From 93c9bc271cd791a3482253c76393c2544d9913e6 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Fri, 7 Apr 2023 23:09:47 +0800 Subject: [PATCH] =?UTF-8?q?=E6=B7=BB=E5=8A=A023=E5=B1=8A=E5=B4=87=E6=98=8E?= =?UTF-8?q?=E4=BA=8C=E6=A8=A1=E8=AF=95=E5=8D=B7=E5=8F=8A=E7=AD=94=E6=A1=88?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具/批量收录题目.py | 4 +- 工具/文本文件/metadata.txt | 68 ++++++- 题库0.3/Problems.json | 399 +++++++++++++++++++++++++++++++++++++ 3 files changed, 466 insertions(+), 5 deletions(-) diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index 87190e4e..aa3b802d 100644 --- a/工具/批量收录题目.py +++ b/工具/批量收录题目.py @@ -1,8 +1,8 @@ #修改起始id,出处,文件名 -starting_id = 14784 +starting_id = 14805 raworigin = "" filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目9.tex" -editor = "20230406\t王伟叶" +editor = "20230407\t王伟叶" indexed = True import os,re,json diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index ef3b7bec..63280355 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,3 +1,65 @@ -remark -14804 -(3) 中证明不可能小于$51$的核心是$max(a_{2n-1},a_{2n})$是严格增数列 \ No newline at end of file + +ans + +14805 +$(1,3)$ + +14806 +$1+\mathrm{i}$ + +14807 +$0$ + +14808 +$2\pi$ + +14809 +$4$ + +14810 +$45$ + +14811 +$90.5$ + +14812 +$40$ + +14813 +$y=-3x-2$ + +14814 +如: 等待时, 前后相邻两辆车的车距都相等; 绿灯亮后, 汽车都是在静止状态匀加速启动; 前一辆车启动后, 下一辆车启动的间隔时间相等; 车辆行驶秩序良好, 不会发生堵塞; 等等 + +14815 +$[\sqrt{2},3\sqrt{2}]$ + +14816 +$(-\dfrac{1}{\mathrm{e}},0)$ + +14817 +D + +14818 +A + +14819 +C + +14820 +D + +14821 +(1) $\arctan\dfrac{\sqrt{3}}2$; (2) $\dfrac{6\sqrt{7}}7$ + +14822 +(1) $B=\dfrac{2\pi}{3}$; (2) 最小值为$-2$, 取到最小值当且仅当$x=\dfrac{7\pi}{12}$ + +14823 +(1) $\dfrac 12$; (2) 分布列为$\begin{pmatrix} 0 & 1 & 2 \\ \dfrac 17 & \dfrac 47 & \dfrac 27\end{pmatrix}$, $E[X]=\dfrac 87$; (3) $3$月$3$日 + +14824 +(1) $m=2$; (2) $\dfrac{2\sqrt{2}-2}3$; (3) 证明略 + +14825 +(1) $(-\dfrac 32,0)$; (2) 证明略; (3) 证明略 diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index b72309c3..c07c339e 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -365156,6 +365156,405 @@ "remark": "(3) 中证明不可能小于$51$的核心是$max(a_{2n-1},a_{2n})$是严格增数列", "space": "12ex" }, + "014805": { + "id": "014805", + "content": "若实数$x$满足$|x-2|<1$, 则$x$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$(1,3)$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题1", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014806": { + "id": "014806", + "content": "设复数$z$满足$(1+\\mathrm{i}) z=2 \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$1+\\mathrm{i}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题2", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014807": { + "id": "014807", + "content": "已知集合$A=\\{1,2\\}$, $B=\\{a, a^2+1\\}$, 若$A \\cap B=\\{1\\}$, 则实数$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$0$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题3", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014808": { + "id": "014808", + "content": "已知函数$y=\\sin (2 \\omega x+\\varphi)$($\\omega>0$)的最小正周期为$1$, 则$\\omega=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$2\\pi$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题4", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014809": { + "id": "014809", + "content": "已知正实数$a$、$b$满足$a b=1$, 则$a+4 b$的最小值等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$4$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题5", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014810": { + "id": "014810", + "content": "在$(x^4+\\dfrac{1}{x})^{10}$的展开式中常数项是\\blank{50}.