From dc9a3ff1c2c9c95fba29085cbfdd942b5e61956f Mon Sep 17 00:00:00 2001 From: WangWeiye Date: Fri, 14 Apr 2023 11:58:50 +0800 Subject: [PATCH] =?UTF-8?q?=E6=94=B6=E5=BD=95=E9=97=B5=E8=A1=8C=E9=95=BF?= =?UTF-8?q?=E5=AE=81=E6=9D=BE=E6=B1=9F2023=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 | 4 +- 题库0.3/Problems.json | 1197 +++++++++++++++++++++++++++++++++++++++++ 2 files changed, 1199 insertions(+), 2 deletions(-) diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index 5d4eed94..dc2c45dc 100644 --- a/工具/批量收录题目.py +++ b/工具/批量收录题目.py @@ -1,8 +1,8 @@ #修改起始id,出处,文件名 -starting_id = 15101 +starting_id = 15164 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" +editor = "20230414\t王伟叶" indexed = True import os,re,json diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 8f6b1918..41874468 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -372894,6 +372894,1203 @@ "remark": "", "space": "12ex" }, + "015164": { + "id": "015164", + "content": "设全集$U=\\{-2,-1,0,1,2\\}$, 集合$A=\\{-2,0,2\\}$, 则$\\overline {A}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题1", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015165": { + "id": "015165", + "content": "若实数$x$、$y$满足$\\lg x=m$、$y=10^{1-m}$, 则$x y=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题2", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015166": { + "id": "015166", + "content": "已知复数$z$满足$z(1-\\mathrm{i})=\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z$的虚部为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题3", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015167": { + "id": "015167", + "content": "已知圆柱的底面积为$9 \\pi$, 侧面积为$12 \\pi$, 则该圆柱的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题4", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015168": { + "id": "015168", + "content": "已知常数$m>0$, $(x+\\dfrac{m}{x})^6$的二项展开式中$x^2$项的系数是$60$, 则$m$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题5", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015169": { + "id": "015169", + "content": "已知事件$A$与事件$B$互斥, 如果$P(A)=0.3$, $P(B)=0.5$, 那么$P(\\overline{A \\cup B})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题6", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015170": { + "id": "015170", + "content": "今年春季流感爆发期间, 某医院准备将$2$名医生和$4$名护士分配到两所学校, 给学校老师和学生接种流感疫苗. 若每所学校分配$1$名医生和$2$名护士, 则不同的分配方法数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题7", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015171": { + "id": "015171", + "content": "$\\displaystyle\\lim _{h \\to 0} \\dfrac{\\ln (h+4)-2 \\ln 2}{h}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题8", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015172": { + "id": "015172", + "content": "若关于$x$的方程$(\\dfrac{1}{2})^x+m=\\sqrt{x+1}$在实数范围内有解, 则实数$m$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题9", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015173": { + "id": "015173", + "content": "已知在等比数列$\\{a_n\\}$中, $a_3$、$a_7$分别是函数$y=x^3-6 x^2+6 x-1$的两个驻点, 则$a_5=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题10", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015174": { + "id": "015174", + "content": "已知抛物线$C_1: y^2=8 x$, 圆$C_2: (x-2)^2+y^2=1$, 点$M$的坐标为$(4,0)$, $P$、$Q$分别为$C_1$、$C_2$上的动点, 且满足$|PM|=|PQ|$, 