diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index d90a86f1..0015b845 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 = 40772 +starting_id = 17400 raworigin = "" -filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目13.tex" -editor = "20230526\t王伟叶" -indexed = False +filename = r"C:\Users\weiye\Documents\wwy sync\待整理word题目\2022全国各高考1.tex" +editor = "20230531\t王伟叶" +indexed = True IndexDescription = "试题" import os,re,json 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 95352c56..5d450499 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 @@ -1,3 +1,16 @@ +20230531 2022届全国各地高考试题汇编 +problems_dict = { +"2022届全国高考新高考I卷":"017244:017265", +"2022届全国高考新高考II卷":"017266:017287", +"2022届全国高考浙江卷":"017288:017309", +"2022届全国高考北京卷":"017310:017330", +"2022届全国高考甲卷理科":"017331:017353", +"2022届全国高考甲卷文科":"017354:017376", +"2022届全国高考乙卷理科":"017377:017399", +"2022届全国高考乙卷文科":"017400:017422" +} + + 20230428 2010年至今秋考试题汇编 problems_dict = { "2010年秋考":"011578:011600", diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 75888202..10a9b002 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -444585,6 +444585,3586 @@ "space": "4em", "unrelated": [] }, + "017244": { + "id": "017244", + "content": "若集合$M=\\{x | \\sqrt{x}<4\\}$, $N=\\{x | 3 x \\geq 1\\}$, 则$M \\cap N=$\n\\bracket{20}.\n\\fourch{$\\{x | 0 \\leq x<2\\}$}{$\\{x | \\dfrac{1}{3} \\leq x<2\\}$}{$\\{x | 3 \\leq x<16\\}$}{$\\{x | \\dfrac{1}{3} \\leq x<16\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题1", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017245": { + "id": "017245", + "content": "若$\\mathrm{i}(1-z)=1$, 则$z+\\overline {z}=$\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$1$}{$2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题2", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017246": { + "id": "017246", + "content": "在$\\triangle ABC$中, 点$D$在边$AB$上, $BD=2DA$. 记$\\overrightarrow{CA}=m$, $\\overrightarrow{CD}=n$, 则$\\overrightarrow{CB}=$\\bracket{20}.\n\\fourch{$3 m-2 n$}{$-2 m+3 n$}{$3 m+2 n$}{$2 m+3 n$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题3", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017247": { + "id": "017247", + "content": "南水北调工程缓解了北方一些地区水资源短缺问题, 其中一部分水蓄入某水库, 已知该水库水位为海拔$148.5 \\mathrm{m}$时, 相应水面的面积为$140 \\text{km}^2$; 水位为海拔$157.5 \\mathrm{m}$时, 相应水面的面积为$180 \\text{km}^2$. 将该水库在这两个水位间的形状看作一个棱台, 则该水库水位从海拔$148.5 \\mathrm{m}$上升到$157.5 \\mathrm{m}$时, 增加的水量约为\\bracket{20}.($\\sqrt{7}\\approx 2.65$)\n\\fourch{$1.0 \\times 10^9 \\mathrm{m}^3$}{$1.2 \\times 10^9 \\mathrm{m}^3$}{$1.4 \\times 10^9 \\mathrm{m}^3$}{$1.6 \\times 10^9 \\mathrm{m}^3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题4", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017248": { + "id": "017248", + "content": "从$2$至$8$的$7$个整数中随机取$2$个不同的数, 则这$2$个数互质的概率为\\bracket{20}\n\\fourch{$\\dfrac{1}{6}$}{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{2}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题5", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017249": { + "id": "017249", + "content": "记函数$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{4})+b$($\\omega>0$)的最小正周期为$T$. 若$\\dfrac{2}{3} \\pi0$)上, 过点$B(0,-1)$的直线交$C$于$P, Q$两点, 则\\blank{50}.\\\\\n\\textcircled{1} $C$的准线为$y=-1$; \\textcircled{2} 直线$AB$与$C$相切; \\textcircled{3} $|OP| \\cdot|OQ|>|OA|^2$; \\textcircled{4} $|BP| \\cdot|BQ|>|BA|^2$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题11", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017255": { + "id": "017255", + "content": "已知函数$f(x)$及其导函数$f'(x)$的定义域均为$\\mathbf{R}$, 记$g(x)=f'(x)$. 若$f(\\dfrac{3}{2}-2 x), g(2+x)$均为偶函数, 则\\blank{50}.\\\\\n\\textcircled{1} $f(0)=0$; \\textcircled{2} $g(-\\dfrac{1}{2})=0$; \\textcircled{3} $f(-1)=f(4)$; \\textcircled{4} $g(-1)=g(2)$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题12", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017256": { + "id": "017256", + "content": "$(1-\\dfrac{y}{x})(x+y)^8$的展开式中$x^2 y^6$的系数为\\blank{50}(用数字作答).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题13", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017257": { + "id": "017257", + "content": "写出与圆$x^2+y^2=1$和$(x-3)^2+(y-4)^2=16$都相切的一条直线的方程: \\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题14", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017258": { + "id": "017258", + "content": "若曲线$y=(x+a) \\mathrm{e}^x$有两条过坐标原点的切线, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题15", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017259": { + "id": "017259", + "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), $C$的上顶点为$A$, 两个焦点为$F_1, F_2$, 离心率为$\\dfrac{1}{2}$, 过$F_1$且垂直于$AF_2$的直线与$C$交于$D, E$两点, $|DE|=6$, 则$\\triangle ADE$的周长是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题16", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017260": { + "id": "017260", + "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 已知$a_1=1$, $\\{\\dfrac{S_n}{a_n}\\}$是公差为$\\dfrac{1}{3}$的等差数列.\\\\\n(1) 求$\\{a_n\\}$得通项公式;\\\\\n(2) 证明: $\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\cdots+\\dfrac{1}{a_n}<2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题17", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017261": { + "id": "017261", + "content": "记$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$, 已知$\\dfrac{\\cos A}{1+\\sin A}=\\dfrac{\\sin 2B}{1+\\cos 2B}$.