diff --git a/工具/修改题目数据库.ipynb b/工具/修改题目数据库.ipynb index ad502b25..c28830b8 100644 --- a/工具/修改题目数据库.ipynb +++ b/工具/修改题目数据库.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 11, + "execution_count": 2, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 11, + "execution_count": 2, "metadata": {}, "output_type": "execute_result" } @@ -19,7 +19,7 @@ "source": [ "import os,re,json\n", "\"\"\"这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭, 随后运行第二个代码块\"\"\"\n", - "problems = \"12505\"\n", + "problems = \"12592\"\n", "\n", "def generate_number_set(string,dict):\n", " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb index 72806d70..8eb21d73 100644 --- a/工具/寻找阶段末尾空闲题号.ipynb +++ b/工具/寻找阶段末尾空闲题号.ipynb @@ -2,14 +2,14 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "首个空闲id: 12529 , 直至 020000\n", + "首个空闲id: 12634 , 直至 020000\n", "首个空闲id: 20227 , 直至 030000\n", "首个空闲id: 30502 , 直至 999999\n" ] diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index fa5766a5..aeda1fee 100644 --- a/工具/添加题目到数据库.ipynb +++ b/工具/添加题目到数据库.ipynb @@ -2,20 +2,20 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 12529\n", - "origin = \"2023届青浦区一模\"\n", - "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目6.tex\"\n", - "editor = \"20221214\\t王伟叶\"" + "starting_id = 12634\n", + "origin = \"2023届徐汇区一模\"\n", + "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目7.tex\"\n", + "editor = \"20221215\\t王伟叶\"" ] }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 10, "metadata": {}, "outputs": [], "source": [ diff --git a/工具/识别题库中尚未标注的题目类型.ipynb b/工具/识别题库中尚未标注的题目类型.ipynb index e19f0157..47c62b9a 100644 --- a/工具/识别题库中尚未标注的题目类型.ipynb +++ b/工具/识别题库中尚未标注的题目类型.ipynb @@ -2,34 +2,118 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "012529 填空题\n", - "012530 填空题\n", - "012531 填空题\n", - "012532 填空题\n", - "012533 填空题\n", - "012534 填空题\n", - "012535 填空题\n", - "012536 填空题\n", - "012537 填空题\n", - "012538 填空题\n", - "012539 填空题\n", - "012540 填空题\n", - "012541 选择题\n", - "012542 选择题\n", - "012543 选择题\n", - "012544 选择题\n", - "012545 解答题\n", - "012546 解答题\n", - "012547 解答题\n", - "012548 解答题\n", - "012549 解答题\n" + "012550 填空题\n", + "012551 填空题\n", + "012552 填空题\n", + "012553 填空题\n", + "012554 填空题\n", + "012555 填空题\n", + "012556 填空题\n", + "012557 填空题\n", + "012558 填空题\n", + "012559 填空题\n", + "012560 填空题\n", + "012561 填空题\n", + "012562 选择题\n", + "012563 选择题\n", + "012564 解答题\n", + "012565 选择题\n", + "012566 解答题\n", + "012567 解答题\n", + "012568 解答题\n", + "012569 解答题\n", + "012570 解答题\n", + "012571 填空题\n", + "012572 填空题\n", + "012573 填空题\n", + "012574 填空题\n", + "012575 填空题\n", + "012576 填空题\n", + "012577 填空题\n", + "012578 填空题\n", + "012579 填空题\n", + "012580 填空题\n", + "012581 填空题\n", + "012582 填空题\n", + "012583 选择题\n", + "012584 选择题\n", + "012585 选择题\n", + "012586 解答题\n", + "012587 解答题\n", + "012588 解答题\n", + "012589 解答题\n", + "012590 解答题\n", + "012591 解答题\n", + "012592 填空题\n", + "012593 填空题\n", + "012594 填空题\n", + "012595 填空题\n", + "012596 填空题\n", + "012597 填空题\n", + "012598 填空题\n", + "012599 填空题\n", + "012600 填空题\n", + "012601 填空题\n", + "012602 填空题\n", + "012603 填空题\n", + "012604 选择题\n", + "012605 选择题\n", + "012606 选择题\n", + "012607 选择题\n", + "012608 解答题\n", + "012609 解答题\n", + "012610 解答题\n", + "012611 解答题\n", + "012612 解答题\n", + "012613 填空题\n", + "012614 填空题\n", + "012615 填空题\n", + "012616 填空题\n", + "012617 填空题\n", + "012618 填空题\n", + "012619 填空题\n", + "012620 填空题\n", + "012621 填空题\n", + "012622 填空题\n", + "012623 填空题\n", + "012624 填空题\n", + "012625 选择题\n", + "012626 填空题\n", + "012627 选择题\n", + "012628 选择题\n", + "012629 解答题\n", + "012630 解答题\n", + "012631 解答题\n", + "012632 解答题\n", + "012633 解答题\n", + "012634 填空题\n", + "012635 填空题\n", + "012636 填空题\n", + "012637 填空题\n", + "012638 填空题\n", + "012639 填空题\n", + "012640 填空题\n", + "012641 填空题\n", + "012642 填空题\n", + "012643 填空题\n", + "012644 填空题\n", + "012645 填空题\n", + "012646 选择题\n", + "012647 选择题\n", + "012648 选择题\n", + "012649 选择题\n", + "012650 解答题\n", + "012651 解答题\n", + "012652 解答题\n", + "012653 解答题\n", + "012654 解答题\n" ] } ], diff --git a/文本处理工具/剪贴板文本整理_word文件.ipynb b/文本处理工具/剪贴板文本整理_word文件.ipynb index 08050fa4..98e610dc 100644 --- a/文本处理工具/剪贴板文本整理_word文件.ipynb +++ b/文本处理工具/剪贴板文本整理_word文件.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 5, + "execution_count": 7, "metadata": {}, "outputs": [], "source": [ @@ -477,6 +477,7 @@ "modified_data = modified_data.replace(\"针角\",\"钝角\")\n", "#mathpix的自由向量修改\n", "modified_data = modified_data.replace(r\"\\vec\",r\"\\overrightarrow \")\n", + "modified_data = modified_data.replace(r\"\\bar\",r\"\\overline \")\n", "#mathpix的极限修改\n", "modified_data = modified_data.replace(r\"\\lim _{n \\rightarrow \\infty}\",r\"\\displaystyle\\lim_{n\\to\\infty}\")\n", "#mathpix的顿号修改\n", @@ -486,6 +487,11 @@ "modified_data = modified_data.replace(r\"\\mid\",\"|\")\n", "modified_data = re.sub(r\"\\\\mathrm\\{\\\\mathrm\\{i\\}\\}\",r\"\\\\mathrm{i}\",modified_data)\n", "modified_data = modified_data.replace(\",$\",\", $\")\n", + "modified_data = modified_data.replace(\" / /\",r\"\\parallel\")\n", + "modified_data = modified_data.replace(\"mathrmR\",r\"mathbf{R}\")\n", + "modified_data = modified_data.replace(r\"^{\\prime}\",\"'\")\n", + "\n", + "\n", "\n", "setCopy(modified_data)\n", "\n", diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index ff3f1d50..6a8eb988 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -309022,6 +309022,2001 @@ "remark": "", "space": "12ex" }, + "012550": { + "id": "012550", + "content": "若集合$M=\\{0,1,2\\}$, $N=\\{x | 2 x-1>0\\}$, 则$M \\cap N=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题1", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012551": { + "id": "012551", + "content": "若$x$满足$\\mathrm{i} x=1+\\mathrm{i}$(其中$\\mathrm{i}$为虚数单位), 则$x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题2", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012552": { + "id": "012552", + "content": "双曲线$x^2-\\dfrac{y^2}8=1$的离心率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题3", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012553": { + "id": "012553", + "content": "在$\\triangle ABC$中, 已知边$AB=4 \\sqrt 3$, 角$A=45^{\\circ}$, $C=60^{\\circ}$, 则边$BC=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题4", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012554": { + "id": "012554", + "content": "已知正实数$x$、$y$满足$\\lg x=m$, $y=10^{m-1}$, 则$\\dfrac xy=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题5", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012555": { + "id": "012555", + "content": "将一颗骰子连掷两次, 每次结果相互独立, 则第一次点数小于$3$且第二次点数大于$3$的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题6", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012556": { + "id": "012556", + "content": "如图, 对于直四棱柱$ABCD-A_1B_1C_1D_1$, 要使$A_1C \\perp B_1D_1$, 则在四边形$ABCD$中, 满足的条件可以是\\blank{50}.(只需写出一个正确的条件)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1.5}\n\\def\\m{1.5}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B1) -- (D1);\n\\draw [dashed] (A1) -- (C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题7", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012557": { + "id": "012557", + "content": "若曲线$\\Gamma: y=\\sqrt x$和直线$l: x-2 y-4=0$的某一条平行线相切, 则切点的横坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题8", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012558": { + "id": "012558", + "content": "已知二次函数$f(x)=a x^2+x+a$的值域为$(-\\infty, \\dfrac 34]$, 则函数$g(x)=2^x+a$的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题9", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012559": { + "id": "012559", + "content": "已知$A(x_1, y_1)$、$B(x_2, y_2)$是圆$x^2+y^2=1$上的两个不同的动点, 且$x_1 y_2=x_2 y_1$, 则$2 x_1+x_2+2 y_1+y_2$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题10", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012560": { + "id": "012560", + "content": "已知函数$f(x)=2 \\sin (\\omega x+\\dfrac{\\pi}4)$($\\omega>0$)在区间$[-1,1]$上的值域为$[m, n]$, 且$n-m=3$, 则$\\omega$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题11", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012561": { + "id": "012561", + "content": "已知平面向量$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$和实数$\\lambda$满足$|\\overrightarrow a|=|\\overrightarrow b|=|\\overrightarrow a+\\overrightarrow b|=2$, $\\overrightarrow a \\cdot \\overrightarrow c+\\overrightarrow b \\cdot \\overrightarrow c=0$, $(\\overrightarrow a-\\lambda \\overrightarrow c) \\cdot(\\overrightarrow b+\\lambda \\overrightarrow c) \\geq 0$, 则$|\\overrightarrow a-\\lambda \\overrightarrow c|+|\\overrightarrow b+\\lambda \\overrightarrow c|$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题12", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012562": { + "id": "012562", + "content": "下列不等式中, 解集为$\\{x |-10$, $a_{n+1} a_n-a_n^2=1$($n \\in \\mathbf{N}$, $n \\geq 1$), 如果$\\dfrac 1{a_1}+\\dfrac 1{a_2}+\\cdots+\\dfrac 1{a_{2022}}=2022$, 那么\\bracket{20}.\n\\twoch{$2022=latex,scale = 1.5]\n\\begin{scope}[x = {(-30:1cm)}, y = {(-150:1cm)}, z = {(90:1cm)}]\n\\draw (0,0,1) circle (1);\n\\draw (-45:1) node [right] {$A$} coordinate (A) arc (-45:135:1);\n\\draw [dashed] (-45:1) arc (-45:-225:1);\n\\draw (A) ++ (0,0,1) node [right] {$A_1$} coordinate (A_1);\n\\draw (75:1) node [below] {$B$} coordinate (B);\n\\draw (195:1) node [above] {$C$} coordinate (C);\n\\draw [dashed] (A) -- (B) -- (C) -- cycle;\n\\draw (A) -- (A_1);\n\\draw (135:1) --++ (0,0,1);\n\\draw [dashed] (B) -- (A_1);\n\\end{scope}\n\\filldraw (0,0) circle (0.01) node [left] {$O$} coordinate (O);\n\\filldraw (0,1) circle (0.01) node [left] {$O_1$} coordinate (O_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$C$到平面$A_1AB$的距离;\\\\\n(2) 在劣弧$\\overset\\frown{BC}$上是否存在一点$D$, 满足$O_1D\\parallel$平面$A_1AB$? 若存在, 求出$\\angle BOD$的大小; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题18", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012568": { + "id": "012568", + "content": "$2022$年, 第二十二届世界杯足球赛在卡塔尔举行, 某国家队$26$名球员的年龄分布茎叶图如图所示:\n\\begin{center}\n\\begin{tabular}{l|l}\n1 & 8 \\ 9 \\\\\n2 & 1 \\ 2 \\ 3 \\ 3 \\ 4 \\ 5 \\ 5 \\ 5 \\ 6 \\ 6 \\ 7 \\ 8 \\ 8 \\ 8 \\ 9 \\ 9 \\ 9 \\\\\n3 & 0 \\ 1 \\ 2 \\ 2 \\ 2 \\ 3 \\ 4 \n\\end{tabular}\n\\end{center}\n(1) 该国家队$25$岁的球员共有几位? 