则点$P$的横坐标的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题11", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015175": { + "id": "015175", + "content": "平面上有一组互不相等的单位向量$\\overrightarrow{OA_1}, \\overrightarrow{OA_2}, \\cdots, \\overrightarrow{OA_n}$, 若存在单位向量$\\overrightarrow{OP}$满足$\\overrightarrow{OP} \\cdot \\overrightarrow{OA_1}+\\overrightarrow{OP} \\cdot \\overrightarrow{OA_2}+\\cdots+\\overrightarrow{OP} \\cdot \\overrightarrow{OA_n}=0$, 则称$\\overrightarrow{OP}$是向量组$\\overrightarrow{OA_1}, \\overrightarrow{OA_2}, \\cdots, \\overrightarrow{OA_n}$的平衡向量. 已知$\\langle\\overrightarrow{OA_1}, \\overrightarrow{OA_2}\\rangle=\\dfrac{\\pi}{3}$, 向量$\\overrightarrow{OP}$是向量组$\\overrightarrow{OA_1}, \\overrightarrow{OA_2}, \\overrightarrow{OA_3}$的平衡向量, 当$\\overrightarrow{OP} \\cdot \\overrightarrow{OA_3}$取得最大值时, $\\overrightarrow{OA_1} \\cdot \\overrightarrow{OA_3}$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题12", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015176": { + "id": "015176", + "content": "下列函数中, 既不是奇函数, 也不是偶函数的为\\bracket{20}.\n\\fourch{$y=0$}{$y=\\dfrac{1}{x}$}{$y=x^2$}{$y=2^x$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题13", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015177": { + "id": "015177", + "content": "在某区高三年级举行的一次质量检测中, 某学科共有$3000$人参加考试. 为了解本次考试学生的成绩情况, 从中抽取了部分学生的成绩 (成绩均为正整数, 满分为$100$分) 作为样本进行统计, 样本容量为$n$. 按照$[50,60),[60,70),[70,80),[80,90),[90,100]$的分组作出频率分布直方图 (如图所示). 已知成绩落在$[50,60)$内的人数为$16$, 则下列结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (40,0) -- (42,0) -- (44,-0.003) -- (46,0.003) -- (48,0)-- (120,0) node [below] {成绩(分)};\n\\draw [->] (40,0) -- (40,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.016,60/0.03,70/0.04,80/0.01,90/0.004}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {50/0.016,60/0.03/x,70/0.04,80/0.01,90/0.004}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}\n\\onech{样本容量$n=1000$}{图中$x=0.025$}{估计全体学生该学科成绩的平均分为$70.6$分}{若将该学科成绩由高到低排序, 前$15 \\%$的学生该学科成绩为A等, 则成绩为$78$分的学生该学科成绩肯定不是A等}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题14", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015178": { + "id": "015178", + "content": "已知$f(x)=\\cos 2 x-a \\sin x$, 若存在正整数$n$, 使函数$y=f(x)$在区间$(0, n \\pi)$内有$2023$个零点, 则实数$a$所有可能的值为\\bracket{20}.\n\\fourch{$1$}{$-1$}{$0$}{$1$或$-1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题15", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015179": { + "id": "015179", + "content": "若数列$\\{b_n\\}$、$\\{c_n\\}$均为严格增数列, 且对任意正整数$n$, 都存在正整数$m$, 使得$b_m \\in[c_n, c_{n+1}]$, 则称数列$\\{b_n\\}$为数列$\\{c_n\\}$的``M数列''. 已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 则下列选项中为假命题的是\\bracket{20}.\n\\onech{存在等差数列$\\{a_n\\}$, 使得$\\{a_n\\}$是$\\{S_n\\}$的``M数列''}{存在等比数列$\\{a_n\\}$, 使得$\\{a_n\\}$是$\\{S_n\\}$的``M数列''}{存在等差数列$\\{a_n\\}$, 使得$\\{S_n\\}$是$\\{a_n\\}$的``M数列''}{存在等比数列$\\{a_n\\}$, 使得$\\{S_n\\}$是$\\{a_n\\}$的``M数列''}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题16", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015180": { + "id": "015180", + "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 已知$\\sin A=\\sin 2B, a=4$, $b=6$.