From 3d4ed6861557bc553247b838793409b54e73e451 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Thu, 13 Apr 2023 23:15:50 +0800 Subject: [PATCH] =?UTF-8?q?=E6=94=B6=E5=BD=95=E8=99=B9=E5=8F=A3=E9=9D=99?= =?UTF-8?q?=E5=AE=89=E6=99=AE=E9=99=802023=E5=B1=8A=E9=AB=98=E4=B8=89?= =?UTF-8?q?=E4=BA=8C=E6=A8=A1=E8=AF=95=E9=A2=98?= 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 +- 题库0.3/Problems.json | 1197 +++++++++++++++++++++++++++++++++++++++++ 2 files changed, 1198 insertions(+), 1 deletion(-) diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index d556a77e..5d4eed94 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 = 15080 +starting_id = 15101 raworigin = "" filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目11.tex" editor = "202304012\t王伟叶" diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index f8eeee1e..f0cab26d 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -370801,6 +370801,1203 @@ "remark": "", "space": "12ex" }, + "015101": { + "id": "015101", + "content": "已知集合$A=\\{x |-20$时, $f(x)=2^x+\\dfrac{9}{2^x+1}$, 则该函数的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题8", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015109": { + "id": "015109", + "content": "端午节吃粽子是我国的传统习俗. 一盘中放有$10$个外观完全相同的粽子, 其中豆沙粽$3$个, 肉粽$3$个, 白米粽$4$个, 现从盘子任意取出$3$个, 则取到白米粽的个数的数学期望为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题9", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015110": { + "id": "015110", + "content": "已知$A, B$是球$O$的球面上两点, $\\angle AOB=60^{\\circ}$, $P$为该球面上的动点, 若三棱锥$P-OAB$体积的最大值为$6$, 则球$O$的表面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题10", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015111": { + "id": "015111", + "content": "过原点的直线$l$与双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a, b>0$)的左、右两支分别交于$M, N$两点, $F(2,0)$为$C$的右焦点, 若$\\overrightarrow{FM} \\cdot \\overrightarrow{FN}=0$, 且$|\\overrightarrow{FM}|+|\\overrightarrow{FN}|=2 \\sqrt{5}$, 则双曲线$C$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题11", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015112": { + "id": "015112", + "content": "已知平面向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}, \\overrightarrow {e}$满足$|\\overrightarrow {a}|=3$, $|\\overrightarrow {e}|=1$, $|\\overrightarrow {b}-\\overrightarrow {a}|=1$, $\\langle\\overrightarrow {a}, \\overrightarrow {e}\\rangle=\\dfrac{2 \\pi}{3}$, 且对任意的实数$t$, 均有$|\\overrightarrow {c}-t \\overrightarrow {e}| \\geq|\\overrightarrow {c}-2 \\overrightarrow {e}|$, 则$|\\overrightarrow {c}-\\overrightarrow {b}|$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题12", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015113": { + "id": "015113", + "content": "已知复数$z=\\dfrac{1}{1-\\mathrm{i}}-\\mathrm{i}$($\\mathrm{i}$为虚数单位$)$, 则$z \\cdot \\overline {z}=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{2}}{2}$}{$\\dfrac{\\sqrt{3}}{2}$}{2}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题13", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015114": { + "id": "015114", + "content": "某同学上学路上有$4$个红绿灯的路口, 