From 330307cd3d209a80c61ccbf820f5d1b20f8b3a02 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Mon, 10 Apr 2023 22:11:18 +0800 Subject: [PATCH] =?UTF-8?q?=E6=94=B6=E5=BD=95=E5=98=89=E5=AE=9A=E4=BA=8C?= =?UTF-8?q?=E6=A8=A1=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 | 443 ++++--------------------------------- 题库0.3/Problems.json | 399 +++++++++++++++++++++++++++++++++ 3 files changed, 448 insertions(+), 398 deletions(-) diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index bdd2ff4d..c7fd3626 100644 --- a/工具/批量收录题目.py +++ b/工具/批量收录题目.py @@ -1,9 +1,9 @@ #修改起始id,出处,文件名 -starting_id = 40570 +starting_id = 14826 raworigin = "" filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目9.tex" editor = "202304010\t王伟叶" -indexed = False +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 0d7fdc60..cb1bf7b1 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,418 +1,69 @@ ans -021441 -错误, 正确, 错误, 错误 - - -021442 -D - - -021443 -C - - -021444 -A - - -021445 -C - - -021446 -D - - -021447 -$-390^\circ$ - - -021448 -$304^\circ$, $-56^\circ$ - - -021449 -$-144^\circ$ - - -021450 -二, 四 - - -021451 -(1) $\{\alpha|\alpha=60^\circ+k\cdot 360^\circ, \ k\in \mathbf{Z}\}$, $-300^\circ$, $60^\circ$, $420^\circ$; (2) $\{\alpha|\alpha = -21^\circ+k\cdot 360^\circ, \ k \in \mathbf{Z}\}$, $-21^\circ$, $339^\circ$, $699^\circ$ - - -021452 -\begin{tikzpicture}[>=latex] -\fill [pattern = north east lines] (30:2) arc (30:60:2) -- (0,0) -- cycle; -\draw (30:2) -- (0,0) -- (60:2); -\draw [->] (-2,0) -- (2,0) node [below] {$x$}; -\draw [->] (0,-2) -- (0,2) node [left] {$y$}; -\draw (0,0) node [below left] {$O$}; -\end{tikzpicture} - - -021453 -$-1290^{\circ}$;第二象限 - - -021454 -(1) $ \{\alpha|\alpha=45^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(2) $\{\alpha|\alpha=135^{\circ}+k\cdot 180^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(3) $\{\alpha|\alpha=45^{\circ}+k\cdot 90^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(4) $\{\alpha|180^{\circ}+k\cdot 360^{\circ}<\alpha<270^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$. - - -021455 -(1) $ \{\beta|\beta=\alpha+180^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(2) $\{\beta|\beta=\alpha+90^{\circ}+k\cdot 180^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(3) $\{\beta|\beta=-\alpha+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(4) $\{\beta|\beta=90^{\circ}-\alpha+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$. - - -021456 -C - - -021457 -B - - -021458 -$\dfrac{\pi}{12}$; $\dfrac{7\pi}{12}$; $\dfrac{5\pi}{4}$; $300^{\circ}$; $324^{\circ}$; $315^{\circ}$; $(\dfrac{270}{\pi})^{\circ}$ - - -021459 -(1)$\frac{50\pi+180}{9}$;(2)$\frac{250\pi}{9}$ - - -021460 -$\sqrt{3}$ - - -021461 -(1)$\frac{\pi}{3}$;(2)$\frac{2\pi}{3}$ - - -021462 -(1)$16\pi+\frac{2\pi}{3}$,二;\\ -(2)$-18\pi+\frac{4\pi}{3}$,三;\\ -(3)$-2\pi+\frac{7\pi}{5}$,三;\\ -(4)$-2\pi+\frac{3\pi}{4}$,二. - - -021463 -$\frac{1}{2}$ - - -021464 -(1) $\{\alpha|-\frac{\pi}{2}+2k\pi<\alpha<2k\pi,\ k \in \mathbf{Z}\}$;\\ -(2) $\{\alpha|\alpha=\frac{k\pi}{2},\ k \in \mathbf{Z}\}$. - - -021465 -(1) $\beta=\alpha+2k\pi,\ k \in \mathbf{Z}$;\\ -(2) $\beta=-\alpha+2k\pi,\ k \in \mathbf{Z}$;\\ -(3) $\beta=-\alpha+\pi+2k\pi,\ k \in \mathbf{Z}$;\\ -(4) $\beta=\alpha+\pi+2k\pi,\ k \in \mathbf{Z}$. - - -021466 -(1) $\{\alpha|-\frac{\pi}{4}+2k\pi \le \alpha \le \frac{\pi}{2}+2k\pi,\ k \in \mathbf{Z}\}$;\\ -(2) $\{\alpha|\frac{\pi}{6}+k\pi \le \alpha \le \frac{5\pi}{6}+k\pi,\ k \in \mathbf{Z}\}$. - - -021467 -(1) 第四象限；第四象限；\\ -(2) 第二象限或者第四象限；第一象限或第二象限或者$y$轴正半轴. - - -021468 -$A\cap B=\{\alpha | 2k \pi+\dfrac{5\pi}{6}<\alpha<2k \pi+\dfrac{7\pi}{6},\ k \in \mathbf{Z} \}$ - - -021469 -\begin{tabular}{|c|c|c|c|c|c|} -\hline &$P(-5,12)$&$P(0,-6)$&$P(6,0)$&$P(-9,-12)$&$P(1,-\sqrt{3})$\\ -\hline$\sin \alpha$&$\dfrac{12}{13}$ &$-1$ & $0$&$-\dfrac{4}{5}$ &$-\dfrac{\sqrt{3}}2$ \\ -\hline$\cos \alpha$&$-\dfrac{5}{13}$ &$0$ & $1$&$-\dfrac{3}{5}$ &$\dfrac 12$ \\ -\hline$\tan \alpha$&$-\dfrac{12}{5}$ &不存在 & $0$&$\dfrac{4}{3}$ &$-\sqrt{3}$ \\ -\hline$\cot \alpha$&$-\dfrac{5}{12}$ &$0$ & 不存在 &$\dfrac {3}{4}$ &$-\dfrac{\sqrt{3}}3$ \\ -\hline -\end{tabular} - - -040018 -(1) $\dfrac{\pi}{4}$; (2) $\dfrac{\pi}{6}$; (3) $\dfrac{\pi}{10}$; (4) $\dfrac{\pi}{3}$; (5) $\dfrac{5\pi}{12}$; (6) $\dfrac{\pi}{15}$ - - -040019 -(1) $60^{\circ}$; (2) $36^{\circ}$; (3) $45^{\circ}$; (4) $75^{\circ}$; (5) $40^{\circ}$; (6) $54^{\circ}$ - - -040020 -(1) $2k\pi+\dfrac{\pi}{2}$; (2) $2k\pi+\dfrac{3\pi}{2}$; (3) $2k\pi+\dfrac{7\pi}{6}$; (4) $k\pi+\dfrac{\pi}{4}$; (5) $\dfrac{k\pi}{2}+\dfrac{\pi}{6}$ - - -040021 -(1) $k \times 360^{\circ}+60^{\circ}$;\\ -(2) $k \times 360^{\circ}+330^{\circ}$; \\ -(3) $k \times 360^{\circ}-210^{\circ}$; \\ -(4) $k \times 180^{\circ}-45^{\circ}$; \\ -(5) $k \times 90^{\circ}+50^{\circ}$ - - -040022 -(1) $330^{\circ}$; (2) $240^{\circ}$; (3) $210^{\circ}$; (4) $300^{\circ}$ - - -040023 -(1) $\dfrac{4\pi}{3}$; (2) $\dfrac{11\pi}{6}$; (3) $10-2\pi$; (4) $-10+4\pi$ - - -040024 -$18$ - - -040025 -$3$,$-2$ - - -040026 -(1) $1037$; (2) $-4k+53$; (3) $500$ - - -040027 -$-2n+10$ - - -040028 -15 - - -040029 -$7$ - - -040030 -$(4,\dfrac{14}{3}]$ - - -040031 -$2n-1$ - - -040032 -$(3,\dfrac{35}{9})$或$(\dfrac{35}{9},3)$ - - -040033 -$200$ - - -040034 -略 - - -040035 -$a_n=\begin{cases}1, & n=1,\\ 2n, & n=2k, \\ 2n-2, & n=2k+1\end{cases}$($k\in \mathbf{N}$, $k\ge 1$) - - -040036 -$6n-3$ - - -040057 -$\dfrac{19}{28}\sqrt{7}$ - - -040058 -$\dfrac{79}{156}$ - - -040059 -$2$ - - -040060 -$-\dfrac{\sqrt{1-m^2}}{m}$ - - -040061 -$-\dfrac{1}{5}, \dfrac{1}{5}$ - - -040062 -$-\dfrac{1}{3}, 3$ - - -040063 -$\dfrac{1}{2}, -2$ - - -040064 -$\dfrac{\sqrt{6}}{3}$ - - -040065 -$\dfrac{1}{3}, -\dfrac{9}{4}$ - - -040066 -$\dfrac{1}{3}, \dfrac{7}{9}$ - - -040067 -$\pm\dfrac{\sqrt{2}}{3}$ - - -040068 -$\dfrac{1}{4}, \dfrac{2}{5}$ - - -040069 -$\dfrac{1-\sqrt{17}}{4}$ - - -040070 -(1) 三； (2) 三 - - -040071 -(1) $[-\dfrac{1}{2},\dfrac{1}{2})\cup\{1\}$; (2) $[-\dfrac{\pi}{3},\dfrac{\pi}{3})$; (3) $\{-\dfrac{1}{2}\}$ - - -040072 -(1) $-\tan \alpha-\cot \alpha$; (2) $-\dfrac{\sqrt{2}}{\sin \alpha}$; (3) $-1$; (4) $0$ - - -040073 -略 - - -040074 -$-\dfrac{10}{9}$ - - -040075 -$a_n=\dfrac{1}{3n-2}$ - - -040076 -$a_n=\dfrac{1}{n}$ - - -040077 -$(n-\dfrac{4}{5})5^n$ - - -040078 -$2^{n+1}-3$ - - -040079 -$1078$ - - -040080 -$S_n=\begin{cases}\dfrac{n^2}{2}+n-\dfrac 23+\dfrac 23\cdot 2^n, & n\text{为偶数},\\ \dfrac{n^2}{2}-\dfrac 76+\dfrac 23\cdot 2^{n+1}, & n\text{为奇数} \end{cases}$ - - -040081 -(1) 略； (2) $n^2$ - - -040082 -(1) 不存在； (2) 存在, 如$c_n=2^{n-1}$ - - -040083 -$\dfrac{\sqrt{3}}{2}$ - - -040084 -$0$ - - -040085 -$\{0,-2\pi\}$ - - -040086 -$-\dfrac{\pi}6,\dfrac 56\pi$ - - -040087 -$\cot \alpha$ - - -040088 -$7+4\sqrt{3}$ - - -040089 -$\dfrac{\sqrt{2}-\sqrt{6}}{4}$ - - -040090 -$\dfrac{\sqrt{3}+\sqrt{35}}{12}$ - - -040091 -$\dfrac 12$ - - -040092 +14826 $5$ +14827 +$\dfrac 43$ -040093 -$-\dfrac 12$ +14828 +$\{1\}$ +14829 +$\pi$ -040094 -$\dfrac{\pi}{12}$ +14830 +$\dfrac 14$ +14831 +$1$ -040095 -$\{x|x=\pm\frac 23 \pi+2k\pi,k \in \mathbf{Z}\}$ +14832 +$3$ +14833 +$\dfrac 52$ -040096 -$\dfrac 43 \pi$ +14834 +$2\pi$ +14835 +$0.9$ -040097 -$\textcircled{4}$ +14836 +$2\sqrt{2}$ +14837 +$(0,4)$ -040098 +14838 +B + +14839 +B + +14840 C +14841 +D -040099 -$\dfrac{-2\sqrt{2}-\sqrt{3}}6$ +14842 +(1) 相交; (2) $5\sqrt{5}+8$ +14843 +(1) $f(x)=\dfrac{\sqrt{2}}2\sin (2x+\dfrac\pi 4)+\dfrac 12$, 最大值为$\dfrac{1+\sqrt{2}}2$, 当且仅当$x=\dfrac\pi 8+k\pi$, $k\in \mathbf{Z}$时取得; (2) $A=\dfrac\pi 4$, $B=\dfrac\pi 3$, $AC=\sqrt{6}$ -040100 -$-\dfrac 7{25}$ - - -040101 -$-\dfrac {\pi}3$ - - -040102 -$(-\dfrac {12}{13}, \dfrac{5}{13})$ - - -040103 -$(\dfrac {5-12\sqrt{3}}{2}, \dfrac{12-5\sqrt{3}}{2})$ - - -040104 -略 +14844 +(1) 中位数$M=42.5$, 列联表如下: \begin{tabular}{|c|c|c|} +\hline & 超过$M$& 不超过$M$\\ +\hline 上班时间 & 10 & 10 \\ +\hline 下班时间 & 11 & 9\\ +\hline +\end{tabular}; (2) $\chi^2=0.1$, 无显著差异 +14845 +(1) $P(4a^{\frac 13},4a^{\frac 23})$; (2) $1$; (3) $2\sqrt{2}$或$\dfrac{\sqrt{2}}4$ +14846 +(1) 证明略 (2) $(\pi,\pi+3\sqrt{3}]$; (3) 证明略, 反之不一定成立, 如取$a_n$是常数$a$, 满足$a+2\sin