From e328cbb710e85abb001b2f08eed1bb3d2c790705 Mon Sep 17 00:00:00 2001 From: wwy_at_gzs Date: Thu, 8 Jun 2023 15:01:29 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A5=E9=A2=98=E7=9B=AE?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 440 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 440 insertions(+) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 13128000..f10902db 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -462778,6 +462778,446 @@ "space": "4em", "unrelated": [] }, + "018039": { + "id": "018039", + "content": "已知集合$M=\\{-2,-1,0,1,2\\}$, $N=\\{x| x^2-x-6 \\geq 0\\}$, 则$M \\cap N=$\\bracket{20}.\n\\fourch{$\\{-2,-1,0,1\\}$}{$\\{-1,-2\\}$}{$\\{-2\\}$}{$\\{2\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学1", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018040": { + "id": "018040", + "content": "已知$z=\\dfrac{1-\\mathrm{i}}{2+2 \\mathrm{i}}$, 则$z-\\overline {z}=$\\bracket{20}.\n\\fourch{$-\\mathrm{i}$}{$\\mathrm{i}$}{$0$}{$1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学2", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018041": { + "id": "018041", + "content": "已知向量$\\overrightarrow{a}=(1,1)$, $\\overrightarrow{b}=(1,-1)$. 若$(\\overrightarrow{a}+\\lambda \\overrightarrow{b}) \\perp(\\overrightarrow{a}+\\mu \\overrightarrow{b})$, 则\\bracket{20}.\n\\fourch{$\\lambda+\\mu=1$}{$\\lambda+\\mu=-1$}{$\\lambda \\mu=1$}{$\\lambda \\mu=-1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学3", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018042": { + "id": "018042", + "content": "设函数$f(x)=2^{x(x-a)}$在区间$(0,1)$单调递减, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty,-2]$}{$[-2,0)$}{$(0,2]$}{$[2,+\\infty)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学4", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018043": { + "id": "018043", + "content": "设椭圆$C_1: \\dfrac{x^2}{a^2}+y^2=1$($a>1$), $C_2: \\dfrac{x^2}{4}+y^2=1$的离心率分别为$e_1, e_2$. 若$e_2=\\sqrt{3} e_1$, 则$a=$\\bracket{20}.\n\\fourch{$\\dfrac{2 \\sqrt{3}}{3}$}{$\\sqrt{2}$}{$\\sqrt{3}$}{$\\sqrt{6}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学5", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018044": { + "id": "018044", + "content": "过点$(0,-2)$与圆$x^2+y^2-4 x-1=0$相切的两条直线的夹角为$\\alpha$, 则$\\sin \\alpha=$\\bracket{20}.\n\\fourch{$1$}{$\\dfrac{\\sqrt{15}}{4}$}{$\\dfrac{\\sqrt{10}}{4}$}{$\\dfrac{\\sqrt{6}}{4}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学6", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018045": { + "id": "018045", + "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 设甲: $\\{a_n\\}$为等差数列: 乙: $\\{\\dfrac{S_n}{n}\\}$为等差数列, 则\\bracket{20}.\n\\twoch{甲是乙的充分条件但不是必要条件}{甲是乙的必要条件但不是充分条件}{甲是乙的充要条件}{甲既不是乙的充分条件也不是乙的必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学7", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018046": { + "id": "018046", + "content": "已知$\\sin (\\alpha-\\beta)=\\dfrac{1}{3}$, $\\cos \\alpha \\sin \\beta=\\dfrac{1}{6}$, 则$\\cos (2 \\alpha+2 \\beta)=$\\bracket{20}.\n\\fourch{$\\dfrac{7}{9}$}{$\\dfrac{1}{9}$}{$-\\dfrac{1}{9}$}{$-\\dfrac{7}{9}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学8", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018047": { + "id": "018047", + "content": "有一组样本数据$x_1, x_2, \\cdots, x_6$, 其中$x_1$是最小值, $x_6$是最大值, 则\\blank{50}.