求该国家队球员年龄的第$75$百分位数;\\\\\n(2) 从这$26$名球员中随机选取$11$名球员参加某项活动, 求这$11$名球员中至少有一位年龄不小于$30$岁的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题19", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012569": { + "id": "012569", + "content": "如图, 点$A$、$B$、$C$分别为椭圆$\\Gamma: \\dfrac{x^2}4+y^2=1$的左、右顶点和上顶点, 点$P$是$\\Gamma$上在第一象限内的动点, 直线$AP$与直线$BC$相交于点$Q$, 直线$CP$与$x$轴相交于点$M$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->, name path = xaxis] (-2.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-1.8) -- (0,1.8) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw [name path = elli] (0,0) ellipse (2 and 1);\n\\draw (-2,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below right] {$B$} coordinate (B);\n\\draw (0,1) node [above left] {$C$} coordinate (C);\n\\draw ({2*cos(23)},{sin(23)}) node [above right] {$P$} coordinate (P);\n\\draw [name path = CB] (C) -- (B);\n\\draw [name path = OP] (O) -- (P);\n\\path [name path = CP] (C) -- ($(C)!2!(P)$);\n\\path [name intersections = {of = CB and OP, by = Q}];\n\\draw (Q) node [below] {$Q$} coordinate (Q);\n\\path [name intersections = {of = CP and xaxis, by = M}];\n\\draw (M) node [below] {$M$} coordinate (M);\n\\draw (A) -- (Q) (C) -- (M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$BC$的方程;\\\\\n(2) 求证:$\\overrightarrow{OQ} \\cdot \\overrightarrow{OM}=4$;\\\\\n(3) 已知直线$l_1$的方程为$x+2 y-1=0$, 线段$QM$的中点为$T$, 是否存在垂直于$y$轴的直线$l_2$, 使得点$T$到$l_1$和$l_2$的距离之积为定值? 若存在, 求出$l_2$的方程; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题20", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012570": { + "id": "012570", + "content": "定义: 如果函数$y=f(x)$和$y=g(x)$的图像上分别存在点$M$和$N$关于$x$轴对称, 则称函数$y=f(x)$和$y=g(x)$具有$C$关系.\\\\\n(1) 判断函数$f(x)=\\log_2(8 x^2)$和$g(x)=\\log_{\\frac 12} x$是否具有$C$关系;\\\\\n(2) 若函数$f(x)=a \\sqrt {x-1}$和$g(x)=-x-1$不具有$C$关系, 求实数$a$的取值范围;\\\\\n(3) 若函数$f(x)=x e^x$和$g(x)=m \\sin x(m<0)$在区间$(0, \\pi)$上具有$C$关系, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区一模试题21", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012571": { + "id": "012571", + "content": "已知集合$A=\\{x||x-1 |<1\\}$, $\\mathbf{Z}$是整数集, 则$A \\cap \\mathbf{Z}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题1", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012572": { + "id": "012572", + "content": "已知复数$z=\\dfrac 1{\\mathrm{i}}$, $\\mathrm{i}$是虚数单位, 则$z$的虚部为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题2", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012573": { + "id": "012573", + "content": "直线$x=1$与直线$\\sqrt 3 x-y+1=0$的夹角大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题3", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012574": { + "id": "012574", + "content": "已知$m \\in \\mathbf{R}$, 若关于$x$的方程$2 m x^2+3 x+m-1=m^2 \\cdot x^2+(m+1) x+1$解集为$\\mathbf{R}$, 则$m$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题4", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012575": { + "id": "012575", + "content": "已知某一个圆锥的侧面积为$20 \\pi$, 底面积为$16 \\pi$, 则这个圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题5", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012576": { + "id": "012576", + "content": "某果园种植了$100$棵苹果树, 随机抽取的$12$棵果树的产量(单位: 千克)分别为: $24,25,36,27,28,32,20,26,29,30,26,33$据此预计, 该果园的总产量为千克以及第$75$百分位数为\\blank{50}千克.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题6", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012577": { + "id": "012577", + "content": "已知常数$m \\in \\mathbf{R}$, 在$(x+m y)^n$的二项展开式中, $x^3 y^3$项的系数等于$160$, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题7", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012578": { + "id": "012578", + "content": "若函数$y=\\dfrac 1{x-1}$的值域是$(-\\infty, 0) \\cup[\\dfrac 12,+\\infty)$, 则此函数的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题8", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012579": { + "id": "012579", + "content": "如图为正六棱柱$ABCDEF-A' B' C' D' E' F'$. 其 6 个侧面的 12 条面对角线所在直线中, 与直线$A' B$异面的共有\\blank{50}条.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, x = {(-135:0.5cm)}, y = {(0:1cm)}, z = {(90:1cm)}]\n\\draw (0,0) node [left] {$A$} coordinate (A) ++ (0,0,1) node [left] {$A'$} coordinate (A');\n\\draw (A) ++ (30:1) node [below] {$B$} coordinate (B) ++ (0,0,1) node [above] {$B'$} coordinate (B');\n\\draw (B) ++ (90:1) node [below] {$C$} coordinate (C) ++ (0,0,1) node [above] {$C'$} coordinate (C');\n\\draw (C) ++ (150:1) node [right] {$D$} coordinate (D) ++ (0,0,1) node [right] {$D'$} coordinate (D');\n\\draw (D) ++ (210:1) node [below] {$E$} coordinate (E) ++ (0,0,1) node [above] {$E'$} coordinate (E');\n\\draw (E) ++ (270:1) node [below] {$F$} coordinate (F) ++ (0,0,1) node [above] {$F'$} coordinate (F');\n\\draw (A) -- (A') (B) -- (B') (C) -- (C') (D) -- (D') (A') -- (B') -- (C') -- (D') -- (E') -- (F') -- cycle (A) -- (B) -- (C) -- (D);\n\\draw [dashed] (E) -- (E') (F) -- (F') (A) -- (F) -- (E) -- (D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题9", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012580": { + "id": "012580", + "content": "关于$x$的方程$|2 x-3|+|-x+2|=|x-1|$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题10", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012581": { + "id": "012581", + "content": "在空间直角坐标系中, 点$A(1,0,0)$, 点$B(5,-4,3)$, 点$C(2,0,1)$, 则$\\overrightarrow{AB}$在$\\overrightarrow{CA}$方向上的投影向量的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题11", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012582": { + "id": "012582", + "content": "已知抛物线$x^2=3 y$, 动点$A$自原点出发, 沿着$y$轴正方向向上匀速运动, 速度大小为$v$. 