\\\\\n(1) 求$\\cos B$的值;\\\\\n(2) 求$\\triangle ABC$的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题17", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015181": { + "id": "015181", + "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$为矩形, $PD \\perp$平面$ABCD$, $PD=AD=2$, $AB=4$, 点$E$在线段$AB$上, 且$BE=\\dfrac{1}{4} AB$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (4,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.75!(B)$) node [below] {$E$} coordinate (E);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(E)(P)--(B);\n\\draw [dashed] (P)--(D)--(A)(D)--(C)(D)--(B)(A)--(C)(C)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CE \\perp$平面$PBD$;\\\\\n(2) 求二面角$P-CE-A$的余弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题18", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015182": { + "id": "015182", + "content": "在临床检测试验中, 某地用某种抗原来诊断试验者是否患有某种疾病. 设事件$A$表示试验者的检测结果为阳性, 事件$B$表示试验者患有此疾病. 据临床统计显示, $P(A | B)$$=0.99, P(\\overline {A} | \\overline {B})=0.98$. 已知该地人群中患有此种疾病的概率为$0.001$. (下列两小题计算结果中的概率值精确到$0.00001$)\\\\\n(1) 对该地某人进行抗原检测, 求事件$A$与$\\overline {B}$同时发生的概率;\\\\\n(2) 对该地$3$个患有此疾病的患者进行抗原检测, 用随机变量$X$表示检测结果为阳性的人数, 求$X$的分布和期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题19", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015183": { + "id": "015183", + "content": "已知$O$为坐标原点, 曲线$C_1: \\dfrac{x^2}{a^2}-y^2=1$($a>0$)和曲线$C_2: \\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$有公共点, 直线$l_1: y=k_1 x+b_1$与曲线$C_1$的左支相交于$A$、$B$两点, 线段$AB$的中点为$M$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.75]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [draw, name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\path [draw, name path = hypbo, samples = 100, domain = -2:3] plot ({-sqrt(\\x*\\x+1)},\\x); \n\\path [draw, name path = hypbo2, samples = 100, domain = -2:2] plot ({sqrt(\\x*\\x+1)},\\x); \n\\path [name path = AB] (-3,3) -- (-1.75,-2);\n\\path [name path = CD] (0,-2) -- (1,2);\n\\path [draw, name path = MN] (-3,0.75) -- (3,-0.75);\n\\path [name intersections = {of = AB and hypbo, by = {B,A}}];\n\\path [name intersections = {of = CD and elli, by = {C,D}}];\n\\path [name intersections = {of = AB and MN, by = M}];\n\\path [name intersections = {of = CD and MN, by = N}];\n\\draw (A) node [above] {$A$} -- (B) node [below] {$B$};\n\\draw (C) node [above] {$C$} -- (D) node [below] {$D$};\n\\draw (M) node [above] {$M$} (N) node [below] {$N$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若曲线$C_1$和$C_2$有且仅有两个公共点, 求曲线$C_1$的离心率和渐近线方程;\\\\\n(2) 若直线$OM$经过曲线$C_2$上的点$T(\\sqrt{2},-1)$, 且$a^2$为正整数, 求$a$的值;\\\\\n(3) 若直线$l_2: y=k_2 x+b_2$与曲线$C_2$相交于$C$、$D$两点, 且直线$OM$经过线段$CD$中点$N$, 求证: $k_1^2+k_2^2>1$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题20", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015184": { + "id": "015184", + "content": "如果曲线$y=f(x)$存在相互垂直的两条切线, 称函数$y=f(x)$是``正交函数''. 已知$f(x)=x^2+a x+2 \\ln x$, 设曲线$y=f(x)$在点$M(x_0, f(x_0))$处的切线为$l_1$.\\\\\n(1) 当$f'(1)=0$时, 求实数$a$的值;\\\\\n(2) 当$a=-8$, $x_0=8$时, 是否存在直线$l_2$满足$l_1 \\perp l_2$, 且$l_2$与曲线$y=f(x)$相切? 