收录高三下学期测验03新题
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20240326-221631 高三下学期周末卷06
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030615,030966,030760,014809,024333,032180,014811,024334,031000,014812,032181,014815,030674,030516,032182,030785,031048,019475,032183,031087,014825
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20240327-192547 高三下学期测验03
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014996:014999,015001,017227,032184:032185,015003,040713,032186,017232,032187:032188,030993,032189:032194
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"20230413\t王伟叶"
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@ -716371,7 +716378,8 @@
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"017229"
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@ -750309,6 +750317,236 @@
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"space": "4em",
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"032184": {
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"id": "032184",
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"content": "设数列 $\\{a_n\\}$ 是首项为 1 , 公比为 $-\\dfrac{1}{2}$ 的无穷等比数列, 则数列 $\\{a_n\\}$ 的所有项的和为\\blank{50}.",
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"genre": "填空题",
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"032185": {
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"id": "032185",
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"content": "已知平面向量 $\\overrightarrow{a}=(2,1), \\overrightarrow{b}$ 为单位向量, 且 $(\\overrightarrow{a}+2 \\overrightarrow{b}) \\perp(\\overrightarrow{a}-\\overrightarrow{b})$, 则向量 $\\overrightarrow{b}$ 在向量 $\\overrightarrow{a}$ 上的投影向量的坐标为\\blank{50}.",
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"032186": {
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"id": "032186",
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"content": "若平面上的三个单位向量 $\\overrightarrow{a}$、$\\overrightarrow{b}$、$\\overrightarrow{c}$ 满足 $|\\overrightarrow{a}\\cdot \\overrightarrow{b}|=\\dfrac{1}{2}$, $|\\overrightarrow{a}\\cdot \\overrightarrow{c}|=\\dfrac{\\sqrt{3}}{2}$, 则 $\\overrightarrow{b}\\cdot \\overrightarrow{c}$ 的所有可能的值组成的集合为\\blank{50}.",
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"032187": {
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"id": "032187",
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"content": "已知 $a$、$b \\in \\mathbf{R}$, $a>b$, 则下列不等式中不一定成立的是\\bracket{20}.\n\\fourch{$a+2>b+2$}{$2 a>2 b$}{$a^2>b^2$}{$2^a>2^b$}",
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"objs": [],
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"genre": "选择题",
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"ans": "",
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"032188": {
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"id": "032188",
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"content": "用最小二乘法求回归方程是为了使\\bracket{20}.\n\\fourch{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)^2$ 最小}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)$ 最小}{$\\displaystyle\\sum_{i=1}^n(y_i-\\overline{y})=0$}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)=0$}",
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"20240327\t余利成"
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"032189": {
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"id": "032189",
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"content": "若直线 $a x+b y=2$ 经过点 $M(\\cos \\theta, \\sin \\theta)$, 其中 $\\theta \\in \\mathbf{R}$, 则\\bracket{20}.\n\\fourch{$a^2+b^2 \\leq 4$}{$a^2+b^2 \\geq 4$}{$\\dfrac{1}{a^2}+\\dfrac{1}{b^2}\\leq 4$}{$\\dfrac{1}{a^2}+\\dfrac{1}{b^2}\\geq 4$}",
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"032190": {
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"id": "032190",
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"content": "如图, 在四棱锥 $P-ABCD$ 中, 底面 $ABCD$ 是边长为 $a$ 的正方形, 侧面 $PAD \\perp$ 底面 $ABCD$,且 $PA=PD=\\dfrac{\\sqrt{2}}{2}a$, 设 $E$、$F$ 分别为 $PC$、$BD$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,1) node [below] {$A$} coordinate (A);\n\\draw (2,0,1) node [below] {$B$} coordinate (B);\n\\draw (2,0,-1) node [right] {$C$} coordinate (C);\n\\draw (0,0,-1) node [left] {$D$} coordinate (D);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(D)$) node [below] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(P)$) node [above] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(P)--(D)--(B)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明:直线 $EF \\parallel $ 平面 $PAD$;\\\\\n(2) 求直线 $PB$ 与平面 $ABCD$ 所成的角的正切值.",
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"objs": [],