a=\pi$(这样的$a$有三个) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index df81f716..ed6d48d9 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -365575,6 +365575,405 @@ "remark": "", "space": "12ex" }, + "014826": { + "id": "014826", + "content": "已知复数$z=3+4 \\mathrm{i}$, 其中$\\mathrm{i}$是虚数单位, 则$|z|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$5$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题1", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014827": { + "id": "014827", + "content": "双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{7}=1$的离心率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$\\dfrac 43$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题2", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014828": { + "id": "014828", + "content": "已知$A=\\{x | \\dfrac{x-1}{x} \\leq 0\\}$, $B=\\{x | x \\geq 1\\}$, 则$A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$\\{1\\}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题3", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014829": { + "id": "014829", + "content": "函数$y=\\sin 2 x$的最小正周期为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$\\pi$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题4", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014830": { + "id": "014830", + "content": "$\\triangle ABC$是边长为$1$的等边三角形, 点$M$为边$AB$的中点, 则$\\overrightarrow{AC} \\cdot \\overrightarrow{AM}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$\\dfrac 14$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题5", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014831": { + "id": "014831", + "content": "已知函数$y=2 x+\\dfrac{1}{8 x}$, 定义域为$(0,+\\infty)$, 则该函数的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$1$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题6", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014832": { + "id": "014832", + "content": "已知$n \\in \\mathbf{N}$, 若$\\mathrm{C}_6^n=\\mathrm{P}_5^2$, 则$n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$3$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题7", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014833": { + "id": "014833", + "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=\\begin{cases}2 n, & n=1, \\\\ 2^{-n}, & n \\geq 2,\\end{cases}$ 前$n$项和为$S_n$, 则$\\displaystyle\\lim _{n \\to+\\infty} S_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$\\dfrac 52$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题8", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014834": { + "id": "014834", + "content": "已知四棱锥$P-ABCD$的底面是边长为$\\sqrt{2}$的正方形, 侧棱长均为$\\sqrt{5}$. 