}{x-a}<0\\}$, 且 $2 \\in A$, $3 \\notin A$, 求实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -652022,7 +652883,9 @@ "id": "024031", "content": "已知全集 $U=\\{0,1,2\\}$ 且 $\\overline{A}=\\{2\\}$, 则集合 $A$ 的真子集共有\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652042,7 +652905,9 @@ "id": "024032", "content": "设 $p, q \\in \\mathbf{R}$. 已知 $A=\\{x | 3 x^2+p x-7=0\\}$, $B=\\{x | 3 x^2-7 x+q=0\\}$, 若 $A \\cap B= \\{-\\dfrac{1}{3}\\}$, 则 $A \\cup B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652062,7 +652927,9 @@ "id": "024033", "content": "若集合 $A=\\{(x, y) | \\dfrac{x^2}{2}+y^2<1\\}$, $B=\\{(x, y) | x \\in \\mathbf{Z}, y \\in \\mathbf{Z}\\}$, 则 $A \\cap B$ 的元素个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652084,7 +652951,10 @@ "id": "024034", "content": "已知集合 $A=\\{y | y=\\sqrt{x^2-1}\\}$, $B=\\{x | y=\\lg (x-2 x^2)\\}$, 则 $\\overline{A \\cap B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652104,7 +652974,9 @@ "id": "024035", "content": "对于任意两集合 $A$、$B$, 定义 $A-B=\\{x | x \\in A$ 且 $x \\notin B\\}$, $A \\triangle B=(A-B) \\cup(B-A)$, 记 $A=\\{y | y \\geq 0\\}$, $B=\\{x |-3 \\leq x \\leq 3\\}$, 则 $A \\triangle B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652127,7 +652999,9 @@ "id": "024036", "content": "设 $a \\in \\mathbf{R}$. 已知集合 $A=\\{a^2, a+1,-3\\}$, $B=\\{a-3,2 a-1, a^2+1\\}$, 若 $A \\cap B=\\{-3\\}$, 则 $a$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652147,7 +653021,9 @@ "id": "024037", "content": "设全集为 $\\mathbf{R}$, 集合 $A=\\{x | \\dfrac{1}{x}\\geq 1\\}$, $B=\\{y | y=x^2+x+1, x \\in \\mathbf{R}\\}$, 则 $A \\cap \\overline{B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652167,7 +653043,9 @@ "id": "024038", "content": "设 $a \\in \\mathbf{R}$, 集合 $A=\\{x | x^2-4 x+3=0,\\ x \\in \\mathbf{R}\\}$, $B=\\{x|x^2-(a+1)x+a=0, \\ x\\in \\mathbf{R}\\}$. 若 $A \\cup B=A$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652189,7 +653067,9 @@ "id": "024039", "content": "设集合 $A=\\{x | a \\leq x \\leq a+3\\}$, $B=\\{x | x<-1$ 或 $x>5\\}$, 其中 $a \\in \\mathbf{R}$.\\\\\n(1) 若 $A \\cap B \\neq \\varnothing$, 则 $a$ 的取值范围为\\blank{50}.\\\\\n(2) 若 $A \\cap B=A$, 则 $a$ 的取值范围为\\blank{50}.\\\\\n(3) 设全集 $\\mathbf{R}$. 若 $A \\cup \\overline{B}=\\overline{B}$, 则 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652209,7 +653089,9 @@ "id": "024040", "content": "命题 $p$: 若 $a$、$b \\in \\mathbf{R}$, 则 $|a|+|b|>1$ 是 $|a+b|>1$ 的充分非必要条件; \n命题 $q$: 函数 $y=\\sqrt{|x-1|-2}$ 的定义域是 $(-\\infty,-1] \\cup[3,+\\infty)$, 则\\bracket{20}.\n\\fourch{``$p$ 或 $q$''为假}{``$p$ 且 $q$''为真}{$p$ 真 $q$ 假}{$p$ 假 $q$ 真}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -652229,7 +653111,10 @@ "id": "024041", "content": "设 $x \\in \\mathbf{R}$, 则``$|x+1|<2$''是``$\\lg x<0$''的条件\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652249,7 +653134,9 @@ "id": "024042", "content": "设 $a$、$b \\in R$, 用 $\\alpha \\Rightarrow \\beta, \\beta \\Rightarrow \\alpha, \\alpha \\Leftrightarrow \\beta$ 这三种符号中的一种填空:\\\\\n(1) $\\alpha: $ 方程 $x^2+a x+b=0$ 有两相异实数根; $\\beta: b<0$. 则\\blank{50};\\\\\n(2) $\\alpha: a^2+b^2=0 ; \\beta: a=b=0$. 则\\blank{50};\\\\\n(3) $\\alpha: a b>0 ; \\beta:|a+b|=|a|+|b|$. 则\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652269,7 +653156,9 @@ "id": "024043", "content": "已知集合 $M=\\{x | x>2\\}$, $P=\\{x | x<3\\}$, 那么``$x \\in M$ 或 $x \\in P$''是``$x \\in M \\cap P$''的条件. (填入``充要''、``充分非必要''、``必要非充分''、``非充分非必要''之一)", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -652292,7 +653181,9 @@ "id": "024044", "content": "设 $x$、$y \\in \\mathbf{R}$, 用反证法证明命题``如果 $x^2+y^2<4$, 那么 $|x|<2$ 且 $|y|<2$''时, 应先假设\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652312,7 +653203,9 @@ "id": "024045", "content": "设 $n \\in \\mathbf{N}$ 且 $n \\geq 1$, 一元二次方程 $x^2-4 x+n=0$ 有整数根的一个充要条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652332,7 +653225,9 @@ "id": "024046", "content": "函数 $f(x)=x^2+m x+1$ 的图像关于直线 $x=1$ 对称的一个充要条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652354,7 +653249,9 @@ "id": "024047", "content": "已知 $P=\\{x | x^2-8 x-20 \\leq 0\\}$, 非空集合 $S=\\{x | 1-m \\leq x \\leq 1+m\\}$, 若 $x \\in P$ 是 $x \\in S$ 的必要条件, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652376,7 +653273,9 @@ "id": "024048", "content": "设 $a$、$b$、$c \\in(0,1)$, 用反证法证明: 下列三个关于 $x$ 的方程$a x^2+x+1-b=0$、$b x^2+x+1-c=0$、$c x^2+x+1-a=0$ 中至少有一个有实数根.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -652396,7 +653295,9 @@ "id": "024049", "content": "若 $a$、$b$、$c \\in \\mathbf{R}$, 且 $a>b$, 则下列不等式中一定成立的序号为\\blank{50}.\\\\\n\\textcircled{1} $a+c \\geq b+c$; \\textcircled{2} $a c \\geq b c$; \\textcircled{3} $\\dfrac{c^2}{a-b}>0$; \\textcircled{4} $(a-b) c^2\\geq 0$; \\textcircled{5} $\\sqrt[3]{a}>\\sqrt[3]{b}$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652416,7 +653317,9 @@ "id": "024050", "content": "若方程 $x^2+p x+q=0$ 的两根均为正整数, 且 $p+q=28$, 则该方程两根为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652436,7 +653339,9 @@ "id": "024051", "content": "已知三个不等式: \\textcircled{1} $a b>0 ;\\textcircled{2}-\\dfrac{c}{a}<-\\dfrac{d}{b}$; \\textcircled{3} $b c>a d$. 以其中两个作为条件, 余下一个作为结论, 则可以组成\\blank{50}个真命题.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652458,7 +653363,9 @@ "id": "024052", "content": "已知 $m>1$, 设 $A=\\sqrt{m+1}-\\sqrt{m}$, $B=\\sqrt{m}-\\sqrt{m-1}$, 则 $A$、$B$ 之间的大小关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652481,7 +653388,9 @@ "id": "024053", "content": "设角 $\\alpha$、$\\beta$ 满足 $-\\dfrac{\\pi}{2}<\\alpha<\\beta<\\dfrac{\\pi}{2}$, 则 $\\alpha-\\beta$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652501,7 +653410,9 @@ "id": "024054", "content": "实验室原有两瓶酒精溶液分别为 $a_1$、$a_2$ (单位: g), 其中对应含有酒精 $b_1, b_2$ (单位: g),\n且两瓶酒精浓度满足 $\\dfrac{b_1}{a_1}<\\dfrac{b_2}{a_2}$, 小明将两瓶酒精溶液全都倒下一个新容器中后, 新溶液的浓度发生变化. 请根据以上事实, 提炼出关于不等式的命题:\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652521,7 +653432,9 @@ "id": "024055", "content": "若 $a\\dfrac{1}{b}$}{$\\dfrac{1}{a-b}>\\dfrac{1}{a}$}{$|a|>|b|$}{$a^2>b^2$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -652541,7 +653454,9 @@ "id": "024056", "content": "下面四个条件中, 使 $a>b$ 成立的充分非必要条件是\\bracket{20}.\n\\fourch{$a>b+1$}{$a>b-1$}{$a^2>b^2$}{$a^3>b^3$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -652563,7 +653478,9 @@ "id": "024057", "content": "已知 $a \\neq 1$ 且 $a \\in \\mathbf{R}$, 试比较 $\\dfrac{1}{1-a}$ 与 $1+a$ 的大小.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -652586,7 +653503,9 @@ "id": "024058", "content": "若不等式 $a x^2+b x+c>0$($a$、$b$、$c \\in \\mathbf{R}$) 的解集是 $\\{x | 2(2 x+3)^3-x$ 的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652688,7 +653615,9 @@ "id": "024063", "content": "若全集 $U=\\mathbf{R}$, $f(x)$、$g(x)$ 均为 $x$ 的二次函数, $P=\\{x | f(x)<0\\}$, $Q=\\{x | g(x) \\geq 0\\}$, 则不等式组 $\\begin{cases}f(x)<0,\\\\g(x)<0\\end{cases}$ 的解集可用 $P$、$Q$ 表示为 \\bracket{20}.\n\\fourch{$P \\cap Q$}{$P \\cap \\overline{Q}$}{$P \\cup Q$}{$P \\cup \\overline{Q}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -652710,7 +653639,9 @@ "id": "024064", "content": "若 $a$、$b$、$c$ 是实常数, 则``$a>0$ 且 $b^2-4 a c<0$''是``对任意 $x \\in \\mathbf{R}$, 有 $a x^2+b x+c>0$''的\\bracket{20};\n\\twoch{充分非必要条件}{必要非充分条件 }{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -652730,7 +653661,9 @@ "id": "024065", "content": "不等式组 $\\begin{cases}|1-2 x|>3,\\\\x^2+x-6 \\leq 0\\end{cases}$ 的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652750,7 +653683,9 @@ "id": "024066", "content": "解关于 $x$ 的不等式: $x^2-(a+a^2) x+a^3>0$($a \\in \\mathbf{R}$).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -652770,7 +653705,9 @@ "id": "024067", "content": "若 $a>0$、$b>0$ 且 $2 a^2+b^2=2$, 则 $a \\sqrt{1+b^2}$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652790,7 +653727,9 @@ "id": "024068", "content": "已知 $a>0$, $b>0$, 若不等式 $(\\dfrac{2}{a}+\\dfrac{1}{b})(2 a+b) \\geq m$ 恒成立, 则实数 $m$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652810,7 +653749,9 @@ "id": "024069", "content": "若 $x \\in \\mathbf{R}$, 则方程 $|x-1|+|3 x-2|=|4 x-3|$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -652834,7 +653775,10 @@ "id": "024070", "content": "下列各函数中, 最小值为 $2$ 的是\\bracket{20}.