From 206e8687de8b2ab412931d6af5ca949f0aac2f3e Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Fri, 14 Apr 2023 21:32:23 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A52023=E5=B1=8A=E5=A5=89?= =?UTF-8?q?=E8=B4=A4=E4=BA=8C=E6=A8=A1=E8=AF=95=E9=A2=98,=20=E5=8D=95?= =?UTF-8?q?=E5=85=83=E6=A0=87=E7=AD=BE=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 | 2 +- 工具/文本文件/metadata.txt | 320 +++---------------- 工具/文本文件/批量题目分类号记录.txt | 5 +- 题库0.3/Problems.json | 441 +++++++++++++++++++++++++++ 4 files changed, 488 insertions(+), 280 deletions(-) diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index dc2c45dc..93f34547 100644 --- a/工具/批量收录题目.py +++ b/工具/批量收录题目.py @@ -1,5 +1,5 @@ #修改起始id,出处,文件名 -starting_id = 15164 +starting_id = 15227 raworigin = "" filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目11.tex" editor = "20230414\t王伟叶" diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 2e9058da..43424ec1 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,320 +1,86 @@ -ans +tags -15122 -$\{0,1,2\}$ +15227 +第一单元 -15123 -$1$ -15124 -$\sqrt{5}$ +15228 +第五单元 -15125 -$\dfrac{x^2}{\frac{11}3}+\dfrac{y^2}{\frac{11}2}=1$($\dfrac{3x^2}{11}+\dfrac{2y^2}{11}=1$) -15126 -$-\dfrac 23$ +15229 +第八单元 -15127 -$3$ -15128 -$(0,\dfrac 12]$ +15230 +第六单元 -15129 -$(\dfrac 14,\dfrac{\sqrt{3}}4)$ -15130 -$y=0.186x+11.571$ +15231 +第八单元 -15131 -$\dfrac{\sqrt{2}}2$ -15132 -$95.4$(或者$95.5$也可以) +15232 +第一单元 -15133 -$1+2\lg 2$ -15134 -C +15233 +第八单元 -15135 -B -15136 -A +15234 +第八单元 -15137 -D -15138 -(1) 证明略; (2) $2^{n+2}-4-3n$ +15235 +第七单元 -15139 -(1) $\dfrac 23$; (2) 证明略 -15140 -(1) $2\sqrt{3}$; (2) $\sqrt{2}$ +15236 +第三单元 -15141 -(1) 分布为$\begin{pmatrix}0 & 1 & 2 \\ \dfrac 15 & \dfrac 35 & \dfrac 15\end{pmatrix}$, 期望为$1$; (2) $\chi^2\approx 0.595$, 脑瘤病患在左右侧的部位与习惯在哪一侧接听手机电话无关 -15142 -(1) $2x-y-2\ln 2=0$; (2) $-a-\dfrac 12$; (3) 有且仅有一个零点 +15237 +第五单元 -15143 -$(-1,1)$ +15238 +第二单元 -15144 -$\pi$ -15145 -$2$ +15239 +第七单元 -15146 -$9$ -15147 -$(-\infty,0)\cup [\dfrac 13,+\infty)$ +15240 +第三单元 -15148 -$-\dfrac 17$ -15149 -$36\pi$ +15241 +第九单元 -15150 -$14$ -15151 -$68$ +15242 +第四单元 -15152 -$3$ -15153 -$(-\dfrac 12,0)$ +15243 +第四单元 -15154 -$3\sqrt{5}$ -15155 -A +15244 +第六单元 -15156 -D -15157 -B +15245 +第二单元 -15158 -D -15159 -(1) 证明略; (2) $8\sqrt{3}$ +15246 +第三单元 -15160 -(1) $[1,+\infty)$; (2) $10216$ -15161 -(1) $\dfrac 13$; (2) $1.2$ +15247 +第七单元 -15162 -(1) $2$; (2) $(-1,\dfrac 74)$; (3) 是定值$\sqrt{6}$ -15163 -(1) $(1,+\infty)$; (2) $[-1,\dfrac 32]$; (3) 证明略 - - -15164 -$\{-1,1\}$ - -15165 -$10$ - -15166 -$\dfrac 12$ - -15167 -$18\pi$ - -15168 -$2$ - -15169 -$0.2$ - -15170 -$12$ - -15171 -$\dfrac 14$ - -15172 -$[-2,+\infty)$ - -15173 -$\sqrt{2}$ - -15174 -$[\dfrac 76,\dfrac{15}2]$ - -15175 -$\dfrac{-3\pm \sqrt{6}}6$ - -15176 -D - -15177 -C - -15178 -B - -15179 -C - -15180 -(1) $\dfrac 13$; (2) $8\sqrt{2}$ - -15181 -(1) 证明略; (2) $\dfrac{4\sqrt{21}}{21}$ - -15182 -(1) $0.01998$; (2) 分布为$\begin{pmatrix}0 & 1 & 2 & 3 \\ 0.00000 & 0.00030 & 0.02940 & 0.97030 \end{pmatrix}$, 期望为$2.97$ - - -15183 -(1) 