假设他走到每个路口遇到绿灯的概率为$\\dfrac{2}{3}$, 且在各个路口遇到红灯或绿灯互不影响, 则该同学上学路上至少遇到$2$次绿灯的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{8}$}{$\\dfrac{3}{8}$}{$\\dfrac{7}{8}$}{$\\dfrac{8}{9}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题14", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015115": { + "id": "015115", + "content": "对于函数$f(x)=\\sqrt{3} \\sin x \\cos x+\\sin ^2 x-\\dfrac{1}{2}$, 给出下列结论:\\\\\n\\textcircled{1} 函数$y=f(x)$的图像关于点$(\\dfrac{5 \\pi}{12}, 0)$对称;\\\\\n\\textcircled{2} 函数$y=f(x)$在区间$[\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3}]$上的值域为$[-\\dfrac{1}{2}, 1]$;\\\\\n\\textcircled{3} 将函数$y=f(x)$的图像向左平移$\\dfrac{\\pi}{3}$个单位长度得到函数$y=-\\cos 2 x$的图像;\\\\\n\\textcircled{4} 曲线$y=f(x)$在$x=\\dfrac{\\pi}{4}$处的切线的斜率为$1$.\n则所有正确的结论是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{2}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题15", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015116": { + "id": "015116", + "content": "在数列$\\{b_n\\}$中, 若有$b_m=b_n$($m, n$均为正整数, 且$m \\neq n$), 就有$b_{m+1}=b_{n+1}$, 则称数列$\\{b_n\\}$为``递等数列''. 已知数列$\\{a_n\\}$满足$a_5=5$, 且$a_n=n(a_{n+1}-a_n)$, 将``递等数列''$\\{b_n\\}$的前$n$项和记为$S_n$, 若$b_1=a_1=b_4$, $b_2=a_2$, $S_5=a_{10}$, 则$S_{2023}=$\\bracket{20}.\n\\fourch{$4720$}{$4719$}{$4718$}{$4716$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题16", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015117": { + "id": "015117", + "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 已知$a_1=2$, $a_{n+1}=S_n$($n$为正整数).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$b_n=\\log _2 a_n$, 若$b_m+b_{m+1}+b_{m+2}+\\cdots+b_{m+9}=145$, 求正整数$m$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题17", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015118": { + "id": "015118", + "content": "如图, 在圆锥$PO$中, $AB$是底面的直径, $C$是底面圆周上的一点, 且$PO=3$, $AB=4$, $\\angle BAC=30^{\\circ}$, $M$是$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,3) node [above] {$P$} coordinate (P);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (A) arc (180:360:2 and 0.5) -- (P)--cycle;\n\\draw [dashed] (A) arc (180:0:2 and 0.5) -- cycle(O)--(P);\n\\draw (-60:2 and 0.5) node [below] {$C$} coordinate (C);\n\\draw (P)--(C);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (B)--(C)(O)--(C)(P)--(M)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$PBC \\perp$平面$POM$;\\\\\n(2) 求二面角$O-PB-C$的余弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题18", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015119": { + "id": "015119", + "content": "电解电容是常见的电子元件之一. 检测组在$85^{\\circ} \\text{C}$的温度条件下对电解电容进行质量检测, 按检测结果将其分为次品、正品, 其中正品分合格品、优等品两类.\\\\\n(1) 铝箔是组成电解电容必不可少的材料. 现检测组在$85^{\\circ} \\text{C}$的温度条件下, 对铝箔质量与电解电容质量进行测试, 得到如下$2 \\times 2$列联表, 那么他们是否有$99.9 \\%$的把握认为电解电容质量与铝箔质量有关? 请说明理由;\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 电解电容为次品 & 电解电容为正品 \\\\\n\\hline 铝箔为次品 & 174 & 76 \\\\\n\\hline 铝箔为正品 & 108 & 142 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(2) 电解电容经检验为正品后才能装箱, 已知两箱电解电容 (每箱$50$个), 第一箱和第二箱中分别有优等品$8$件与$9$件. 现用户从两箱中随机挑选出一箱, 并从该箱中先后随机抽取两个元件, 求在第一次取出的是优等品的情况下, 第二次取出的是合格品的概率.