\n\\twoch{$y=x+\\dfrac{1}{x}$}{$y=\\sin x+\\dfrac{1}{\\sin x}$, $x \\in(0, \\dfrac{\\pi}{2})$}{$y=\\dfrac{x^2+2}{\\sqrt{x^2+1}}$}{$y=3^x+3^{-x}$, $x>0$}", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -652857,7 +653801,9 @@ "id": "024071", "content": "``$|x-a|0$''是``$\\{S_n\\}$ 是严格增数列''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653003,7 +653961,9 @@ "id": "024078", "content": "已知数列 $\\{a_n\\}$ 中, $a_{n+1}=\\dfrac{a_n^2}{2 a_n-5}$, 若该数列既是等差数列又是等比数列, 则该数列前 $20$ 项和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653023,7 +653983,9 @@ "id": "024079", "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=\\dfrac{9^n(n+1)}{10^n},$($n \\geq 1$, $n \\in \\mathbf{N}$), 则数列 $\\{a_n\\}$ 的最大项是第\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653043,7 +654005,9 @@ "id": "024080", "content": "已知 $a_n=\\dfrac{n}{n^2+156},$($n \\geq 1$, $n \\in \\mathbf{N}$) 则数列 $\\{a_n\\}$ 的最大项是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653063,7 +654027,9 @@ "id": "024081", "content": "已知数列 $\\{a_n\\}$ 的通项公式是 $a_n=n^2+k n+4$. 若数列 $\\{a_n\\}$ 为严格增数列, 求实数 $k$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -653086,7 +654052,9 @@ "id": "024082", "content": "若在等差数列 $\\{a_n\\}$ 中, $a_4+a_8=10$, $a_{10}=6$, 则公差 $d=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653106,7 +654074,9 @@ "id": "024083", "content": "在等差数列 $\\{a_n\\}$ 中, 若 $S_n$ 为 $\\{a_n\\}$ 的前 $n$ 项和, $2 a_7=a_8+5$, 则 $S_{11}$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653126,7 +654096,9 @@ "id": "024084", "content": "设等差数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_m=4$, $S_m=0$, $S_{m+2}=14,$($m \\geq 2$, $m \\in \\mathbf{N}$), 则 $a_{2023}$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653146,7 +654118,9 @@ "id": "024085", "content": "已知等差数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $S_7=a_4$, 则以下结论中正确的是\\blank{50}.(填序号)\\\\\n\\textcircled{1} $a_1+a_3=0$; \\textcircled{2} $a_3+a_5=0$; \\textcircled{3} $S_3=S_4$; \\textcircled{4} $S_4=S_5$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653166,7 +654140,9 @@ "id": "024086", "content": "等差数列 $\\{a_n\\}$ 中, 若 $S_n$ 是其前 $n$ 项和, $a_1=-9$, $\\dfrac{S_9}{9}-\\dfrac{S_7}{7}=2$, 则 $a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653186,7 +654162,9 @@ "id": "024087", "content": "等差数列 $\\{a_n\\}$ 的各项均不为零, 其前 $n$ 项和为 $S_n$. 若 $a_{n+1}^2=a_{n+2}+a_n$, 则 $S_{2 n+1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653206,7 +654184,9 @@ "id": "024088", "content": "已知 $S_n$ 是等差数列 $\\{a_n\\}$ 的前 $n$ 项和, 且 $S_6>S_7>S_5$, 给出下列五个命题: \\textcircled{1} $d<0$; \\textcircled{2} $S_{11}>0$; \\textcircled{3} $S_{12}>0$; \\textcircled{4} 数列 $\\{S_n\\}$ 中的最大项为 $S_{11}$; \\textcircled{5} $|a_6|>|a_7|$, 其中正确命题的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653226,7 +654206,9 @@ "id": "024089", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_1=1$, $a_n+a_{n+1}=2 n+1$($n \\in \\mathbf{N}$, $n \\geq 1$), 则 $S_{21}$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653246,7 +654228,9 @@ "id": "024090", "content": "记 $S_n$ 为等差数列 $\\{a_n\\}$ 的前 $n$ 项和. 已知 $S_9=-a_5$.\\\\\n(1) 若 $a_3=4$, 求 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 若 $a_1>0$, 求使得 $S_n\\geq a_n$ 的 $n$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -653266,7 +654250,9 @@ "id": "024091", "content": "等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_3+4S_2=0$, 则公比 $q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653286,7 +654272,9 @@ "id": "024092", "content": "若等比数列 $\\{a_n\\}$ 的各项均为正数, $a_1+2 a_2=3$, $a_3^2=4 a_2a_6$, 则 $a_4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653306,7 +654294,9 @@ "id": "024093", "content": "在正项等比数列 $\\{a_n\\}$ 中, 已知 $a_1a_2a_3=4$, $a_4a_5a_6=12$, $a_{n-1}a_na_{n+1}=324$($n \\geq 2$, $n \\in \\mathbf{N}$), 则 $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653326,7 +654316,9 @@ "id": "024094", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, $a_1=1$, $S_n=2 a_{n+1}$, 则 $S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653346,7 +654338,9 @@ "id": "024095", "content": "已知 $\\{a_n\\}$ 是递减的等比数列, 且 $a_2=2$,. $a_1+a_3=5$, 则 $\\{a_n\\}$ 的通项公式为 $a_1a_2+a_2a_3+\\cdots+a_na_{n+1}=$\\blank{50}($n \\in \\mathbf{N}$, $n \\geq 1$)", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653366,7 +654360,9 @@ "id": "024096", "content": "已知等比数列 $\\{a_n\\}$ 为严格减数列, 且 $a_5^2=a_{10}$, $2(a_n+a_{n+2})=5 a_{n+1}$, 则数列 $\\{a_n\\}$ 的通项公式 $a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653388,7 +654384,9 @@ "id": "024097", "content": "无穷等比数列 $\\{a_n\\}$ 的各项和为 $S$, 若数列 $\\{b_n\\}$ 满足 $b_n=a_{3 n-2}+a_{3 n-1}+a_{3 n}$, 则数列 $\\{b_n\\}$的各项和为\\bracket{20}.\n\\fourch{$S$}{$3S$}{$S^2$}{$S^3$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -653408,7 +654406,9 @@ "id": "024098", "content": "设等比数列 $\\{a_n\\}$ 的公比为 $q$, 则下列结论中正确的是\\blank{50}.\\\\\n\\textcircled{1} 数列 $\\{a_na_{n+1}\\}$ 是公比为 $q^2$ 的等比数列;\\\\\n\\textcircled{2} 数列 $\\{a_n+a_{n+1}\\}$ 是公比为 $q$ 的等比数列;\\\\\n\\textcircled{3} 数列 $\\{a_n-a_{n+1}\\}$ 是公比为 $q$ 的等比数列;\\\\\n\\textcircled{4} 数列 $\\{\\dfrac{1}{a_n}\\}$ 是公比为 $\\dfrac{1}{q}$ 的等比数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653428,7 +654428,9 @@ "id": "024099", "content": "已知数列 $\\{a_n\\}$ 的首项 $a_1>0$, $a_{n+1}=\\dfrac{3 a_n}{2 a_n+1}$($n \\in N$, $n \\geq 1$), 且 $a_1=\\dfrac{2}{3}$.\\\\\n(1) 求证: $\\{\\dfrac{1}{a_n}-1\\}$ 是等比数列, 并求出 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 求数列 $\\{\\dfrac{1}{a_n}\\}$ 的前 $n$ 项和 $T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -653448,7 +654450,9 @@ "id": "024100", "content": "若数列 $\\{a_n\\}$ 的前 $n$ 项的和 $S_n=n^2+3 n$, 则 $a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653468,7 +654472,9 @@ "id": "024101", "content": "数列 $\\{a_n\\}$ 中, 若 $a_1=2$, 且 $a_{n+1}=\\dfrac{1}{2}(a_1+a_2+a_3+\\cdots+a_n)$, 则其前 $n$ 项和 $S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653488,7 +654494,9 @@ "id": "024102", "content": "数列 $\\{a_n\\}$ 的通项公式为 $a_n=\\dfrac{1}{\\sqrt{n}+\\sqrt{n-1}}$, 若该数列的前 $k$ 项之和等于 $9$ , 则 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653510,7 +654518,9 @@ "id": "024103", "content": "若数列 $1, \\dfrac{1}{1+2}, \\dfrac{1}{1+2+3}, \\cdots, \\dfrac{1}{1+2+3+\\cdots+n}, \\cdots$, 前 $n$ 项和为 $\\dfrac{47}{24}$, 则项数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653530,7 +654540,9 @@ "id": "024104", "content": "计算 $1 \\dfrac{1}{3}+2 \\dfrac{1}{9}+3 \\dfrac{1}{27}+\\cdots+(n+\\dfrac{1}{3^n})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653550,7 +654562,9 @@ "id": "024105", "content": "若数列 $\\{a_n\\}$ 满足 $a_1=1$, 且对任意的 $n \\in N$, $n \\geq 1$, 都有 $a_{n+1}=a_n+n+1$, 则以下命题中是真命题有\\blank{50}. (填序号)\\\\\n\\textcircled{1} $a_n=\\dfrac{n(n+1)}{2}$;\\\\\n\\textcircled{2} 数列 $\\{\\dfrac{1}{a_n}\\}$ 的前 $100$ 项和为 $\\dfrac{200}{101}$;\\\\\n\\textcircled{3} 数列 $\\{\\dfrac{1}{a_n}\\}$ 的前 $100$ 项和为 $\\dfrac{99}{100}$;\\\\\n\\textcircled{4} 数列 $\\{a_n\\}$ 的第 $100$ 项为 $50050$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653570,7 +654584,9 @@ "id": "024106", "content": "已知数列 $\\{a_n\\}$ 满足 $a_1+4 a_2+4^2a_3+\\cdots+4^{n-1}a_n=\\dfrac{n}{4}$($n \\in N$, $n \\geq 1$).\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 设 $b_n=\\dfrac{4^na_n}{2 n+1}$, 求数列 $\\{b_nb_{n+1}\\}$ 的前 $n$ 项和 $T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -653590,7 +654606,9 @@ "id": "024107", "content": "记 $S_n$ 为数列 $\\{a_n\\}$ 的前 $n$ 项和, 若 $a_1=19$, $S_n=n a_{n+1}+n(n+1)$.\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 设 $b_n=|a_n|$, 设数列 $\\{b_n\\}$ 的前 $n$ 项和为 $T_n$, 求 $T_{20}$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -653610,7 +654628,9 @@ "id": "024108", "content": "若 $f(n)=1^2+2^2+\\cdots+n^2+(n+1)^2+n^2+\\cdots+2^2+1^2$, 则 $f(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653630,7 +654650,9 @@ "id": "024109", "content": "用数学归纳法证明``$1+\\dfrac{1}{2}+\\dfrac{1}{3}+\\cdots+\\dfrac{1}{2^n-1}\\dfrac{n}{n+1}$ 对任意 $n \\geq k$($n, k \\in \\mathbf{N}$, $n$、$k \\geq 1$) 的自然数都成立, 则 $k$ 的最小值为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -653673,7 +654697,9 @@ "id": "024111", "content": "一个与正整数 $n$ 有关的命题, 当 $n=2$ 时命题成立, 且由 $n=k$($k \\geq 2$, $k \\in \\mathbf{N}$) 时命题成立可以推得 $n=k+2$ 时命题也成立, 则\\bracket{20}.