\\\\\n\\textcircled{1} $x_2, x_3, x_4, x_5$的平均数等于$x_1, x_2, \\cdots, x_6$的平均数;\\\\\n\\textcircled{2} $x_2, x_3, x_4, x_5$的中位数等于$x_1, x_2, \\cdots, x_6$的中位数;\\\\\n\\textcircled{3} $x_2, x_3, x_4, x_5$的标准差不小于$x_1, x_2, \\cdots, x_6$的标准差;\\\\\n\\textcircled{4} $x_2, x_3, x_4, x_5$的极差不大于$x_1, x_2, \\cdots, x_6$的极差.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学9", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018048": { + "id": "018048", + "content": "噪声污染问题越来越受到重视. 用声压级来度量声音的强弱, 定义声压级$L_p=20 \\times \\lg \\dfrac{p}{p_0}$, 其中常数$p_0$($p_0>0$) 是听觉下限阈值, $p$是实际声压. 下表为不同声源的声压级:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline 声源 & 与声源的距离/m & 声压级/dB \\\\\n\\hline 燃油轮 & 10 &$60 \\sim 90$\\\\\n\\hline 混合动力汽车 & 10 &$50 \\sim 60$\\\\\n\\hline 电动汽车 & 10 & 40 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n已知在距离燃油汽车、混合动力汽车、电动汽车$10 \\text{m}$处测得实际声压分别为$p_1$, $p_2$, $p_3$, 则\\blank{50}.\\\\\n\\textcircled{1} $p_1 \\geq p_2$; \\textcircled{2} $p_2>10 p_1$; \\textcircled{3} $p_3=100 p_0$; \\textcircled{4} $p_1 \\leq 100 p_2$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学10", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018049": { + "id": "018049", + "content": "已知函数$f(x)$的定义域为$\\mathbf{R}$, $f(x y)=y^2 f(x)+x^2 f(y)$, 则\\blank{50}.\\\\\n\\textcircled{1} $f(0)=0$; \\textcircled{2} $f(1)=0$; \\textcircled{3} $f(x)$是偶函数; \\textcircled{4} $x=0$为$f(x)$的极小值点.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学11", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018050": { + "id": "018050", + "content": "下列物体中, 能够被整体放入棱长为$1$ (单位: m ) 的正方体容器 (容器壁厚度忽略不计) 内的有\\blank{50}.\\\\\n\\textcircled{1} 直径为$0.99 \\mathrm{m}$的球体;\\\\\n\\textcircled{2} 所有棱长均为$1.4 \\mathrm{m}$的四面体;\\\\\n\\textcircled{3} 底面直径为$0.01 \\mathrm{m}$, 高为$1.8 \\mathrm{m}$的圆柱体;\\\\\n\\textcircled{4} 底面直径为$1.2 \\mathrm{m}$, 高为$0.01 \\mathrm{m}$的圆柱体.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学12", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018051": { + "id": "018051", + "content": "某学校开设了$4$门体育类选修课和$4$门艺术类选修课, 学生需从这$8$门课中选修$2$门或$3$门课, 并且每类选修课至少选修$1$门, 则不同的选课方案共有\\blank{50}种(用数字作答).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学13", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018052": { + "id": "018052", + "content": "在正四棱台$ABCD-A_1B_1C_1D_1$中, $AB=2$, $A_1B_1=1$, $AA_1=\\sqrt{2}$, 则该棱台的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学14", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018053": { + "id": "018053", + "content": "已知函数$f(x)=\\cos \\omega x-1$($\\omega>0$) 在区间$[0,2 \\pi]$有且仅有$3$个零点, 则$\\omega$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学15", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018054": { + "id": "018054", + "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的左、右焦点分别为$F_1, F_2$. 点$A$在$C$上, 点$B$在$y$轴上, $\\overrightarrow{F_1A} \\perp \\overrightarrow{F_1B}$, $\\overrightarrow{F_2A}=-\\dfrac{2}{3}\\overrightarrow{F_2B}$, 则$C$的离心率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学16", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018055": { + "id": "018055", + "content": "已知在$\\triangle ABC$中, $A+B=3C$, $2 \\sin (A-C)=\\sin B$. \\\\\n(1) 求$\\sin A$;\\\\\n(2) 设$AB=5$, 求$AB$边上的高.