(用数字作答)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$45$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题6", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014811": { + "id": "014811", + "content": "以下数据为参加某次数学竞赛的$15$人的成绩 (单位: 分), 分数从低到高依次是: 56、70、72、78、79、80、81、83、84、86、88、90、91、94、98, 则这$15$人成绩的第$80$百分位数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$90.5$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题7", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014812": { + "id": "014812", + "content": "某单位为了解用电量$y$度与气温$x^{\\circ} \\text{C}$之间的关系, 随机统计了某$4$天的用电量与当天气温.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 气温$({ }^{\\circ} \\text{C})$& 14 & 12 & 8 & 6 \\\\\n\\hline 用电量 (度) & 22 & 26 & 34 & 38 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n由表中数据所得回归直线方程为$y=-2 x+\\hat{b}$, 据此预测当气温为$5^{\\circ} \\text{C}$时, 用电量的度数约为\\blank{50}${ }^{\\circ} \\text{C}$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$40$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题8", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014813": { + "id": "014813", + "content": "已知抛物线$x^2=2 y$上的两个不同的点$A$、$B$的横坐标恰好是方程$x^2+6 x+4=0$的根, 则直线$AB$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$y=-3x-2$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题9", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014814": { + "id": "014814", + "content": "在一个十字路口, 每次亮绿灯的时长为$30$秒, 那么, 每次绿灯亮时, 在一条直行道路上能有多少汽车通过? 这个问题涉及车长、车距、车速、堵塞的干扰等多种因素, 不同型号车的车长是不同的, 驾驶员的习惯不同也会使车距、车速不同, 行人和非机动车的干扰因素则复杂且不确定. 面对这些不同和不确定, 需要作出假设, 例如小明发现虽然通过路口的车辆各种各样, 但多数是小轿车, 因此小明给出如下假设: 通过路口的车辆长度都相等, 请写出一个你认为合理的假设\\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "如: 等待时, 前后相邻两辆车的车距都相等; 绿灯亮后, 汽车都是在静止状态匀加速启动; 前一辆车启动后, 下一辆车启动的间隔时间相等; 车辆行驶秩序良好, 不会发生堵塞; 等等", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题10", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014815": { + "id": "014815", + "content": "设平面向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$满足: $|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=|\\overrightarrow {c}|$, $|\\overrightarrow {a}-\\overrightarrow {b}|=1$, $\\overrightarrow {b} \\perp \\overrightarrow {c}$, 则$|\\overrightarrow {b}-\\overrightarrow {c}|$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$[\\sqrt{2},3\\sqrt{2}]$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题11", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014816": { + "id": "014816", + "content": "若函数$y=\\begin{cases}\\dfrac{x^3}{\\mathrm{e}^x}, & x \\geq 0, \\\\ a x^2, & x<0\\end{cases}$的图像上点$A$与点$B$, 点$C$与点$D$分别关于原点对称, 除此之外, 不存在函数图像上的其它两点关于原点对称, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$(-\\dfrac{1}{\\mathrm{e}},0)$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题12", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014817": { + "id": "014817", + "content": "下列函数在其定义域上既是严格增函数, 又是奇函数的是\\bracket{20}.\n\\fourch{$f(x)=\\tan x$}{$f(x)=-\\dfrac{1}{x}$}{$f(x)=x-\\cos x$}{$f(x)=\\mathrm{e}^x-\\mathrm{e}^{-x}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "D", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题13", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014818": { + "id": "014818", + "content": "设两个正态分布$N(\\mu_1, \\sigma_1^2)$($\\sigma_1>0$)和$N(\\mu_2, \\sigma_2^2)$($\\sigma_2>0$)的正态密度函数图像如图所示, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\def\\s{0.3}\n\\def\\m{0.5}\n\\draw [domain = -0.3:1.3, samples = 100] plot (\\x,{1/sqrt(2*pi)/\\s*exp(-(\\x-\\m)*(\\x-\\m)/\\s/\\s/2)});\n\\def\\s{0.2}\n\\def\\m{-0.1}\n\\draw [domain = -1:0.8, samples = 100] plot (\\x,{1/sqrt(2*pi)/\\s*exp(-(\\x-\\m)*(\\x-\\m)/\\s/\\s/2)});\n\\draw (-0.1,2) node [left] {$N(\\mu_1,\\sigma_1^2)$};\n\\draw (0.5,1.5) node [right] {$N(\\mu_2,\\sigma_2^2)$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\mu_1<\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1<\\mu_2$, $\\sigma_1>\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1>\\sigma_2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "A", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题14", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014819": { + "id": "014819", + "content": "《九章算术》中将底面为直角三角形且侧棱垂直于底面的三棱柱称为``堑堵''; 底面为矩形, 一条侧棱垂直于底面的四棱锥称之为``阳马''; 四个面均为直角三角形的四面体称为``鳖臑''. 如图, 在堑堵$ABC-A_1B_1C_1$中, $AC \\perp BC$, 且$AA_1=AB=2$. 下列说法错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1.5,0,{sqrt(3)/2}) node [below] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)!0.5!(B)$) node [above] {$E$} coordinate (E);\n\\draw ($(C)!{2.5/7}!(A_1)$) node [right] {$F$} coordinate (F);\n\\draw (A_1)--(C)(A_1)--(A)--(C)--(B)--(B_1)--cycle(A_1)--(C_1)--(B_1)(C)--(C_1)(A)--(F);\n\\draw [dashed] (A_1)--(B)(A)--(B)(A)--(E)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\onech{四棱锥$B-A_1ACC_1$为``阳马''}{四面体$A_1C_1CB$为``鳖臑''}{四棱锥$B-A_1ACC_1$体积的最大值为$\\dfrac{2}{3}$}{过$A$点作$AE \\perp A_1B$于点$E$, 过$E$点作$EF \\perp A_1B$并交$A_1C$于点$F$, 则$A_1B \\perp$平面$AEF$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "C", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题15", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014820": { + "id": "014820", + "content": "已知数列$\\{a_n\\}$是各项为正数的等比数列, 公比为$q$, 在$a_1$、$a_2$之间插入$1$个数, 使这$3$个数成等差数列, 记公差为$d_1$, 在$a_2$、$a_3$之间插入$2$个数, 使这$4$个数成等差数列, 公差为$d_2$, $\\cdots$, 在$a_n$、$a_{n+1}$之间插入$n$个数, 使这$n+2$个数成等差数列, 公差为$d_n$, 则\\bracket{20}.\n\\twoch{当$01$时, 数列$\\{d_n\\}$严格增}{当$d_1>d_2$时, 数列$\\{d_n\\}$严格减}{当$d_1=latex]\n\\filldraw (0,0) node [above left] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2.3) node [above] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O_1) ellipse (1.5 and 0.5);\n\\draw (O) ++ (1.5,0) node [right] {$B$} coordinate (B) --++ (0,2.3) node [right] {$B_1$} coordinate (B_1);\n\\draw (O) ++ (-1.5,0) node [left] {$A$} coordinate (A) --++ (0,2.3) node [left] {$A_1$} coordinate (A_1) -- (B_1);\n\\draw (A) arc (180:360:1.5 and 0.5);\n\\draw [dashed] (A) arc (180:0:1.5 and 0.5);\n\\draw (-60:1.5 and 0.5) node [below] {$P$} coordinate (P);\n\\draw [dashed] (A_1)--(P)(A_1)--(B)--(P)--(A)--(B)(O)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$A_1P$与平面$ABP$所成角的大小;\\\\\n(2) 求点$A$到平面$A_1BP$的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) $\\arctan\\dfrac{\\sqrt{3}}2$; (2) $\\dfrac{6\\sqrt{7}}7$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题17", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014822": { + "id": "014822", + "content": "在$\\triangle ABC$中, $a$、$b$、$c$分别是内角$A$、$B$、$C$的对边, $\\overrightarrow {m}=(2 a+c, b)$, $\\overrightarrow {n}=(\\cos B, \\cos C)$, $\\overrightarrow {m} \\cdot \\overrightarrow {n}=0$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 设$f(x)=2 \\cos x \\sin (x+\\dfrac{\\pi}{3})-2 \\sin ^2 x \\sin B+2 \\sin x \\cos x \\cos (A+C)$, 当$x \\in[\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3}]$时, 求$f(x)$的最小值及相应的$x$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) $B=\\dfrac{2\\pi}{3}$; (2) 最小值为$-2$, 取到最小值当且仅当$x=\\dfrac{7\\pi}{12}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题18", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014823": { + "id": "014823", + "content": "某校工会开展健身健步走活动, 要求教职工上传$3$月$1$日至$3$月$7$日的微信运动步数信息, 下图是职工甲和职工乙微信运动步数情况:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\foreach \\i in {1,2,...,7} {\\draw ({\\i*0.7},0) node [below] {$\\i$};};\n\\draw (0,0) node [below] {$3$月};\n\\draw (0,0)--(5.5,0);\n\\foreach \\i/\\j in {1/8566,2/19891,3/16820,4/5207,5/13022,6/11860,7/15524}\n{\\draw ({\\i*0.7},2) node [above] {\\tiny$\\j$};};\n\\draw (0.7,0) -- (0.7,0.8566) -- (1.4,1.9891) -- (2.1,1.6820) -- (2.8,0.5207) -- (3.5,1.3022) -- (4.2,1.1860) -- (4.9,1.5524) -- (4.9,0);\n\\foreach \\i/\\j in {0.7/0.8566,1.4/1.9891,2.1/1.6820,2.8/0.5207,3.5/1.3022,4.2/1.1860,4.9/1.5524}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (2.75,-0.5) node [below] {职工甲};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\foreach \\i in {1,2,...,7} {\\draw ({\\i*0.7},0) node [below] {$\\i$};};\n\\draw (0,0) node [below] {$3$月};\n\\draw (0,0)--(5.5,0);\n\\foreach \\i/\\j in {1/11845,2/10577,3/9780,4/4872,5/17022,6/9655,7/12396}\n{\\draw ({\\i*0.7},2) node [above] {\\tiny$\\j$};};\n\\draw (0.7,0) -- (0.7,1.1845) -- (1.4,1.0577) -- (2.1,0.9780) -- (2.8,0.4872) -- (3.5,1.7022) -- (4.2,0.9655) -- (4.9,1.2396) -- (4.9,0);\n\\foreach \\i/\\j in {0.7/1.1845,1.4/1.0577,2.1/0.9780,2.8/0.4872,3.5/1.7022,4.2/0.9655,4.9/1.2396}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (2.75,-0.5) node [below] {职工乙};\n\\end{tikzpicture}\n\\end{center}\n(1) 从$3$月$2$日至$3$月$7$日中任选一天, 求这一天职工甲和职工乙微信运动步数都不低于$10000$的概率;\\\\\n(2) 从$3$月$1$日至$3$月$7$日中任选两天, 记职工乙在这两天中微信运动步数不低于$10000$的天数为$X$, 求$X$的分布列及数学期望;\\\\\n(3) 下图是校工会根据$3$月$1$日至$3$月$7$日某一天的数据制作的全校$200$名教职工微信运动步数的频率分布直方图, 已知这一天甲和乙微信记步数在单位$200$名教职工中排名(按照从大到小排序)分别为第$68$和第$142$, 请指出这是根据哪一天的数据制作的频率分布直方图. (不用说明理由)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.1, yscale = 30]\n\\draw [->] (0,0) -- (48,0) node [below] {微信计步数(单位:千步)};\n\\draw [->] (0,0) -- (0,0.08) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {0/0.02,5/0.04,10/0.06,15/0.05,20/0.03}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (5,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0/0.02,5/0.04,10/0.06,15/0.05,20/0.03}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (25,0) node [below] {$25$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) $\\dfrac 12$; (2) 分布列为$\\begin{pmatrix} 0 & 1 & 2 \\\\ \\dfrac 17 & \\dfrac 47 & \\dfrac 27\\end{pmatrix}$, $E[X]=\\dfrac 87$; (3) $3$月$3$日", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题19", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014824": { + "id": "014824", + "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{m^2}+\\dfrac{y^2}{2}=1$($m>0$, $m \\neq \\sqrt{2}$), 点$A$、$B$分别是椭圆$\\Gamma$与$y$轴的交点(点$A$在点$B$的上方), 过点$D(0,1)$且斜率为$k$的直线$l$交椭圆$\\Gamma$于$E$、$G$两点.\\\\\n(1) 若椭圆$\\Gamma$焦点在$x$轴上, 且其离心率是$\\dfrac{\\sqrt{2}}{2}$, 求实数$m$的值;\\\\\n(2) 若$m=k=1$, 求$\\triangle BEG$的面积;\\\\\n(3) 设直线$AE$与直线$y=2$交于点$H$, 证明: $B$、$G$、$H$三点共线.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) $m=2$; (2) $\\dfrac{2\\sqrt{2}-2}3$; (3) 证明略", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题20", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014825": { + "id": "014825", + "content": "已知定义域为$D$的函数$y=f(x)$, 其导函数为$y'=f'(x)$, 满足对任意的$x \\in D$都有$|f'(x)|<1$.\\\\\n(1) 若$f(x)=a x+\\ln x$, $x \\in[1,2]$, 求实数$a$的取值范围;\\\\\n(2) 证明: 方程$f(x)-x=0$至多只有一个实根;\\\\\n(3) 若$y=f(x)$, $x \\in \\mathbf{R}$是周期为$2$的周期函数, 证明: 对任意的实数$x_1$、$x_2$, 都有$|f(x_1)-f(x_2)|<1$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) $(-\\dfrac 32,0)$; (2) 证明略; (3) 证明略", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三崇明区二模试题21", + "edit": [ + "20230407\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, "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.",