\\\\\n(1) 若$C=\\dfrac{2 \\pi}{3}$, 求$B$;\\\\\n(2) 求$\\dfrac{a^2+b^2}{c^2}$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题18", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017262": { + "id": "017262", + "content": "如图, 直三棱柱$ABC-A_1B_1C_1$的体积为$4$, $\\triangle A_1BC$的面积为$2 \\sqrt{2}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)!0.5!(C)$) node [above] {$D$} coordinate (D);\n\\draw (A)--(A_1)(B)--(B_1)(C)--(C_1)(A)--(B)--(C)(A_1)--(B_1)--(C_1)--cycle(A_1)--(B);\n\\draw [dashed] (A)--(C)--(A_1)(A)--(D)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$A$到平面$A_1BC$的距离;\\\\\n(2) 设$D$为$A_1C$的中点, $AA_1=AB$, 平面$A_1BC \\perp$平面$ABB_1A_1$, 求二面角$A-BD-C$的正弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题19", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017263": { + "id": "017263", + "content": "一医疗团队为研究某地的一种地方性疾病与当地居民的卫生习惯 (卫生习惯分为良好和不够良好两类) 的关系, 在已患该疾病的病例中随机调查了$100$例 (称为病例组), 同时在未患该疾病的人群中随机调査了$100$人 (称为对照组), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 不够良好 & 良好 \\\\\n\\hline 病例组 & 40 & 60 \\\\\n\\hline 对照组 & 10 & 90 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 能否有$99 \\%$的把握认为患该疾病群体与未患该疾病群体的卫生习惯有差异?\\\\\n(2) 从该地的人群中任选一人, $A$表示事件``选到的人卫生习惯不够良好'', $B$表示事件``选到的人患有该疾病'', $\\dfrac{P(B | A)}{P(\\overline {B} | A)}$与$\\dfrac{P(B | \\overline {A})}{P(\\overline {B} | \\overline {A})}$的比值是卫生习惯不够良好对患该疾病风险程度的一项度量指标, 记该指标为$R$.\\\\\n(I) 证明: $R=\\dfrac{P(A | B)}{P(\\overline {A} | B)} \\cdot \\dfrac{P(\\overline {A} | \\overline {B})}{P(A | \\overline {B})}$;\\\\\n(II) 利用该调査数据, 给出$P(A | B)$, $P(A | \\overline {B})$的估计值, 并利用(I)的结果给出$R$的估计值.\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline$P(\\chi^2 \\geq k)$& 0.050 & 0.010 & 0.001 \\\\\n\\hline$k$& 3.841 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题20", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017264": { + "id": "017264", + "content": "已知点$A(2,1)$在双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{a^2-1}=1$($a>1$)上, 直线$l$交$C$于$P, Q$两点, 直线$AP, AQ$的斜率之和为$0$.\\\\\n(1) 求$l$的斜率;\\\\\n(2) 若$\\tan \\angle PAQ=2 \\sqrt{2}$, 求$\\triangle PAQ$的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题21", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017265": { + "id": "017265", + "content": "已知函数$f(x)=\\mathrm{e}^x-a x$和$g(x)=a x-\\ln x$有相同的最小值.\\\\\n(1) 求$a$;\\\\\n(2) 证明: 存在直线$y=b$, 其与两条曲线$y=f(x)$和$y=g(x)$共有三个不同的交点, 并且从左到右的三个交点的横坐标成等差数列.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考I卷试题22", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017266": { + "id": "017266", + "content": "已知集合$A=\\{-1,1,2,4\\}$, $B=\\{x \\| x-1 | \\leq 1\\}$, 则$A \\cap B=$\\bracket{20}.\n\\fourch{$\\{-1,2\\}$}{$\\{1,2\\}$}{$\\{1,4\\}$}{$\\{-1,4\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题1", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017267": { + "id": "017267", + "content": "$(2+2 \\mathrm{i})(1-2 \\mathrm{i})=$\\bracket{20}.\n\\fourch{$-2+4 \\mathrm{i}$}{$-2-4 \\mathrm{i}$}{$6+2 \\mathrm{i}$}{$6-2 \\mathrm{i}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题2", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017268": { + "id": "017268", + "content": "中国的古建筑不仅是挡风遮雨的住处, 更是美学和哲学的体现. 如图是某古建筑物的剖面图, 其中$DD_1$, $CC_1$, $BB_1$, $AA_1$是举, $OD_1$, $DC_1$, $CB_1$, $BA_1$是相等的步, 相邻桁的举步之比分别为$\\dfrac{DD_1}{OD_1}=0.5$, $\\dfrac{CC_1}{DC_1}=k_1$, $\\dfrac{BB_1}{CB_1}=k_2$, $\\dfrac{AA_1}{BA_1}=k_3$, 已知$k_1, k_2, k_3$成公差为$0.1$的等差数列, 且直线$OA$的斜率为$0.725$, 则$k_3=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (9.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) node [below] {$D_1$} coordinate (D_1);\n\\draw (D_1) ++ (0,0.5) node [below right] {$D$} coordinate (D);\n\\draw (D) ++ (1,0) node [below] {$C_1$} coordinate (C_1);\n\\draw (C_1) ++ (0,0.7) node [below right] {$C$} coordinate (C);\n\\draw (C) ++ (1,0) node [below] {$B_1$} coordinate (B_1);\n\\draw (B_1) ++ (0,0.8) node [below right] {$B$} coordinate (B);\n\\draw (B) ++ (1,0) node [below] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (0,0.9) node [above] {$A$} coordinate (A);\n\\draw (D_1)--(D) (C_1)--(C) (B_1)--(B) (A_1) -- (A);\n\\draw (D) --++ (6,0) coordinate (D_2) (C) --++ (4,0) coordinate (C_2) (B) --++ (2,0) coordinate (B_2);\n\\draw (B_2) --++ (0,-0.8) (C_2) --++ (0,-0.7) (D_2) --++ (0,-0.5);\n\\draw (O)--(D)--(C)--(B)--(A)--(B_2)--(C_2)--(D_2)--(8,0);\n\\draw [dashed] (O)--(A);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$0.75$}{$0.8$}{$0.85$}{$0.9$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题3", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017269": { + "id": "017269", + "content": "已知$\\overrightarrow {a}=(3,4)$, $\\overrightarrow {b}=(1,0)$, $\\overrightarrow {c}=\\overrightarrow {a}+t \\overrightarrow {b}$, $\\langle\\overrightarrow {a}, \\overrightarrow {c}\\rangle=\\langle \\overrightarrow {b}, \\overrightarrow {c}\\rangle$, 则$t=$\\bracket{20}.\n\\fourch{$-6$}{$-5$}{$5$}{$6$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题4", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017270": { + "id": "017270", + "content": "有甲乙丙丁戊$5$名同学站成一排参加文艺汇演, 若甲不站在两端, 丙和丁相邻的不同排列方式有多少种\\bracket{20}.\n\\fourch{$12$种}{$24$种}{$36$种}{$48$种}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题5", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017271": { + "id": "017271", + "content": "角$\\alpha, \\beta$满足$\\sin (\\alpha+\\beta)+\\cos (\\alpha+\\beta)=2 \\sqrt{2} \\cos (\\alpha+\\dfrac{\\pi}{4}) \\sin \\beta$, 则\\bracket{20}.\n\\fourch{$\\tan (\\alpha+\\beta)=1$}{$\\tan (\\alpha+\\beta)=-1$}{$\\tan (\\alpha-\\beta)=1$}{$\\tan (\\alpha-\\beta)=-1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题6", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017272": { + "id": "017272", + "content": "正三棱台高为$1$, 上下底边长分别是$3 \\sqrt{3}$和$4 \\sqrt{3}$, 所有顶点在同一球面上, 则球的表面积是\\bracket{20}.