离心率为$\dfrac{\sqrt{5}}2$, 渐近线方程为$y=\pm \dfrac 12 x$; (2) $1$; (3) 证明略 - -15184 -(1) $-4$; (2) 存在切线$l_2$与$l_1$垂直, 理由略; (3) $(\dfrac{3-\sqrt{5}}{2},\dfrac 12]\cup [2,\dfrac{3+\sqrt{5}}{2})$ - -15185 -$\{2,4\}$ - -15186 -$(-\infty,1)$ - -15187 -$0.2$ - -15188 -$(1,+\infty)$ - -15189 -$\dfrac{2\sqrt{2}}3\pi$ - -15190 -$1$ - -15191 -$0.8$ - -15192 -$4$ - -15193 -$3$ - -15194 -$[-1,1]$ - -15195 -$3$ - -15196 -$\dfrac{\sqrt{21}}{14}$ - -15197 -D - -15198 -A - -15199 -C - -15200 -B - -15201 -(1) 证明略; (2) 分布为$\begin{pmatrix} 0 & 1 & 2 \\ \dfrac 3{10} & \dfrac 35 & \dfrac{1}{10}\end{pmatrix}$, 期望为$\dfrac 45$ - -15202 -(1) 证明略; (2) $60^\circ$ - -15203 -(1) $\lambda \approx 29.21$, 估计到$2023$年底该地新能源汽车保有量约$40.3$万辆; (2) $\hat{\lambda}\approx 29.67$, $\hat{r}\approx 0.32$ - -15204 -(1) $\dfrac{x^2}4+\dfrac{y^2}3=1$; (2) $1$; (3) 值恒为$-3$, 证明略 - -15205 -(1) 振幅为$\sqrt{2}$, 周期为$2\pi$, 初相位为$\dfrac{\pi}{4}$; (2) $(\dfrac{\pi}{3},\dfrac{5\pi}3]$; (3) $(0,1)$ - - -15206 -$\{1\}$ - -15207 -$5$ - -15208 -$-1$ - -15209 -$0.94$ - -15210 -$-\dfrac{24}{7}$ - -15211 -$28$ - -15212 -$50$ - -15213 -$\dfrac 13$ - -15214 -$\dfrac 8{11}$ - -15215 -$9$ - -15216 -$\dfrac{\sqrt{2}}2$ - -15217 -$\sqrt{15}a$ - -15218 -B - -15219 -B - -15220 -C - -15221 -B - -15222 -(1) $\dfrac \pi 3$; (2) $\dfrac 32$ - -15223 -(1) 证明略; (2) $\arctan\dfrac{2\sqrt{5}}5$ - -15224 -(1) $43.3$亿元; (2) (i) 全款购车两年后资产总额为$0.0322a$万元, 分期付款购车两年后资产总额为$0.0233a$万元, 应选择全款购车; (ii) $a<21.2134$, 这一措施对购买A, B, C车型有效 - -15225 -(1) $C_1:\dfrac{x^2}2+y^2=1$, $C_2:\dfrac{x^2}2-y^2=1$; (2) $y=\dfrac 12 x$或$y=-\dfrac 12 x$; (3) $2$ - -15226 -(1) $g(x)=f(h(x))$; (2) 导函数为$2(2x)^{2x}(1+\ln (2x))$, 最小值为$(\dfrac 1{\mathrm{e}})^{\frac 1{\mathrm{e}}}$; (3) 证明略 diff --git a/工具/文本文件/批量题目分类号记录.txt b/工具/文本文件/批量题目分类号记录.txt index 1b5dfc39..57c2874e 100644 --- a/工具/文本文件/批量题目分类号记录.txt +++ b/工具/文本文件/批量题目分类号记录.txt @@ -11,7 +11,7 @@ problems_dict = { "第九单元":"014791,014797,014811,014812,014823,014844,015009,015026,015035,015051,015077,015094,015098,015106,015119,015130,015141,015145,015151,015177,015197,015203,015219" } -20230414 2023届高三二模(14区, 缺奉贤徐汇) +20230414 2023届高三二模(15区, 缺徐汇) problems_dict = { "2023届高三杨浦区二模试题":"14784:14804", "2023届高三崇明区二模试题":"14805:14825", @@ -26,7 +26,8 @@ problems_dict = { "2023届普陀区高三二模试题":"15143:15163", "2023届闵行区高三二模试题":"15164:15184", "2023届长宁区高三二模试题":"15185:15205", -"2023届松江区高三二模试题":"15206:15226" +"2023届松江区高三二模试题":"15206:15226", +"2023届奉贤区高三二模试题":"15227:15247", } 20230414 2022学年第二学期高一高二材料收集 diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 4e092928..139f0880 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -374685,6 +374685,447 @@ "remark": "", "space": "12ex" }, + "015227": { + "id": "015227", + "content": "已知集合$A=\\{1,2\\}, B=\\{a, 3\\}$, 若$A \\cap B=\\{2\\}$, 则$a=$\\blank{50}.", + "objs": [], + "tags": [ + "第一单元" + ], + "genre": "填空题", + "ans": "$2$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题1", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015228": { + "id": "015228", + "content": "已知$x \\in \\mathbf{R}$, $y \\in \\mathbf{R}$, 