\n\\twoch{该命题对于 $n>2$ 的自然数 $n$ 都成立}{该命题对于所有的正偶数都成立}{该命题何时成立与 $k$ 取值无关}{以上答案都不对}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -653693,7 +654719,9 @@ "id": "024112", "content": "用数学归纳法证明: $7^n+3^{n-1}$($n \\in \\mathbf{N}$, $n \\geq 1$) 能被 $4$ 整除.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -653713,7 +654741,9 @@ "id": "024113", "content": "已知 $\\{a_n\\}$ 是各项均为正数的等比数列, 且 $a_1+a_2=3$, $a_3-a_2=2$, 则数列 $\\{a_n\\}$ 的通项公式 $a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653733,7 +654763,9 @@ "id": "024114", "content": "等差数列 $\\{b_n\\}$ 的前 $n$ 项和为 $S_n$, 且 $b_3=5$, $S_4=16$, 则数列 $\\{b_n\\}$ 的通项公式 $b_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653753,7 +654785,9 @@ "id": "024115", "content": "若等比数列前三项之和为 $6$, 前六项之和为 $-42$, 则首项 $a_1=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653773,7 +654807,9 @@ "id": "024116", "content": "某人从 2017 年 1 月 2 日起, 每年 1 月 2 日到银行存人一万元定期储蓄, 若年利率为 $p$, 且保持不变, 并约定每年到期存款均自动转为新一年的定期存款, 到 2025 年 1 月 2 日将所有存款和利息全部取回, 则可取回的钱的总数为\\blank{50}万元.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653795,7 +654831,9 @@ "id": "024117", "content": "等差数列 $\\{a_n\\}$ 中, 若 $|a_3|=|a_9|$, $d<0$, 则使得它的前 $n$ 项之和 $S_n$ 取得最大值时的自然数 $n$ 是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653815,7 +654853,9 @@ "id": "024118", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n=n^2-8 n$.\\\\\n(1) 求数列 $\\{|a_n|\\}$ 的通项公式;\\\\\n(2) 求数列 $\\{|a_n|\\}$ 的前 $n$ 项和 $H_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -653838,7 +654878,9 @@ "id": "024119", "content": "抛物线 $y^2+4 x=0$ 的准线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653860,7 +654902,9 @@ "id": "024120", "content": "抛物线 $y=a x^2$ 的焦点坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653880,7 +654924,9 @@ "id": "024121", "content": "抛物线以原点为顶点, 以坐标轴为对称轴, 且焦点在直线 $x-2 y-4=0$ 上, 则抛物线的标准方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653902,7 +654948,9 @@ "id": "024122", "content": "若抛物线的顶点是原点, 焦点是椭圆 $4 x^2+y^2=1$ 的一个焦点, 则此抛物线的焦点到准线的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653922,7 +654970,9 @@ "id": "024123", "content": "在平面直角坐标系 $x O y$ 中, 双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的右支与焦点为 $F$ 的抛物线 $x^2=2 p y$($p>0$) 交于 $A$、$B$ 两点. 若 $AF+BF=4OF$, 则该双曲线的渐近线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653942,7 +654992,9 @@ "id": "024124", "content": "若抛物线 $y^2=4 x$ 内有一点 $A(2,1)$, 在抛物线上找一点 $P$, 使得 $|PF|+|PA|$ 取得最小值, 则点 $P$ 的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653964,7 +655016,9 @@ "id": "024125", "content": "已知抛物线 $C$ 的顶点是椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$ 的中心, 焦点与该椭圆的右焦点 $F_2$ 重合, 若抛物线 $C$ 与该椭圆在第一象限的交点为 $P$, 椭圆的左焦点为 $F_1$, 则 $|PF_1|$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -653984,7 +655038,9 @@ "id": "024126", "content": "抛物线 $y^2=4 x$ 的焦点为 $F$, 准线为 $l$, 经过点 $F$ 且斜率为 $\\sqrt{3}$ 的直线与抛物线在 $x$ 轴上方的部分相交于点 $A$, 若 $AK \\perp l$, 垂足为 $K$, 则 $\\triangle AKF$ 的面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654004,7 +655060,9 @@ "id": "024127", "content": "已知抛物线 $y^2=8 x$, 过动点 $M(a, 0)$ 且斜率为 $1$ 的直线 $l$ 与抛物线交于不同的两点\n$A$、$B$, 且 $|AB| \\leq 8$.\\\\\n(1) 求实数 $a$ 的取值范围;\\\\\n(2) 若线段 $AB$ 的垂直平分线交 $x$ 轴于点 $N$, 求 $\\triangle NAB$ 的面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654026,7 +655084,9 @@ "id": "024128", "content": "设 $O$ 为坐标原点, 若抛物线 $y^2=2 x$ 与过焦点的直线交于点 $A$、$B$, 则 $\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654046,7 +655106,9 @@ "id": "024129", "content": "设 $F$ 为抛物线 $y^2=4 x$ 的焦点, $A$、$B$、$C$ 为该抛物线上三点, 若 $A$、$B$、$C$ 三点坐标分别为 $(1,2)$、$(x_1, y_1)$、$(x_2, y_2)$, 且 $|\\overrightarrow{FA}|+|\\overrightarrow{FB}|+|\\overrightarrow{FC}|=10$, 则 $x_1+x_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654066,7 +655128,9 @@ "id": "024130", "content": "已知点 $P$ 是抛物线 $y^2=2 x$ 上动点, $A(\\dfrac{7}{2}, 4)$, 若点 $P$ 到 $y$ 轴的距离为 $d_1$, 点 $P$ 到点 $A$ 的距离为 $d_2$, 则 $d_1+d_2$ 的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654086,7 +655150,9 @@ "id": "024131", "content": "已知 $F$ 为抛物线 $C: y^2=6 x$ 的焦点, 过点 $F$ 的直线 $l$ 与抛物线 $C$ 相交于 $A$、$B$ 两点, 若 $|AF|=3|BF|$, 则 $|AB|$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654106,7 +655172,9 @@ "id": "024132", "content": "已知抛物线 $C: y^2=2 x$, 点 $A(2,0)$、$B(-2,0)$, 过点 $A$ 的直线 $l$ 与抛物线 $C$ 交于 $M$、$N$两点. 求证: $\\angle ABM=\\angle ABN$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654126,7 +655194,9 @@ "id": "024133", "content": "若 $a>1$, 则双曲线 $\\dfrac{x^2}{a^2}-y^2=1$ 的离心率的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654146,7 +655216,9 @@ "id": "024134", "content": "若椭圆 $C: \\dfrac{x^2}{2}+y^2=1$, 直线 $l: y=x+3$, 则椭圆 $C$ 上的点到直线 $l$ 距离最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654168,7 +655240,9 @@ "id": "024135", "content": "椭圆 $\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$ 的焦点为 $F_1$、$F_2$, 点 $P$ 为椭圆上一动点, 当 $\\angle F_1PF_2$ 为钝角时, 点 $P$ 的横坐标的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654190,7 +655264,9 @@ "id": "024136", "content": "椭圆 $C_1: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的焦点 $F_1$、$F_2$ 是等轴双曲线 $C_2: \\dfrac{x^2}{2}-\\dfrac{y^2}{2}=1$ 的顶点, 若椭圆 $C_1$ 与双曲线 $C_2$ 的一个交点是 $P, \\triangle PF_1F_2$ 的周长为 $4+2 \\sqrt{2}$.\\\\\n(1) 求椭圆 $C_1$ 的标准方程;\\\\\n(2) 点 $M$ 是双曲线 $C_2$ 上任意不同于其顶点的动点, 设直线 $MF_1$、$MF_2$ 的斜率分别为 $k_1$、$k_2$, 求证: $k_1$、$k_2$ 的乘积为定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654210,7 +655286,9 @@ "id": "024137", "content": "若双曲线 $m x^2+n y^2=1$ 的一个焦点与抛物线 $y=\\dfrac{1}{8}x^2$ 的焦点相同, 离心率为 $2$ , 则抛物线的焦点到双曲线的一条渐近线的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654230,7 +655308,9 @@ "id": "024138", "content": "若经过椭圆 $\\dfrac{x^2}{2}+y^2=1$ 的一个焦点作倾斜角为 $45^{\\circ}$ 的直线 $l$, 交椭圆于 $A$、$B$ 两点. 设 $O$为坐标原点, 则 $\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654250,7 +655330,9 @@ "id": "024139", "content": "抛物线 $x^2=-2 y$ 与过点 $P(0,-1)$ 的直线 $l$ 交于 $A$、$B$ 两点, 若 $OA$ 与 $OB$ 的斜率之和为 $1$ , 则直线 $l$ 的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654270,7 +655352,9 @@ "id": "024140", "content": "已知动点 $P$ 在双曲线 $C: x^2-\\dfrac{y^2}{3}=1$ 上, 双曲线 $C$ 的左、右焦点分别为 $F_1$、$F_2$,\n下列结论中错误的是\\blank{50}.\\\\\n\\textcircled{1} 双曲线 $C$ 的离心率为 $\\sqrt{2}$;\\\\\n\\textcircled{2} 双曲线 $C$ 的渐近线方程为 $y= \\pm \\dfrac{\\sqrt{3}}{3}x$;\\\\\n\\textcircled{3} 双曲线 $\\dfrac{y^2}{3}-x^2=1$ 与已知双曲线 $C$ 的渐近线并不相同;\\\\\n\\textcircled{4} 动点 $P$ 到两条渐近线的距离之积为定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654290,7 +655374,9 @@ "id": "024141", "content": "已知椭圆 $C: \\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1, F_1$、$F_2$ 分别为它的左、右焦点, $A$、$B$ 分别为它的左、右顶点, 已知定点 $Q(4,2)$, 点 $P$ 是椭圆上的一个动点, 下列结论中错误的是\\blank{50}.\\\\\n\\textcircled{1} 存在点 $P$, 使得 $\\angle F_1PF_2=120^{\\circ}$;\\\\\n\\textcircled{2} 直线 $PA$ 与直线 $PB$ 斜率乘积为定值;\\\\\n\\textcircled{3} $\\dfrac{1}{|PF_1|}+\\dfrac{25}{|PF_2|}$ 有最小值 $\\dfrac{18}{5}$;\\\\\n\\textcircled{4} $|PQ|+|PF_1|$ 的范围为 $[2 \\sqrt{17}, 12]$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654310,7 +655396,9 @@ "id": "024142", "content": "若抛物线 $C: y^2=2 p x$($p \\geq 0$) 的焦点 $F$ 与椭圆 $E: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$ 的一个焦点重合, 过坐标原点 $O$ 作两条互相垂直的射线 $OM$、$ON$, 与抛物线 $C$ 分别交于 $M$、$N$ 两点, 求证: 直线 $MN$ 恒过定点, 并求出该定点坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654330,7 +655418,9 @@ "id": "024143", "content": "一枚炮弹被发射后, 其升空高度 $h$ 与时间 $t$ 的函数关系为 $h=130 t-5 t^2$, 则该函数的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654350,7 +655440,9 @@ "id": "024144", "content": "某种商品进价为 $4$ 元/件, 当日均零售价为 $6$ 元/件, 日均销售 $100$ 件, 当单价每增加 $1$ 元, 日均销量减少 $10$ 件, 试计算该商品在销售过程中, 若每天固定成本为 $20$ 元, 则预计单价为\\blank{50}元/件时, 利润最大.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654370,7 +655462,9 @@ "id": "024145", "content": "某市出租车收费标准如下: 起步价为 $8$ 元, 起步里程为 $3 \\mathrm{km}$ (不超过 $3 \\mathrm{km}$ 按起步价付费); 超过 $3 \\mathrm{km}$ 但不超过 $8 \\mathrm{km}$ 时, 超过部分按每千米 $2.15$ 元收费; 超过 $8 \\mathrm{km}$ 时, 超过部分按每千米 $2.85$ 元收费, 另每次乘坐需付燃油附加费 $1$ 元. 