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学17", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018056": { + "id": "018056", + "content": "如图, 在正四楼柱$ABCD-A_1B_1C_1D_1$中, $AB=2$, $AA_1=4$. 点$A_2, B_2, C_2, D_2$分别在棱$AA_1, BB_1, CC_1, DD_1$上, $AA_2=1$, $BB_2=DD_2=2$, $CC_2=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$D$} coordinate (D);\n\\draw (D) ++ (\\l,0,0) node [below right] {$A$} coordinate (A);\n\\draw (D) ++ (\\l,0,-\\l) node [right] {$B$} coordinate (B);\n\\draw (D) ++ (0,0,-\\l) node [left] {$C$} coordinate (C);\n\\draw (D) -- (A) -- (B);\n\\draw [dashed] (D) -- (C) -- (B);\n\\draw (D) ++ (0,{2*\\l},0) node [left] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,{2*\\l},0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,{2*\\l},0) node [above right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,{2*\\l},0) node [above left] {$C_1$} coordinate (C_1);\n\\draw (D_1) -- (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw (D) -- (D_1) (A) -- (A_1) (B) -- (B_1);\n\\draw [dashed] (C) -- (C_1);\n\\draw ($(A)!0.25!(A_1)$) node [right] {$A_2$} coordinate (A_2);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$B_2$} coordinate (B_2);\n\\draw ($(C)!0.75!(C_1)$) node [left] {$C_2$} coordinate (C_2);\n\\draw ($(D)!0.5!(D_1)$) node [left] {$D_2$} coordinate (D_2);\n\\draw (D_2)--(A_2);\n\\draw [dashed] (B_2)--(C_2);\n\\draw ($(B)!0.75!(B_1)$) node [right] {$P$} coordinate (P);\n\\draw (P)--(A_2);\n\\draw [dashed] (P)--(C_2)(C_2)--(D_2)(C_2)--(A_2);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $B_2C_2\\parallel A_2D_2$;\\\\\n(2) 点$P$在棱$BB_1$上, 当二面角$P-A_2C_2-D_2$为$150^{\\circ}$时, 求$B_2P$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学18", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018057": { + "id": "018057", + "content": "已知函数$f(x)=a(\\mathrm{e}^x+a)-x$.\\\\\n(1) 讨论$f(x)$的单调性;\\\\\n(2) 证明: 当$a>0$时, $f(x)>2 \\ln a+\\dfrac{3}{2}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学19", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018058": { + "id": "018058", + "content": "设等差数列$\\{a_n\\}$的公差为$d$, 且$d>1$. 令$b_n=\\dfrac{n^2+n}{a_n}$, 记$S_n, T_n$分别为数列$\\{a_n\\},\\{b_n\\}$的前$n$项和.\\\\\n(1) 若$3 a_2=3 a_1+a_3$, $S_3+T_3=21$, 求$\\{a_n\\}$的通项公式;\\\\\n(2) 若$\\{b_n\\}$为等差数列, 且$S_{99}-T_{99}=99$, 求$d$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学20", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018059": { + "id": "018059", + "content": "甲乙两人投篮, 每次由其中一人投篮, 规则如下: 若命中则此人继续投篮, 若未命中则换为对方投篮. 无论之前投篮情况如何, 甲每次投篮的命中率均为$0.6$, 乙每次投篮的命中率均为$0.8$, 由抽签确定第$1$次投篮的人选, 第一次投篮的人是甲, 乙的概率各为$0.5$.\\\\\n(1) 求第$2$次投篮的人是乙的概率;\\\\\n(2) 求第$i$次投篮的人是甲的概率;\\\\\n(3) 已知: 若随机变量$X_i$服从两点分布, 且$P(X_i=1)=1-P(X_i=0)=q_i$, $i=1,2, \\cdots, n$, 则$E[\\displaystyle\\sum_{i=1}^n X_i]=\\displaystyle\\sum_{i=1}^n q_i$. 记前$n$次(即从第$1$次到第$n$次投篮)中甲投篮的次数为$Y$, 求$E[Y]$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学21", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018060": { + "id": "018060", + "content": "在直角坐标系$x O y$中, 点$P$到$x$轴的距离等于点$P$到点$(0, \\dfrac{1}{2})$的距离, 记动点$P$的轨迹为$W$.\\\\\n(1) 求$W$的方程;\\\\\n(2) 已知矩形$ABCD$有三个顶点在$W$上, 证明: 矩形$ABCD$的周长大于$3 \\sqrt{3}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考新高考I卷数学22", + "edit": [ + "20230608\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",