请说明理由;\\\\\n(3) 当$a \\geq-5$时, 如果函数$y=f(x)$是``正交函数'', 求满足要求的实数$a$的集合$D$; 若对任意$a \\in D$, 曲线$y=f(x)$都不存在与$l_1$垂直的切线$l_2$, 求$x_0$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届闵行区高三二模试题21", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015185": { + "id": "015185", + "content": "已知集合$A=\\{1,2,3,4,5\\}$, $B=\\{2,4,6,8\\}$, 则$A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题1", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015186": { + "id": "015186", + "content": "若``$x=1$''是``$x>a$''的充分条件, 则实数$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题2", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015187": { + "id": "015187", + "content": "已知事件$A$与事件$B$相互独立, 如果$P(A)=0.5$, $P(A \\cap \\overline {B})=0.4$, 那么$P(B)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题3", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015188": { + "id": "015188", + "content": "当$x \\in[a,+\\infty)$时, 幂函数$y=x^2$的图像总在$y=x^{\\frac{1}{2}}$的图像上方, 则$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题4", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015189": { + "id": "015189", + "content": "已知圆锥侧面展开图的圆心角为$\\dfrac{2 \\pi}{3}$, 底面周长为$2 \\pi$. 则这个圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题5", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015190": { + "id": "015190", + "content": "若函数$y=\\ln (1+x)-a \\ln (1-x)$为奇函数, 则实数$a$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题6", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015191": { + "id": "015191", + "content": "设随机变量$X$服从正态分布$N(2, \\sigma^2)$, 若$P(X \\leq 1)=0.2$, 则$P(X<3)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题7", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015192": { + "id": "015192", + "content": "某小学开展劳动教育, 欲在围墙边用栅栏围成一个$2$平方米的矩形植物种植园, 矩形的一条边为围墙, 如图. 则至少需要\\blank{50}米栅栏.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [pattern = north east lines] (-0.2,0.2) rectangle (3.2,0);\n\\draw (-0.2,0) -- (3.2,0);\n\\draw (0,0) rectangle (3,-1.5);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题8", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015193": { + "id": "015193", + "content": "已知函数$y=f(x)$, $y=g(x)$满足$f(x)+x g(x)=x^2-1$, 若$f(1)=1$, 则$f'(1)+g'(1)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题9", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015194": { + "id": "015194", + "content": "若对任意$x \\in[1,2]$, 均有$|x^2-a|+|x+a|=|x^2+x|$, 则实数$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题10", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015195": { + "id": "015195", + "content": "已知空间向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$、$\\overrightarrow {d}$满足: $|\\overrightarrow {a}-\\overrightarrow {b}|=1$, $|\\overrightarrow {b}-\\overrightarrow {c}|=2$, $(\\overrightarrow {a}-\\overrightarrow {b})\\parallel(\\overrightarrow {b}-\\overrightarrow {c})$, $(\\overrightarrow {a}-\\overrightarrow {d}) \\cdot(\\overrightarrow {b}-\\overrightarrow {d})=0$, 则$|\\overrightarrow {c}-\\overrightarrow {d}|$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题11", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015196": { + "id": "015196", + "content": "已知$F_1$、$F_2$是双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点, $l$是$\\Gamma$的一条渐近线, 以$F_2$为圆心的圆与$l$相切于点$P$. 若双曲线$\\Gamma$的离心率为$2$, 则$\\sin \\angle PF_1F_2=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题12", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015197": { + "id": "015197", + "content": "在下列统计量中, 用来描述一组数据离散程度的量是\\bracket{20}.