且$x+\\mathrm{i}=y+y \\mathrm{i}, \\mathrm{i}$是虚数单位, 则$x+y=$\\blank{50}.", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "$2$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题2", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015229": { + "id": "015229", + "content": "$(2 x+1)^5$的二项展开式中$x^2$项的系数为\\blank{50}(用数值回答).", + "objs": [], + "tags": [ + "第八单元" + ], + "genre": "填空题", + "ans": "$40$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题3", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015230": { + "id": "015230", + "content": "已知圆柱的上、下底面的中心分别为$O_1$, $O_2$, 过直线$O_1O_2$的平面截该圆柱所得的截面是面积为$8$的正方形, 则该圆柱的侧面积为\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "$8\\pi$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题4", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015231": { + "id": "015231", + "content": "某校高中三年级$1600$名学生参加了区第一次高考模拟统一考试, 已知数学考试成绩量$X$服从正态分布$N(100, \\sigma^2)$(试卷满分为$150$分). 统计结果显示, 数学考试成绩在$80$分到$120$分之间的人数约为总人数的$\\dfrac{3}{4}$, 则此次统考中成绩不低于$120$分的学生人数约为\\blank{50}人.", + "objs": [], + "tags": [ + "第八单元" + ], + "genre": "填空题", + "ans": "$200$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题5", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015232": { + "id": "015232", + "content": "已知两个正数$a, b$的几何平均值为$1$, 则$a^2+b^2$的最小值为\\blank{50}.", + "objs": [], + "tags": [ + "第一单元" + ], + "genre": "填空题", + "ans": "$2$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题6", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015233": { + "id": "015233", + "content": "设某种动物活到$20$岁的概率为$0.8$, 活到$25$岁的概率为$0.4$. 现有一只$20$岁的该种动物, 它活到$25$岁的概率是\\blank{50}.", + "objs": [], + "tags": [ + "第八单元" + ], + "genre": "填空题", + "ans": "$0.5$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题7", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015234": { + "id": "015234", + "content": "已知随机变量$X$的分布为$\\begin{pmatrix}1 & 2 & 3 \\\\ \\dfrac{1}{2} & \\dfrac{1}{3} & \\dfrac{1}{6}\\end{pmatrix}$, 且$Y=a X+3$, 若$E[Y]=-2$, 则实数$a=$\\blank{50}.", + "objs": [], + "tags": [ + "第八单元" + ], + "genre": "填空题", + "ans": "$-3$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题8", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015235": { + "id": "015235", + "content": "设圆$x^2+y^2-2 x-4 y+4=0$与双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$的渐近线相切, 则该双曲线的渐近线方程为\\blank{50}.", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "填空题", + "ans": "$3x\\pm 4y=0$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题9", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015236": { + "id": "015236", + "content": "$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$, 若$\\triangle ABC$的面积为$\\dfrac{a^2+b^2-c^2}{4}$, 则$C=$\\blank{50}.", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "填空题", + "ans": "$\\dfrac\\pi 4$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题10", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015237": { + "id": "015237", + "content": "在集合$\\{1,2,3,4\\}$中任取一个偶数$a$和一个奇数$b$构成一个以原点为起点的向量$\\overrightarrow {\\alpha}=(a, b)$, 