\n\\fourch{$100 \\pi$}{$128 \\pi$}{$144 \\pi$}{$192 \\pi$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题7", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017273": { + "id": "017273", + "content": "若函数$f(x)$的定义域为$\\mathbf{R}$, 且$f(x+y)+f(x-y)=f(x) f(y)$, $f(1)=1$, 则$\\displaystyle\\sum_{k=1}^{22} f(k)=$\\bracket{20}.\n\\fourch{$-3$}{$-2$}{$0$}{$1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题8", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017274": { + "id": "017274", + "content": "函数$f(x)=\\sin (2 x+\\varphi)$($0<\\varphi<\\pi$)的图像关于$(\\dfrac{2 \\pi}{3}, 0)$中心对称, 则\\blank{50}.\\\\\n\\textcircled{1} $y=f(x)$在$(0, \\dfrac{5 \\pi}{12})$单调递减;\\\\ \\textcircled{2} $y=f(x)$在$(-\\dfrac{\\pi}{12}, \\dfrac{11 \\pi}{12})$有$2$个极值点;\\\\\n\\textcircled{3} 直线$x=\\dfrac{7 \\pi}{6}$是一条对称轴;\\\\\n\\textcircled{4} 直线$y=\\dfrac{\\sqrt{3}}{2}-x$是一条切线.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题9", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017275": { + "id": "017275", + "content": "已知$O$为坐标原点, 过抛物线$C: y^2=2 p x (p>0)$的焦点$F$的直线与$C$交于$A, B$两点, 点$A$在第一象限, 点$M(p, 0)$, 若$|AF|=|AM|$, 则\\blank{50}.\\\\\n\\textcircled{1} 直线$AB$的斜率为$2 \\sqrt{6}$; \\textcircled{2} $|OB|=|OF|$; \\textcircled{3} $|AB|>4|OF|$; \\textcircled{4} $\\angle OAM+\\angle OBM<180^{\\circ}$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题10", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017276": { + "id": "017276", + "content": "如图, 四边形$ABCD$为正方形, $ED \\perp$平面$ABCD$, $FB\\parallel ED$, $AB=ED=2FB$, 记三棱锥$E-ACD$, $F-ABC$, $F-ACE$的体积分别为$V_1$, $V_2$, $V_3$, 则\\blank{50}.\\\\\n\\textcircled{1} $V_3=2V_2$; \\textcircled{2} $V_3=V_1$; \\textcircled{3} $V_3=V_1+V_2$; \\textcircled{4} $2V_3=3V_1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$E$} coordinate (E);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (D) ++ (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (B) ++ (0,1,0) node [left] {$F$} coordinate (F);\n\\draw (E)--(A)--(B)--(C)--cycle;\n\\draw (B)--(F)--(E)(A)--(F)--(C);\n\\draw [dashed] (E)--(D)--(A)(D)--(C)(A)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题11", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017277": { + "id": "017277", + "content": "若实数$x, y$满足$x^2+y^2-x y=1$, 则\\blank{50}.\\\\\n\\textcircled{1} $x+y \\leq 1$; \\textcircled{2} $x+y \\geq-2$; \\textcircled{3} $x^2+y^2 \\leq 2$; \\textcircled{4} $x^2+y^2 \\geq 1$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题12", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017278": { + "id": "017278", + "content": "已知随机变量$X$服从正态分布$N(2, \\sigma^2)$, 且$P(22.5)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题13", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017279": { + "id": "017279", + "content": "写出曲线$y=\\ln |x|$过坐标原点的切线方程: \\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题14", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017280": { + "id": "017280", + "content": "已知点$A(-2,3)$, $B(0, a)$, 若直线$AB$关于$y=a$的对称直线与圆$(x+3)^2+(y+2)^2=1$存在公共点, 则实数$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题15", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017281": { + "id": "017281", + "content": "已知椭圆$\\dfrac{x^2}{6}+\\dfrac{y^2}{3}=1$, 直线$l$与椭圆在第一象限交于$A, B$, 与$x$轴, $y$轴分别交于$M, N$, 且$|MA|=|NB|$, $|MN|=2 \\sqrt{3}$, 则直线$l$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题16", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017282": { + "id": "017282", + "content": "已知$\\{a_n\\}$为等差数列, $\\{b_n\\}$是公比为$2$的等比数列, 且$a_2-b_2=a_3-b_3=b_4-a_4$.\\\\\n(1) 证明: $a_1=b_1$;\\\\\n(2) 求集合$\\{k | b_k=a_m+a_1,\\ 1 \\leq m \\leq 500\\}$中元素的个数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题17", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017283": { + "id": "017283", + "content": "记$\\triangle ABC$的三个内角分别为$A$、$B$、$C$, 其对边分别为$a, b, c$, 分别以$a, b, c$为边长的三个正三角形的面积依次为$S_1, S_2, S_3$, 已知$S_1-S_2+S_3=\\dfrac{\\sqrt{3}}{2}$, $\\sin B=\\dfrac{1}{3}$.\\\\\n(1) 求$\\triangle ABC$的面积;\\\\\n(2) 若$\\sin A \\sin C=\\dfrac{\\sqrt{2}}{3}$, 求$b$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题18", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017284": { + "id": "017284", + "content": "在某地区进行流行病调查, 随机调查了$100$名某种疾病患者的年龄, 得到如下的样本数据频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.06, yscale = 180]\n\\draw [->] (0,0) -- (105,0) node [below] {年龄/岁};\n\\draw [->] (0,0) -- (0,0.03) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0/0.001,10/0.002,20/0.012,30/0.017,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0/0.001,20/0.012,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (90,0) node [below] {$90$};\n\\end{tikzpicture}\n\\end{center}\n(1) 估计该地区这种疾病患者的平均年龄 (同一组中的数据用该组区间的中点值代表);\\\\\n(2) 估计该地区一人患这种疾病年龄在区间$[20,70)$的概率.\\\\\n(3) 已知该地区这种疾病的患病率为$0.1 \\%$, 该地区的年龄位于区间$[40,50)$的人口占该地区总人口的$16 \\%$, 从该地区任选一人, 若此人年龄位于区间$[40,50)$, 求此人患该种疾病的概率. (样本数据中的患者年龄位于各地区的频率作为患者年龄位于该区间的概率, 精确到 $0.0001$)\\\\", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题19", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017285": { + "id": "017285", + "content": "如图, $PO$是三棱锥$P-ABC$的高, $PA=PB$, $AB \\perp AC$, $E$是$PB$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(135:0.5cm)}, scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({4*sqrt(3)},0,0) node [right] {$B$} coordinate (B);\n\\draw ({2*sqrt(3)},0,2) node [right] {$O$} coordinate (O);\n\\draw (0,0,12) node [left] {$C$} coordinate (C);\n\\draw (O) ++ (0,3,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(B)$) node [right] {$E$} coordinate (E);\n\\draw (C)--(A)--(B)--(P)--cycle(A)--(E)(A)--(P);\n\\draw [dashed] (P)--(O)(C)--(E)(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $OE\\parallel$平面$PAC$;\\\\\n(2) 若$\\angle ABO=\\angle CBO=30^{\\circ}$, $PO=3$, $PA=5$, 求二面角$C-AE-B$的正弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题20", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017286": { + "id": "017286", + "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的右焦点为$F(2,0)$, 渐近线方程为$y= \\pm \\sqrt{3} x$.