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"genre": "解答题",
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"ans": "",
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"20240327\t余利成"
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"space": "4em",
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"032191": {
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"id": "032191",
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"content": "在 $\\triangle ABC$ 中, 角 $A$、$B$、$C$ 所对边的边长分别为 $a$、$b$、$c$, 且 $a-2 c \\cos B=c$.\\\\\n(1) 若 $\\cos B=\\dfrac{1}{3}$, $c=3$, 求 $b$ 的值;\\\\\n(2) 若 $\\triangle ABC$ 为锐角三角形, 求 $\\sin C$ 的取值范围.",
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"objs": [],
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"genre": "解答题",
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"032192": {
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"id": "032192",
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"content": "某无人飞机研发中心最近研发了一款新能源无人飞机, 在投放市场前对 100 架新能源无人飞机进行了单次最大续航里程的测试现对测试数据进行分析, 得到如图所示的频率分布直方图:\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.0143, yscale = 333.3333]\n\\draw [->] (130,0) -- (146.67,0) -- (150.83,0.0006) -- (159.17,-0.0006) -- (163.33,0)-- (480,0) node [below right] {单次最大续航里程(千米)};\n\\draw [->] (130,0) -- (130,0.012) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (130,0) node [below] {$0$};\n\\foreach \\i/\\j in {180/0.002,230/0.004,280/0.009,330/0.004,380/0.001}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (50,0) --++ (0,-\\j);};\n\\draw (430,0) node [below] {$430$};\n\\foreach \\i/\\j/\\k in {180/0.002/0.002,330/0.004/0.004,280/0.009/0.009,380/0.001/0.001}\n{\\draw [dashed] (\\i,\\j) -- (130,\\j) node [left] {$\\k$};};\n\\end{tikzpicture}\n\\end{center}\n(1) 估计这 100 架新能源无人飞机的单次最大续航里程的平均值 $\\overline{x}$ (同一组中的数据用该组区间的中点值代表); \\\\\n(2) 经计算第 (1) 问中样本标准差 $s$ 的近似值为 50 , 根据大量的测试数据, 可以认为这款新能源无人飞机的单次最大续航里程 $X$ 近似地服从正态分布 $N(\\mu, \\sigma^2)$ (用样本平均数 $\\overline{x}$ 和标准差 $s$ 分别作为 $\\mu$ 和 $\\sigma$ 的近似值), 现任取一架新能源无人飞机, 求它的单次最大续航里程 $X \\in[250,400]$ 的概率;\n(参考数据: 若随机变量 $X \\sim N(\\mu, \\sigma^2)$, 则 $P(\\mu-\\sigma \\leq X \\leq \\mu+\\sigma) \\approx 0.6827$, $P(\\mu-2 \\sigma \\leq X \\leq \\mu+2 \\sigma) \\approx 0.9545$, $P(\\mu-3 \\sigma \\leq X \\leq \\mu+3 \\sigma) \\approx 0.9973$)\n(3) 该无人飞机研发中心依据新能源无人飞机的载重量和续航能力分为卓越 $A$ 型、卓越 $B$ 型和卓越 $C$ 型, 统计分析可知卓越 $A$ 型、卓越 $B$ 型和卓越 $C$ 型的分布比例为 $3: 2: 1$, 研发中心在投放市场前决定分别按卓越 $A$ 型、卓越 $B$ 型和卓越 $C$ 型的分布比例分层随机共抽取 6 架, 然后再从这 6 架中随机抽取 3 架进行综合性能测试, 记随机变量 $Y$ 是综合性能测试的 3 架中卓越 $A$ 型的架数, 求随机变量 $Y$ 的分布和数学期望.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"20240327\t余利成"
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"032193": {
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"id": "032193",
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"content": "椭圆 $C$ 的方程为 $x^2+3 y^2=4$, $A$、$B$ 为椭圆的左右顶点, $F_1$、$F_2$ 为左右焦点, $P$ 为椭圆上的动点.\\\\\n(1) 求椭圆的离心率;\\\\\n(2) 若 $\\triangle PF_1F_2$ 为直角三角形, 求 $\\triangle PF_1F_2$ 的面积;\\\\\n(3) 若 $Q$、$R$ 为椭圆上异于 $P$ 的点, 直线 $PQ$、$PR$ 均与圆 $x^2+y^2=\\dfrac{1}{4}$ 相切, 记直线 $PQ$、$PR$ 的斜率分别为 $k_1$、$k_2$, 是否存在位于第一象限的点 $P$, 使得 $k_1 k_2=1$ ? 若存在, 求出点 $P$ 的坐标, 若不存在, 请说明理由.",
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"objs": [],
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"genre": "解答题",
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"ans": "",
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"20240327\t余利成"
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"032194": {
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"id": "032194",
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"content": "已知 $a \\in \\mathbf{R}$, $f(x)=(a-2) x^3-x^2+5 x+(1-a) \\ln x$.\\\\\n(1) 若 1 为函数 $y=f(x)$ 的驻点, 求实数 $a$ 的值;\\\\\n(2) 若 $a=0$, 试问曲线 $y=f(x)$ 是否存在切线与直线 $x-y-1=0$ 互相垂直? 说明理由;\\\\\n(3) 若 $a=2$, 是否存在等差数列 $x_1, x_2, x_3$($0<x_1<x_2<x_3$), 使得曲线 $y=f(x)$ 在点 $(x_2, f(x_2))$处的切线与过两点 $(x_1, f(x_1))$、$(x_3, f(x_3))$ 的直线互相平行? 若存在, 求出所有满足条件的等差数列; 若不存在, 说明理由.",
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"objs": [],
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"ans": "",
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"20240327\t余利成"
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"040001": {
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"id": "040001",
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"content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\ y=t^2-2\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{一支双曲线}{线段}{圆弧}{射线}",
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"017229"
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