\\\\\n附录: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中 $n=a+b+c+d$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline $P\\left(\\chi^2 \\geq k\\right)$ & 0.100 & 0.050 & 0.025 & 0.010 & 0.001 \\\\\n\\hline $k$ & 2.706 & 3.841 & 5.024 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题19", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015120": { + "id": "015120", + "content": "已知动点$R(x, y)$到点$F(1,0)$的距离和它到直线$x=2$的距离之比等于$\\dfrac{\\sqrt{2}}{2}$, 动点$M$的轨迹记为曲线$C$, 过点$F$的直线$l$与曲线$C$相交于$P, Q$两点.\\\\\n(1) 求曲线$C$的方程;\\\\\n(2) 若$\\overrightarrow{FP}=-2 \\overrightarrow{FQ}$, 求直线$l$的方程;\\\\\n(3) 已知$A(-\\sqrt{2}, 0)$, 直线$AP, AQ$分别与直线$x=2$相交于$M, N$两点, 求证: 以$MN$为直径的圆经过点$F$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届虹口区高三二模试题20", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015121": { + "id": "015121", + "content": "设$f(x)=\\mathrm{e}^x$, $g(x)=\\ln x$, $h(x)=\\sin x+\\cos x$.\\\\\n(1) 求函数$y=\\dfrac{h(x)}{f(x)}$, $x \\in(-\\pi, 3 \\pi)$的单调区间和极值;\\\\\n(2) 若关于$x$不等式$f(x)+h(x) \\geq a x+2$在区间$[0,+\\infty)$上恒成立, 求实数$a$的取值范围;\\\\\n(3) 若存在直线$y=t$, 其与曲线$y=\\dfrac{x}{f(x)}$和$y=\\dfrac{g(x)}{x}$共有$3$个不同交点$A(x_1, t)$, $B(x_2, t)$, $C(x_3, t)$($x_10$)为偶函数, 则函数$f(x)$的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题7", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015129": { + "id": "015129", + "content": "已知向量$\\overrightarrow {a}=(1, \\sqrt{3})$, 且$\\overrightarrow {a}, \\overrightarrow {b}$的夹角为$\\dfrac{\\pi}{3}$, $(\\overrightarrow {a}+\\overrightarrow {b}) \\cdot(2 \\overrightarrow {a}-3 \\overrightarrow {b})=4$, 则$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影\n向量等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题8", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015130": { + "id": "015130", + "content": "某运动生理学家在一项健身活动中选择了$10$名男性参与者, 以他们的皮下脂肪厚度来估计身体的脂肪含量, 其中脂肪含量以占体重 (单位: $\\text{kg}$) 的百分比表示. 得到脂肪含量和体重的数据如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n个体编号 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\\hline\n体重$x$($\\text{kg}$) & 89 & 88 & 66 & 59 & 93 & 73 & 82 & 77 & 100 & 67\\\\\\hline\n脂肪含量$y$($\\%$) & 28 & 27 & 24 & 23 & 29 & 25 & 29 & 25 & 30 & 23\\\\\\hline\n\\end{tabular}\n\\end{center}\n建立男性体重与脂肪含量的回归方程为: \\blank{100}(结果中回归系数保留三位小数).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题9", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015131": { + "id": "015131", + "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$中, $E$为$AB$的中点, $F$为正方形$BCC_1B_1$的中心, 则直线$EF$与侧面$BB_1C_1C$所成角的正切值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (\\l,0,0) node [below right] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (\\l,0,-\\l) node [right] {$C_1$} coordinate (C_1);\n\\draw (A_1) ++ (0,0,-\\l) node [left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1);\n\\draw [dashed] (A_1) -- (D_1) -- (C_1);\n\\draw (A_1) ++ (0,\\l,0) node [left] {$A$} coordinate (A);\n\\draw (B_1) ++ (0,\\l,0) node [right] {$B$} coordinate (B);\n\\draw (C_1) ++ (0,\\l,0) node [above right] {$C$} coordinate (C);\n\\draw (D_1) ++ (0,\\l,0) node [above left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A_1) -- (A) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(B)$) node [above] {$E$} coordinate (E);\n\\draw ($(B_1)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\filldraw (E) circle (0.03) (F) circle (0.03);\n\\draw [dashed] (E)--(F);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题10", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015132": { + "id": "015132", + "content": "今年是农历癸卯兔年, 一种以兔子形象命名的牛奶糖深受顾客欢迎. 标识质量为$500 \\text{g}$的这种袋装奶糖的质量指标$X$是服从正态分布$N(500,2.5^2)$的随机变量. 若质量指标介于$495 \\text{g}$(含) 至$505 \\text{g}$(含) 之间的产品包装为合格包装, 则随意买一包这种袋装奶糖, 是合格包装的可能性大小为\\blank{50}$\\%$. (结果保留一位小数)\\\\\n(已知$\\Phi(1) \\approx 0.8413$, $\\Phi(2) \\approx 0.9772$, $\\Phi(3) \\approx 0.9987$. $\\Phi(x)$表示标准正态分布的密度函数从$-\\infty$到$x$的累计面积)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题11", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015133": { + "id": "015133", + "content": "若$10^x-10^y=10$, 其中$x, y \\in \\mathbf{R}$, 则$2 x-y$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题12", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015134": { + "id": "015134", + "content": "若直线$l$的方向向量为$\\overrightarrow {a}$, 平面$\\alpha$的法向量为$\\overrightarrow {n}$, 则能使$l\\parallel \\alpha$的是\\bracket{20}.\n\\twoch{$\\overrightarrow {a}=(1,0,0)$, $\\overrightarrow {n}=(-2,0,0)$}{$\\overrightarrow {a}=(1,3,5)$, $\\overrightarrow {n}=(1,0,1)$}{$\\overrightarrow {a}=(1,-1,3)$, $\\overrightarrow {n}=(0,3,1)$}{$\\overrightarrow {a}=(0,2,1)$, $\\overrightarrow {n}=(-1,0,-1)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题13", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015135": { + "id": "015135", + "content": "摩天轮常被当作一个城市的地标性建筑, 如静安大悦城的``Sky Ring''摩天轮是上海首个悬臂式屋顶摩天轮. 摩天轮最高点离地面高度$106$米, 转盘直径$56$米, 轮上设置$30$个极具时尚感的$4$人轿舱, 拥有$360$度的绝佳视野. 游客从离楼顶屋面最近的平台位置进入轿舱, 开启后按逆时针匀速旋转$t$分钟后, 游客距离地面的高度为$h$米, $h=$$-28 \\cos (\\dfrac{\\pi t}{6})+78$. 若在$t_1, t_2$时刻, 游客距离地面的高度相等, 则$t_1+t_2$的最小值为\\bracket{20}.\n\\fourch{$6$}{$12$}{$18$}{$24$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题14", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015136": { + "id": "015136", + "content": "设直线$l_1: x-2 y-2=0$与$l_2$关于直线$l: 2 x-y-4=0$对称, 则直线$l_2$的方程是\\bracket{20}.\n\\fourch{$11 x+2 y-22=0$}{$11 x+y+22=0$}{$5 x+y-11=0$}{$10 x+y-22=0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题15", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015137": { + "id": "015137", + "content": "函数$y=x \\ln x$\\bracket{20}.\n\\onech{是严格增函数}{在$(0, \\dfrac{1}{\\mathrm{e}})$上是严格增函数, 在$(\\dfrac{1}{\\mathrm{e}},+\\infty)$上是严格减函数}{是严格减函数}{在$(0, \\dfrac{1}{\\mathrm{e}})$上是严格减函数, 在$(\\dfrac{1}{\\mathrm{e}},+\\infty)$上是严格增函数}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题16", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015138": { + "id": "015138", + "content": "已知各项均为正数的数列$\\{a_n\\}$满足$a_1=1$, $a_n=2 a_{n-1}+3$(正整数$n \\geq 2$).