从所有得到的以原点为起点的向量中任取两个向量为邻边作平行四边形, 面积不超过$4$的平行四边形的个数是\\blank{50}.", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "$3$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题11", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015238": { + "id": "015238", + "content": "已知$y=f(x)$为$\\mathbf{R}$上的奇函数, 且当$x \\geq 0$时, $f(x)=\\dfrac{x^2}{2}+\\dfrac{25}{4} \\ln (x+1)+\\dfrac{12}{\\pi} \\cos \\dfrac{\\pi}{3} x+a$, 则$y=f(x)$的驻点为\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "$\\pm \\dfrac 32$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题12", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015239": { + "id": "015239", + "content": "``$a=2$''是``直线$y=-a x+2$与直线$y=\\dfrac{a}{4} x-1$垂直''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": [ + "第七单元" + ], + "genre": "选择题", + "ans": "A", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题13", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015240": { + "id": "015240", + "content": "下列函数中, 以$\\pi$为周期且在区间$(\\dfrac{\\pi}{2}, \\pi)$上是严格增函数的是\\bracket{20}.\n\\fourch{$f(x)=|\\cos 2 x|$}{$f(x)=|\\sin 2 x|$}{$f(x)=|\\cos x|$}{$f(x)=|\\sin x|$}", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "选择题", + "ans": "C", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题14", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015241": { + "id": "015241", + "content": "某校一个课外学习小组为研究某作物种子的发芽率$y$和温度$x$(单位: ${ }^{\\circ} \\text{C}$)的关系, 在$20$个不同的温度条件下进行种子发芽实验, 由实验数据$(x_i, y_i)$($i=1,2, \\cdots, 20$)得到下面的散点图:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.2, yscale = 0.5]\n\\foreach \\i/\\j in {24.584/4.712,34.971/5.412,28.833/5.164,17.519/4.140,26.540/4.970,33.338/5.315,23.496/4.898,10.435/1.571,10.260/1.424,30.130/5.160,22.395/4.805,23.658/4.630,22.717/4.700,27.019/5.071,16.696/3.939,19.019/4.382,31.128/5.205,19.833/4.356,12.683/3.085,20.028/4.331}\n{\\filldraw (\\i,\\j) circle (0.15 and 0.06);};\n\\foreach \\i in {0,10,20,30,40}\n{\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$} coordinate (\\i);};\n\\foreach \\i in {20,40,60,80,100}\n{\\draw (40,{\\i/16}) -- (0,{\\i/16}) node [left] {$\\i\\%$};};\n\\draw (42,0) node [below right] {温度/${}^\\circ\\text{C}$} (40,0) -- (0,0) node [left] {$0$} -- (0,{100/16});\n\\draw (-6,3) node [rotate = 90] {发芽率};\n\\end{tikzpicture}\n\\end{center}\n由此散点图, 在$10^{\\circ} \\text{C}$至$40^{\\circ} \\text{C}$之间, 下面四个回归方程类型中最适合作为发芽率$y$和温度$x$的回归方程类型的是\\bracket{20}.\n\\fourch{$y=a+b x$}{$y=a+b x^2$}{$y=a+b \\mathrm{e}^x$}{$y=a+b \\ln x$}", + "objs": [], + "tags": [ + "第九单元" + ], + "genre": "选择题", + "ans": "D", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题15", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015242": { + "id": "015242", + "content": "设$S_n$是一个无穷数列$\\{a_n\\}$的前$n$项和, 若一个数列满足对任意的正整数$n$, 不等式$\\dfrac{S_n}{n}<\\dfrac{S_{n+1}}{n+1}$恒成立, 则称数列$\\{a_n\\}$为和谐数列, 有下列$3$个命题:\\\\\n\\textcircled{1} 若对任意的正整数$n$均有$a_n=latex]\n\\draw (0,0,-1) node [below] {$D$} coordinate (D);\n\\draw (0,0,1) node [below] {$A$} coordinate (A);\n\\draw (2,0,1) node [below] {$B$} coordinate (B);\n\\draw (2,0,-1) node [right] {$C$} coordinate (C);\n\\draw (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$PAB \\perp$平面$PAD$;\\\\\n(2) 若$PA=PD=AB=DC, \\angle APD=90^{\\circ}$, 且四棱锥$P-ABCD$的体积为$\\dfrac{8}{3}$, 求$PB$与平面$ABCD$所成的线面角的大小.