\\\\\n(1) 求$C$的方程;\\\\\n(2) 过$F$的直线与$C$的两条渐近线分别交于$A, B$两点, 点$P(x_1, y_1), Q(x_2, y_2)$在$C$上, 且$x_1>x_2>0$, $y_1>0$, 过$P$且斜率为$-\\sqrt{3}$的直线与过$Q$且斜率为$\\sqrt{3}$的直线交于点$M$. 请从下面\\textcircled{1}\\textcircled{2}\\textcircled{3}中选取两个作为条件, 证明另外一个条件成立:\n\\textcircled{1}$M$在$AB$上; \\textcircled{2}$PQ\\parallel AB$; \\textcircled{3}$|MA|=|\\mathrm{MB}|$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题21", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017287": { + "id": "017287", + "content": "已知函数$f(x)=x \\mathrm{e}^{a x}-\\mathrm{e}^x$.\\\\\n(1) 当$a=1$时, 讨论$f(x)$的单调性;\\\\\n(2) 当$x>0$时, $f(x)<-1$, 求$a$的取值范围;\\\\\n(3) 设$n \\in \\mathbf{N}$, $n\\ge 1$, 证明: $\\dfrac{1}{\\sqrt{1^2+1}}+\\dfrac{1}{\\sqrt{2^2+2}}+\\cdots+\\dfrac{1}{\\sqrt{n^2+n}}>\\ln (n+1)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考新高考II卷试题22", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017288": { + "id": "017288", + "content": "设集合$A=\\{1,2\\}$, $B=\\{x | 2,4,6\\}$则$A \\cup B=$\\bracket{20}.\n\\fourch{$\\{2\\}$}{$\\{1,2\\}$}{$\\{2,4,6\\}$}{$\\{1,2,4,6\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题1", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017289": { + "id": "017289", + "content": "已知$a, b \\in \\mathbf{R}$, $a+3 \\mathrm{i}=(b+\\mathrm{i}) \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$a=$\\bracket{20}.\n\\fourch{$a=1$, $b=-3$}{$a=-1$, $b=3$}{$a=-1$, $b=-3$}{$a=1$, $b=3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题2", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017290": { + "id": "017290", + "content": "若实数$x, y$满足约束条件$\\begin{cases}x-2 \\geq 0,\\\\ 2 x+y-7 \\leq 0,\\\\ x-y-2 \\leq 0,\\end{cases}$ 则$z=3 x+4 y$的最大值是 \\bracket{20}.\n\\fourch{$20$}{$18$}{$13$}{$6$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题3", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017291": { + "id": "017291", + "content": "设$x \\in \\mathbf{R}$, 则``$\\sin x=1$''是``$\\cos x=0$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题4", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017292": { + "id": "017292", + "content": "某几何体的三视图如图所示(单位: $\\text{cm}$), 则该几何体的体积(单位: $\\text{cm}^3)$是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0) -- (4,0) -- (3,2) -- (1,2) -- cycle (1,2) -- (1,4) arc (180:0:1) -- (3,2) (1,4) -- (3,4);\n\\foreach \\i in {0,1,3,4}\n{\\draw (\\i,-0.3) --++ (0,-0.4);};\n\\draw [->] (-0.5,-0.5) -- (0,-0.5);\n\\draw [->] (4.5,-0.5) -- (4,-0.5);\n\\draw [<->] (1,-0.5) -- (3,-0.5);\n\\draw (0.5,-0.5) node {$1$} (3.5,-0.5) node {$1$};\n\\draw (2,-0.5) node [fill = white] {$2$};\n\\draw (2,-1.5) node {正视图};\n\\draw (6,0) -- (10,0) -- (9,2) -- (7,2) -- cycle (7,2) -- (7,4) arc (180:0:1) -- (9,2) (7,4) -- (9,4);\n\\foreach \\i in {6,7,9,10}\n{\\draw (\\i,-0.3) --++ (0,-0.4);};\n\\draw [->] (5.5,-0.5) -- (6,-0.5);\n\\draw [->] (10.5,-0.5) -- (10,-0.5);\n\\draw [<->] (7,-0.5) -- (9,-0.5);\n\\draw (6.5,-0.5) node {$1$} (9.5,-0.5) node {$1$};\n\\draw (8,-0.5) node [fill = white] {$2$};\n\\draw (8,-1.5) node {侧视图};\n\\foreach \\i in {0,2,4,5}\n{\\draw (4.7,\\i) -- (5.3,\\i);};\n\\draw [<->] (5,0) -- (5,2) node [midway, fill = white] {$2$};\n\\draw [<->] (5,2) -- (5,4) node [midway, fill = white] {$2$};\n\\draw (5,4.5) node {$1$};\n\\draw [->] (5,5.5) -- (5,5);\n\\draw (2,-4) circle (1) circle (2);\n\\draw (2,-6.5) node {俯视图};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$22 \\pi$}{$8 \\pi$}{$\\dfrac{22}{3} \\pi$}{$\\dfrac{16}{3} \\pi$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题5", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017293": { + "id": "017293", + "content": "为了得到$y=2 \\sin 3 x$的图像, 只要把函数$y=2 \\sin (3 x+\\dfrac{\\pi}{5})$图像上所有点\\bracket{20}.\n\\twoch{向左平移$\\dfrac{\\pi}{5}$个单位长度}{向右平移$\\dfrac{\\pi}{5}$个单位长度}{向左平移$\\dfrac{\\pi}{15}$个单位长度}{向右平移$\\dfrac{\\pi}{15}$个单位长度}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题6", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017294": { + "id": "017294", + "content": "已知$2^a=5$, $\\log _83=b$, 则$4^{a-3 b}=$\\bracket{20}.\n\\fourch{$25$}{$5$}{$\\dfrac{25}{9}$}{$\\dfrac{25}{9}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题7", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017295": { + "id": "017295", + "content": "如图, 已知正三棱柱$ABC-A_1B_1C_1$, $AC=AA_1$, $E, F$分别是棱$BC, A_1C_1$上的点. 记$EF$与$AA_1$所成的角为$\\alpha$, $EF$与平面$ABC$所成的角为$\\beta$, 二面角$F-BC-A$的平面角为$\\gamma$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\draw ($(B)!0.6!(C)$) node [below right] {$E$} coordinate (E);\n\\draw ($(A_1)!0.3!(C_1)$) node [above] {$F$} coordinate (F);\n\\draw [dashed] (E)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\alpha \\leq \\beta \\leq \\gamma$}{$\\beta \\leq \\alpha \\leq \\gamma$}{$\\beta \\leq \\gamma \\leq \\alpha$}{$\\alpha \\leq \\gamma \\leq \\beta$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题8", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017296": { + "id": "017296", + "content": "已知$a, b \\in \\mathbf{R}$, 若对任意$x \\in \\mathbf{R}$, $a|x-b|+|x-4|-|2 x-5| \\geq 0$, 则\\bracket{20}.