若点$A$、$B$、$C$、$D$在圆柱的一个底面圆周上, 点$P$在圆柱的另一个底面内, 则该圆柱的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$2\\pi$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题9", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014835": { + "id": "014835", + "content": "已知某产品的一类部件由供应商$A$和$B$提供, 占比分别为$\\dfrac{1}{3}$和$\\dfrac{2}{3}$, 供应商$A$提供的部件的良品率为$0.96$, 若该部件的总体良品率为$0.92$, 则供应商$B$提供的部件的良品率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$0.9$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题10", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014836": { + "id": "014836", + "content": "如图, 线段$AB$的长为$8$, 点$C$在线段$AB$上, $AC=2$. 点$P$为线段$CB$上任意一点, 点$A$绕着点$C$顺时针旋转, 点$B$绕着点$P$逆时针旋转. 若它们恰重合于点$D$, 则$\\triangle CDP$的面积的最大值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (8,0) node [right] {$B$} coordinate (B);\n\\draw (5.5,0) node [below] {$P$} coordinate (P);\n\\draw ({24/7},{4*sqrt(6)/7}) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)(C)--(D)--(P);\n\\draw [dashed] (A) arc (180:{atan(2*sqrt(6)/5)}:2);\n\\draw [dashed] (B) arc (0:{180-atan(8*sqrt(6)/29)}:2.5);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$2\\sqrt{2}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题11", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014837": { + "id": "014837", + "content": "若关于$x$的函数$y=\\dfrac{x^3+a}{\\mathrm{e}^x}$在$\\mathbf{R}$上存在极小值($\\mathrm{e}$为自然对数的底数), 则实数$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$(0,4)$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题12", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014838": { + "id": "014838", + "content": "设$a \\in \\mathbf{R}$, 则``$a<1$''是``$a^2=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{{sqrt(5)}}\n\\draw (0,0,0) node [below left] {$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$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(C1)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (A)--(E)--(D1)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 判断直线$AE$与$D_1F$的关系, 并说明理由;\\\\\n(2) 若直线$D_1E$与底面$ABCD$所成角为$\\dfrac{\\pi}{4}$, 求四棱柱$ABCD-A_1B_1C_1D_1$的全面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) 相交; (2) $5\\sqrt{5}+8$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题17", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014843": { + "id": "014843", + "content": "已知向量$\\overrightarrow {a}=(\\sin x, 1+\\cos 2 x)$, $\\overrightarrow {b}=(\\cos x, \\dfrac{1}{2})$, $f(x)=\\overrightarrow {a} \\cdot \\overrightarrow {b}$.\\\\\n(1) 求函数$y=f(x)$的最大值及相应$x$的值;\\\\\n(2) 在$\\triangle ABC$中, 角$A$为锐角, 且$A+B=\\dfrac{7 \\pi}{12}$, $f(A)=1$, $BC=2$, 求边$AC$的长.