\n\\fourch{平均数}{众数}{百分位数}{标准差}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题13", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015198": { + "id": "015198", + "content": "设复平面上表示$2-\\mathrm{i}$和$3+4 \\mathrm{i}$的点分别为点$A$和点$B$, 则表示向量$\\overrightarrow{AB}$的复数在复平面上所对应的点位于\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题14", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015199": { + "id": "015199", + "content": "已知正方体$ABCD-A_1B_1C_1D_1$, 点$P$在直线$AD_1$上, $Q$为线段$BD$的中点. 则下列说法不正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,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 ($(A)!0.6!(D1)$) node [left] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(D)$) node [left] {$Q$} coordinate (Q);\n\\draw [dashed] (B)--(D)(A)--(D1)(P)--(Q);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{存在点$P$, 使得$PQ \\perp A_1C_1$}{存在点$P$, 使得$PQ\\parallel A_1B$}{直线$PQ$始终与直线$CC_1$异面}{直线$PQ$始终与直线$BC_1$异面}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题15", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015200": { + "id": "015200", + "content": "设各项均为实数的等差数列$\\{a_n\\}$和$\\{b_n\\}$的前$n$项和分别为$S_n$和$T_n$, 对于方程\\textcircled{1} $2023 x^2-S_{2023} x+T_{2023}=0$, \\textcircled{2} $x^2-a_1 x+b_1=0$, \\textcircled{3} $x^2+a_{2023} x+b_{2023}=0$.\n下列判断正确的是\\bracket{20}.\n\\twoch{若\\textcircled{1}有实根, \\textcircled{2}有实根, 则\\textcircled{3}有实根}{若\\textcircled{1}有实根, \\textcircled{2}无实根, 则\\textcircled{3}有实根}{若\\textcircled{1}无实根, \\textcircled{2}有实根, 则\\textcircled{3}无实根}{若\\textcircled{1}无实根, \\textcircled{2}无实根, 则\\textcircled{3}无实根}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题16", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015201": { + "id": "015201", + "content": "盒子中有$5$个乒乓球, 其中$2$个次品, $3$个正品. 现从中不放回地随机摸取$2$次小球, 每次一个.\\\\\n(1) 记``第二次摸出的小球是正品''为事件$B$, 求证: $P(B)=\\dfrac{3}{5}$;\\\\\n(2) 用$X$表示摸出的$2$个小球中次品的个数, 求$X$的分布和期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题17", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015202": { + "id": "015202", + "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$为直角梯形, $AD\\parallel BC$, $AB \\perp BC$, $AB=AD$, $BC=2AB, E$、$F$分别为棱$BC$、$BP$中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (-2,0,2) node [below] {$D$} coordinate (D);\n\\draw (-4,0,0) node [left] {$C$} coordinate (C);\n\\draw (-1,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(P)$) node [right] {$F$} coordinate (F);\n\\draw (C)--(D)--(A)--(B)--(P)--cycle(P)--(D)(P)--(A)--(F);\n\\draw [dashed] (C)--(B)(E)--(A)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$AEF\\parallel$平面$DCP$;\\\\\n(2) 若平面$PBC \\perp$乎面$ABCD$, 直线$AP$与平面$PBC$所成的角为$45^{\\circ}$, 且$CP \\perp PB$, 求二面角$P-AB-C$的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题18", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015203": { + "id": "015203", + "content": "某地新能源汽车保有量符合阻滞型增长模型$x(t)=\\dfrac{M}{1+\\lambda e^{-r t}}$, 其中$x(t)$为自统计之日起, 经过$t$年后该地新能源汽车保有量, $\\lambda$和$r$为增长系数, $M$为饱和量. 下表是该地近$6$年年底的新能源汽车的保有量 (万辆) 的统计数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 年份 & 2018 & 2019 & 2020 & 2021 & 2022 \\\\\n\\hline $t$& 0 & 1 & 2 & 3 & 4 \\\\\n\\hline 保有量$x(t)$& 9.6 & 12.9 & 17.1 & 23.2 & 31.4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n假设该地新能源汽车饱和量$M=290$万辆.