过$A$作$y$轴的垂线交抛物线于$B$点, 再过$B$作$x$轴的垂线交$x$轴于$C$点. 当$A$运动至$(0,100)$时, 点$C$的瞬时速度的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题12", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012583": { + "id": "012583", + "content": "已知$\\triangle ABC$, 那么``$|\\overrightarrow{AC}|^2+|\\overrightarrow{AB}|^2-|\\overrightarrow{BC}|^2<0$''是``$\\triangle ABC$为钝角三角形''的\\bracket{20}.\n\\twoch{充分条件但非必要条件}{必要条件但非充分条件}{充要条件}{以上皆非}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题13", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012584": { + "id": "012584", + "content": "已知四条双曲线, $\\Gamma_1: x^2-y^2=1$, $\\Gamma_2: \\dfrac{x^2}9-\\dfrac{y^2}4=1$, $\\Gamma_3: \\dfrac{y^2}4-\\dfrac{x^2}9=1$, $\\Gamma_4: \\dfrac{x^2}{16}-\\dfrac{y^2}{16}=1$, 关于下列三个结论的正确选项为\\bracket{20}.\\\\\n\\textcircled{1} $\\Gamma_4$的开口最为开阔; \\textcircled{2} $\\Gamma_1$的开口比$\\Gamma_3$的更为开阔; \\textcircled{3} $\\Gamma_2$和$\\Gamma_3$的开口的开阔程度相同.\n\\fourch{只有一个正确}{只有两个正确}{均正确}{均不正确}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题14", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012585": { + "id": "012585", + "content": "甲、乙两人弈棋, 根据以往总共$20$次的对弈记录, 甲取胜$10$次, 乙取胜$10$次. 两人进行一场五局三胜的比赛, 最终胜者赢得$200$元奖金. 第一局、第二局比赛都是甲胜, 现在比赛因意外中止. 鉴于公平, 奖金应该分给甲\\bracket{20}.\n\\fourch{$100$元}{$150$元}{$175$元}{$200$元}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题15", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012586": { + "id": "012586", + "content": "中国古代数学家用圆内接正$6 n$边形的周长来近似计算圆周长, 以估计圆周率$\\pi$的值. 若据此证明$\\pi>3.14$, 则正整数$n$至少等于\n\\fourch{$8$}{$9$}{$10$}{$11$}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题16", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012587": { + "id": "012587", + "content": "如图, 已知正四棱柱$ABCD-A_1B_1C_1D_1$, 底面正方形$ABCD$的边长为$2$,$AA_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1) -- (C1) (A1) -- (B);\n\\draw [dashed] (B) -- (D) -- (A1) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$AA_1CC_1 \\perp$平面$A_1BD$;\\\\\n(2) 求点$A$到平面$A_1BD$的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题17", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012588": { + "id": "012588", + "content": "若数列$\\{\\dfrac 1{a_n}\\}$是等差数列, 则称数列$\\{a_n\\}$为调和数列. 若实数$a$、$b$、$c$依次成调和数列, 则称$b$是$a$和$c$的调和中项.\\\\\n(1) 求$\\dfrac 13$和$1$的调和中项;\\\\\n(2) 已知调和数列$\\{a_n\\}$, $a_1=6$, $a_4=2$, 求$\\{a_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题18", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012589": { + "id": "012589", + "content": "李先生属于一年工作$250$天的上班族, 计划购置一辆新车用以通勤. 大致推断每天早八点从家出发, 晩上六点回家, 往返总距离为$40$公里. 考虑从$A$、$B$两款车型中选择其一, $A$款车是燃油车, $B$款车是电动车, 售价均为$30$万元. 现提供关于两种车型的相关信息:\\\\\n$A$款车的油耗为$6$升/百公里, 油价为每升$8$至$9$元. 车险费用$4000$元/年. 购置税为售价的$10\\%$. 购车后, 车价每年折旧率为$12\\%$. 保养费用平均$2000$元/万公里;\\\\\n$B$款车的电耗为$20$度/百公里, 电费为每度$0.6$至$0.7$元. 车险费用$6000$元/年. 国务院$2022$年出台文件, 宣布保持免除购置税政策. 电池使用寿命为$5$年, 更换费用为$10$万元. 购车后, 车价每年折旧率为$15\\%$. 保养费用平均$1000$元/万公里.\\\\\n(1) 除了上述了解到的情况, 还有哪些因素可能需要考虑? 写出这些因素(至少 3 个, 不超过 5 个);\\\\\n(2) 为了简化问题, 请对相关因素做出合情假设, 由此为李先生作出买车的决策, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题19", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012590": { + "id": "012590", + "content": "如图所示, 由半椭圆$C_1: \\dfrac{x^2}4+\\dfrac{y^2}{b^2}=1$($y \\leq 0$)和两个半圆$C_2:(x+1)^2+y^2=1$($y \\geq 0$)、$C_3:(x-1)^2+y^2=1$($y \\geq 0$)组成曲线$C: F(x, y)=0$, 其中点$A_1$、$A_2$依次为$C_1$的左、 右顶点, 点$B$为$C_1$的下顶点, 点$F_1$、$F_2$依次为$C_1$的左、右焦点. 若点$F_1$、$F_2$分别为曲线$C_2$、$C_3$的圆心.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below right] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (1,0) circle (0.03) node [below] {$F_2$} coordinate (F_2);\n\\filldraw (-1,0) circle (0.03) node [below] {$F_1$} coordinate (F_1);\n\\draw (0,0) arc (0:180:1) node [below left] {$A_1$} coordinate (A_1) arc (180:270:2 and {sqrt(3)}) node [below right] {$B$} coordinate (B) arc (270:360:2 and {sqrt(3)}) node [below right] {$A_2$} coordinate (A_2) arc (0:180:1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$C_1$的方程;\\\\\n(2) 若点$P$、$Q$分别在$C_2$、$C_3$上运动, 求$|BP|+|BQ|$的最大值, 并求出此时点$P$、$Q$的坐标;\\\\\n(3) 若点$M$在曲线$C: F(x, y)=0$上运动, 点$N(0,-1)$, 求$|NM|$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届嘉定区一模试题20", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012591": { + "id": "012591", + "content": "已知$f(x)=\\dfrac{\\ln x}x$.\\\\\n(1) 求函数$y=f(x)$的导数, 并证明: 函数$y=f(x)$在$[\\mathrm{e},+\\infty)$上是严格减函数(常数$\\mathrm{e}$为自然对数的底);\\\\\n(2) 根据(1), 判断并证明$89^{99}$与$99^{89}$的大小关系, 并请推广至一般的结论(无须证明);\\\\\n(3) 已知$a$、$b$是正整数, $a\\ln x$的充要条件是\\bracket{20}.\n\\fourch{$x>0$}{$x>1$}{$x>10$}{$00$, 若向量$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$满足$|\\overrightarrow a|:|\\overrightarrow b|:|\\overrightarrow c|=1: k: 3$, 且$\\overrightarrow b-\\overrightarrow a=2(\\overrightarrow c-\\overrightarrow b)$, 则满足条件的$k$的取值可以是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区一模试题15", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012607": { + "id": "012607", + "content": "设$A_1$、$A_2$、$A_3$、$\\cdots$、$A_7$是均含有$2$个元素的集合, 且$A_1 \\cap A_7=\\varnothing$, $A_i \\cap A_{i+1}=\\varnothing$($i=1,2,3, \\cdots, 6$), 记$B=A_1 \\cup A_2 \\cup A_3 \\cup \\cdots \\cup A_7$, 则$B$中元素个数的最小值是\\bracket{20}.