\n\\fourch{$a \\leq 1$, $b \\geq 3$}{$a \\leq 1$, $b \\leq 3$}{$a \\geq 1$, $b \\geq 3$}{$a \\geq 1$, $b \\leq 3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题9", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017297": { + "id": "017297", + "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=a_n-\\dfrac{1}{3} a_n^2$($n \\in \\mathbf{N}$, $n\\ge 1$), 则\\bracket{20}.\n\\fourch{$2<100 a_{100}<\\dfrac{5}{2}$}{$\\dfrac{5}{2}<100 a_{100}<3$}{$3<100 a_{100}<\\dfrac{7}{2}$}{$\\dfrac{7}{2}<100 a_{100}<4$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题10", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017298": { + "id": "017298", + "content": "我国南宋著名数学家秦九韶, 发现了从三角形三边求面积的公式, 他把这种方法称为``三斜求积'', 它填补了我国传统数学的一个空白. 如果把这个方法写成公式, 就是$S=$$\\sqrt{\\dfrac{1}{4}[c^2 a^2-(\\dfrac{c^2+a^2-b^2}{2})^2]}$, 其中$a, b, c$是三角形的三边, $S$是三角形的面积. 设某三角形的三边$a=\\sqrt{2}$, $b=\\sqrt{3}$, $c=2$, 则该三角形的面积$S=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题11", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017299": { + "id": "017299", + "content": "已知多项式$(x+2)(x-1)^4=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4+a_5 x^5$, 则$a_2=$\\blank{50}, $a_1+a_2+a_3+a_4+a_5=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题12", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017300": { + "id": "017300", + "content": "若$3 \\sin \\alpha-\\cos \\beta=\\sqrt{10}$, $\\alpha+\\beta=\\dfrac{\\pi}{2}$, 则$\\sin \\alpha=$\\blank{50}, $\\cos 2 \\beta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题13", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017301": { + "id": "017301", + "content": "已知$f(x)=\\begin{cases}-x^2+2, & x \\leq 1, \\\\ x+\\dfrac{1}{x}-1, & x>1,\\end{cases}$则$f(f(\\dfrac{1}{2}))=$\\blank{50}; 若当$x \\in[a, b]$时, $1 \\leq f(x) \\leq 3$, 则$b-a$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题14", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017302": { + "id": "017302", + "content": "现有$7$张卡片, 分别写上数字$1,2,2,3,4,5,6$. 从这$7$张卡片中随机抽取$3$张, 记所抽取卡片上数字的最小值为$\\xi$, 则$P(\\xi=2)=$\\blank{50}, $E[\\xi ]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题15", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017303": { + "id": "017303", + "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左焦点为$F$, 过$F$且斜率为$\\dfrac{b}{4 a}$的直线交双曲线于点$A(x_1, y_1)$, 交双曲线的渐近线于点$B(x_2, y_2)$且$x_1<0=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (5,0,0) node [below] {$B$} coordinate (B);\n\\draw (5,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (C) ++ (-3,0,0) node [below] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(C)$) node [right] {$N$} coordinate (N);\n\\draw (N) ++ (0,1.5,0) node [above] {$F$} coordinate (F);\n\\draw (F) ++ (-1,0,0) node [above] {$E$} coordinate (E);\n\\draw ($(E)!0.5!(A)$) node [left] {$M$} coordinate (M);\n\\draw (A)--(B)--(F)--(E)--cycle;\n\\draw (B)--(C)--(F)(N)--(F)(B)--(M);\n\\draw [dashed] (C)--(D)--(A)(D)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $FN \\perp AD$;\\\\\n(2) 求直线$BM$与平面$ADE$所成角的正弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题19", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017307": { + "id": "017307", + "content": "已知等差数列$\\{a_n\\}$的首项$a_1=-1$, 公差$d>1$, 记$\\{a_n\\}$的前$n$项和为$S_n$($n \\in \\mathbf{N}$, $n\\ge 1$).\\\\\n(1) 若$S_4-2 a_2 a_3+6=0$, 求$S_n$;\\\\\n(2) 若对于每个$n \\in \\mathbf{N}$, $n\\ge 1$, 存在实数$c_n$, 使$a_n+c_n, a_{n+1}+4 c_n, a_{n+2}+15 c_n$成等比数列, 求$d$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题20", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017308": { + "id": "017308", + "content": "如图, 已知椭圆$\\dfrac{x^2}{12}+y^2=1$. 设$A, B$是椭圆上异于$P(0,1)$的两点, 且点$Q(0, \\dfrac{1}{2})$在线段$AB$上, 直线$PA, PB$分别交直线$y=-\\dfrac{1}{2} x+3$于$C, D$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-4,0) -- (6.5,0) node [above] {$x$};\n\\draw [->] (0,-1.5) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [name path = elli, draw] (0,0) ellipse ({2*sqrt(3)} and 1);\n\\draw (110:{sqrt(12)} and 1) node [above] {$P$} coordinate (P);\n\\draw (0,0.5) node [below right] {$Q$} coordinate (Q);\n\\path [name path = line, draw] ($(6,0)!1.1!(0,3)$) -- ($(6,0)!-0.1!(0,3)$);\n\\draw ({-2*sqrt(3)},0) node [below left] {$A$} coordinate (A);\n\\path [name path = AB] (A) -- ($(A)!2!(Q)$);\n\\path [name intersections = {of = elli and AB, by = B}];\n\\draw (B) node [below] {$B$};\n\\path [name path = AP] (A) -- ($(A)!2.5!(P)$);\n\\path [name path = BP] (P) -- ($(P)!2!(B)$);\n\\path [name intersections = {of = AP and line, by = C}];\n\\path [name intersections = {of = BP and line, by = D}];\n\\draw (A)--(C) node [above] {$C$} (P) -- (D) node [above] {$D$};\n\\draw (A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$P$到椭圆点的距离的最大值;\\\\\n(2) 求$|CD|$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考浙江卷试题21", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017309": { + "id": "017309", + "content": "设函数$f(x)=\\dfrac{\\mathrm{e}}{2 x}+\\ln x$($x>0$).\\\\\n(1) 求$f(x)$的单调区间;\\\\\n(2) 已知$a, b \\in \\mathbf{R}$, 曲线$y=f(x)$上不同的三点$(x_1, f(x_1)),(x_2, f(x_2)),(x_3, f(x_3))$处的切线都经过点$(a, b)$. 证明:\\\\\n(I) 若$a>\\mathrm{e}$, 则$0N_0$时, $a_n>0$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题6", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017316": { + "id": "017316", + "content": "在北京冬奥会上, 国家速滑馆``冰丝带''使用高效环保的二氧化碳跨临界直制冰技术, 为实现绿色东奥作出了贡献, 如图描述了一定条件下二氧化碳所处的状态与$T$和$\\lg P$的关系, 其中$T$表示温度, 单位是$\\text{K}$, $P$表示压强, 单位是$\\text{bar}$. 