\\\\\n(1) 求证: 数列$\\{a_n+3\\}$是等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的前$n$项和$S_n$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题17", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015139": { + "id": "015139", + "content": "如图, 在五面体$ABCDEF$中, $FA \\perp$平面$ABCD$, $AD\\parallel BC\\parallel FE$, $AB \\perp AD$, 若$AD=2$, $AF=AB=BC=FE=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$F$} coordinate (F);\n\\draw (1,1,0) node [above] {$E$} coordinate (E);\n\\draw (1,0,1) node [below] {$C$} coordinate (C);\n\\draw ($(C)!0.5!(E)$) node [left] {$M$} coordinate (M);\n\\draw (B)--(C)--(D)--(E)--(F)--cycle(E)--(C)(M)--(D);\n\\draw [dashed] (F)--(A)--(B)(A)--(D)(A)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求五面体$ABCDEF$的体积;\\\\\n(2) 若$M$为$EC$的中点, 求证: 平面$CDE \\perp$平面$AMD$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题18", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015140": { + "id": "015140", + "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$(其中$a>0, b>0$)的左、右焦点分别为$F_1(-c, 0)$、$F_2(c, 0)$(其中$c>0$).\\\\\n(1) 若双曲线$\\Gamma$过点$(2,1)$且一条渐近线方程为$y=\\dfrac{\\sqrt{2}}{2} x$; 直线$l$的倾斜角为$\\dfrac{\\pi}{4}$, 在$y$轴上的截距为$-2$. 直线$l$与该双曲线$\\Gamma$交于两点$A$、$B, M$为线段$AB$的中点, 求$\\triangle MF_1F_2$的面积;\\\\\n(2) 以坐标原点$O$为圆心, $c$为半径作圆, 该圆与双曲线$\\Gamma$在第一象限的交点为$P$. 过$P$作圆的切线, 若切线的斜率为$-\\sqrt{3}$, 求双曲线$\\Gamma$的离心率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题19", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015141": { + "id": "015141", + "content": "概率统计在生产实践和科学实验中应用广泛. 请解决下列两个问题.\\\\\n(1) 随着中小学``双减''政策的深入人心, 体育教学和各项体育锻炼迎来时间充沛的春天. 某初中学校学生篮球队从开学第二周开始每周进行训练, 第一次训练前共有$6$个篮球, 其中$3$个是新球(即没有用过的球), $3$个是旧球(即至少用过一次的球). 每次训练, 都是从中不放回任意取出$2$个篮球, 训练结束后放回原处. 设第一次训练时取到的新球个数为$X$, 求随机变量$X$的分布和期望;\\\\\n(2) 由于手机用微波频率信号传递信息, 那么长时间使用手机是否会增加得脑瘤的概率? 研究者针对这个问题, 对脑瘤病人进行问卷调查, 询问他们是否总是习惯在固定的一侧接听电话? 如果是, 是哪边? 结果有$88$人喜欢用固定的一侧接电话. 其中脑瘤部位在左侧的病人习惯固定在左侧接听电话的有$14$人, 习惯固定在右侧接听电话的有$28$人; 脑瘤部位在右侧的病人习惯固定在左侧接听电话的有$19$人, 习惯固定在右侧接听电话的有$27$人.\n根据上述信息写出下面这张$2 \\times 2$列联表中字母所表示的数据, 并对患脑瘤在左右侧的部位是否与习惯在该侧接听手机电话相关进行独立性检验. (显著性水平$\\alpha=0.05$)\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 习惯固定在左侧接听电话 & 习惯固定在右侧接听电话 & 总计 \\\\\n\\hline\n脑瘤部位在左侧的病人 & $a$ & $b$ & $42$ \\\\\n\\hline\n脑瘤部位在右侧的病人 & $c$ & $d$ & $46$ \\\\\n\\hline\n总计 & $a+c$ & $b=d$ & $88$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n参考公式及数据: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中$n=a+b+c+d$, $P(\\chi^2 \\geq 3.841) \\approx 0.05$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题20", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015142": { + "id": "015142", + "content": "已知函数$f(x)=\\dfrac{1}{2} x^2-(a+1) x+a \\ln x$. (其中$a$为常数)\\\\\n(1) 若$a=-2$, 求曲线$y=f(x)$在点$(2, f(2))$处的切线方程;\\\\\n(2) 当$a<0$时, 求函数$y=f(x)$的最小值;\\\\\n(3) 当$0 \\leq a<1$时, 试讨论函数$y=f(x)$的零点个数, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届静安区高三二模试题21", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015143": { + "id": "015143", + "content": "设全集$U=\\mathbf{R}$, 若集合$A=\\{x \\| x | \\geq 1, x \\in \\mathbf{R}\\}$, 则$\\overline {A}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题1", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015144": { + "id": "015144", + "content": "函数$y=\\cos ^2 x-\\sin ^2 x$的最小正周期为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题2", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015145": { + "id": "015145", + "content": "现有一组数$1,1,2,2,3,5,6,7,9,9$, 则该组数的第$25$百分位数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题3", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015146": { + "id": "015146", + "content": "设$3 \\mathrm{i}$($\\mathrm{i}$为虚数单位)是关于$x$的方程$x^2+m=0$($m \\in \\mathbf{R})$的根, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题4", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015147": { + "id": "015147", + "content": "函数$y=\\sqrt{3-\\dfrac{1}{x}}$的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题5", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015148": { + "id": "015148", + "content": "若$\\pi<\\theta<\\dfrac{3 \\pi}{2}$且$\\sin \\theta=-\\dfrac{3}{5}$, 则$\\tan (\\theta-\\dfrac{\\pi}{4})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题6", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015149": { + "id": "015149", + "content": "现有一个底面半径为$2 \\text{cm}$、高为$9 \\text{cm}$的圆柱形铁料, 若将其熔铸成一个球形实心工件, 则该工件的表面积为\\blank{50}$\\text{cm}^2$(损耗忽略不计).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题7", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015150": { + "id": "015150", + "content": "设$\\triangle ABC$的三边$a, b, c$满足$a: b: c=7: 5: 3$, 且$S_{\\triangle ABC}=15 \\sqrt{3}$, 则此三角形最长的边长为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题8", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015151": { + "id": "015151", + "content": "``民生''供电公司为了分析``康居''小区的用电量$y$(单位: $\\text{kW}\\cdot\\text{h}$)与气温$x$(单位: ${}^\\circ\\text{C})$之间的关系, 随机统计了$4$天的用电量与当天的气温, 这两者之间的对应关系见下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 气温$x$& 18 & 13 & 10 & -1 \\\\\n\\hline 用电量$y$& 24 & 34 & 38 & 64 \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n若上表中的数据可用回归方程$y=-2 x+b$($b \\in \\mathbf{R}$)来预测, 则当气温为$-4^{\\circ} \\text{C}$时该小区相应的用电量约为$\\text{kW} \\cdot \\text{h}$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题9", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015152": { + "id": "015152", + "content": "设$F_1$、$F_2$为双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{9}=1(a>0)$左、右焦点, 且$\\Gamma$的离心率为$\\sqrt{5}$, 若点$M$在$\\Gamma$的右支上, 直线$F_1M$与$\\Gamma$的左支相交于点$N$, 且$|MF_2|=|MN|$, 则$|F_1N|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题10", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015153": { + "id": "015153", + "content": "设$a>0$且$a \\neq 1$, 若在平面直角坐标系$xOy$中, 函数$y=\\log _a(a x+2)$与$y=\\log _a \\dfrac{1}{2 x+a}$的图像关于直线$l$对称, 则$l$与这两个函数图像的公共点的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题11", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015154": { + "id": "015154", + "content": "设$x$、$y \\in \\mathbf{R}$, 若向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$满足$\\overrightarrow {a}=(x, 1)$, $\\overrightarrow {b}=(2, y)$, $\\overrightarrow {c}=(1,1)$, 且向量$\\overrightarrow {a}-\\overrightarrow {b}$与$\\overrightarrow {c}$互相平行, 则$|\\overrightarrow {a}|+2|\\overrightarrow {b}|$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题12", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015155": { + "id": "015155", + "content": "设$a$、$b$为实数, 则``$a>b>0$''的一个充分非必要条件是\\bracket{20}.