\n\\fourch{$5$}{$6$}{$7$}{$8$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区一模试题16", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012608": { + "id": "012608", + "content": "如图所示, $BD$为四边形$ABCD$的对角线, 设$AB=AD=1$, $\\triangle BCD$为等边三角形. 记$\\angle BAD=\\theta$($0<\\theta<\\pi$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A) node [right] {$\\theta$};\n\\draw (-70:1) node [below] {$B$} coordinate (B);\n\\draw (60:1) node [above] {$D$} coordinate (D);\n\\draw ($(B)!1!-60:(D)$) node [right] {$C$} coordinate (C);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw [dashed] (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$BD=\\sqrt 3$时, 求$\\theta$的值;\\\\\n(2) 设$S$为四边形$ABCD$的面积, 用含有$\\theta$的关系式表示$S$, 并求$S$的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区一模试题17", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012609": { + "id": "012609", + "content": "设$a$、$b$均为正整数, $\\{a_n\\}$为首项为$a$、公差为$b$的等差数列, $\\{b_n\\}$为首项为$b$、公比为$a$的等比数列.\\\\\n(1) 设$t$为正整数, 当$a=3$, $b=1$, $a_7=latex]\n\\draw [dashed] (-2,-1,0) -- (2,-1,0) (0,-1,-2) -- (0,-1,2);\n\\draw (-2,-1,-0.3) -- (-0.3,-1,-0.3) -- (-0.3,-1,-2);\n\\draw (-2,-1,0.3) -- (-0.3,-1,0.3) -- (-0.3,-1,2);\n\\draw (2,-1,-0.3) -- (0.3,-1,-0.3) -- (0.3,-1,-2);\n\\draw (2,-1,0.3) -- (0.3,-1,0.3) -- (0.3,-1,2);\n\\fill [domain = 0:360, white] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\fill [domain = 0:360, pattern = north east lines] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\fill [domain = 0:360, white] plot ({cos(\\x)},0,{sin(\\x)});\n\\draw [domain = 0:360,ultra thick,samples = 100] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\draw [domain = 0:360,thick] plot ({cos(\\x)},0,{sin(\\x)});\n\\filldraw (0,0) circle (0.03) node [left] {$O$} coordinate (O);\n\\draw [domain = 0:180,thick] plot ({-sqrt(2)*cos(\\x)},{2*sin(\\x)},{sqrt(2)*cos(\\x)});\n\\draw [domain = 0:180,thick] plot ({sqrt(2)*cos(\\x)},{2*sin(\\x)},{sqrt(2)*cos(\\x)});\n\\draw (0,2,0) node [above] {$S$} coordinate (S);\n\\draw ({cos(45)},0,{sin(45)}) --++ (0,{sqrt(3)},0);\n\\draw ({cos(135)},0,{sin(135)}) node [right] {$D$} coordinate (D) --++ (0,{sqrt(3)},0) node [above left] {$C$} coordinate (C);\n\\draw ({cos(225)},0,{sin(225)}) --++ (0,{sqrt(3)},0);\n\\draw ({cos(315)},0,{sin(315)}) --++ (0,{sqrt(3)},0);\n\\draw [thick] ({2*cos(45)},0,{2*sin(45)}) node [below left] {$A$} coordinate (A) -- ({3*cos(45)},-1,{3*sin(45)}) node [below] {$B$} coordinate (B);\n\\draw [thick] ({2*cos(45)},0,{2*sin(45)}) node [below left] {$A$} coordinate (A) -- ({3*cos(45)},-1,{3*sin(45)}) node [below] {$B$} coordinate (B);\n\\draw [thick] ({2*cos(135)},0,{2*sin(135)}) -- ({3*cos(135)},-1,{3*sin(135)});\n\\draw [thick] ({2*cos(315)},0,{2*sin(315)}) -- ({3*cos(315)},-1,{3*sin(315)});\n\\draw [thick,dashed] ({2*cos(225)},0,{2*sin(225)}) -- ({3*cos(225)},-1,{3*sin(225)});\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:$CD\\parallel$平面$SOA$;\\\\\n(2) 设$AB$为经过$A$的一条步道, 其长度为$12$米且与地面所成角的大小为$30^{\\circ}$. 桥面小圆与大圆的半径之比为$4: 5$, 当桥面大圆半径为$20$米时, 求点$C$到地面的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区一模试题19", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012611": { + "id": "012611", + "content": "在$xOy$坐标平面内, 已知椭圆$\\Gamma: \\dfrac{x^2}9+\\dfrac{y^2}5=1$的左、右焦点分别为$F_1$、$F_2$, 直线$y=k_1 x$($k_1 \\neq 0$)与$\\Gamma$相交于$A$、$B$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (3 and {sqrt(5)});\n\\filldraw (-2,0) circle (0.05) node [below] {$F_1$} coordinate (F_1);\n\\filldraw (2,0) circle (0.05) node [below] {$F_2$} coordinate (F_2);\n\\path [name path = line] (-3,{-5/sqrt(7)}) -- (3,{5/sqrt(7)});\n\\draw [name intersections = {of = elli and line, by = {A,B}}];\n\\draw (A) node [above] {$A$} -- (F_2) -- (B) node [below left] {$B$} -- cycle;\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$} coordinate (O);\n\\draw [name path = elli] (0,0) ellipse (3 and {sqrt(5)});\n\\path [name path = line] (-3,{-5/sqrt(7)}) -- (3,{5/sqrt(7)});\n\\draw [name intersections = {of = elli and line, by = {A,B}}];\n\\draw (A) node [above] {$A$} -- (B) node [below left] {$B$};\n\\draw (B) -- ($(-3,0)!(B)!(3,0)$) node [above] {$M$} coordinate (M);\n\\draw ($(O)!0.5!(M)$) node [above] {$N$} coordinate (N);\n\\path [name path = AP] (A) -- ($(N)!-0.8!(A)$);\n\\draw [name intersections= {of = AP and elli, by = P}];\n\\draw (A) -- (P) node [below left] {$P$} coordinate (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 记$d$为$A$到直线$2 x+9=0$的距离, 当$k_1$变化时, 求证:$\\dfrac{|AF_1|}d$为定值;\\\\\n(2) 当$\\angle AF_2B =120^\\circ$时, 求$|AF_2|\\cdot|BF_2|$的值;\\\\\n(3) 过$B$作$BM \\perp x$轴, 垂足为$M$, $OM$的中点为$N$, 延长$AN$交$\\Gamma$于另一点$P$, 记直线$PB$的斜率为$k_2$, 当$k_1$取何值时, $|k_1-k_2|$有最小值? 并求出此最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区一模试题20", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012612": { + "id": "012612", + "content": "若函数$y=f(x)$($x \\in D$)同时满足下列两个条件, 则称$y=f(x)$在$D$上具有性质$M$.