下列结论中正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (4.5,0) node [below] {$T$};\n\\draw [->] (0,0) -- (0,4.5) node [left] {$\\lg P$};\n\\foreach \\i/\\j in {0/200,1/250,2/300,3/350,4/400}\n{\\draw (\\i,0) --++ (0,-0.2) node [below] {$\\j$};\n\\draw (0,\\i) --++ (-0.2,0) node [left] {$\\i$};};\n\\draw (4,0) -- (4,4) -- (0,4);\n\\draw [domain = 0:2] plot (\\x,{2*ln(\\x+1.1)/ln(3.1)});\n\\draw (0.4,{2*ln(0.4+1.1)/ln(3.1)}) coordinate (A);\n\\draw (A) -- (0.5,2.5);\n\\draw [domain = 0.5:4] plot (\\x,{1.5/ln(8)*(ln(\\x)-ln(4))+4});\n\\draw [dashed] (2,2) --++ (2,0) (2,2) -- (2,{1.5/ln(8)*(ln(2)-ln(4))+4});\n\\draw (2.5,1) node {气态};\n\\draw (3,3) node {超临界状态};\n\\draw (0.5,3.5) node {固态};\n\\draw (1.2,2.2) node {液态}; \n\\end{tikzpicture}\n\\end{center}\n\\onech{当$T=220$, $P=1026$时, 二氧化碳处于液态}{当$T=270$, $P=128$时, 二氧化碳处于气态}{当$T=300$, $P=9987$时, 二氧化碳处于超临界状态}{当$T=360$, $P=729$时, 二氧化碳处于超临界状态}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题7", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017317": { + "id": "017317", + "content": "若$(2 x-1)^4=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$, 则$a_0+a_2+a_4=$\\bracket{20}.\n\\fourch{$40$}{$41$}{$-40$}{$-41$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题8", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017318": { + "id": "017318", + "content": "已知正三棱锥$P-ABC$的$6$条棱长均为$6$, $S$是$\\triangle ABC$及其内部的点构成的集合, 设集合$T=\\{Q \\in S | PQ \\leq 5\\}$, 则$T$表示的区域的面积为\\bracket{20}.\n\\fourch{$\\dfrac{3 \\pi}{4}$}{$\\pi$}{$2 \\pi$}{$3 \\pi$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题9", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017319": { + "id": "017319", + "content": "在$\\triangle ABC$中, $AC=3$, $BC=4$, $\\angle C=90^{\\circ}$. $P$为$\\triangle ABC$所在平面内的动点, 且$PC=1$, 则$\\overrightarrow{PA} \\cdot \\overrightarrow{PB}$的取值范围是\\bracket{20}.\n\\fourch{$[-5,3]$}{$[-3,5]$}{$[-6,4]$}{$[-4,6]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题10", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017320": { + "id": "017320", + "content": "函数$f(x)=\\dfrac{1}{x}+\\sqrt{1-x}$的定义域是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题11", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017321": { + "id": "017321", + "content": "已知双曲线$y^2+\\dfrac{x^2}{m}=1$的渐近线方程为$y= \\pm \\dfrac{\\sqrt{3}}{3} x$, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题12", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017322": { + "id": "017322", + "content": "若函数$f(x)=A \\sin x-\\sqrt{3} \\cos x$的一个零点为$\\dfrac{\\pi}{3}$, 则$A=$\\blank{50}, $f(\\dfrac{\\pi}{12})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题13", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017323": { + "id": "017323", + "content": "设函数$f(x)=\\begin{cases}-a x+1, & x=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (0,0,2) node [below] {$C$} coordinate (C);\n\\draw (1.95,{sqrt(4-1.95*1.95)}) node [right] {$A$} coordinate (A);\n\\draw (A) ++ (0,2,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [left] {$C_1$} coordinate (C_1);\n\\draw ($(A)!0.5!(C)$) node [below] {$N$} coordinate (N);\n\\draw ($(B_1)!0.5!(A_1)$) node [above] {$M$} coordinate (M);\n\\draw (C)--(A)--(A_1)--(B_1)--(C_1)--cycle(A_1)--(C_1);\n\\draw [dashed] (C)--(B)--(A)(B)--(B_1)(B)--(N)--(M)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN\\parallel$平面$BCC_1B_1$;\\\\\n(2) 若$AB \\perp MN$, 求直线$AB$与平面$BMN$所成角的正弦值;\\\\\n(3) 若$BM=MN$, 求直线$AB$与平面$BMN$所成角的正弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题17", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017327": { + "id": "017327", + "content": "在校运会上, 只有甲、乙、丙三名同学参加铅球比赛, 比赛成绩达到$9.50 \\mathrm{m}$(含$9.50 \\text{m}$)以上的同学获优秀奖. 为预测优秀奖的人数及冠军得主, 收集了甲、乙、丙以往的比赛成绩, 并整理得到如下数据 (单位: $\\text{m}$):\\\\\n甲: $9.80,9.70,9.55,9.54,9.48,9.42,9.40,9.35,9.30,9.25$\\\\\n乙: $9.78,9.56,9.51,9.36,9.32,9.23$\\\\\n丙: $9.85,9,65,9.20,9.16$\\\\\n假设用频率估计概率, 且甲、乙、丙的比赛成绩相互独立.\\\\\n(1) 估计甲在校运动会铅球比赛中获得优秀奖的概率;\\\\\n(2) 设$X$是甲、乙、丙在校运动会铅球比赛中获得优秀奖的总人数, 估计$X$的数学期望$E[X[$;\\\\\n(3) 在校运动会铅球比赛中, 甲、乙、丙谁获得冠军的概率估计值最大? (结论不要求证明)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题18", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017328": { + "id": "017328", + "content": "已知椭圆$E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的一个顶点为$A(0,1)$, 焦距为$2 \\sqrt{3}$.\\\\\n(1) 求椭圆$E$的方程;\\\\\n(2) 过点$P(-2,1)$作斜率为$k$的直线与椭圆$E$交于不同的两点$B, C$, 直线$AB, AC$分别与$x$轴交于点$M, N$. 当$|MN|=2$时, 求$k$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题19", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017329": { + "id": "017329", + "content": "已知函数$f(x)=\\mathrm{e}^x \\ln (1+x)$.\\\\\n(1) 求曲线$y=f(x)$在$(0, f(0))$处的切线方程;\\\\\n(2) 设$g(x)=f'(x)$, 讨论$g(x)$在$[0,+\\infty)$上的单调性;\\\\\n(3) 证明: 对任意的$s, t \\in(0,+\\infty)$, 有$f(s+t)>f(s)+f(t)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题20", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017330": { + "id": "017330", + "content": "已知$Q: a_1, a_2, \\cdots, a_k$为有穷整数数列. 给定正整数$m$, 若对任意的$n \\in\\{1,2, \\cdots, m\\}$, 在$Q$中存在$a_1, a_{i+1}, a_{i+2}, \\cdots, a_{i+j}$($j \\geq 0$), 使得$a_i+a_{i+1}+a_{i+2}+\\cdots+a_{i+j}=n$, 则称$Q$为$m-$连续可表数列.\\\\\n(1) 判断$Q: 2,1,4$是否为$5-$连续可表数列? 是否为$6-$连续可表数列? 说明理由;\\\\\n(2) 若$Q: a_1, a_2, \\cdots, a_k$为$8-$连续可表数列, 求证: $k$的最小值为 $4$;\\\\\n(3) 若$Q: a_1, a_2, \\cdots, a_k$为$20-$连续可表数列, $a_1+a_2+\\cdots+a_k<20$, 求证: $k \\geq 7$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考北京卷试题21", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017331": { + "id": "017331", + "content": "若$z=-1+\\sqrt{3} \\mathrm{i}$, 则$\\dfrac{z}{z \\overline {z}-1}=$\\bracket{20}.