\n\\fourch{$\\sqrt{a-1}>\\sqrt{b-1}$}{$a^2>b^2$}{$\\dfrac{1}{b}>\\dfrac{1}{a}$}{$a-b>b-a$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题13", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015156": { + "id": "015156", + "content": "设$a$、$b$表示空间的两条直线, $\\alpha$表示平面, 给出下列结论:\\\\\n\\textcircled{1} 若$a\\parallel b$且$b \\subset \\alpha$, 则$a\\parallel \\alpha$;\\\\\n\\textcircled{2} 若$a\\parallel \\alpha$且$b \\subset \\alpha$, 则$a\\parallel b$;\\\\\n\\textcircled{3} 若$a\\parallel b$且$a\\parallel \\alpha$, 则$b\\parallel \\alpha$;\\\\\n\\textcircled{4} 若$a\\parallel \\alpha$且$b\\parallel \\alpha$, 则$a\\parallel b$.\\\\\n其中不正确的个数是\\bracket{20}.\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题14", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015157": { + "id": "015157", + "content": "设$P$为曲线$C: y^2=4 x$上的任意一点, 记$P$到$C$的准线的距离为$d$. 若关于点集$A=\\{M| |MP |=d\\}$和$B=\\{(x, y) |(x-1)^2+(y-1)^2=r^2\\}$, 给出如下结论:\\\\\n\\textcircled{1} 任意$r \\in(0,+\\infty)$, $A \\cap B$中总有$2$个元素;\\\\ \\textcircled{2} 存在$r \\in(0,+\\infty)$, 使得$A \\cap B=\\varnothing$.\\\\\n其中正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}成立, \\textcircled{2}成立}{\\textcircled{1}不成立, \\textcircled{2}成立}{\\textcircled{1}成立, \\textcircled{2}不成立}{\\textcircled{1}不成立, \\textcircled{2}不成立}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题15", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015158": { + "id": "015158", + "content": "设$\\omega>0$, 若在区间$[\\pi, 2 \\pi)$上存在$a, b$且$a=latex, scale = 0.6]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (3,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,4) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,3) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,3) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,3) node [above] {$C_1$} coordinate (C_1);\n\\draw (A)--(B)--(B_1)--(C_1)--(A_1)--cycle(A_1)--(B_1);\n\\draw [dashed] (A)--(C)--(B)--(C_1)--(C)(A)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AC \\perp BC_1$;\\\\\n(2) 设$AC_1$与底面$ABC$所成角的大小为$60^{\\circ}$, 求三梭锥$C-ABC_1$的体积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题17", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015160": { + "id": "015160", + "content": "已知$a>b$均为不是$1$的正实数, 设函数$y=f(x)$的表达式为$f(x)=a \\cdot b^x$($x \\in \\mathbf{R}$).\\\\\n(1) 设$a>b$且$f(x) \\leq b \\cdot a^x$, 求$x$的取值范围;\\\\\n(2) 设$a=\\dfrac{1}{16}$, $b=4$, 记$a_n=\\log _2 f(n)$, $b_n=f(n)$, 现将数列$\\{a_n\\}$中剔除$\\{b_n\\}$的项后、不改变其原来顺序所组成的数列记为$\\{c_n\\}$, 求$\\displaystyle\\sum_{i=1}^{100} c_i$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题18", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015161": { + "id": "015161", + "content": "现有$3$个盒子, 其中第一个盒子中装有$1$个白球、$4$个黑球; 第二个盒子装有$2$个白球、$3$个黑球; 第三个盒子装有$3$个白球、$2$个黑球. 现任取一个盒子, 从中任取$3$个球.