现某人乘坐一次出租车付费 $22.6$ 元, 则此次出租车行驶了\\blank{50}$\\mathrm{km}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654390,7 +655484,9 @@ "id": "024146", "content": "拟定甲、乙两地通话 $m$ 分钟的电话费 (单位: 元) 由 $f(m)=1.06(0.5[m]+1)$ 给出, 其中 $m>0$, $[m]$ 是不超过 $m$ 的最大整数 (如 $[3]=3$, $[3.7]=3$, $[3.1]=3$ ), 则甲、乙两地通话 $6.5$ 分钟的电话费为\\blank{50}元.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654410,7 +655506,9 @@ "id": "024147", "content": "已知正方形 $ABCD$ 的边长为 $4$ , 动点 $P$ 从点 $B$ 开始沿折线 $BCDA$ 向点 $A$ 运动. 若点 $P$ 运动的路程为 $x$, $\\triangle ABP$ 的面积为 $S$, 则函数 $S=f(x)$ 的图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, xscale = 0.12, yscale = 0.2]\n\\draw [->] (0,0) -- (16,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,10) node [left] {$S$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {4,8,12}\n{\\draw (\\i,0.5) -- (\\i,0) node [below] {$\\i$};};\n\\foreach \\i in {4,8}\n{\\draw (0.5,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw [domain = 0:12, samples = 100] plot (\\x,{8-(\\x-6)*(\\x-6)/36*8});\n\\draw [dashed] (6,0) -- (6,8);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.12, yscale = 0.2]\n\\draw [->] (0,0) -- (16,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,10) node [left] {$S$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {4,8,12}\n{\\draw (\\i,0.5) -- (\\i,0) node [below] {$\\i$};};\n\\foreach \\i in {4,8}\n{\\draw (0.5,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw (0,0) sin (4,8) -- (8,8) cos (12,0);\n\\draw [dashed] (4,0) -- (4,8) (8,0) -- (8,8);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.12, yscale = 0.2]\n\\draw [->] (0,0) -- (16,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,10) node [left] {$S$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {4,8,12}\n{\\draw (\\i,0.5) -- (\\i,0) node [below] {$\\i$};};\n\\foreach \\i in {4,8}\n{\\draw (0.5,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw (0,0) cos (4,8) -- (8,8) sin (12,0);\n\\draw [dashed] (4,0) -- (4,8) (8,0) -- (8,8);\n\\end{tikzpicture}}{\n\\begin{tikzpicture}[>=latex, xscale = 0.12, yscale = 0.2]\n\\draw [->] (0,0) -- (16,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,10) node [left] {$S$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {4,8,12}\n{\\draw (\\i,0.5) -- (\\i,0) node [below] {$\\i$};};\n\\foreach \\i in {4,8}\n{\\draw (0.5,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw (0,0) -- (4,8) -- (8,8) -- (12,0);\n\\draw [dashed] (4,0) -- (4,8) (8,0) -- (8,8);\n\\end{tikzpicture}\n}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -654430,7 +655528,9 @@ "id": "024148", "content": "科学家以里氏震级来度量地震的强度, 若设 $I$ 为地震时所散发出来的相对能量程度, 则里氏震级度量 $r$ 可定义为 $r=\\dfrac{2}{3}\\lg I+2$, 则每增加一个震级, 相对能量程度扩大到\\bracket{20}.($\\sqrt{10}\\approx 3.16$)\n\\fourch{$31.6$ 倍}{$13.16$ 倍}{$6.32$ 倍}{$3.16$ 倍}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -654452,7 +655552,9 @@ "id": "024149", "content": "小王大学毕业后, 决定利用所学专业进行自主创业. 经过市场调查, 生产某小型电子产品需投入年固定成本 $3$ 万元, 每生产 $x$ 万件, 需另投入流动成本 $W(x)$ 万元, 在年产量不足 $8$ 万件时, $W(x)=\\dfrac{1}{3}x^2+x$ (万元). 在年产量不小于 $8$ 万件时, $W(x)=6 x+\\dfrac{100}{x}-38$ (万元). 每件产品售价 $5$ 元. 通过市场分析, 小王生产的商品当年能全部售完.\\\\\n(1) 写出年利润 $L(x)$ (万元) 关于年产量 $x$ (万件) 的函数解析式;(注: 年利润$=$年销售收入$-$固定成本$-$流动成本);\\\\\n(2) 年产量为多少万件时, 小王在这一商品的生产中所获利润最大? 最大利润是多少?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654472,7 +655574,9 @@ "id": "024150", "content": "若对任意的实数 $x>0$, $x \\ln x-x-a \\geq 0$ 恒成立, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654494,7 +655598,9 @@ "id": "024151", "content": "已知函数 $f(x)=x^2-2 \\ln x$, 若在定义域内存在 $x_0$, 使得不等式 $f(x_0)-m \\leq 0$ 成立, 则实数 $m$ 的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654516,7 +655622,9 @@ "id": "024152", "content": "已知函数 $f(x)=\\dfrac{2}{2^x+1}-1$, 若 $f(4^x-1)>f(3)$, 则实数 $x$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654536,7 +655644,9 @@ "id": "024153", "content": "已知定义在 $(0,+\\infty)$ 上的函数 $f(x)$ 满足 $x f'(x)-f(x)<0$, 其中 $f'(x)$ 是函数 $f(x)$ 的导函数, 若 $f(m-2021)>(m-2021) f(1)$, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654556,7 +655666,9 @@ "id": "024154", "content": "已知函数 $f(x)=(x-3) \\mathrm{e}^x$. 求 $f(x)$ 过 $(-1,0)$ 的切线方程.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654576,7 +655688,9 @@ "id": "024155", "content": "设 $f'(x)$ 是奇函数 $f(x)$ 的导函数, $f(-2)=-3$, 且对任意 $x \\in \\mathbf{R}$ 都有 $f'(x)<2$,\\\\\n(1) 求 $f(2)$;\\\\\n(2) 求解关于 $x$ 的不等式 $f(\\mathrm{e}^x)<2 \\mathrm{e}^x-1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654596,7 +655710,9 @@ "id": "024156", "content": "已知函数 $f(x)=a^x+x^2-x \\ln a$($a>0$, $a \\neq 1$). 求函数 $f(x)$ 的极小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654616,7 +655732,9 @@ "id": "024157", "content": "已知函数 $f(x)=\\dfrac{1}{3}x^3-\\dfrac{1+a}{2}x^2+a x+1$($a \\in \\mathbf{R}$). 求函数 $y=f(x)$ 的单调区间.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654636,7 +655754,9 @@ "id": "024158", "content": "若直线 $l_1:(2-m) x+3 y=4-2 m$ 与 $l_2: 2 x+(3+m) y=4$ 垂直, 则实数 $m$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654660,7 +655780,9 @@ "id": "024159", "content": "直线 $(m+2) x-(2 m-1) y-(3 m-4)=0$($m \\in \\mathbf{R}$) 恒过点\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654682,7 +655804,9 @@ "id": "024160", "content": "若在直线 $y=-2$ 上有点 $P$, 它到点 $A(-3,1)$ 和点 $B(5,-1)$ 的距离之和最小, 则点 $P$ 的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654702,7 +655826,9 @@ "id": "024161", "content": "若函数 $f(x)=x^3+a x^2-x-9$ 在 $x=-1$ 处取得极值, 则 $f(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654724,7 +655850,10 @@ "id": "024162", "content": "设曲线 $y=x^{n+1}$($n \\in \\mathbf{N}$, $n \\geq 1$) 在点 $(1,1)$ 处的切线与 $x$ 轴的交点的横坐标为 $x_n$, 则 $x_1 \\cdot x_2 \\cdot x_3 \\cdot x_4 \\cdots \\cdot x_{2022}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654744,7 +655873,9 @@ "id": "024163", "content": "当 $x=m$ 时, 函数 $f(x)=x^3-x^2+3 x-2 \\ln x$ 取得最小值, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654764,7 +655895,9 @@ "id": "024164", "content": "函数 $y=f(x)$ 在定义域 $(-\\dfrac{3}{2}, 3)$ 内的图像如下图所示. 记 $y=f(x)$ 的导函数为 $y=f'(x)$, 则不等式 $f'(x) \\leq 0$ 的解集为\\blank{50}.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-1.1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (-1.5,-1) -- (-1,0) sin ({-1/3},0.8) cos (0.5,0) sin (1,-0.5) cos ({4/3},0) sin (2,1.5) cos ({8/3},0) -- (3,-1);\n\\foreach \\i/\\j in {-1.5/-1,{-1/3}/0.8,1/-0.5,2/1.5,3/-1}\n{\\draw [dashed] (\\i,\\j) -- (\\i,0);};\n\\filldraw [fill = white] (-1.5,-1) circle (0.03) (3,-1) circle (0.03);\n\\draw (-1.5,0) node [above] {$-\\frac{3}{2}$};\n\\draw (-1,0) node [below] {$-1$};\n\\draw ({-1/3},0) node [below] {$-\\frac{1}{3}$};\n\\draw (1,0) node [above] {$1$};\n\\draw ({4/3},0) node [below] {$\\frac{4}{3}$};\n\\draw (2,0) node [below] {$2$};\n\\draw ({8/3},0) node [below] {$\\frac{8}{3}$};\n\\draw (3,0) node [above] {$3$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654786,7 +655919,9 @@ "id": "024165", "content": "设 $F_1$、$F_2$ 分别为椭圆 $\\Gamma: \\dfrac{x^2}{3}+y^2=1$ 的左、右焦点, 点 $A$、$B$ 在椭圆 $\\Gamma$ 上, 且不是椭圆的顶点. 若 $\\overrightarrow{F_1A}+\\lambda \\overrightarrow{F_2B}=\\overrightarrow{0}$, 且 $\\lambda>0$, 则实数 $\\lambda$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654806,7 +655941,9 @@ "id": "024166", "content": "在平面直角坐标系 $x O y$ 中, 过点 $P(-3, a)$ 作圆 $x^2+y^2-2 x=0$ 的两条切线, 切点分别为 $M(x_1, y_1)$、$N(x_2, y_2)$. 若 $(x_2-x_1)(x_2+x_1)+(y_2-y_1)(y_2+y_1-2)=0$, 则实数 $a$ 的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654826,7 +655963,9 @@ "id": "024167", "content": "已知双曲线 $C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>b>0$) 的离心率为 $\\sqrt{5}$, 虚轴长为 $4$ .\\\\\n(1) 求双曲线标准方程;\\\\\n(2) 过点 $(0,1)$ 、倾斜角为 $45^{\\circ}$ 的直线 $l$ 与双曲线 $C$ 相交于 $A$、$B$ 两点. 求 $\\triangle AOB$ 面积.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654846,7 +655985,9 @@ "id": "024168", "content": "已知函数 $f(x)=x^2+\\ln x-a x$.