\\\\\n\\textcircled{1} $y=f(x)$在$D$上的导数$f'(x)$存在;\\\\\n\\textcircled{2} $y=f'(x)$在$D$上的导数$f''(x)$存在, 且$f''(x)>0$(其中$f''(x)=[f'(x)]'$)恒成立.\\\\\n(1) 判断函数$y=\\lg \\dfrac 1x$在区间$(0,+\\infty)$上是否具有性质$M$? 并说明理由;\\\\\n(2) 设$a$、$b$均为实常数, 若奇函数$g(x)=2 x^3+a x^2+\\dfrac bx$在$x=1$处取得极值, 是否存在实数$c$, 使得$y=g(x)$在区间$[c,+\\infty)$上具有性质$M$? 若存在, 求出$c$的取值范围; 若不存在, 请说明理由;\\\\\n(3) 设$k \\in \\mathbf{Z}$且$k>0$, 对于任意的$x \\in(0,+\\infty)$, 不等式$\\dfrac{1+\\ln (x+1)}x>\\dfrac k{x+1}$成立, 求$k$的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区一模试题21", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012613": { + "id": "012613", + "content": "设全集$U=\\{1,2,3,4\\}$, $A=\\{1,3\\}$, 则$\\overline A=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题1", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012614": { + "id": "012614", + "content": "不等式$x^2-3 x+2<0$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题2", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012615": { + "id": "012615", + "content": "复数$z$满足$\\overline z=\\dfrac 1{1+\\mathrm{i}}$(其中$\\mathrm{i}$为虚数单位), 则复数$z$在复平面上所对应的点$Z$到原点$O$的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题3", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012616": { + "id": "012616", + "content": "设向量$\\overrightarrow a$、$\\overrightarrow b$满足$|\\overrightarrow a|=1 , \\overrightarrow a \\cdot \\overrightarrow b=2$, 则$\\overrightarrow a \\cdot(\\overrightarrow a+\\overrightarrow b)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题4", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012617": { + "id": "012617", + "content": "如图, 在三棱台$ABC-A_1B_1C_1$的$9$条棱所在直线中, 与直线$A_1B$是异面直线的共有\\blank{50}条.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,-0.5) node [below] {$B$} coordinate (B);\n\\draw (3,0.3) node [right] {$C$} coordinate (C);\n\\path (1.4,2) coordinate (P);\n\\draw (A) -- ($(A)!0.5!(P)$) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) -- ($(B)!0.5!(P)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) -- ($(C)!0.5!(P)$) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw (A_1) -- (B);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题5", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012618": { + "id": "012618", + "content": "甲、乙两城市某月初连续$7$天的日均气温数据如图所示, 则在这$7$天中, \\\\\n\\textcircled{1} 甲城市日均气温的中位数与平均数相等;\\\\\n\\textcircled{2}甲城市的日均气温比乙城市的日均气温稳定;\\\\\n\\textcircled{3} 乙城市日均气温的极差为$3^{\\circ} \\text{C}$;\\\\\n\\textcircled{4} 乙城市日均气温的众数为$5^{\\circ} \\text{C}$.\n以上判断正确的是\\blank{50}.(写出所有正确判断的序号)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\foreach \\i in {1,2,...,7} {\n\\draw [gray] (\\i,0) -- (\\i,7) (0,\\i) -- (7,\\i); \n\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.2,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw [->] (0,0) -- (8,0) node [below] {日期};\n\\draw [->] (0,0) -- (0,8) node [left] {气温$^\\circ\\text{C}$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,5) -- (2,3) -- (3,6) -- (4,3) -- (5,7) -- (6,5) -- (7,6);\n\\draw [dashed] (1,5) -- (2,4) -- (3,6) -- (4,5) -- (5,5) -- (6,4) -- (7,6);\n\\draw (7.5,5.5) -- (9.5,5.5) node [right] {甲};\n\\draw [dashed] (7.5,4) -- (9.5,4) node [right] {乙};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题6", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012619": { + "id": "012619", + "content": "有甲、乙、丙三项任务, 其中甲需$2$人承担, 乙、丙各需$1$人承担 . 现从$6$人中任选$4$人承担这三项任务, 则共有\\blank{50}种不同的选法.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题7", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012620": { + "id": "012620", + "content": "研究发现, 某昆虫释放信息素$t$秒后, 在距释放处$x$米的地方测得的信息素浓度$y$满足$\\ln y=-\\dfrac 12 \\ln t-\\dfrac kt x^2+a$, 其中$k, a$为非零常数. 已知释放$1$秒后, 在距释放处$2$米的地方测得信息素浓度为$m$, 则释放信息素$4$秒后, 距释放处的\\blank{50}米的位置, 信息素浓度为$\\dfrac m2$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题8", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012621": { + "id": "012621", + "content": "若$\\overrightarrow{OA}=(1,-2,0)$, $\\overrightarrow{OB}=(2,1,0)$, $\\overrightarrow{OC}=(1,1,3)$, 则三棱锥$O-ABC$的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题9", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012622": { + "id": "012622", + "content": "已知函数$y=2 \\sin (\\omega x+\\dfrac{\\pi}6)(\\omega>0)$的图像向右平移$\\varphi(0<\\varphi<\\dfrac{\\pi}2)$个单位, 可得到函数$y=\\sin 2 x-a \\cos 2 x$的图像, 则$\\varphi=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题10", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012623": { + "id": "012623", + "content": "已知$AA_1$是圆柱的一条母线, $AB$是圆柱下底面的直径, $C$是圆柱下底面圆周上异于$A$、$B$的点. 若圆柱的侧面积为$4 \\pi$, 则三棱锥$A_1-ABC$外接球体积的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题11", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012624": { + "id": "012624", + "content": "已知$F_1$、$F_2$为椭圆$\\Gamma: \\dfrac{x^2}{a^2}+y^2=1(a>1)$的左右焦点, $A$为$\\Gamma$的上顶点, 直线$l$经过点$F_1$且与$\\Gamma$交于$B$、$C$两点. 若$l$垂直平分线段$AF_2$, 则$\\triangle ABC$的周长是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题12", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012625": { + "id": "012625", + "content": "若$\\alpha$为第四象限角, 则\\bracket{20}.\n\\fourch{$\\sin 2\\alpha<0$}{$\\cos 2\\alpha<0$}{$\\sin 2\\alpha>0$}{$\\cos 2\\alpha>0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题13", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012626": { + "id": "012626", + "content": "设$f(x)=x^\\alpha$($\\alpha \\in(-2,-1, \\dfrac 12, 1,2,3))$, 则``函数$y=f(x)$的图像经过点$(-1,-1)$''是``函数$y=f(x)$为奇函数''的\\blank{50}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题14", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012627": { + "id": "012627", + "content": "掷两颗骰子, 观察掷得的点数. 设事件$A$为: 至少一个点数是奇数; 事件$B$为: 点数之和是偶数, 事件$A$的概率为$P(A)$, 事件$B$的概率为$P(B)$. 则$1-P(A \\cap B)$是下列哪个事件的概率\\bracket{20}.\n\\fourch{两个点数都是偶}{至多有一个点数是偶数}{两个点数都是奇数}{至多有一个点数是奇数}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题15", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012628": { + "id": "012628", + "content": "函数$f(x)=(\\mathrm{e}^{a x}-b)^2$的大致图像如图, 在下列选项中, 则实数$a$、$b$的取值只可能是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw [->] (-8,0) -- (8,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,10) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3.1:8,samples = 100] plot (\\x,{pow(exp(-0.5*\\x)-1.5,2)});\n\\draw [dashed] (-8,2.25) -- (8,2.25);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$a>0$、$b>1$}{$a>0$、$01$}{$a<0$、$0=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,1.6,0) node [above] {$D$} coordinate (D);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D)$) node [above] {$F$} coordinate (F);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (D) -- (B) (F) -- (E);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线$EF\\parallel$平面$ABD$;\\\\\n(2) 求证: 直线$BC \\perp$平面$ABD$;\\\\\n(3) 若直线$CD$与平面$ABC$所成的角为$45^{\\circ}$, 直线$CD$与平面$ABD$所成角为$30^{\\circ}$, 求二面角$B-AD-C$的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题19", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012632": { + "id": "012632", + "content": "已知抛物线$\\Gamma: y^2=4 x$的焦点为$F$, 准线为$l$.\\\\\n(1) 若$F$为双曲线$C: \\dfrac{x^2}{a^2}-2 y^2=1(a>0)$的一个焦点, 求双曲线$C$的离心率$e$;\\\\\n(2) 设$l$与$x$轴的交点为$E$, 点$P$在第一象限, 且在$\\Gamma$上, 若$\\dfrac{|PF|}{|PE|}=\\dfrac{\\sqrt 2}2$, 求直线$EP$的方程;\\\\\n(3) 经过点$F$且斜率为$k$($k \\neq 0$)的直线$l'$与$\\Gamma$相交于$A$、$B$两点, $O$为坐标原点, 直线$OA$、$OB$分别与$l$相交于点$M$、$N$. 试探究: 以线段$MN$为直径的圆$C$是否过定点, 若是, 求出定点的坐标; 若不是, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区一模试题20", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012633": { + "id": "012633", + "content": "已知函数$y=f(x)$的定义域为$(0,+\\infty)$.\\\\\n(1) 若$f(x)=\\ln x$;\\\\\n\\textcircled{1} 求曲线$y=f(x)$在点$(1,0)$处的切线方程;\\\\\n\\textcircled{2} 求函数$g(x)=f(x)+x^2-3 x$的单调减区间和极小值;\\\\\n(2) 若对任意$a, b \\in(1,+\\infty)$($a0\\}$, 则$\\overline A=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题1", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012635": { + "id": "012635", + "content": "在复平面内, 复数$z$所对应的点的坐标为$(1,-1)$, 则$z \\cdot \\overline z=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题2", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012636": { + "id": "012636", + "content": "不等式$\\dfrac{x+5}{x^2+2 x+3} \\geq 1$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题3", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012637": { + "id": "012637", + "content": "函数$y=\\tan x$在区间$(\\dfrac{\\pi}2, \\dfrac{3 \\pi}2)$上的零点是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题4", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012638": { + "id": "012638", + "content": "已知$f(x)$是定义域为$\\mathbf{R}$的奇函数, 且$x \\leq 0$时, $f(x)=e^x-1$, 则$f(x)$的值域是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题5", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012639": { + "id": "012639", + "content": "在$(x-\\dfrac 2x)^9$的二项展开式中, $x^3$项的系数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题6", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012640": { + "id": "012640", + "content": "已知圆锥的侧面积为$2 \\pi$, 且侧面展开图为半圆, 则该圆锥底面半径为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题7", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012641": { + "id": "012641", + "content": "在数列$\\{a_n\\}$中, $a_1=2$, 且$a_n=a_{n-1}+\\lg \\dfrac n{n-1}$($n \\geq 2$), 则$a_{100}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题8", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012642": { + "id": "012642", + "content": "某中学从甲、乙两个班中各选出$15$名学生参加知识竞赛, 将他们的成绩(满分$100$分)进行统计分析, 绘制成如图所示的茎叶图. 设成绩在$88$分以上(含$88$分)的学生为优秀学生, 现从甲、乙两班的优秀学生中各取$1$人, 记甲班选取的学生成绩不低于乙班选取得学生成绩记为事件$A$, 则事件$A$发生的概率$P(A)=$\\blank{50}.\n\\begin{center}\n\\begin{tabular}{r|c|l}\n甲 & & 乙\\\\\n& 5 & 8\\\\\n8\\ 0& 6 & 6\\ 9\\\\\n9 \\ 8 \\ 5 & 7 & 0 \\ 5 \\ 6 \\ 6 \\ 6 \\ 8 \\ 8\\\\\n8 \\ 7 \\ 6 \\ 4 \\ 1 & 8 & 6 \\ 6 \\\\\n8 \\ 6 \\ 2 \\ 2 \\ 1 & 9 & 5 \\ 8 \\ 8\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题9", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012643": { + "id": "012643", + "content": "在$\\triangle ABC$中, $AC=4$, 且$\\overrightarrow{AC}$在$\\overrightarrow{AB}$方向上的数量投影是$-2$, 则$|\\overrightarrow{BC}-\\lambda \\overrightarrow{BA}|$($\\lambda \\in \\mathbf{R}$)的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题10", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012644": { + "id": "012644", + "content": "设$k \\in \\mathbf{R}$, 函数$y=|x^2-4 x+3|$的图像与直线$y=k x+1$有四个交点, 且这些交点的横坐标分别为$x_1, x_2, x_3, x_4(x_10$, 则``$a>b$''是``$\\dfrac 1a<\\dfrac 1b$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分也非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题13", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012647": { + "id": "012647", + "content": "已知圆$C_1$的半径为$3$, 圆$C_2$的半径为$7$, 若两圆相交, 则两圆的圆心距可能是\\bracket{20}.