\\\\\n(1) 求取到的白球数不少于$2$个的概率;\\\\\n(2) 设$X$为所取到的白球数, 求取到的白球数的期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题19", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015162": { + "id": "015162", + "content": "在$xOy$平面上, 设椭圆$\\Gamma: \\dfrac{x^2}{m^2}+y^2=1$($m>1$), 梯形$ABCD$的四个顶点均在$\\Gamma$上, 且$AB\\parallel CD$. 设直线$AB$的方程为$y=k x$($k \\in \\mathbf{R}$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (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 ({sqrt(3)},0.5) node [above] {$C$} coordinate (C);\n\\draw ({-sqrt(3)},0.5) node [above] {$D$} coordinate (D);\n\\filldraw ({sqrt(3)},0) node [below] {$F$} coordinate (F) circle (0.03);\n\\draw (B)--(C)--(D)--(A);\n\\draw (0,-1.5) node [below] {第(1)小题}; \n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$} coordinate (O);\n\\path [draw, name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\filldraw (0,2) node [right] {$P$} coordinate (P) circle (0.03);\n\\path [name path = DP] (P) --++ (-2,-4);\n\\path [name intersections = {of = DP and elli, by = {C,D}}];\n\\path [name path = BA] (1,2) -- (-0.6,-1.2);\n\\path [name intersections = {of = BA and elli, by = {B,A}}];\n\\draw (C) node [above] {$C$};\n\\draw (D) node [below] {$D$};\n\\draw (A) node [below] {$A$};\n\\draw (B) node [above] {$B$};\n\\draw (P)--(D);\n\\draw (B)--(C)--(D)--(A)--cycle;\n\\draw [->] (O)--(C);\n\\draw [->] (O)--(D);\n\\draw (0,-1.5) node [below] {第(2)小题}; \n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$} coordinate (O);\n\\path [draw, name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\draw ({2/sqrt(3)},{1/sqrt(3)}) node [above] {$B$} coordinate (B);\n\\draw ($(B)!2!(O)$) node [below] {$A$} coordinate (A);\n\\draw ({-sqrt(2)},{sqrt(2)}) node [left] {$M$} coordinate (M);\n\\draw ($(A)!0.5!(M)$) node [left] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(M)$) node [above] {$C$} coordinate (C);\n\\draw (M)--(A)--(B)--cycle(C)--(D);\n\\draw (0,-1.5) node [below] {第(3)小题}; \n\\end{tikzpicture}\n\\end{center}\n(1) 若$AB$为$\\Gamma$的长轴, 梯形$ABCD$的高为$\\dfrac{1}{2}$, 且$C$在$AB$上的射影为$\\Gamma$的焦点, 求$m$的值;\\\\\n(2) 设$m=\\sqrt{2}$, 直线$CD$经过点$P(0,2)$, 求$\\overrightarrow{OC} \\cdot \\overrightarrow{OD}$的取值范围;\\\\\n(3) 设$m=\\sqrt{2}$, $|AB|=2|CD|$, $AD$与$BC$的延长线相交于点$M$, 当$k$变化时, $\\triangle MAB$的面积是否为定值? 若是, 求出该定值; 若不是, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题20", + "edit": [ + "202304012\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015163": { + "id": "015163", + "content": "已知$a$、$b \\in \\mathbf{R}$, 设函数$y=f(x)$的表达式为$f(x)=a \\cdot x^2-b \\cdot \\ln x$(其中$x>0$).\\\\\n(1) 设$a=1$, $b=0$, 当$f(x)>x^{-1}$时, 求$x$的取值范围;\\\\\n(2) 设$a=2$, $b>4$, 集合$D=(0,1]$, 记$g(x)=2 c x-\\dfrac{1}{x^2}$($c \\in \\mathbf{R}$), 若$y=g(x)$在$D$上为严格增函数且对$D$上的任意两个变量$s, t$, 均有$f(s) \\geq g(t)$成立, 求$c$的取值范围;\\\\\n(3) 当$a=0$, $b<0$, $x>1$时, 记$h_n(x)=[f(x)]^n+\\dfrac{1}{[f(x)]^n}$, 其中$n$为正整数. 求证: $[h_1(x)]^n+2 \\geq h_n(x)+2^n$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届普陀区高三二模试题21", + "edit": [ + "202304012\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.",