\\\\\n(1) 当 $a=3$ 时, 求 $f(x)$ 的单调增区间;\\\\\n(2) 若 $f(x)$ 在 $(0,1)$ 上是严格减函数, 求实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654866,7 +656007,9 @@ "id": "024169", "content": "如图所示, $ABCD$ 是边长为 $60 \\mathrm{cm}$ 的正方形硬纸片, 切去阴影部分所示的四个全等的等腰直角三角形, 再沿虚线折起, 使得 $A$、$B$、$C$、$D$ 四个点重合于图中的点 $P$, 正好形成一个正四棱柱形状的包装盒, $E$、$F$ 在 $AB$ 上是被切去的等腰直角三角形斜边的两个端点, 设 $AE=FB=x \\mathrm{cm}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A) -- (2,0) node [right] {$B$} coordinate (B) -- (2,2) node [right] {$C$} coordinate (C) -- (0,2) node [left] {$D$} coordinate (D) -- cycle;\n\\filldraw [pattern = north east lines] (0.7,0) --++ (0.3,0.3) --++ (0.3,-0.3);\n\\filldraw [pattern = north east lines] (0.7,2) --++ (0.3,-0.3) --++ (0.3,0.3);\n\\filldraw [pattern = north east lines] (0,0.7) --++ (0.3,0.3) --++ (-0.3,0.3);\n\\filldraw [pattern = north east lines] (2,0.7) --++ (-0.3,0.3) --++ (0.3,0.3);\n\\draw [dashed] (0.7,0) -- (2,1.3) (2,0.7) -- (0.7,2) (1.3,2) -- (0,0.7) (0,1.3) -- (1.3,0);\n\\draw [dashed] (1.3,0) -- (2,0.7) (2,1.3) -- (1.3,2) (0.7,2) -- (0,1.3) (0,0.7) -- (0.7,0);\n\\draw (0.7,0) node [below] {$E$} coordinate (E);\n\\draw (1.3,0) node [below] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(E)$) node [below] {$x$};\n\\draw ($(B)!0.5!(F)$) node [below] {$x$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) rectangle ({0.7*sqrt(2)},{0.3*sqrt(2)}) coordinate (T);\n\\draw (T) --++ (0.35,0.35) coordinate (D) --++ (0,{-0.3*sqrt(2)}) --++ (-0.35,-0.35);\n\\draw (T) ++ (0.35,0.35) --++ ({-0.7*sqrt(2)},0) coordinate (A) --++ (-0.35,-0.35) coordinate (B);\n\\path [name path = AT, draw] (A)--(T);\n\\path [name path = BD, draw] (B)--(D);\n\\path [name intersections = {of = AT and BD, by = P}];\n\\filldraw (P) circle (0.03) node [above] {$P$} coordinate (P);\n\\path (1,-1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求包装盒的容积 $V(x)$ 关于 $x$ 的函数表达式, 并求出函数的定义域;\\\\\n(2) 当 $x$ 为多少时, 包装盒的容积 $V(x)(\\mathrm{cm})^3$ 最大? 并求出此时包装盒的高与底面边长的比值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654888,7 +656031,9 @@ "id": "024170", "content": "已知椭圆 $\\Gamma: \\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$.\\\\\n(1) 若抛物线 $C$ 的焦点与椭圆 $\\Gamma$ 的焦点重合, 求抛物线 $C$ 的标准方程;\\\\\n(2) 若椭圆 $\\Gamma$ 的上顶点 $A$ 、右焦点 $F$ 及 $x$ 轴上一点 $M$ 构成直角三角形, 求点 $M$ 的坐标;", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -654910,7 +656055,9 @@ "id": "024171", "content": "某市 $A, B, C$ 三个区共有高中学生 $20000$ 人, 其中 $A$ 区高中学生 $7000$ 人, 若采用分层抽样的方法从这三个区所有高中学生中抽取一个容量为 $600$ 人的样本进行学习兴趣调查, 则 $A$ 区应抽取\\blank{50}人.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654930,7 +656077,9 @@ "id": "024172", "content": "某班 $48$ 名学生参加建校 $100$ 周年知识竞赛, 若成绩都在区间 $[40,100]$ 上, 其频率分布直方图如图所示, 则成绩不低于 $60$ 分的人数为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.06, yscale = 80]\n\\draw [->] (30,0) -- (32,0) -- (33,-0.002) -- (35,0.002) -- (36,0)-- (115,0) node [below] {成绩/分};\n\\draw [->] (30,0) -- (30,0.04) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (30,0) node [below left] {$O$};\n\\foreach \\i/\\j in {40/0.01,50/0.015,60/0.015,70/0.03,80/0.025,90/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {40/0.01,60/0.015,70/0.03,80/0.025,90/0.005}\n{\\draw [dashed] (\\i,\\j) -- (30,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654950,7 +656099,9 @@ "id": "024173", "content": "若某班级要从 $4$ 名男生和 $3$ 名女生中选取 $3$ 名同学参加志愿者活动, 则选出的 $3$ 人中既有男生又要有女生的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654974,7 +656125,9 @@ "id": "024174", "content": "某副食品店对某月的前 $11$ 天内每天的顾客人数进行统计得到样本数据的茎叶图如图所示, 则样本的中位数和方差 (方差的结果保留一位小数)分别是\\blank{50}.\n\\begin{center}\n\\begin{tabular}{l|llllll}\n3 & 1 & 2 & 4\\\\\n4 & 4 & 5 & 5 & 7 & 7 & 8 \\\\\n5 & 0 & 0 \n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -654994,7 +656147,9 @@ "id": "024175", "content": "在二项式 $(1+x)^5$ 的展开式中任取两项, 则所取两项中至少有一项的系数为偶数的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655016,7 +656171,9 @@ "id": "024176", "content": "若从一副 $52$ 张的扑克牌中随机抽取 $1$ 张, 放回后再抽取 $1$ 张, 则两张牌都是``K''的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655036,7 +656193,9 @@ "id": "024177", "content": "若在含有 $3$ 件次品的 $10$ 件产品中任取 $4$ 件, $X$ 表示取到的次品数, 则 $P(X=2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655056,7 +656215,9 @@ "id": "024178", "content": "非空集合 $A$ 中所有元素乘积记为 $T(A)$. 已知集合 $M=\\{1,4,5,7,8\\}$, 若从集合 $M$ 的所有非空子集中任选一个子集 $A$, 则 $T(A)$ 为偶数的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655079,7 +656240,9 @@ "id": "024179", "content": "某校计划在秋季运动会期间开展``运动与健康''知识大赛. 为此某班开展了 $10$ 次模拟测试, 以此选拔选手代表班级参赛. 下表为甲、乙两名学生的历次模拟测试成绩.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n场次 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline\n甲 & 98 & 94 & 97 & 97 & 95 & 93 & 93 & 95 & 93 & 95 \\\\\n\\hline\n乙 & 92 & 94 & 93 & 94 & 95 & 94 & 96 & 97 & 97 & 98 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n甲、乙两名学生测试成绩的平均数分别记作 $\\overline{x}, \\overline{y}$, 方差分别记作 $S_1^2, S_2^2$.\\\\\n(1) 求 $\\overline{x}, \\overline{y}, S_1^2, S_2^2$;\\\\\n(2) 以这 $10$ 次模拟测试成绩及(1)中的结果为参考, 请你从甲、乙两名学生中选出一人代表班级参加比赛, 并说明你作出选择的理由.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655099,7 +656262,9 @@ "id": "024180", "content": "从甲地到乙地要经过 $3$ 个十字路路, 设各路路信号灯工作相互独立, 且在各路路遇到红灯的概率分别为 $\\dfrac{1}{2}, \\dfrac{1}{3}, \\dfrac{1}{4}$.\\\\\n(1) 设 $X$ 表示一辆车从甲地到乙地遇到红灯的个数, 求随机变量 $X$ 的分布;\\\\\n(2) 若有两辆车独立地从甲地到乙地, 求这两辆车共遇到 $1$ 个红灯的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655119,7 +656284,9 @@ "id": "024181", "content": "已知 $(2 x-1)^n=a_0+a_1(x-1)+a_2(x-1)^2+\\cdots+a_n(x-1)^n$, $n$ 为正整数, $a_2=60$.\\\\\n(1) 求 $a_0+a_1+a_2+\\cdots+a_n$ 的值;\\\\\n(2) 设 $r \\in\\{0,1,2, \\cdots, n\\}$, 求 $a_r$ 的最大值.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655139,7 +656306,10 @@ "id": "024182", "content": "某学校进行体检, 现得到所有男生的身高数据, 从中随机抽取 $50$ 人进行统计 (已知这 $50$ 人身高介于 $155 \\mathrm{cm}$ 到 $195 \\mathrm{cm}$ 之间), 现将抽取结果按如下方式分成八组: 第一组 $[155,160)$ 、第二组 $[160,165)$、$\\cdots$ 、第八组 $[190,195)$, 并按此分组绘制如图所示的频率分布直方图, 其中第六组和第七组还没有绘制完成, 已知第一组与第八组人数相同, 第六组和第七组人数的比为 $5: 2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.15, yscale = 70]\n\\draw [->] (150,0) -- (151,0) -- (151.5,-0.003) -- (152.5,0.003) -- (153,0)-- (205,0) node [below] {身高/cm};\n\\draw [->] (150,0) -- (150,0.07) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (150,0) node [below left] {$O$};\n\\foreach \\i/\\j in {155/0.008,160/0.016,165/0.04,170/0.04,175/0.06,180/0,185/0,190/0.008}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (5,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {160/0.016,165/0.04,175/0.06,190/0.008}\n{\\draw [dashed] (\\i,\\j) -- (150,\\j) node [left] {$\\k$};};\n\\draw (195,0) node [below] {$195$};\n\\end{tikzpicture}\n\\end{center}\n(1) 计算第六组和第七组的频率;\\\\\n(2) 用分层抽样的方法在身高为 $[170,180)$ 内抽取一个容量为 $5$ 的样本, 从样本中任意抽取 $2$ 位男生, 求这两位男生身高都在 $[175,180)$ 内的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655159,7 +656329,10 @@ "id": "024183", "content": "某部门为了解一企业在生产过程中的用水量情况, 对其每天的用水量做了记录, 得到了大量该企业的日用水量的统计数据, 从这些统计数据中随机抽取 $12$ 天的数据作为样本, 得到如图所示的茎叶图(单位: 吨). 若用水量不低于 $95$ 吨, 则称这一天的用水量超标.\n\\begin{center}\n\\begin{tabular}{c|cccccc}\n7&3&1\\\\\n8&3&5&6&7&8&9\\\\\n9&5&7&8&9\n\\end{tabular}\n\\end{center}\n(1) 从这 $12$ 天的数据中随机抽取 $3$ 个, 求至多有 $1$ 天的用水量超标的概率;\\\\\n(2) 以这 $12$ 天的样本数据中用水量超标的频率作为概率, 估计该企业未来 $3$ 天中用水量超标的天数, 记随机变量 $X$ 为未来这 $3$ 天中用水量超标的天数, 求 $X$ 的分布、数学期望和方差.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655179,7 +656352,9 @@ "id": "024184", "content": "若角 $\\alpha$ 的终边经过点 $P(2,-3)$, 则角 $\\alpha$ 的余弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655201,7 +656376,9 @@ "id": "024185", "content": "已知 $\\mathrm{i}$ 为虚数单位, 若复数 $z=\\dfrac{\\mathrm{i}}{\\sqrt{2}+\\mathrm{i}}$, 则 $z \\cdot \\overline{z}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655221,7 +656398,9 @@ "id": "024186", "content": "化简: $\\dfrac{\\sin (2 \\pi-\\alpha) \\tan (\\pi+\\alpha) \\tan (-\\alpha-\\dfrac{\\pi}{2})}{\\cos (\\pi-\\alpha) \\tan (3 \\pi-\\alpha)}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655241,7 +656420,9 @@ "id": "024187", "content": "若函数 $f(x)=\\sin (k x+\\dfrac{\\pi}{5})$ 的最小正周期为 $\\dfrac{2 \\pi}{3}$, 则实数 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655261,7 +656442,9 @@ "id": "024188", "content": "已知向量 $\\overrightarrow{a}$、$\\overrightarrow{b}$ 满足 $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, 且向量 $\\overrightarrow{a}$、$\\overrightarrow{b}$ 的夹角为 $\\dfrac{\\pi}{4}$, 若 $\\overrightarrow{a}-\\lambda \\overrightarrow{b}$ 与 $\\overrightarrow{b}$ 垂直, 则实数 $\\lambda$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655281,7 +656464,10 @@ "id": "024189", "content": "设 $\\theta \\in(0, \\dfrac{\\pi}{2})$. 