\\\\\n(1) 若$r=0.31$, 假定$2018$年数据满足公式$x(t)=\\dfrac{M}{1+\\lambda e^{-r t}}$, 计算$\\lambda$的值(精确到$0.01$), 并估算$2023$年年底该地新能源汽车保有量(精确到$0.1$万辆);\\\\\n(2) 设$y=\\dfrac{M}{x(t)}-1$, 则$\\ln y$与$t$线性相关, 请依据以上表格中相关数据, 利用线性回归分析确定$\\lambda$和$r$的值(精确到$0.01$).\\\\\n附: 线性回归方程$y=\\hat{a} x+\\hat{b}$中回归系数计算公式如下:\n$\\hat{a}=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})(y_i-\\overline {y})}{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})^2}$, $\\hat{b}=\\overline {y}-\\hat{a} \\overline {x}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题19", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015204": { + "id": "015204", + "content": "已知拋物线$\\Gamma: y^2=4 x$的焦点为$F$, 准线为$l$, 直线$l'$经过点$F$且与$\\Gamma$交于点$A$、$B$.\\\\\n(1) 求以$F$为焦点, 坐标轴为对称轴, 离心率为$\\dfrac{1}{2}$的椭圆的标准方程;\\\\\n(2) 若$|AB|=5$, 求线段$AB$的中点到$x$轴的距离;\\\\\n(3) 设$O$为坐标原点, $M$为$\\Gamma$上的动点, 直线$AM$、$BM$分别与准线$l$交于点$C$、$D$. 求证: $\\overrightarrow{OC} \\cdot \\overrightarrow{OD}$为常数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题20", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015205": { + "id": "015205", + "content": "(1) 求简谐振动$y=\\sin x+\\cos x$的振幅、周期和初相位$\\varphi$($\\varphi \\in[0,2 \\pi)$);\\\\\n(2) 若函数$y=\\sin \\dfrac{1}{2} x+\\dfrac{1}{2} \\cos x$在区间$(0, m)$上有唯一的极大值点, 求实数$m$的取值范围;\\\\\n(3) 设$a>0$, $f(x)=\\sin a x-a \\sin x$, 若函数$y=f(x)$在区间$(0, \\pi)$上是严格增函数, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届长宁区高三二模试题21", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015206": { + "id": "015206", + "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{x | \\dfrac{2}{x}>1\\}$, 则$A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题1", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015207": { + "id": "015207", + "content": "若复数$z$满足$\\mathrm{i} \\cdot z=3-4 \\mathrm{i}$, 则$|\\overline {z}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题2", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015208": { + "id": "015208", + "content": "已知空间向量$\\overrightarrow {a}=(1,2,3)$, $\\overrightarrow {b}=(2,-2,0)$, $\\overrightarrow {c}=(1,1, \\lambda)$, 若$\\overrightarrow {c} \\perp(2 \\overrightarrow {a}+\\overrightarrow {b})$, 则$\\lambda=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题3", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015209": { + "id": "015209", + "content": "已知随机变量$X$服从正态分布$N(0,1)$, 若$P(X<-1.96)=0.03$, 则$P(|X|<1.96)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题4", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015210": { + "id": "015210", + "content": "已知$\\dfrac{\\pi}{2}<\\theta<\\pi$, 且$\\cos \\theta=-\\dfrac{4}{5}$, 则$\\tan 2 \\theta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题5", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015211": { + "id": "015211", + "content": "在二项式$(x-\\dfrac{1}{x})^8$的展开式中, 含$x^4$的项的系数是\\blank{50}.(用数字作答)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题6", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015212": { + "id": "015212", + "content": "将右图所示的圆锥形容器内的液体全部倒入底面半径为$50 \\text{mm}$的直立的圆柱形容器内, 则液面高度为\\blank{50}$\\text{mm}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [pattern = north east lines] (-0.5,1.5) arc (180:0:0.5 and 0.125) -- (0,0)--cycle;\n\\draw (0,0) -- (1,3) (0,0) -- (-1,3) -- (1,3);\n\\draw (0,3) ellipse (1 and 0.25);\n\\draw (-0.5,1.5) arc (180:360:0.5 and 0.125);\n\\draw [dashed] (-0.5,1.5) arc (180:0:0.5 and 0.125);\n\\draw (-1.1,0) -- (-1.5,0) (-1.1,3) -- (-1.5,3);\n\\draw [<->] (-1.3,0) -- (-1.3,3) node [midway, fill = white] {\\tiny$300\\text{mm}$};\n\\draw (-1,3.1) -- (-1,3.7) (1,3.1) -- (1,3.7);\n\\draw [<->] (-1,3.5) -- (1,3.5) node [midway, fill=white] {\\tiny$200\\text{mm}$};\n\\draw (0.8,1.5) -- (1.2,1.5) (0.8,0) -- (1.2,0);\n\\draw [<->] (1,1.5) -- (1,0) node [midway, fill=white] {\\tiny$150\\text{mm}$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题7", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015213": { + "id": "015213", + "content": "从$4$名男生和$3$名女生中抽取两人加入志愿者服务队. 用$A$表示事件``抽到的两名学生性别相同'', 用$B$表示事件``抽到的两名学生都是女生'', 则$P(B | A)=$\\blank{50}.(结果用最简分数表示)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题8", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015214": { + "id": "015214", + "content": "参考《九章算术》中``竹九节''问题, 提出: 一根$9$节的竹子, 自上而下各节的容积成等差数列, 上面$4$节的容积共$2$升, 下面$3$节的容积共$3$升, 则第$5$节的容积为\\blank{50}升.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题9", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015215": { + "id": "015215", + "content": "已知$x \\in(0, \\dfrac{\\pi}{2})$, 则$\\dfrac{1}{\\sin ^2 x}+\\dfrac{4}{\\cos ^2 x}$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题10", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015216": { + "id": "015216", + "content": "已知函数$y=f(x)$为$\\mathbf{R}$上的奇函数, 且$f(x)+f(2-x)=0$, 当$-10$). 若存在$m, n \\in \\mathbf{R}$, 使得$m \\overrightarrow{AB}+\\overrightarrow{OA}$与$n \\overrightarrow{AB}+\\overrightarrow{OB}$垂直, 且$|(m \\overrightarrow{AB}+\\overrightarrow{OA})-(n \\overrightarrow{AB}+\\overrightarrow{OB})|=a$, 则$|\\overrightarrow{AB}|$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题12", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015218": { + "id": "015218", + "content": "已知直线$l_1: a x+y+1=0$与直线$l_2: x+a y-2=0$, 则``$l_1\\parallel l_2$''是``$a=1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题13", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015219": { + "id": "015219", + "content": "为了解某社区居民的家庭年收入与年支出的关系, 随机调查了该社区$5$户家庭, 得到如下统计数据表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 收入$x$(万元) & 8.2 & 8.6 & 10.0 & 11.3 & 11.9 \\\\\n\\hline 支出$y$(万元) & 6.2 & 7.5 & 8.0 & 8.5 & 9.8 \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n根据上表可得回归直线方程$y=\\hat{a} x+\\hat{b}$, 其中$\\hat{a}=0.76$, $\\hat{b}=\\overline {y}-\\hat{a} \\overline {x}$, 据此估计, 该社区一户收入为$15$万元家庭年支出为\\bracket{20}.\n\\fourch{$11.4$万元}{$11.8$万元}{$12.0$万元}{$12.2$万元}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题14", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015220": { + "id": "015220", + "content": "若方程$f(x) \\cdot g(x)=0$的解集为$M$, 则以下结论一定正确的是\\bracket{20}.\\\\\n\\textcircled{1} $M=\\{x | f(x)=0\\} \\cup\\{x | g(x)=0\\}$;\\\\\n\\textcircled{2} $M=\\{x | f(x)=0\\} \\cap\\{x | g(x)=0\\}$;\\\\\n\\textcircled{3} $M \\subseteq\\{x | f(x)=0\\} \\cup\\{x | g(x)=0\\}$;\\\\\n\\textcircled{4} $M \\supseteq\\{x | f(x)=0\\} \\cap\\{x | g(x)=0\\}$.\n\\fourch{\\textcircled{1}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题15", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "015221": { + "id": "015221", + "content": "已知函数$y=\\dfrac{1}{3} x^3-x^2-3 x+a$, $a \\in \\mathbf{R}$, 在区间$(t-3, t+5)$上有最大值, 则实数$t$的取值范围是\\bracket{20}.