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "(1) 证明略; (2) $30^\\circ$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题18", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015245": { + "id": "015245", + "content": "设函数$y=f(x)$的定义域是$\\mathrm{R}$, 它的导数是$f'(x)$. 若存在常数$m$($m \\in \\mathbf{R}$), 使得$f(x+m)=-f'(x)$对一切$x$恒成立, 那么称函数$y=f(x)$具有性质$P(m)$.\\\\\n(1) 求证: 函数$y=\\mathrm{e}^x$(其中$\\mathrm{e}$为自然对数的底数)不具有性质$P(m)$;\\\\\n(2) 判别函数$y=\\sin x$是否具有性质$P(m)$. 若具有求出$m$的取值集合; 若不具有请说明理由.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "解答题", + "ans": "(1) 证明略; (2) 当且仅当$m=2k\\pi-\\dfrac\\pi 2$, $k\\in \\mathbf{Z}$时, $y=\\sin x$具有性质$P(m)$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题19", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015246": { + "id": "015246", + "content": "某小区有块绿地, 绿地的平面图大致如下图所示, 并铺设了部分人行通道. 小区物业根据居民需求, 决定在绿地修建一个休息亭.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.8]\n\\draw (0,0) coordinate (O);\n\\draw (1,0) coordinate (A);\n\\draw (2,0) coordinate (B);\n\\draw (2,{sqrt(3)}) coordinate (C);\n\\draw (-1,{sqrt(3)}) coordinate (D);\n\\draw (-0.5,{sqrt(3)/2}) coordinate (E);\n\\fill [pattern = north east lines] (A) arc (0:120:1) -- (D)--(C)--(B)--cycle;\n\\draw [dashed] (A) arc (0:120:1) -- (O) -- cycle;\n\\draw (A)--++ (0,-0.1) --++ (1.1,0) --++ (0,{sqrt(3)+0.2}) --++ ({-3.1-0.1*sqrt(3)},0) --++ (-60:{1+0.1*sqrt(3)}) coordinate (E1) -- (E);\n\\draw [dashed] (A) ++ (0,-0.1) --++ ({-1-0.1/sqrt(3)},0) -- (E1);\n\\draw (A)--(B)--(C)--(D)--(E);\n\\end{tikzpicture}\n\\end{center}\n为了简单起见, 现作如下假设:\n假设\\textcircled{1} 绿地是由线段$AB, BC, CD, DE$和弧$\\overset\\frown{EA}$围成的, 其中$\\overset\\frown{EA}$是以$O$点为圆心, 圆心角为$\\dfrac{2 \\pi}{3}$的扇形的弧, 见图 1;\\\\\n假设\\textcircled{2} 线段$AB, BC, CD, DE$所在的路行人是可通行的, 圆弧$\\overset\\frown{EA}$暂时未修路;\\\\ \n假设\\textcircled{3} 路的宽度在这里暂时不考虑;\\\\\n假设\\textcircled{4} 路用线段或圆弧表示, 休息亭用点表示.\n图1至图3中的相关边、角满足以下条件: 直线$BA$与$DE$的交点是$O$, $AB\\parallel CD$, $\\angle ABC=\\dfrac{\\pi}{2}$. $DE=EO=OA=AB=200$米. 根据不同的设计方案解决相应问题, 结果精确到米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,{sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (-1,{sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (-0.5,{sqrt(3)/2}) node [below left] {$E$} coordinate (E);\n\\draw [dashed] (A) arc (0:120:1) -- (O) -- cycle;\n\\draw (O) node [above right] {\\tiny$\\dfrac{2\\pi}3$};\n\\path (O) -- (A) node [midway, below] {\\tiny$200$};\n\\path (A) -- (B) node [midway, below] {\\tiny$200$};\n\\path (E) -- (D) node [midway, sloped, below] {\\tiny$200$};\n\\draw (A)--(B)--(C)--(D)--(E);\n\\draw (A) node [below = 0.4] {图1};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,{sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (-1,{sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (-0.5,{sqrt(3)/2}) node [below left] {$E$} coordinate (E);\n\\draw [dashed] (A) arc (0:120:1) -- (O) -- cycle;\n\\draw [very thick] (A) arc (0:120:1);\n\\draw [very thick] (60:1) node [below] {$Q$} coordinate (Q) -- (C);\n\\draw (O) node [above right] {\\tiny$\\dfrac{2\\pi}3$};\n\\path (O) -- (A) node [midway, below] {\\tiny$200$};\n\\path (A) -- (B) node [midway, below] {\\tiny$200$};\n\\path (E) -- (D) node [midway, sloped, below] {\\tiny$200$};\n\\draw (A)--(B)--(C)--(D)--(E);\n\\draw (A) node [below = 0.4] {图2};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,{sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (-1,{sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (-0.5,{sqrt(3)/2}) node [below left] {$E$} coordinate (E);\n\\draw [dashed] (A) arc (0:120:1) -- (O) -- cycle;\n\\draw [very thick] (75:1) node [below] {$P$} coordinate (P) arc (75:120:1);\n\\draw [very thick] (P) -- ($(B)!(P)!(C)$) node [right] {$M$} coordinate (M);\n\\draw [very thick] (P) -- ($(C)!(P)!(D)$) node [above] {$N$} coordinate (N);\n\\draw (O) node [above right] {\\tiny$\\dfrac{2\\pi}3$};\n\\path (O) -- (A) node [midway, below] {\\tiny$200$};\n\\path (A) -- (B) node [midway, below] {\\tiny$200$};\n\\path (E) -- (D) node [midway, sloped, below] {\\tiny$200$};\n\\draw (A)--(B)--(C)--(D)--(E);\n\\draw (A) node [below = 0.4] {图3};\n\\end{tikzpicture}\n\\end{center}\n(1) 假设休息亭建在弧$\\overset\\frown{EA}$的中点, 记为$Q$, 沿$\\overset\\frown{EA}$和线段$QC$修路, 如图2所示. 求$QC$的长;\\\\\n(2) 假设休息亭建在弧$\\overset\\frown{EA}$上的某个位置, 记为$P$, 作$PM \\perp BC$交$BC$于$M$, 作$PN \\perp CD$交$DC$于$N$. 沿$\\overset\\frown{EP}$、线段$PM$和线段$PN$修路, 如图3所示. 求修建的总路长$\\overset\\frown{EP}+PM+PN$的最小值;\\\\\n(3) 请你对(1)和(2)涉及到的两种设计方案做个简明扼要的评价.", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "解答题", + "ans": "(1) 约$346$米($200\\sqrt{3}$米); (2) 约$651$米($\\angle AOP=\\dfrac\\pi 2$); (3) 方案1涉及到的设计方案总路径是$\\dfrac{400\\pi}{3}+200 \\sqrt{3} \\approx 765$米, 比起方案2显然不是最优(短)路径; 方案2涉及到的设计方案显然相对于方案1是相对不便捷(不利于$AB$段附近居民前往)等等. 说明: 可以从多个角度考虑, 但以下两个指标是主要的衡量指标: 一修的路相对短, 二修的路相对便于居民出行, 若学生自己有一个评价标准, 并根据自己的标准并给予自圆其说适当给予评分\n譬如说: 有的同学直接连接$OC$, 休息亭建立在$OC$与$\\overset\\frown{EA}$的交点处, 将$\\overset\\frown{EA}$与$QC$全部修好, 这样更短($\\dfrac{400}{3} \\pi+200 \\approx 619$米)也相对便捷等等", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题20", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015247": { + "id": "015247", + "content": "已知椭圆$C: \\dfrac{x^2}{4}+\\dfrac{y^2}{b^2}=1$($b>0$), $A(0, b)$, $B(0,-b)$, 椭圆$C$内部的一点$T(t, \\dfrac{1}{2})$($t>0$). 过点$T$作直线$AT$交椭圆于$M$, 作直线$BT$交椭圆于$N$, $M$、$N$是不同的两点.\\\\\n(1) 若椭圆$C$的离心率是$\\dfrac{\\sqrt{3}}{2}$, 求$b$的值;\\\\\n(2) 设$\\triangle BTM$的面积是$S_1$, $\\triangle ATN$的面积是$S_2$, 若$\\dfrac{S_1}{S_2}=5$, 当$b=1$时, 求$t$的值;\\\\\n(3) 若点$U(x_u, y_u), V(x_v, y_v)$满足$x_uy_v$, 则称点$U$在点$V$的左上方. 求证: 当$b>\\dfrac{1}{2}$时, 点$N$在点$M$的左上方.", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "解答题", + "ans": "(1) $1$或$4$; (2) $1$; (3) 证明略", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届奉贤区高三二模试题21", + "edit": [ + "20230414\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.",