若关于 $x$ 的方程 $(\\sin \\theta+\\cos \\theta)^2=2^x+2^{-x}$ 有解, 则 $\\dfrac{1}{\\sin \\theta}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655301,7 +656487,9 @@ "id": "024190", "content": "锐角 $\\triangle ABC$ 中, 若 $\\tan C=2$, 则 $\\dfrac{\\sin A}{\\sin B}$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655321,7 +656509,9 @@ "id": "024191", "content": "计算: $1+2 \\mathrm{i}+3 \\mathrm{i}^2+4 \\mathrm{i}^3+5 \\mathrm{i}^4+\\cdots+2021 \\mathrm{i}^{2020}+2022 \\mathrm{i}^{2021}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655343,7 +656533,9 @@ "id": "024192", "content": "已知 $\\triangle ABC$ 是边长为 $2 \\sqrt{3}$ 的正三角形, $PQ$ 为 $\\triangle ABC$ 外接圆 $O$ 的一条直径, 若 $M$ 为 $\\triangle ABC$ 边上的动点, 则 $\\overrightarrow{PM}\\cdot \\overrightarrow{MQ}$ 的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655363,7 +656555,9 @@ "id": "024193", "content": "下列命题中, 正确的是\\bracket{20}.\n\\onech{复数与它的共轭复数的差是纯虚数}{$z_1^2+z_2^2=0$ 是复数 $z_1=z_2=0$ 的充要条件}{复数 $Z$ 为纯虚数的必要非充分条件是 $Z+\\overline{Z}=0$}{任何两个复数都不可以比较大小}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -655383,7 +656577,9 @@ "id": "024194", "content": "点 $O$ 在 $\\triangle ABC$ 所在平面内, 给出下列关系式:\\\\\n\\textcircled{1} $\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}=\\overrightarrow{0}$;\\\\\n\\textcircled{2} $\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=\\overrightarrow{OB}\\cdot \\overrightarrow{OC}=\\overrightarrow{OC}\\cdot \\overrightarrow{OA}$;\\\\\n\\textcircled{3} $\\overrightarrow{OA}\\cdot(\\dfrac{1}{|\\overrightarrow{AC}|}\\overrightarrow{AC}-\\dfrac{1}{|\\overrightarrow{AB}|}\\overrightarrow{AB})=\\overrightarrow{OB}\\cdot(\\dfrac{1}{|\\overrightarrow{BC}|}\\overrightarrow{BC}-\\dfrac{1}{|\\overrightarrow{BA}|}\\overrightarrow{BA})=0$;\\\\\n\\textcircled{4} $(\\overrightarrow{OA}+\\overrightarrow{OB}) \\cdot \\overrightarrow{AB}=(\\overrightarrow{OB}+\\overrightarrow{OC}) \\cdot \\overrightarrow{BC}=0$.\\\\\n则点 $O$ 依次为 $\\triangle ABC$ 的\\bracket{20}.\n\\twoch{内心、外心、重心、垂心}{重心、外心、内心、垂心}{重心、垂心、内心、外心}{外心、内心、垂心、重心}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -655403,7 +656599,9 @@ "id": "024195", "content": "已知 $\\overrightarrow{OP}=(2,1)$, $\\overrightarrow{OA}=(1,7)$, $\\overrightarrow{OB}=(5,1)$, 设 $M$ 是直线 $OP$ 上一点 ($O$ 为坐标原点).\\\\\n(1) 求使 $\\overrightarrow{MA}\\cdot \\overrightarrow{MB}$ 取最小值时的 $\\overrightarrow{OM}$;\\\\\n(2) 对 (1) 中的点 $M$, 求 $\\angle AMB$ 的值 (结果用反三角函数值表示).", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655423,7 +656621,9 @@ "id": "024196", "content": "如图, 某快递小哥从 $A$ 地出发, 沿小路 $AB \\to BC$ 以平均时速 $20 \\mathrm{km}/ \\mathrm{h}$, 送快件到 $C$处, 已知 $BD=10 \\mathrm{km}$, $\\angle DCB=45^{\\circ}$, $\\angle CDB=30^{\\circ}, \\triangle ABD$ 是等腰三角形, $\\angle ABD=120^{\\circ}$.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0) node [right] {$D$} coordinate (D);\n\\draw (-120:2) node [left] {$A$} coordinate (A);\n\\draw (105:{2/sin(45)*sin(30)}) node [left] {$C$} coordinate (C);\n\\draw (C)--(B)--(A)--(D)--cycle(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 试问, 快递小哥能否在 $50 \\mathrm{min}$ 内将快件送到 $C$ 处?\\\\\n(2) 快递小哥出发 $15 \\mathrm{min}$ 后, 快递公司发现快件有重大问题, 由于通信不畅, 公司只能派车沿大路 $AD arrow DC$ 追赶, 若汽车平均时速 $60 \\mathrm{km}/ \\mathrm{h}$, 问汽车能否先到达 $C$ 处?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655443,7 +656643,9 @@ "id": "024197", "content": "如图, 点 $P$ 在直径 $AB=1$ 的半圆上移动 (点 $P$ 不与 $A$、$B$ 重合), 过 $P$ 作圆的切线 $PT$, 且 $PT=1$, $\\angle PAB=\\alpha$. 过点 $B$ 作 $BC \\perp PT$ 于点 $C$.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2.5]\n\\def\\t{110}\n\\def\\s{60}\n\\draw (\\t:0.5) node [above] {$A$} coordinate (A);\n\\draw ({\\t-180}:0.5) node [below] {$B$} coordinate (B);\n\\draw (\\s:0.5) node [above] {$P$} coordinate (P);\n\\draw (P) ++ ({\\s-90}:1) node [right] {$T$} coordinate (T);\n\\draw ($(P)!(B)!(T)$) node [above right] {$C$} coordinate (C);\n\\draw (A)--(B)--(T)--(P)--cycle(P)--(B)--(C);\n\\draw (B) arc ({\\t-180}:\\t:0.5);\n\\draw [dashed] (B) arc ({\\t+180}:\\t:0.5);\n\\draw pic [draw, \"$\\alpha$\", scale = 0.5, angle eccentricity = 1.8] {angle = B--A--P};\n\\end{tikzpicture}\n\\end{center}\n(1) 求三角形 $PAB$ 的面积 (用 $\\alpha$ 表示);\\\\\n(2) 当 $\\alpha$ 为何值时, 四边形 $ABTP$ 的面积最大?\\\\\n(3) 求 $PA+PB+PC$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655463,7 +656665,9 @@ "id": "024198", "content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=BC=1$, $AA_1=\\sqrt{3}$, 则异面直线 $AD_1$ 与 $DB_1$ 所成角的余弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655483,7 +656687,9 @@ "id": "024199", "content": "平面 $M$ 上有 $4$ 个点, 平面 $N$ 上有 $3$ 个点, 这 $7$ 个点最多可确定\\blank{50}个平面.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655503,7 +656709,9 @@ "id": "024200", "content": "已知 $A(1,1,1)$、$B(-1,0,4)$、$C(2,-2,3)$, 则以 $AB$、$AC$ 为邻边的平行四边形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655527,7 +656735,9 @@ "id": "024201", "content": "若正三棱锥的高为 $1$, 底面边长为 $2 \\sqrt{3}$, 内有一个球与四个面都相切, 则棱锥的内切球的半径为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655549,7 +656759,9 @@ "id": "024202", "content": "如图, 在棱长为 $1$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $M$、$N$ 分别是 $A_1D_1$、$A_1B_1$ 的中点, 过直线 $BD$ 的平面 $\\alpha \\parallel $ 平面 $AMN$, 则平面 $\\alpha$ 截该正方体所得截面的面积为\\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 ($(A_1)!0.5!(D_1)$) node [left] {$M$} coordinate (M);\n\\draw ($(A_1)!0.5!(B_1)$) node [below] {$N$} coordinate (N);\n\\draw (M)--(N)--(A);\n\\draw [dashed] (B)--(D)(A)--(M);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655569,7 +656781,9 @@ "id": "024203", "content": "我们把平面内与直线垂直的非零向量称为直线的法向量, 在平面直角坐标系中, 利用求动点轨迹方程的方法, 可以求出过点 $A(-3,4)$, 且法向量为 $\\overrightarrow{n}=(1,-2)$ 的直线方程为$1 \\times(x+3)+(-2) \\times(y-4)=0$, 即 $x-2 y+11=0$. 类比以上方法, 在空间直角坐标系中, 经过点 $A(1,2,3)$, 且法向量为 $\\overrightarrow{m}=(-1,-2,1)$ 的平面的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655589,7 +656803,9 @@ "id": "024204", "content": "在正方体 $ABCD-A_1B_1C_1D_1$ 中, $M$、$N$、$Q$ 分别是棱 $D_1C_1$、$A_1D_1$、$BC$ 的中点, 点 $P$ 在 $BD_1$ 上且 $BP=\\dfrac{2}{3}BD_1$, 则下面所有说法中正确的序号是\\blank{50}.\\\\\n\\textcircled{1} $MN \\parallel $ 平面 $APC$;\\\\\n\\textcircled{2} $C_1Q \\parallel $ 平面 $APC$;\\\\\n\\textcircled{3} $A$、$P$、$M$ 三点共线;\\\\\n\\textcircled{4} 平面 $MNQ \\parallel $ 平面 $APC$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655609,7 +656825,9 @@ "id": "024205", "content": "已知圆柱的上底面圆周经过正三棱锥 $P-ABC$ 的三条侧棱的中点, 下底面圆心为此三棱锥底面中心 $O$. 若三棱锥 $P-ABC$ 的高为该圆柱外接球半径的 $2$ 倍, 则该三棱锥的外接球与圆柱外接球的半径之比为\\bracket{20}.\n\\fourch{$2: 1$}{$7: 4$}{$3: 1$}{$5: 3$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -655629,7 +656847,9 @@ "id": "024206", "content": "如图, 已知圆锥 $SO$ 底面圆的半径 $r=1$, 直径 $AB$ 与直径 $CD$ 垂直, 母线 $SA$ 与底面所成角的大小为 $\\dfrac{\\pi}{3}$.\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (0,{2*sqrt(3)}) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(B)$) node [above right] {$E$} coordinate (E);\n\\draw (100:2 and 0.5) node [above] {$D$} coordinate (D);\n\\draw (-80:2 and 0.5) node [below] {$C$} coordinate (C);\n\\draw (A)--(S)--(B)(A) arc (180:360:2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw [dashed] (A)--(B)(C)--(D)(C)--(E)--(D)(O)--(S);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆锥 $SO$ 的侧面积;\\\\\n(2) 若 $E$ 为母线 $SA$ 的中点, 求二面角 $E-CD-B$ 的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655649,7 +656869,9 @@ "id": "024207", "content": "如图, 在四面体 $P-ABC$ 中, $PA=PC=AB=BC=5$, $AC=6$, $PB=4 \\sqrt{2}$, 线段 $AC$、$PA$ 的中点分别为 $O$、$Q$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw (-3,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,4,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,4) node [below] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(P)$) node [above left] {$Q$} coordinate (Q);\n\\draw (A)--(P)--(C)--(B)--cycle(Q)--(B)(P)--(B);\n\\draw [dashed] (A)--(C)(B)--(O)--(P)(O)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:平面 $PAC \\perp$ 平面 $ABC$;\\\\\n(2) 求四面体 $P-OBQ$ 的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655669,7 +656891,9 @@ "id": "024208", "content": "如图, 在三棱锥 $P-ABC$ 中, $PA \\perp$ 底面 $ABC, \\angle BAC=90^{\\circ}$. 