\n\\fourch{$-1+\\sqrt{3} \\mathrm{i}$}{$-1-\\sqrt{3} \\mathrm{i}$}{$-\\dfrac{1}{3}+\\dfrac{\\sqrt{3}}{3} \\mathrm{i}$}{$-\\dfrac{1}{3}-\\dfrac{\\sqrt{3}}{3} \\mathrm{i}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题1", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017332": { + "id": "017332", + "content": "某社区通过公益讲座宣传中国非物质文化遗产保护知识. 为了解讲座效果, 随机抽取$10$位社区居民, 让他们在讲座前和讲座后各回答一份相关知识问卷, 这$10$位社区居民在讲座前和讲座后问卷答题的正确率如下图. 则下列选项正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.7, yscale = 0.08]\n\\draw [->] (0,55) -- (11,55);\n\\draw [->] (0,55) -- (0,56) -- (0.2,57) -- (-0.2,58) -- (0,59) -- (0,105);\n\\draw (0,55) node [below left] {$O$};\n\\foreach \\i in {1,2,...,10}\n{\\draw (\\i,55.5) -- (\\i,55) node [below] {$\\i$};};\n\\foreach \\i in {60,65,...,100}\n{\\draw [dotted] (10.5,\\i) -- (0,\\i) node [left] {$\\i\\%$};};\n\\draw (5.5,45) node {居民编号};\n\\draw (-2,80) node [rotate = 90] {正确率};\n\\filldraw (12,70) circle (0.05 and {7/16}) ++ (1,0) node {讲座后};\n\\filldraw (12,80) node {\\tiny$\\times$} node {\\tiny$+$} ++ (1,0) node {讲座前};\n\\foreach \\i/\\j/\\k in {1/65/90,2/60/85,3/70/80,4/60/90,5/65/85,6/75/85,7/90/95,8/85/100,9/80/85,10/95/100}\n{\\filldraw (\\i,\\j) node {\\tiny$\\times$} node {\\tiny$+$} (\\i,\\k) circle (0.05 and {7/16});};\n\\end{tikzpicture}\n\\end{center}\n\\onech{讲座前问卷答题的正确率的中位数小于$70 \\%$}{讲座后问卷答题的正确率的平均数大于$85 \\%$}{讲座前问卷答题的正确率的标准差小于讲座后正确率的标准差}{讲座后问卷答题的正确率的极差大于讲座前正确率的极差}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题2", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017333": { + "id": "017333", + "content": "设全集$U=\\{-2,-1,0,1,2,3\\}$, 集合$A=\\{-1,2\\}$, $B=\\{x | x^2-4 x+3=0\\}$, 则$\\overline{A \\cup\nB}=$\\bracket{20}.\n\\fourch{$\\{1,3\\}$}{$\\{0,3\\}$}{$\\{-2,1\\}$}{$\\{-2,0\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题3", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017334": { + "id": "017334", + "content": "如图, 网格纸上绘制的是一个多面体的三视图, 网格小正方形的边长为$1$, 则该多面体的体积为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\foreach \\i in {0,1,2,3,4,5,6}\n{\\draw [dashed,thin] (\\i,0) -- (\\i,4) (\\i+8,0) -- (\\i+8,4) (\\i,-6) -- (\\i,-2);};\n\\foreach \\i in {0,1,2,3,4}\n{\\draw [dashed,thin] (0,\\i) -- (6,\\i) (8,\\i) -- (14,\\i) (0,\\i-6) -- (6,\\i-6);};\n\\draw [ultra thick] (1,1) -- (5,1) -- (3,3) -- (1,3) -- cycle;\n\\draw [ultra thick] (10,1) rectangle (12,3);\n\\draw [ultra thick] (1,-5) rectangle (5,-3) (3,-5) -- (3,-3);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$8$}{$12$}{$16$}{$20$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题4", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017335": { + "id": "017335", + "content": "函数$y=(3^x-3^{-x}) \\cos x$在区间$[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}]$的图像大致为\\bracket{20}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{(pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{abs((pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180))});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{-(pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{-abs((pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180))});\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题5", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017336": { + "id": "017336", + "content": "当$x=1$时, 函数$f(x)=a \\ln x+\\dfrac{b}{x}$取得最大值$-2$, 则$f'(2)=$\\bracket{20}.\n\\fourch{$-1$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}{$1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题6", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017337": { + "id": "017337", + "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 已知$B_1D$与平面$ABCD$和平面$AA_1B_1B$所成的角均为$30^{\\circ}$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\def\\l{{sqrt(3)}}\n\\def\\m{1}\n\\def\\n{1}\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 (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (B_1)--(D)--(B);\n\\draw (A)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$AB=2AD$}{$AB$与平面$AB_1C_1D$所成的角为$30^{\\circ}$}{$AC=CB_1$}{$B_1D$与平面$BB_1C_1C$所成的角为$45^{\\circ}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题7", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017338": { + "id": "017338", + "content": "沈括的《梦溪笔谈》是中国古代科技史上的杰作, 其中收录了计算圆弧长度的``会圆术'', 如图, $\\overset\\frown{AB}$是以为$O$圆心, $OA$为半径的圆弧, $C$是$AB$的中点, $D$在$\\overset\\frown{AB}$上, $CD \\perp AB$. ``会圆术''给出$\\overset\\frown{AB}$的弧长的近似值$s$的计算公式: $s=AB+\\dfrac{CD^2}{OA}$. 当$OA=2$, $\\angle AOB=60^{\\circ}$时, $s=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (60:2) node [right] {$B$} coordinate (B);\n\\draw (120:2) node [left] {$A$} coordinate (A);\n\\draw (90:2) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [below] {$C$} coordinate (C);\n\\draw (A)--(O)--(B) arc (60:120:2) -- (B) (C)--(D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{11-3 \\sqrt{3}}{2}$}{$\\dfrac{11-4 \\sqrt{3}}{2}$}{$\\dfrac{9-3 \\sqrt{3}}{2}$}{$\\dfrac{9-4 \\sqrt{3}}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题8", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017339": { + "id": "017339", + "content": "甲、乙两个圆锥的母线长相等, 侧面展开图的圆心角之和为$2\\pi$, 侧面积分别为$S_{\\text{甲}}$和$S_{\\text{乙}}$, 体积分别为$V_{\\text{甲}}$和$V_{\\text{乙}}$, 若$\\dfrac{S_{\\text{甲}}}{{S_{\\text{乙}}}}=2$, 则$\\dfrac{V_{\\text{甲}}}{{V_{\\text{乙}}}}=$\\bracket{20}.\n\\fourch{$\\sqrt{5}$}{$2 \\sqrt{2}$}{$\\sqrt{10}$}{$\\dfrac{5 \\sqrt{10}}{4}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题9", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017340": { + "id": "017340", + "content": "椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左顶点为$A$, 点$P, Q$均在$C$上, 且关于$y$轴对称. 