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) $f(x)=\\dfrac{\\sqrt{2}}2\\sin (2x+\\dfrac\\pi 4)+\\dfrac 12$, 最大值为$\\dfrac{1+\\sqrt{2}}2$, 当且仅当$x=\\dfrac\\pi 8+k\\pi$, $k\\in \\mathbf{Z}$时取得; (2) $A=\\dfrac\\pi 4$, $B=\\dfrac\\pi 3$, $AC=\\sqrt{6}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题18", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014844": { + "id": "014844", + "content": "李先生是一名上班族, 为了比较上下班的通勤时间, 记录了$20$天个工作日内, 家里到单位的上班时间以及同路线返程的下班时间(单位: 分钟), 如下茎叶图显示两类时间的共$40$个记录:\n\\begin{center}\n\\begin{tabular}{cccccccccccc|c|ccccccccccc}\n\\multicolumn{12}{r|}{上班时间} & & \\multicolumn{11}{l}{下班时间} \\\\\n& & & & & & & & 9 & 8 & 8 & 7 & 3 & 6 & 7 & 8 & 8 & 8 & 9 \\\\\n6 & 5 & 4 & 4 & 3 & 3 & 2 & 2 & 2 & 1 & 1 & 0 & 4 & 0 & 0 & 1 & 3 & 3 & 3 & 3 & 4 & 4 & 5 & 5 \\\\\n& & & & & & & & 4 & 2 & 2 & 1 & 5 & 1 & 7\\\\\n& & & & & & & & & & & & 6 & 4\n\\end{tabular}\n\\end{center}\n(1) 求出这$40$个通勤记录的中位数$M$, 并完成下列$2 \\times 2$列联表:\n\\begin{center}\n\\begin{tabular}{|l|l|l|}\n\\hline & 超过$M$& 不超过$M$\\\\\n\\hline 上班时间 & & \\\\\n\\hline 下班时间 & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(2) 根据列联表中的数据, 请问上下班的通勤时间是否有显著差异? 并说明理由.\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, $n=a+b+c+d$, $P(\\chi^2 \\geq 3.841) \\approx 0.05$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) 中位数$M=42.5$, 列联表如下: \\begin{tabular}{|c|c|c|}\n\\hline & 超过$M$& 不超过$M$\\\\\n\\hline 上班时间 & 10 & 10 \\\\\n\\hline 下班时间 & 11 & 9\\\\\n\\hline\n\\end{tabular}; (2) $\\chi^2=0.1$, 无显著差异", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题19", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014845": { + "id": "014845", + "content": "若直线和抛物线的对称轴不平行且与抛物线只有一个公共点, 则称该直线是抛物线在该点处的切线, 该公共点为切点. 已知抛物线$C_1: y^2=4 a x$和$C_2: x^2=4 y$, 其中$a>0$. $C_1$与$C_2$在第一象限内的交点为$P$. $C_1$与$C_2$在点$P$处的切线分别为$l_1$和$l_2$, 定义$l_1$和$l_2$的夹角为曲线$C_1$、$C_2$的夹角.\\\\\n(1) 求点$P$的坐标;\\\\\n(2) 若$C_1$、$C_2$的夹角为$\\arctan \\dfrac{3}{4}$, 求$a$的值;\\\\\n(3) 若直线$l_3$既是$C_1$也是$C_2$的切线, 切点分别为$Q$、$R$, 当$\\triangle PQR$为直角三角形时, 求出相应的$a$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) $P(4a^{\\frac 13},4a^{\\frac 23})$; (2) $1$; (3) $2\\sqrt{2}$或$\\dfrac{\\sqrt{2}}4$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题20", + "edit": [ + "202304010\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014846": { + "id": "014846", + "content": "已知$f(x)=x+2 \\sin x$, 等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 记$T_n=\\displaystyle\\sum_{i=1}^n f(a_1)$.\\\\\n(1) 求证: 函数$y=f(x)$的图像关于点$(\\pi, \\pi)$中心对称;\\\\\n(2) 若$a_1$、$a_2$、$a_3$是某三角形的三个内角, 求$T_3$的取值范围;\\\\\n(3) 若$S_{100}=100 \\pi$, 求证: $T_{100}=100 \\pi$. 反之是否成立? 并请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) 证明略 (2) $(\\pi,\\pi+3\\sqrt{3}]$; (3) 证明略, 反之不一定成立, 如取$a_n$是常数$a$, 满足$a+2\\sin a=\\pi$(这样的$a$有三个)", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届高三嘉定区二模试题21", + "edit": [ + "202304010\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.",