点 $D$、$E$、$N$ 分别为棱 $PA$、$PC$、$BC$ 的中点, $M$ 是线段 $AD$ 的中点, $PA=AC=4$, $AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, x = {(-7:1cm)}, z = {(-135:0.5cm)}, scale = 0.7]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,4,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [left] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(P)$) node [above right] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) node [below] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(D)$) node [left] {$M$} coordinate (M);\n\\draw (B)--(C)--(P)--cycle(B)--(E)(N)--(E);\n\\draw [dashed] (A)--(B)(A)--(C)(A)--(P)(B)--(D)--(E)(E)--(M)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN \\parallel $ 平面 $BDE$;\\\\\n(2) 已知点 $H$ 在棱 $PA$ 上, 且直线 $NH$ 与直线 $BE$ 所成角的余弦值为 $\\dfrac{\\sqrt{7}}{21}$, 求线段 $AH$ 的长.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655689,7 +656913,9 @@ "id": "024209", "content": "已知 $\\overrightarrow{a}=(x_1, y_1, z_1)$, $\\overrightarrow{b}=(x_2, y_2, z_2)$, $\\overrightarrow{c}=(x_3, y_3, z_3)$, 定义一种运算: $(\\overrightarrow{a}\\times \\overrightarrow{b}) \\cdot \\overrightarrow{c}=x_1y_2z_3+x_2y_3z_1+x_3y_1z_2-x_1y_3z_2-x_2y_1z_3-x_3y_2z_1$, 已知四棱锥 $P-ABCD$中, 底面 $ABCD$ 是一个平行四边形, $\\overrightarrow{AB}=(2,-1,4)$, $\\overrightarrow{AD}=(4,2,0)$, $\\overrightarrow{AP}=(-1,2,1)$.\\\\\n(1) 试计算 $(\\overrightarrow{AB}\\times \\overrightarrow{AD}) \\cdot \\overrightarrow{AP}$ 的绝对值的值, 并求证 $PA \\perp$ 面 $ABCD$;\\\\\n(2) 求四棱锥 $P-ABCD$ 的体积, 说明 $(\\overrightarrow{AB}\\times \\overrightarrow{AD}) \\cdot \\overrightarrow{AP}$ 的绝对值的值与四棱锥 $P-ABCD$体积的关系, 并由此猜想向量这一运算 $(\\overrightarrow{AB}\\times \\overrightarrow{AD}) \\cdot \\overrightarrow{AP}$ 的绝对值的几何意义.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655709,7 +656935,9 @@ "id": "024210", "content": "如图, 在三棱锥 $P-ABC$ 中, $AB=BC=2 \\sqrt{2}$, $PA=PB=PC=AC=4$, $O$ 为 $AC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0) node [above left] {$O$} coordinate (O);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw ($(B)!{1/3}!(C)$) node [below right] {$M$} coordinate (M);\n\\draw (0,{2*sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(M)(P)--(B);\n\\draw [dashed] (A)--(M)(A)--(C)(P)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $PO \\perp$ 平面 $ABC$;\\\\\n(2) 若点 $M$ 在棱 $BC$ 上, 且二面角 $M-PA-C$ 为 $30^{\\circ}$, 求 $PC$ 与平面 $PAM$ 所成角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655731,7 +656959,9 @@ "id": "024211", "content": "函数 $y=\\dfrac{(x-2)^0}{x+1}+\\log _x(x+2)$ 的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655751,7 +656981,9 @@ "id": "024212", "content": "``$a=1$''是``函数 $f(x)=|x-a|$ 在区间 $[1,+\\infty)$ 上为严格增函数''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655774,7 +657006,9 @@ "id": "024213", "content": "函数 $y=|x+1|+\\sqrt{x^2-4 x+4}$ 的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655794,7 +657028,9 @@ "id": "024214", "content": "若函数 $f(x)=3^x+\\dfrac{a}{3^x+1}$ 最小值为 $\\dfrac{5}{3}$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655817,7 +657053,9 @@ "id": "024215", "content": "函数 $y=\\log _{0.7}(6-x-x^2)$ 的严格增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655839,7 +657077,9 @@ "id": "024216", "content": "$f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 且 $f(x+1)=\\dfrac{1+f(x)}{1-f(x)}$, ($f(x) \\neq 0,1$). 若 $f(1)=3$, 则 $f(2025)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655859,7 +657099,10 @@ "id": "024217", "content": "若 $p$: $\\log _2x<0$, $q$: $x<1$, 则 $p$ 是 $q$ 成立的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -655879,7 +657122,9 @@ "id": "024218", "content": "若定义在 $\\mathbf{R}$ 上的奇函数 $f(x)$ 在 $(0,+\\infty)$ 上是严格增函数, 又 $f(-3)=0$, 则不等式 $x f(x)<0$ 的解集为\\bracket{20}.\n\\fourch{$(-3,0) \\cup(0,3)$}{$(-\\infty,-3) \\cup$$(3,+\\infty)$}{$(-3,0) \\cup$$(3,+\\infty)$}{$(-\\infty,-3) \\cup(0,3)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -655899,7 +657144,9 @@ "id": "024219", "content": "(1) 已知 $x \\in[\\dfrac{1}{27}, 9]$, 求函数 $y=\\log _3\\dfrac{x}{27}\\cdot \\log _33 x$ 的最大值和最小值;\\\\\n(2) 已知 $a>0$ 且 $a \\neq 1$, 关于 $x$ 的方程 $a^{2 x}-2 k a^x+k+2=0$ 有实数解, 求实数 $k$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655919,7 +657166,9 @@ "id": "024220", "content": "设 $a>0$ 且 $a \\neq 1$, $t \\in R$, 已知函数 $f(x)=\\log _a(x+1)$, $g(x)=2 \\log _a(2 x+t)$.\\\\\n(1) 当 $t=-1$ 时, 求不等式 $f(x) \\leq g(x)$ 的解;\\\\\n(2) 若函数 $F(x)=a^{f(x)}+t x^2-2 t+1$ 在区间 $(-1,2]$ 上有零点, 求实数 $t$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655939,7 +657188,9 @@ "id": "024221", "content": "某生态基地种植某中药材的年固定成本为 $250$ 万元, 每产出 $x$ 吨需另外投入可变成本 $h(x)$ 万元, 已知 $h(x)=\\begin{cases}a x^2+49 x,& x \\in(0,50],\\\\51 x+\\dfrac{13635}{2 x+1}-860,& x \\in(50,100] .\\end{cases}$ 通过市场分析, 该中药材可以每吨 $50$ 万元的价格全部售完. 设基地种植该中药材年利润为 $y$ 万元, 当基地产出该中药材 $40$ 吨时, 年利润为 $190$ 万元.\\\\\n(1) 求实数 $a$ 的值;\\\\\n(2) 求年利润 $y$ 的最大值 (精确到 $0.1$ 万元), 并求此时的年产量 (精确到 $0.1$ 吨).", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655959,7 +657210,9 @@ "id": "024222", "content": "对于定义域为 $D$ 的函数 $y=f(x)$, 如果存在区间 $[m, n] \\subseteq D$, 同时满足:\\\\\n(1) $f(x)$ 在 $[m, n]$ 内是单调函数;\\\\\n(2) 当定义域是 $[m, n]$ 时, $f(x)$ 的值域也是 $[m, n]$. 则称 $[m, n]$ 是该函数的``和谐区间''.\\\\\n(1) 求证: 函数 $y=g(x)=3-\\dfrac{5}{x}$ 不存在``和谐区间'';\\\\\n(2) 已知: 函数 $y=\\dfrac{(a^2+a) x-1}{a^2x},$($a \\in \\mathbf{R}$, $a \\neq 0$) 有``和谐区间''$[m, n]$, 当 $a$ 变化时, 求出 $n-m$ 的最大值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -655979,7 +657232,9 @@ "id": "024223", "content": "不等式 $(\\dfrac{1}{3})^{x^2-3 x-4}>(\\dfrac{1}{3})^{2 x+10}$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -655999,7 +657254,9 @@ "id": "024224", "content": "已知集合 $A=\\{1,2\\}$, $B=\\{1,2,3,4\\}$, 满足 $A \\cup C=B \\cap C$ 的集合 $C$ 的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -656019,7 +657276,9 @@ "id": "024225", "content": "已知不等式 $|x-m|<1$ 成立的充分不必要条件是 $\\dfrac{1}{3}0$}{$(x+y) x y \\leq 0$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -656187,7 +657460,10 @@ "id": "024233", "content": "若 $aa^2$; \\textcircled{2} $\\dfrac{1}{b}<\\dfrac{1}{a}$; \\textcircled{3} $3^b>3^a$; \\textcircled{4} $\\lg \\dfrac{a}{b}<0$; \\textcircled{5} $a^{\\frac{1}{5}}\\log _a(-x^2+2 x+3)$ 的解, 求此不等式的解集.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -656227,7 +657505,9 @@ "id": "024235", "content": "已知集合 $A=\\{x | x^2+x-6 \\leq 0\\}$, $B=\\{x|| x-a |<1\\}$.\\\\\n(1) 若 $B \\subseteq A$, 求实数 $a$ 的取值范围;\\\\\n(2) 若 $A \\cap B=\\varnothing$, 求实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -656247,7 +657527,9 @@ "id": "024236", "content": "某商场对全年购销某种品牌电脑的策略调整如下: 分批购入价值 $4000$ 元的电脑共 $1800$ 台, 每批都购入 $x$ 台 ($x \\in \\mathbf{Z}$), 且每批均需支付运费 $400 $元, 储存购入的电脑全年保管费与每批购入的电脑的总价值 (不含运费) 成正比, 若每批购 $200$ 台, 则全年需要用去运费和保管费 $43600$ 元.\\\\\n(1) 试将全年所需运费和保管费 $y($ 元 $)$ 表示为每批购人台数 $x$ 的函数;\\\\\n(2) 现全年只有 $24000$ 元资金可用于支付运费和保管费, 试分析是否能够恰当安排每批进货数量, 使资金够用?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -656267,7 +657549,9 @@ "id": "024237", "content": "已知非空集合 $S \\subset\\{x | x \\in \\mathbf{N}, x \\geq 1\\}$, 且若 $x \\in S$, 则 $\\dfrac{36}{x}\\in S$.\\\\\n(1) 写出所有只含三个元素的集合 $S$;\\\\\n(2) 写出所有只含四个元素的集合 $S$;\\\\\n(3) 满足题设条件的集合 $S$ 共有几个?", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -656287,7 +657571,9 @@ "id": "024238", "content": "已知 $a$ 和 $b$ 是任意非零实数.\\\\\n(1) 求 $\\dfrac{|2 a+b|+|2 a-b|}{|a|}$ 的最小值;\\\\\n(2) 若不等式 $|2 a+b|+|2 a-b| \\geq|a|(|2+x|+|2-x|)$ 恒成立, 求实数 $x$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -656307,7 +657593,9 @@ "id": "024239", "content": "在等差数列 $\\{a_n\\}$ 中, 若 $a_1, a_2, a_4$ 成等比数列, 其公比为 $q$, 则 $q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -656327,7 +657615,9 @@ "id": "024240", "content": "等差数列 $\\{a_n\\}, a_1=\\dfrac{5}{6}$, 公差 $d=-\\dfrac{1}{6},\\{a_n\\}$ 的前 $n$ 项的和为 $S_n$. 若 $S_k=-5$, 则 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -656347,7 +657637,9 @@ "id": "024241", "content": "等差数列前 $4$ 项和为 $124$, 最后 $4$ 项和为 $156$, 若它的各项和为 $210$, 则该数列的项数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -656370,7 +657662,10 @@ "id": "024242", "content": "若 $\\alpha:(\\dfrac{1}{4})^x,(\\dfrac{1}{2})^x, 2^{x-4}$ 成等比数列; $\\beta: \\lg x, \\lg (x+2), \\lg (2 x+1)$ 成等差数列. 