\n\\fourch{$-6=latex]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw (0,0,{2*sin(22.5)}) node [below] {$A$} coordinate (A);\n\\draw (0,0,{-2*sin(22.5)}) node [below] {$C$} coordinate (C);\n\\draw ({-2*cos(22.5)},0,0) node [left] {$D$} coordinate (D);\n\\draw ({2*cos(22.5)},0,0) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(D)$) node [left] {$M$} coordinate (M);\n\\draw (P)--(D)--(A)--(B)--cycle(P)--(A)(M)--(A);\n\\draw [dashed] (D)--(C)--(B)(D)--(B)(A)--(C)(P)--(O)(P)--(C)(C)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $PB\\parallel$平面$ACM$;\\\\\n(2) 求直线$AM$与平面$ABCD$所成角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题18", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015224": { + "id": "015224", + "content": "某城市响应国家号召, 积极调整能源结构, 推出多种价位的新能源电动汽车. 根据前期市场调研, 有购买新能源车需求的约有$2$万人, 他们的选择意向统计如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 车型 &A&B&C&D&E&F\\\\\n\\hline 价格 & 9 万元 & 12 万元 & 18 万元 & 24 万元 & 30 万元 & 40 万元 \\\\\n\\hline 占比 &$5 \\%$&$15 \\%$&$25 \\%$&$35 \\%$&$15 \\%$&$5 \\%$\\\\\n\\hline\n\\end{tabular} \n\\end{center}\n(1) 如果有购车需求的这些人今年都购买了新能源车, 今年新能源车的销售额预计约为多少亿元?\\\\\n(2) 车企推出两种付款方式: 全款购车: 购车时一次性付款可优惠车价的$3 \\%$; 分期付款: 无价格优惠, 购车时先付车价的一半, 余下的每半年付一次, 分$4$次付完, 每次付车价的$\\dfrac{1}{8}$.\\\\\n(i) 某位顾客现有$a$万元现金, 欲购买价值$a$万元的某款车, 付款后剩余的资金全部用于购买半年期的理财产品(该理财产品半年期到期收益率为$1.8 \\%$), 到期后, 可用资金(含理财收益)继续购买半年期的理财产品. 问: 顾客选择哪一种付款方式收益更多? (计算结果精确到$0.0001$)\\\\\n(ii) 为了激励购买理财产品, 银行对采用分期付款方式的顾客, 赠送价值$1888$元的大礼包, 试问: 这一措施对哪些车型有效? (计算结果精确到$0.0001$)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题19", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015225": { + "id": "015225", + "content": "已知椭圆$C_1: \\dfrac{x^2}{2}+\\dfrac{y^2}{b^2}=1$的左、右焦点分别为$F_1$、$F_2$, 离心率为$e_1$; 双曲线\n$C_2: \\dfrac{x^2}{2}-\\dfrac{y^2}{b^2}=1$的左、右焦点分别为$F_3$、$F_4$, 离心率为$e_2$, $e_1 \\cdot e_2=\\dfrac{\\sqrt{3}}{2}$. 过点$F_1$作不垂直于$y$轴的直线$l$交曲线$C_1$于点$A$、$B$, 点$M$为线段$AB$的中点, 直线$OM$交曲线$C_2$于$P$、$Q$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse ({sqrt(2)} and 1);\n\\draw [domain = -1.6:1.6, samples = 100] plot ({sqrt(2*\\x*\\x+2)},\\x);\n\\draw [domain = -1.6:1.6, samples = 100] plot ({-sqrt(2*\\x*\\x+2)},\\x);\n\\filldraw (-1,0) circle (0.03) node [below] {$F_1$} coordinate (F_1);\n\\filldraw (1,0) circle (0.03) node [below] {$F_2$} coordinate (F_2);\n\\filldraw ({-sqrt(3)},0) circle (0.03) node [below] {$F_3$} coordinate (F_3);\n\\filldraw ({sqrt(3)},0) circle (0.03) node [below] {$F_4$} coordinate (F_4);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$C_1$、$C_2$的方程;\\\\\n(2) 若$\\overrightarrow{AF_1}=3 \\overrightarrow{F_1B}$, 求直线$PQ$的方程;\\\\\n(3) 求四边形$APBQ$面积的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题20", + "edit": [ + "20230414\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "015226": { + "id": "015226", + "content": "已知$x>0$, 记$f(x)=e^x$, $g(x)=x^x$, $h(x)=\\ln g(x)$.\\\\\n(1) 试将$y=f(x)$、$y=g(x)$、$y=h(x)$中的一个函数表示为另外两个函数复合而成的复合函数;\\\\\n(2) 借助 (1) 的结果, 求函数$y=g(2 x)$的导函数和最小值;\\\\\n(3) 记$H(x)=\\dfrac{f(x)-h(x)}{x}+x+a$, $a$是实常数, 函数$y=H(x)$的导函数是$y'=H'(x)$. 已知函数$y=H(x) \\cdot H'(x)$有三个不相同的零点$x_1$、$x_2$、$x_3$. 求证: $x_1 \\cdot x_2 \\cdot x_3<1$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届松江区高三二模试题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.",