\n\\fourch{$0$}{$4$}{$8$}{$12$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题14", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012648": { + "id": "012648", + "content": "已知平面$\\alpha$、$\\beta$、$\\gamma$两两垂直, 直线$a$、$b$、$c$满足: $a \\subset \\alpha$, $b \\subset \\beta$, $c \\subset \\gamma$, 则直线$a$、$b$、$c$位置关系不可能是\\bracket{20}.\n\\fourch{两两垂直}{两两平行}{两两相交}{两两异面}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题15", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012649": { + "id": "012649", + "content": "设数列$\\{a_n\\}$为: $1$, $\\dfrac 12$, $\\dfrac 12$, $\\dfrac 14$, $\\dfrac 14$, $\\dfrac 14$, $\\dfrac 14$, $\\dfrac 18$, $\\dfrac 18$, $\\dfrac 18$, $\\dfrac 18$, $\\dfrac 18$, $\\dfrac 18$, $\\dfrac 18$, $\\dfrac 18$, $\\cdots$, 其中第$1$项为$\\dfrac 11$, 接下来$2$项均为$\\dfrac 12$, 再接下来$4$项均为$\\dfrac 14$, 再接下来$8$项均为$\\dfrac 18$, $\\cdots$, 以此类推, 记$S_n=\\displaystyle\\sum_{i=1}^n a_i$, 现有如下命题: \\textcircled{1} 存在正整数$k$, 使得$a_k<\\dfrac 1k$; \\textcircled{2} 数列$\\{\\dfrac{S_n}n\\}$是严格减数列. 下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}均为真命题}{\\textcircled{1}和\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题16", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012650": { + "id": "012650", + "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $AB=AC=2$, $AA_1=4$, $AB \\perp AC$, $BE \\perp AB_1$交$AA_1$于点$E$, $D$为$CC_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) node [left] {$B$} coordinate (B) ++ (0,4) node [left] {$B_1$} coordinate (B_1);\n\\draw ({sqrt(2)},0,{-sqrt(2)}) node [right] {$A$} coordinate (A) ++ (0,4)node [above] {$A_1$} coordinate (A_1);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C) ++ (0,4) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$D$} coordinate (D);\n\\draw ($(A)!0.25!(A_1)$) node [right] {$E$} coordinate (E);\n\\draw (B) -- (C) -- (C_1) -- (A_1) -- (B_1) -- cycle;\n\\draw (B_1) -- (C_1) (B_1) -- (D) (B_1) -- (C);\n\\draw [dashed] (B) -- (A) -- (C) (B) -- (E) (A) -- (A_1) (B_1) -- (A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:$BE \\perp$平面$AB_1C$;\\\\\n(2) 求直线$B_1D$与平面$AB_1C$所成角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题17", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012651": { + "id": "012651", + "content": "已知$f(x)=\\ln x-(a+1) x+\\dfrac 12 a x^2$($a \\in \\mathbf{R}$).\\\\\n(1) 当$a=0$时, 求函数$y=f(x)$在点$(1, f(1))$处的切线方程;\\\\\n(2) 当$a \\in(0,1]$时, 求函数$y=f(x)$的单调区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题18", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012652": { + "id": "012652", + "content": "近年来, 为``加大城市公园绿地建设力度, 形成布局合理的公园体系'', 许多城市陆续建起众多``口袋公园''. 现计划在一块边长为$200$米的正方形的空地上按以下要求建造``口袋公园''. 如图所示, 以$EF$中点$A$为圆心, $FG$为半径的扇形草坪区$ABC$, 点$P$在弧$BC$上(不与端点重合),\n$AB$、弧$BC$、$CA$、$PQ$、$PR$、$RQ$为步行道, 其中$PQ$与$AB$垂直, $PR$与$AC$垂直. 设$\\angle PAB=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$E$} coordinate (E) rectangle (3,3) node [above right] {$G$} coordinate (G);\n\\draw (3,0) node [below right] {$F$} coordinate (F) (0,3) node [above left] {$H$} coordinate (H);\n\\draw ($(E)!0.5!(F)$) node [below] {$A$} coordinate (A) --++ (60:3) node [right] {$B$} coordinate (B) arc (60:120:3) node [left] {$C$} coordinate (C) -- cycle;\n\\draw (1.5,0) ++ (80:3) node [above] {$P$} coordinate (P);\n\\draw (P) -- ($(A)!(P)!(B)$) node [below left] {$Q$} coordinate (Q);\n\\draw (P) -- ($(A)!(P)!(C)$) node [below left] {$R$} coordinate (R);\n\\draw (R) -- (Q);\n\\end{tikzpicture}\n\\end{center} \n(1) 如果点$P$位于弧$BC$的中点, 求三条步行道$PQ$、$PR$、$RQ$的总长度;\\\\\n(2) ``地摊经济''对于``拉动灵活就业、增加多源收入、便利居民生活''等都有积极作用. 为此街道允许在步行道$PQ$、$PR$、$RQ$开辟临时摊点, 积极推进``地摊经济''发展, 预计每年能产生的经济效益分别为每米$5$万元、$5$万元及$5.9$万元 . 则这三条步行道每年能产生的经济总效益最高为多少?(精确到 1 万元)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题19", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012653": { + "id": "012653", + "content": "已知曲线$C_i$的方程为$x^2+\\lambda_i y^2=1$($\\lambda_i \\in \\mathbf{R}$, $i=1,2,3$), 直线$l: y=k(x+1)$, $k \\in \\mathbf{R}$.\\\\\n(1) 若曲线$C_1$是焦点在$x$轴上且离心率为$\\dfrac{\\sqrt 2}2$的椭圆, 求$\\lambda_1$的值;\\\\ \n(2) 若$k=1$, $\\lambda_2 \\neq-1$时, 直线$l$与曲线$C_2$相交于两点$M, N$, 且$|MN|=\\sqrt 2$, 求曲线$C_2$的方程;\\\\ \n(3) 若直线$l$与曲线$C_i$在第一象限交于点$P_i(x_i, y_i)$, 是否存在不全相等的$\\lambda_1, \\lambda_2, \\lambda_3$满足$\\lambda_1+\\lambda_3=2 \\lambda_2$, 且使得$x_2^2=x_1 x_3$成立. 若存在, 求出$x_2$的值; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届徐汇区一模试题20", + "edit": [ + "20221215\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012654": { + "id": "012654", + "content": "对于数列$\\{x_n\\}$, $\\{y_n\\}$, 其中$y_n \\in \\mathbf{Z}$, 对任意正整数$n$都有$|x_n-y_n|<\\dfrac 12$, 则称数列$\\{y_n\\}$为数列$\\{x_n\\}$的``接近数列''. 已知$\\{b_n\\}$为数列$\\{a_n\\}$的``接近数列'', 且$A_n=\\sum_{i=1}^n a_i$, $B_n=\\sum_{i=1}^n b_i$.\\\\\n(1) 若$a_n=n+\\dfrac 14$($n$是正整数), 求$b_1, b_2, b_3, b_4$的值;\\\\ \n(2) 若$a_n=\\dfrac 32+(-\\dfrac 9{10})^{n+1}$($n$是正整数), 是否存在$k$($k$是正整数), 使得$A_k