则 $\\alpha$ 是 $\\beta$ 的\\blank{50}条件. (填入``充分不必要''、``必要不充分''、``充要''、``不充分不必要''之一)", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -656391,7 +657686,9 @@ "id": "024243", "content": "已知\\textcircled{1}、\\textcircled{2}、\\textcircled{3}是三个陈述句, 其中\\\\\n\\textcircled{1}: 数列 $\\{a_n\\}$ 是等差数列, 且公差大于 $0$ ;\\\\\n\\textcircled{2}: 数列 $\\{a_n\\}$ 是等比数列, 且公比大于 $1$ ;\\\\\n\\textcircled{3}: 数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 且 $2S_n1$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "20240116\t2024届高三07班\t0.971", "solution": "", @@ -656619,7 +657936,9 @@ "id": "024254", "content": "设向量 $\\overrightarrow{a}=(1,-2)$, $\\overrightarrow{b}=(-1, m)$, 若 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "20240116\t2024届高三07班\t1.000", "solution": "", @@ -656642,7 +657961,9 @@ "id": "024255", "content": "将 $4$ 个人排成一排, 若甲和乙必须排在一起, 则共有\\blank{50}种不同排法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "20240116\t2024届高三07班\t0.914", "solution": "", @@ -656664,7 +657985,9 @@ "id": "024256", "content": "物体位移 $s$ 和时间 $t$ 满足函数关系 $s=100 t-5 t^2$($00$, 使得 $x^2+a x+1<0$''是假命题, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "20240116\t2024届高三07班\t0.714", "solution": "", @@ -656775,7 +658106,9 @@ "id": "024261", "content": "若函数 $f(x)=\\sin x+a \\cos x$ 在 $(\\dfrac{2 \\pi}{3}, \\dfrac{7 \\pi}{6})$ 上是严格单调函数, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "20240116\t2024届高三07班\t0.486", "solution": "", @@ -656795,7 +658128,9 @@ "id": "024262", "content": "设 $f(x)=|\\log _2 x+a x+b|$($a>0$), 记函数 $y=f(x)$ 在区间 $[t, t+1]$($t>0$) 上的最大值为 $M_t(a, b)$, 若对任意 $b \\in \\mathbf{R}$, 都有 $M_t(a, b) \\geq a+1$, 则实数 $t$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "20240116\t2024届高三07班\t0.257", "solution": "", @@ -656815,7 +658150,9 @@ "id": "024263", "content": "下列函数中既是奇函数又是增函数的是\\bracket{20}.\n\\fourch{$f(x)=2 x$}{$f(x)=x^2$}{$f(x)=\\ln x$}{$f(x)=\\mathrm{e}^x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "20240116\t2024届高三07班\t0.971", "solution": "", @@ -656837,7 +658174,9 @@ "id": "024264", "content": "``$P(A \\cap B)=P(A) P(B)$''是``事件 $A$ 与事件 $\\overline{B}$ 互相独立''\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "20240116\t2024届高三07班\t0.857", "solution": "", @@ -656857,7 +658196,9 @@ "id": "024265", "content": "设点 $P$ 是以原点为圆心的单位圆上的动点, 它从初始位置 $P_0(1,0)$ 出发, 沿单位圆按逆时针方向转动角 $\\alpha$($0<\\alpha<\\dfrac{\\pi}{2}$) 后到达点 $P_1$, 然后继续沿单位圆按逆时针方向转动角 $\\dfrac{\\pi}{4}$ 到达 $P_2$. 若点 $P_2$ 的横坐标为 $-\\dfrac{3}{5}$, 则点 $P_1$ 的纵坐标 \\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{2}}{10}$}{$\\dfrac{\\sqrt{2}}{5}$}{$\\dfrac{3 \\sqrt{2}}{5}$}{$\\dfrac{7 \\sqrt{2}}{10}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "20240116\t2024届高三07班\t0.914", "solution": "", @@ -656879,7 +658220,9 @@ "id": "024266", "content": "豆腐发酵后衣面长出一层白线线的长毛就成了毛豆腐. 将三角形豆腐 $ABC$ 悬空挂在发酵空间内, 记发酵后毛豆腐所构成的儿何体为 $T$. 若忽略三角形豆腐 $ABC$ 的原度, 设 $AB=3$, $BC=4$, $AC=5$, 点 $P$ 在 $\\triangle ABC$ 内部. 假设对于任意点 $P$, 满足 $PQ \\leq 1$ 的点 $Q$ 都在 $T$ 内,且对于 $T$ 内任意一点 $Q$, 都存在点 $P$, 满足 $PQ \\leq 1$, 则 $T$ 的体积为\\bracket{20}.\n\\fourch{$12+7 \\pi$}{$12+\\dfrac{22 \\pi}{3}$}{$14+7 \\pi$}{$14+\\dfrac{22 \\pi}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "20240116\t2024届高三07班\t0.686", "solution": "", @@ -656899,7 +658242,9 @@ "id": "024267", "content": "已知等差数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 公差 $d=2$.\\\\\n(1) 若 $S_{10}=100$, 求 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 从集合 $\\{a_1, a_2, a_3, a_4, a_5, a_6\\}$ 中任取 3 个元素, 记这 3 个元索能成等差数列为事件 $A$,求事件 $A$ 发生的概率 $P(A)$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "20240116\t2024届高三07班\t0.914\t0.943", "solution": "", @@ -656919,7 +658264,9 @@ "id": "024268", "content": "如图, 在三棱锥 $A-BCD$ 中, 平面 $ABD \\perp$ 平面 $BCD$, $AB=AD$, $O$为 $BD$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0,0) node [left] {$B$} coordinate (B);\n\\draw (1,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,1,0) node [above] {$A$} coordinate (A);\n\\draw (1,0,2) node [below] {$C$} coordinate (C);\n\\draw (B)--(C)--(D)--(A)--cycle(A)--(C);\n\\draw ($(B)!0.5!(D)$) node [above left] {$O$} coordinate (O);\n\\draw [dashed] (B)--(D)(A)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AO \\perp CD$;\\\\\n(2) 若 $BD \\perp DC$, $BD=DC$, $AO=BO$, 求异面直线 $BC$ 与 $AD$ 所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "20240116\t2024届高三07班\t0.986\t0.743", "solution": "", @@ -656941,7 +658288,9 @@ "id": "024269", "content": "汽车转弯时遵循阿克曼转向几何原理, 即转向时所有车轮中垂线交于一点, 该点称为转向中心: 如图 1, 某汽车四轮中心分别为 $A$、$B$ 、 $C$、$D$, 向左转向, 左前轮转向角为 $\\alpha$, 右前轮转向角为 $\\beta$, 转向中心为 $O$. 设该汽车左右轮距 $AB$ 为 $w$ 米, 前后轴距 $AD$ 为 $l$ 米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{1.57}\n\\def\\w{2.68}\n\\draw (0,0) node [below left] {$D$} coordinate (D);\n\\draw (\\l,0) node [below right] {$C$} coordinate (C);\n\\draw (C)++(0,\\w) node [above right] {$B$} coordinate (B);\n\\draw (D)++(0,\\w) node [above left] {$A$} coordinate (A);\n\\draw (A)--(B)(C)--(D);\n\\draw ($(C)!0.5!(D)$) coordinate (MB) ($(A)!0.5!(B)$) coordinate (MU);\n\\draw (MB)--(MU);\n\\draw (D)++ ({-sqrt(3)*\\w},0) node [below] {$O$} coordinate (O);\n\\draw [dashed] (O)--(D)(O)--(A)(O)--(B)(A)--(D)(B)--(C);\n\\def\\t{atan(\\w/(sqrt(3)*\\w+\\l))}\n\\draw [ultra thick] (D) ++ (0,0.4) --++ (0,-0.8);\n\\draw [ultra thick] (C) ++ (0,0.4) --++ (0,-0.8);\n\\draw [ultra thick] (A) ++ (120:0.4) --++ (-60:0.8);\n\\draw [ultra thick] (B) ++ ({90+\\t}:0.4) --++ ({\\t-90}:0.8);\n\\draw ($(O)!0.5!(C)$) ++ (0,-1) node [below] {图 1};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{1.57}\n\\def\\w{2.68}\n\\draw (0,0) node [below left] {$D$} coordinate (D);\n\\draw (\\l,0) node [below] {$C$} coordinate (C);\n\\draw (C)++(0,\\w) node [above] {$B$} coordinate (B);\n\\draw (D)++(0,\\w) node [above left] {$A$} coordinate (A);\n\\draw (A)--(B)(C)--(D);\n\\draw ($(C)!0.5!(D)$) coordinate (MB) ($(A)!0.5!(B)$) coordinate (MU);\n\\draw (MB)--(MU);\n\\def\\t{atan(\\w/(sqrt(3)*\\w+\\l))}\n\\draw [ultra thick] (D) ++ (0,0.4) --++ (0,-0.8);\n\\draw [ultra thick] (C) ++ (0,0.4) --++ (0,-0.8);\n\\draw [ultra thick] (A) ++ (120:0.4) --++ (-60:0.8);\n\\draw [ultra thick] (B) ++ ({90+\\t}:0.4) --++ ({\\t-90}:0.8);\n\\draw (-1.5,-1) node [left] {$T$} coordinate (T);\n\\draw (-1.5,1.5) node [below left] {$M$} coordinate (M);\n\\draw (-3.5,1.5) node [below] {$N$} coordinate (N);\n\\draw (N) ++ (0,3.5) node [above] {$E$} coordinate (E);\n\\draw (M) ++ (3.5,3.5) node [above right] {$F$} coordinate (F);\n\\draw (T) ++ (3.5,0) node [right] {$S$} coordinate (S);\n\\draw (T)--(M)--(N)(S)--(F)--(E);\n\\draw ($(O)!0.5!(C)$) ++ (1,-1.5) node [below] {图 2};\n\\end{tikzpicture}\n\\end{center}\n(1) 试用 $w$、$l$ 和 $\\alpha$ 表示 $\\tan \\beta$;\\\\\n(2) 如图 2, 有一直角弯道, $M$ 为内直角顶点, $EF$ 为上路边, 路宽均为 $3.5$ 米, 汽车行驶其中, 左轮 $A$、$D$ 与路边 $FS$ 相距 2 米. 试依据如下假设, 对问题*做出判断, 并说明理由.\n假设: \\textcircled{1} 转向过程中, 左前轮转向角 $\\alpha$ 的值始终为 $30^{\\circ}$; \\textcircled{2}设转向中心 $O$ 到路边 $EF$ 的距离为 $d$, 若 $OB0$), $g(x)=\\sin x$, 求实数 $a$ 的取值范围;\\\\\n(3) 若 $y=g(x)$ 为严格减函数, $f(0)=latex,scale = 1.5]\n \\draw (-1.414,0,1.414) node [left] {$A$} coordinate (A);\n \\draw (1.414,0,1.414) node [right] {$B$} coordinate (B);\n \\draw (1.414,0,-1.414) node [right] {$C$} coordinate (C);\n \\draw (-1.414,0,-1.414) node [below] {$D$} coordinate (D);\n \\draw (0,2,0) node [above] {$P$} coordinate (P);\n \\draw (P) -- (A) (P) -- (B) (P) -- (C) (A) -- (B) -- (C);\n \\draw [dashed] (P) -- (D) (A) -- (D) -- (C);\n \\draw ($(P)!0.5!(D)$) node [below] {$F$} coordinate (F);\n \\draw ($(P)!0.5!(B)$) node [right] {$E$} coordinate (E);\n \\draw ($(P)!0.25!(C)$) node [right] {$G$} coordinate (G);\n \\draw ($(A)!0.5!(D)$) node [below] {$M$} coordinate (M);\n \\draw ($(A)!0.5!(B)$) node [below] {$N$} coordinate (N);\n \\draw ($(B)!0.5!(E)$) node [right] {$Q$} coordinate (Q);\n \\draw [dashed] (G) -- (F) -- (A) (M) -- (N);\n \\draw (G) -- (E) -- (A) (N) -- (Q);\n \\end{tikzpicture} \n \\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "",