From 9e54e3bba5e801d7d99d8f9c10d1ab5ca46a0646 Mon Sep 17 00:00:00 2001 From: WangWeiye Date: Wed, 14 Dec 2022 17:11:33 +0800 Subject: [PATCH] =?UTF-8?q?20221214=20afternoon=20=E6=94=B6=E5=BD=95?= =?UTF-8?q?=E8=99=B9=E5=8F=A3=E9=9D=92=E6=B5=A6=E4=B8=80=E6=A8=A1?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具/寻找阶段末尾空闲题号.ipynb | 4 +- 工具/批量添加题库字段数据.ipynb | 43 +- 工具/文本文件/metadata.txt | 119 ++- 工具/添加题目到数据库.ipynb | 10 +- 工具/识别题库中尚未标注的题目类型.ipynb | 44 +- 文本处理工具/剪贴板文本整理_word文件.ipynb | 8 +- 题库0.3/Problems.json | 883 +++++++++++++++++++++ 7 files changed, 998 insertions(+), 113 deletions(-) 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 39f64255..72806d70 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": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "首个空闲id: 12487 , 直至 020000\n", + "首个空闲id: 12529 , 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/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb index a523acd6..bc59c921 100644 --- a/工具/批量添加题库字段数据.ipynb +++ b/工具/批量添加题库字段数据.ipynb @@ -9,28 +9,27 @@ "name": "stdout", "output_type": "stream", "text": [ - "题号: 012487 , 字段: tags 中已添加数据: 第一单元\n", - "题号: 012488 , 字段: tags 中已添加数据: 第一单元\n", - "题号: 012489 , 字段: tags 中已添加数据: 第二单元\n", - "题号: 012490 , 字段: tags 中已添加数据: 第三单元\n", - "题号: 012491 , 字段: tags 中已添加数据: 第五单元\n", - "题号: 012492 , 字段: tags 中已添加数据: 第五单元\n", - "题号: 012493 , 字段: tags 中已添加数据: 第九单元\n", - "题号: 012494 , 字段: tags 中已添加数据: 第七单元\n", - "题号: 012495 , 字段: tags 中已添加数据: 第一单元\n", - "题号: 012496 , 字段: tags 中已添加数据: 第八单元\n", - "题号: 012497 , 字段: tags 中已添加数据: 第四单元\n", - "题号: 012498 , 字段: tags 中已添加数据: 第一单元\n", - "题号: 012499 , 字段: tags 中已添加数据: 第八单元\n", - "题号: 012500 , 字段: tags 中已添加数据: 第六单元\n", - "题号: 012501 , 字段: tags 中已添加数据: 第三单元\n", - "题号: 012502 , 字段: tags 中已添加数据: 第二单元\n", - "题号: 012503 , 字段: tags 中已添加数据: 第三单元\n", - "题号: 012504 , 字段: tags 中已添加数据: 第六单元\n", - "题号: 012505 , 字段: tags 中已添加数据: 第二单元\n", - "题号: 012506 , 字段: tags 中已添加数据: 第七单元\n", - "题号: 012507 , 字段: tags 中已添加数据: 第二单元\n", - "题号: 012507 , 字段: tags 中已添加数据: 第四单元\n" + "题号: 012529 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 012530 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 012531 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 012532 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 012533 , 字段: tags 中已添加数据: 第九单元\n", + "题号: 012534 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 012535 , 字段: tags 中已添加数据: 第八单元\n", + "题号: 012536 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 012537 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 012538 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 012539 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 012540 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 012541 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 012542 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 012543 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 012544 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 012545 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 012546 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 012547 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 012548 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 012549 , 字段: tags 中已添加数据: 第二单元\n" ] } ], diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 1cc124b4..0f28ed9c 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,88 +1,87 @@ tags -12487 +12529 第一单元 -12488 -第一单元 +12530 +第五单元 -12489 +12531 +第四单元 + + +12532 第二单元 -12490 -第三单元 - - -12491 -第五单元 - - -12492 -第五单元 - - -12493 +12533 第九单元 -12494 -第七单元 +12534 +第二单元 -12495 -第一单元 - - -12496 +12535 第八单元 -12497 -第四单元 - - -12498 -第一单元 - - -12499 -第八单元 - - -12500 -第六单元 - - -12501 -第三单元 - - -12502 +12536 第二单元 -12503 -第三单元 - - -12504 -第六单元 - - -12505 -第二单元 - - -12506 +12537 第七单元 -12507 -第二单元 +12538 +第三单元 + + +12539 +第六单元 + + +12540 第四单元 +12541 +第一单元 + + +12542 +第六单元 + + +12543 +第二单元 + + +12544 +第七单元 + + +12545 +第三单元 + + +12546 +第六单元 + + +12547 +第四单元 + + +12548 +第七单元 + + +12549 +第二单元 + + 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 e5a834dc..fa5766a5 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": 2, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 12487\n", - "origin = \"2023届杨浦区一模\"\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 = \"20221213\\t王伟叶\"" + "editor = \"20221214\\t王伟叶\"" ] }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 2, "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 423fd265..e19f0157 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 +2,34 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "012487 填空题\n", - "012488 填空题\n", - "012489 填空题\n", - "012490 填空题\n", - "012491 填空题\n", - "012492 填空题\n", - "012493 填空题\n", - "012494 填空题\n", - "012495 填空题\n", - "012496 填空题\n", - "012497 填空题\n", - "012498 填空题\n", - "012499 选择题\n", - "012500 选择题\n", - "012501 选择题\n", - "012502 选择题\n", - "012503 解答题\n", - "012504 解答题\n", - "012505 解答题\n", - "012506 解答题\n", - "012507 解答题\n" + "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" ] } ], 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 ab79244b..08050fa4 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": 1, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ @@ -481,7 +481,11 @@ "modified_data = modified_data.replace(r\"\\lim _{n \\rightarrow \\infty}\",r\"\\displaystyle\\lim_{n\\to\\infty}\")\n", "#mathpix的顿号修改\n", "modified_data = modified_data.replace(r\" 、 \",r\"$、$\")\n", - "\n", + "#改slant等\n", + "modified_data = modified_data.replace(r\"slant\",\"\")\n", + "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(\",$\",\", $\")\n", "\n", "setCopy(modified_data)\n", "\n", diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index abbef128..ff3f1d50 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -308139,6 +308139,889 @@ "remark": "", "space": "12ex" }, + "012508": { + "id": "012508", + "content": "不等式$\\dfrac x{x+2} \\leq 0$的解集为\\blank{50}.", + "objs": [], + "tags": [ + "第一单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题1", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012509": { + "id": "012509", + "content": "对于正实数$x$, 代数式$x+\\dfrac 4x$的最小值为\\blank{50}.", + "objs": [], + "tags": [ + "第一单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题2", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012510": { + "id": "012510", + "content": "已知一个球的半径为$3$, 则这个球的体积为\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题3", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012511": { + "id": "012511", + "content": "在$(x+\\dfrac 1{\\sqrt x})^7$的二项展开式中$x$项的系数为\\blank{50}.", + "objs": [], + "tags": [ + "第八单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题4", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012512": { + "id": "012512", + "content": "设$m, n \\in \\mathbf{R}$, $\\mathrm{i}$为虚数单位, 若$1-\\sqrt 3 \\mathrm{i}$是关于$x$的二次方程$x^2+m x+n=0$的一个虚根, 则$m+n=$\\blank{50}.", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题5", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012513": { + "id": "012513", + "content": "已知首项为$2$的等比数列$\\{b_n\\}$的公比为$\\dfrac 13$, 则这个数列所有项的和为\\blank{50}.", + "objs": [], + "tags": [ + "第四单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题6", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012514": { + "id": "012514", + "content": "设曲线$y=\\ln x+2 x$的斜率为$3$的切线为$l$, 则$l$的方程为\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题7", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012515": { + "id": "012515", + "content": "第$5$届中国国际进口博览会在上海举行, 某高校派出了包括甲同学在内的$4$名同学参加了连续$5$天的志愿者活动. 已知甲同学参加了$2$天的活动, 其余同学各参加了$1$天的活动, 则甲同学参加连续两天活动的概率为\\blank{50}. (结果用分数表示)", + "objs": [], + "tags": [ + "第八单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题8", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012516": { + "id": "012516", + "content": "设$a, b \\in \\mathbf{R}$, 若函数$f(x)=\\lg|a+\\dfrac 4{2-x}|+b$为奇函数, 则$a+b=$\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题9", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012517": { + "id": "012517", + "content": "设函数$f(x)=\\cos (\\omega x+\\varphi)$(其中$\\omega>0$, $|\\varphi|<\\dfrac{\\pi}2)$, 若函数$y=f(x)$图像的对称轴$x=\\dfrac{\\pi}6$与其对称中心的最小距离为$\\dfrac{\\pi}8$, 则$f(x)=$\\blank{50}.", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题10", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012518": { + "id": "012518", + "content": "在$\\triangle ABC$中, $AB=5$, $AC=6$, $\\cos A=\\dfrac 15$, $O$是$\\triangle ABC$的外心, 若$\\overrightarrow{OP}=x \\overrightarrow{OB}+y \\overrightarrow{OC}$, 其中$x, y \\in[0,1]$, 则动点$P$的轨迹所覆盖图形的面积为\\blank{50}.", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题11", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012519": { + "id": "012519", + "content": "已知$F_1, F_2$是双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a, b>0$)的左、右焦点, 过$F_2$的直线交双曲线的右支于$A, B$两点, 且$|AF_1|=2|AF_2|$, $\\angle AF_1F_2=\\angle F_1BF_2$, 则在下列结论中, 正确结论的序号为\\blank{50}.\\\\\n\\textcircled{1} 双曲线$C$的离心率为$2$;\\\\\n\\textcircled{2} 双曲线$C$的一条渐近线的斜率为$\\sqrt 2$;\\\\\n\\textcircled{3} 线段$AB$的长为$6 a$;\\\\\n\\textcircled{4} $\\triangle AF_1F_2$的面积为$\\sqrt {15} a^2$.", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题12", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012520": { + "id": "012520", + "content": "设$m \\in \\mathbf{R}$, 已知直线$l: y=m x+1$与圆$C: x^2+y^2=1$, 则``$m>0$''是``直线$l$与圆$C$相交''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题13", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012521": { + "id": "012521", + "content": "若复数$z$满足$|z|<1$且$|\\bar z+\\dfrac 1z|=\\dfrac 52$, 则$|z|=$\\bracket{20}.\n\\fourch{$\\dfrac 45$}{$\\dfrac 34$}{$\\dfrac 23$}{$\\dfrac 12$}", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题14", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012522": { + "id": "012522", + "content": "已知$F$是椭圆$C_1: \\dfrac{x^2}4+\\dfrac{y^2}3=1$与抛物线$C_2: y^2=2 p x(p>0)$的一个共同焦点, $C_1$与$C_2$相交于$A, B$两点, 则线段$AB$的长等于\\bracket{20}.\n\\fourch{$\\dfrac 23 \\sqrt 6$}{$\\dfrac 43 \\sqrt 6$}{$\\dfrac 53$}{$\\dfrac{10}3$}", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题15", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012523": { + "id": "012523", + "content": "已知函数$f(x)=\\sin \\dfrac{\\pi x}3$, 数列$\\{a_n\\}$满足$a_1=1$, 且$a_{n+1}=(1+\\dfrac 1n) a_n+\\dfrac 1n$($n$为正整数). 则$f(a_{2022})=$\\bracket{20}.\n\\fourch{$-1$}{$1$}{$-\\dfrac{\\sqrt 3}2$}{$\\dfrac{\\sqrt 3}2$}", + "objs": [], + "tags": [ + "第四单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题16", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012524": { + "id": "012524", + "content": "设$\\triangle ABC$的内角$A, B, C$所对的边分别为$a, b, c$, 已知$2 \\cos (\\pi+A)+\\sin (\\dfrac{\\pi}2+2A)+\\dfrac 32=0$.\\\\\n(1) 求角$A$;\\\\\n(2) 若$c-b=\\dfrac{\\sqrt 3}3 a$, 求证: $\\triangle ABC$是直角三角形.", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题17", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012525": { + "id": "012525", + "content": "在等差数列$\\{a_n\\}$中, $a_1=2$, 且$a_2, a_3+2, a_8$构成等比数列.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 令$b_n=2^{a_n}+9$, 记$S_n$为数列$\\{b_n\\}$的前$n$项和, 若$S_n \\geq 2022$, 求正整数$n$的最小值.", + "objs": [], + "tags": [ + "第四单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题18", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012526": { + "id": "012526", + "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, 底面$ABC$是以$AC$为斜边的等腰直角三角形, 侧面$AA_1C_1C$为菱形, 点$A_1$在底面上的投影为$AC$的中点$D$, 且$AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (-1,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(3)},0) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(B)+(A_1)-(A)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(C)+(A_1)-(A)$) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)!0.5!(B_1)$) node [above right] {$E$} coordinate (E);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1);\n\\draw (A) -- (A_1) -- (C_1) -- (C) (B) -- (B_1);\n\\draw [dashed] (B) -- (D) -- (A_1) (D) -- (E) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}(1) 求证: $BD \\perp CC_1$;\\\\\n(2) 求点$C$到侧面$AA_1B_1B$的距离;\\\\\n(3) 在线段$A_1B_1$上是否存在点$E$, 使得直线$DE$与侧面$AA_1B_1B$所成角的正弦值为$\\dfrac{\\sqrt 6}7$? 若存在, 请求出$A_1E$的长; 若不存在, 请说明理由.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题19", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012527": { + "id": "012527", + "content": "本市某区对全区高中生的身高(单位: 厘米)进行统计, 得到如下的频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.6, yscale = 100]\n\\draw [->] (-0,0) -- (0.1,0) -- (0.2,0.0008) -- (0.4,-0.0008) -- (0.5,0) -- (10,0) node [below] {身高/厘米};\n\\draw [->] (0,-0) -- (0,0.035) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {150/0.022,160/0.027,170/0.025,180/0.015,190/0.01,200/0.001} \n{\\draw ({(\\i-130)/10},0) node [below] {\\small $\\i$} --++ (0,\\j) --++ (1,0) --++ (0,{-\\j});\n\\draw [dashed] (0,\\j) node [left] {\\small $\\j$} -- ({(\\i-130)/10},\\j);};\n\\draw (8,0) node [below] {\\small $210$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若数据分布均匀, 记随机变量$X$为各区间中点所代表的身高, 写出$X$的分布及期望;\\\\\n(2) 已知本市身高在区间$[180,210]$的市民人数约占全市总人数的$10 \\%$, 且全市高中生约占全市总人数的$1.2 \\%$. 现在要以该区本次统计数据估算全市高中生身高情况, 从本市市民中任取$1$人, 若此人的身高位于区间$[180,210]$, 试估计此人是高中生的概率;\\\\\n(3) 现从身高在区间$[170,190)$的高中生中分层抽样抽取一个$80$人的样本. 若身高在区间$[170,180)$中样本的均值为$176$厘米, 方差为$10$; 身高在区间$[180,190)$中样本的均值为$184$厘米, 方差为$16$, 试求这$80$人的方差.", + "objs": [], + "tags": [ + "第八单元", + "第九单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题20", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012528": { + "id": "012528", + "content": "设$a>0$, 已知函数$f(x)=(x-2)^3-a x$.\\\\\n(1) 求函数$y=f(x)$的单调区间;\\\\\n(2) 对于函数$y=f(x)$的极值点$x_0$, 存在$x_1$($x_1 \\neq x_0$), 使得$f(x_1)=f(x_0)$, 试问对任意的正数$a$, $x_1+2 x_0$是否为定值? 若是, 求出这个定值; 若不是, 请说明理由;\\\\\n(3) 若函数$g(x)=|f(x)|$在区间$[0,6]$上的最大值为$40$, 试求$a$的取值集合.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区一模试题21", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012529": { + "id": "012529", + "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{x |(x-1)(x-5)<0\\}$, 则$A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [ + "第一单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题1", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012530": { + "id": "012530", + "content": "若复数$z=\\dfrac{a+\\mathrm{i}}{\\mathrm{i}}$(其中$\\mathrm{i}$为虚数单位)的实部与虚部相等, 则实数$a=$\\blank{50}.", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题2", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012531": { + "id": "012531", + "content": "从等差数列$84,80,76,72, \\cdots$的第\\blank{50}项起, 各项均为负值.", + "objs": [], + "tags": [ + "第四单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题3", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012532": { + "id": "012532", + "content": "不等式$2^{x^2-2 x-3}<(\\dfrac 12)^{3(x-1)}$的解集为\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题4", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012533": { + "id": "012533", + "content": "在一次射击训练中, 某运动员$5$次射击的环数依次是$9,10,9,7,10$, 则该组数据的方差是\\blank{50}.", + "objs": [], + "tags": [ + "第九单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题5", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012534": { + "id": "012534", + "content": "已知函数$f(x)=x^3-2 x$, 则$f(x)$在点$(1, f(1))$处的切线的倾斜角为\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题6", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012535": { + "id": "012535", + "content": "若$(x+\\dfrac{\\sqrt a}{x^2})^6$的展开式的常数项是$45$, 则常数$a$的值为\\blank{50}.", + "objs": [], + "tags": [ + "第八单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题7", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012536": { + "id": "012536", + "content": "若函数$y=f(x)$的定义域和值域分别为$A=\\{1,2,3\\}$和$B=\\{1,2\\}$, 则$y=f(x)$是单调函数的概率是\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题8", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012537": { + "id": "012537", + "content": "已知空间三点$A(-1,3,1)$, $B(2,4,0)$, $C(0,2,4)$, 则以$\\overrightarrow{AB}$、$\\overrightarrow{AC}$为一组邻边的平行四边形的面积大小为\\blank{50}.", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题9", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012538": { + "id": "012538", + "content": "在平面直角坐标系中, $A(0,0)$, $B(1,2)$两点绕定点$P$按顺时针方向旋转$\\theta$角后, 分别到$A'(4,4)$, $B'(5,2)$两点位置, 则$\\cos \\theta$的值为\\blank{50}.", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题10", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012539": { + "id": "012539", + "content": "已知圆柱的轴截面是边长为$2$的正方形, $P$为上底面圆的圆心, $AB$为下底面圆的直径, $C$为下底面圆周上一点, 则三棱锥$P-ABC$外接球的体积为\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题11", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012540": { + "id": "012540", + "content": "已知数列$\\{a_n\\}$中, $a_2=3 a_1$, 记$\\{a_n\\}$的前$n$项和为$S_n$, 且满足$S_{n+1}+S_n+S_{n-1}=3 n^2+2$($n \\geq 2$, $n \\in \\mathbf{N}$). 若对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 都有$a_nb$''是``$\\dfrac 1a<\\dfrac 1b$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", + "objs": [], + "tags": [ + "第一单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题13", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012542": { + "id": "012542", + "content": "已知$m, n$是两条不同直线,$\\alpha, \\beta$是两个不同平面, 则下列命题错误的是\\bracket{20}.\n\\onech{若$\\alpha, \\beta$不平行, 则在$\\alpha$内不存在与$\\beta$平行的直线}{若$m , n$平行于同一平面, 则$m$与$n$可能异面}{若$m, n$不平行, 则$m$与$n$不可能垂直于同一平面}{若$\\alpha, \\beta$垂直于同一平面, 则$\\alpha$与$\\beta$可能相交}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题14", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012543": { + "id": "012543", + "content": "已知函数$y=f(x)$定义域为$\\mathbf{R}$, 下列论断:\\\\ \n\\textcircled{1} 若对任意实数$a$, 存在实数$b$, 使得$f(a)=f(b)$, 且$b=-a$, 则$f(x)$是偶函数.\\\\\n\\textcircled{2} 若对任意实数$a$, 存在实数$b$, 使得$f(a)0$, 若对任意实数$a$, 存在实数$b$, 使得$f(a)=f(b)$, 且$|a-b|=T$, 则$f(x)$是周期函数.\n其中正确的论断的个数是\\bracket{20}.\n\\fourch{$0$个}{$1$个}{$2$个}{$3$个}", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题15", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012544": { + "id": "012544", + "content": "在直角坐标平面$xOy$中, 已知两定点$F_1(-2,0)$与$F_2(2,0)$, $F_1$, $F_2$到直线$l$的距离之差的绝对值等于$2 \\sqrt 2$, 则平面上不在任何一条直线$l$上的点组成的图形面积是\\bracket{20}.\n\\fourch{$4 \\pi$}{$8$}{$2 \\pi$}{$4+\\pi$}", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题16", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012545": { + "id": "012545", + "content": "已知函数$f(x)=\\sqrt 3 \\sin x \\cos x-\\cos ^2 x$, $x \\in \\mathbf{R}$.\\\\\n(1) 求$f(x)$的单调递增区间;\\\\ \n(2) 求$f(x)$在区间$[-\\dfrac{\\pi}4, \\dfrac{\\pi}4]$上的最大值和最小值.", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题17", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012546": { + "id": "012546", + "content": "如图, 在正三棱柱$ABC-A_1B_1C_1$中, $E, F$分别为$BB_1$, $AC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ({\\l/2},0,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (C) ++ (0,\\h) node [below right] {$C_1$} coordinate (C_1);\n\\draw (B) ++ (0,\\h) node [right] {$B_1$} coordinate (B_1);\n\\draw (A) -- (C) -- (B) (A) -- (A_1) (C) -- (C_1) (B) -- (B_1) (A_1) -- (C_1) -- (B_1) (A_1) -- (B_1);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [below left] {$F$} coordinate (F);\n\\draw [dashed] (F) -- (B) (E) -- (A_1);\n\\draw (A_1) -- (C) (C) -- (E);\n\\draw [dashed] (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BF\\parallel$平面$A_1EC$;\\\\ \n(2) 求证: 平面$A_1EC \\perp$平面$ACC_1A_1$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题18", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012547": { + "id": "012547", + "content": "流行性感冒是由流感病毒引起的急性呼吸道传染病. 某市去年$11$月份曾发生流感, 据统计, $11$月$1$日该市的新感染者有$30$人, 以后每天的新感染者比前一天的新感染者增加$50$人. 由于该市医疗部门采取措施, 使该种病毒的传播得到控制, 从$11$月$k+1$($9 \\leq k \\leq 29$, $k \\in \\mathbf{N}$)日起每天的新感染者比前一天的新感染者减少$20$人.\\\\\n(1) 若$k=9$, 求$11$月$1$日至$11$月$10$日新感染者总人数;\\\\\n(2) 若到$11$月$30$日止, 该市在这$30$天内的新感染者总人数为$11940$人, 问$11$月几日, 该市新感染者人数最多? 并求这一天的新感染者人数.", + "objs": [], + "tags": [ + "第四单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题19", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012548": { + "id": "012548", + "content": "在平面直角坐标系$xOy$中, 已知椭圆$\\Gamma: \\dfrac{x^2}2+y^2=1$, 过右焦点$F$作两条互相垂直的弦$AB$, $CD$, 设$AB$, $CD$中点分别为$M$, $N$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,1.5) coordinate (S) (2,{2/3}) coordinate (T) (1,0) coordinate (F) node [below] {$F$};\n\\draw [name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\path [name path = l1] (S) -- ($(S)!1.5!(F)$);\n\\path [name path = l2] (T) -- ($(T)!2.5!(F)$);\n\\path [name intersections = {of = elli and l1, by = {A,B}}];\n\\draw (A) node [above] {$A$} coordinate (A)-- (B) node [below] {$B$} coordinate (B);\n\\path [name intersections = {of = elli and l2, by = {C,D}}];\n\\draw (C) node [right] {$C$} coordinate (C)-- (D) node [below] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [above] {$M$} coordinate (M) -- ($(C)!0.5!(D)$) node [below] {$N$} coordinate (N);\n\\end{tikzpicture}\n\\end{center}\n(1) 写出椭圆右焦点$F$的坐标及该椭圆的离心率;\\\\ \n(2) 证明: 直线$MN$必过定点, 并求出此定点坐标; \\\\\n(3) 若弦$AB, CD$的斜率均存在, 求$\\triangle FMN$面积的最大值.", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题20", + "edit": [ + "20221214\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "012549": { + "id": "012549", + "content": "设函数$f_1(x)=x^2+a \\mathrm{e}^x$(其中$a$是非零常数, $\\mathrm{e}$是自然对数的底), 记$f_n(x)=f_{n-1}'(x)$($n \\geq 2$, $n \\in \\mathbf{N}$).\\\\\n(1) 求对任意实数$x$, 都有$f_n(x)=f_{n-1}(x)$成立的最小整数$n$的值($n \\geq 2$, $n \\in \\mathbf{N}$);\\\\ \n(2) 设函数$g_n(x)=f_2(x)+f_3(x)+\\cdots+f_n(x)$, 若对任意$n \\geq 3$, $n \\in \\mathbf{N}$, $y=g_n(x)$都存在极值点$x=t_n$, 求证: 点$A_n(t_n, g_n(t_n))$($n \\geq 3$, $n \\in \\mathbf{N}$)在一定直线上, 并求出该直线方程;\\\\\n(3) 是否存在正整数$k$($k \\geq 2$)和实数$x_0$, 使$f_k(x_0)=f_{k-1}(x_0)=0$且对于任意$n \\in \\mathbf{N}$, $n\\ge 1$, $f_n(x)$至多有一个极值点, 若存在, 求出所有满足条件的$k$和$x_0$, 若不存在, 说明理由.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届青浦区一模试题21", + "edit": [ + "20221214\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.",