若直线$AP, AQ$的斜率之积为$\\dfrac{1}{4}$, 则$C$的离心率为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{3}}{2}$}{$\\dfrac{\\sqrt{2}}{2}$}{$\\dfrac{1}{2}$}{$\\dfrac{1}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题10", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017341": { + "id": "017341", + "content": "已知$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$区间在$(0, \\pi)$上恰有三个极值点, 两个零点, 则$\\omega$的取值范围是\\bracket{20}.\n\\fourch{$[\\dfrac{5}{3}, \\dfrac{13}{6})$}{$[\\dfrac{5}{3}, \\dfrac{19}{6})$}{$(\\dfrac{13}{6}, \\dfrac{8}{3}]$}{$(\\dfrac{13}{6}, \\dfrac{19}{6}]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题11", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017342": { + "id": "017342", + "content": "已知$a=\\dfrac{31}{32}$, $b=\\cos \\dfrac{1}{4}$, $c=4 \\sin \\dfrac{1}{4}$, 则\\bracket{20}.\n\\fourch{$c>b>a$}{$b>a>c$}{$a>b>c$}{$a>c>b$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题12", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017343": { + "id": "017343", + "content": "设向量$\\overrightarrow {a}, \\overrightarrow {b}$的夹角的余弦值为$\\dfrac{1}{3}$, 且$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=3$, 则$(2 \\overrightarrow {a}+\\overrightarrow {b}) \\cdot \\overrightarrow {b}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题13", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017344": { + "id": "017344", + "content": "若双曲线$y^2-\\dfrac{x^2}{m^2}=1$($m>0$)的渐近线与圆$x^2+y^2-4 y+3=0$相切, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题14", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017345": { + "id": "017345", + "content": "从正方体的$8$个顶点中任选$4$个, 则这$4$个点在同一平面上的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题15", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017346": { + "id": "017346", + "content": "已知$\\triangle ABC$中, 点$D$在边$BC$上, $\\angle ADB=120^{\\circ}$, $AD=2$, $CD=2BD$. 当$\\dfrac{AC}{AB}$取得最小值时, $BD=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题16", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017347": { + "id": "017347", + "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和. 已知$\\dfrac{2S_n}{n}+n=2 a_n+1$.\\\\\n(1) 证明: $\\{a_n\\}$是等差数列;\\\\\n(2) 若$a_4, a_7, a_9$成等比数列, 求$S_n$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题17", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017348": { + "id": "017348", + "content": "在四棱锥$P-ABCD$中, $PD \\perp$底面$ABCD$, $CD\\parallel AB$, $AD=DC=CB=1$, $AB=2$, $PD=\\sqrt{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5, z = {(245:0.5cm)}]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (0.5,0,{-0.5*sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (D) ++ (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (D) ++ (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(P)--cycle;\n\\draw [dashed] (A)--(D)--(C)(D)--(P)(D)--(B)(P)--(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $BD \\perp PA$;\\\\\n(2) 求$PD$与平面$PAB$的所成的角的正弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题18", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017349": { + "id": "017349", + "content": "甲、乙两个学校进行体育比赛, 比赛共设三个项目, 每个项目胜方得$10$分, 负方得$0$分, 没有平局. 三个项目比赛结束后, 总得分高的学校获得冠军. 已知甲学校在三个项目中获胜的概率分别为$0.5,0.4,0.8$, 各项目的比赛结果相互独立.\\\\\n(1) 求甲学校获得冠军的概率;\\\\\n(2) 用$X$表示乙学校的总得分, 求$X$的分布列与期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题19", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017350": { + "id": "017350", + "content": "设抛物线$C: y^2=2 p x$($p>0$)的焦点为$F$, 点$D(p, 0)$, 过$F$的直线交$C$于$M, N$两点, 当直线$MD \\perp x$轴时, $|MF|=3$.\\\\\n(1) 求$C$的方程;\\\\\n(2) 设直线$MD$、$ND$与$C$的另一个交点分别为$A, B$, 记直线$MN$、$AB$的倾斜角分别为$\\alpha$, $\\beta$, 当$\\alpha-\\beta$取得最大值时, 求直线$AB$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题20", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017351": { + "id": "017351", + "content": "已知函数$f(x)=\\dfrac{\\mathrm{e}^x}{x}-\\ln x+x-a$.\\\\\n(1) 若$f(x) \\geq 0$, 求$a$的取值范围;\\\\\n(2) 证明: 若$f(x)$有两个零点$x_1, x_2$, 则$x_1 x_2<1$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题21", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017352": { + "id": "017352", + "content": "在直角坐标系$xOy$中, 曲线$C_1$的参数方程为$\\begin{cases}x=\\dfrac{2+t}{6},\\\\ y=\\sqrt{t}\\end{cases}$($t$是参数), 曲线$C_2$的参数方程为$\\begin{cases}x=-\\dfrac{2+s}{6}, \\\\ y=-\\sqrt{s}\\end{cases}$($s$是参数).\\\\\n(1) 写出$C_1$的普通方程;\\\\\n(2) 以坐标原点为极点, $x$轴正半轴建立极坐标系, 曲线$C_3$的极坐标方程为$2 \\cos \\theta-\\sin \\theta=0$, 求$C_3$与$C_1$交点的直角坐标, 及$C_3$与$C_2$交点的直角坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题22", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017353": { + "id": "017353", + "content": "已知实数$a, b, c$均为正数, 满足$a^2+b^2+4 c^2=3$, 证明:\\\\\n(1) $a+b+2 c \\leq 3$;\\\\\n(2) 若$b=2 c$, 则$\\dfrac{1}{a}+\\dfrac{1}{c} \\geq 3$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷理科试题23", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017354": { + "id": "017354", + "content": "设集合$A=\\{-2,-1,0,1,2\\}$, $B=\\{x | 0 \\leq x<\\dfrac{5}{2}\\}$, 则$A \\cap B=$\\bracket{20}.\n\\fourch{$\\{0,1,2\\}$}{$\\{-2,-1,0\\}$}{$\\{0,1\\}$}{$\\{1,2\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题1", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017355": { + "id": "017355", + "content": "某社区通过公益讲座宣传中国非物质文化遗产保护知识. 为了解讲座效果, 随机抽取$10$位社区居民, 让他们在讲座前和讲座后各回答一份相关知识问卷, 这$10$位社区居民在讲座前和讲座后问卷答题的正确率如下图. 则下列选项正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.7, yscale = 0.08]\n\\draw [->] (0,55) -- (11,55);\n\\draw [->] (0,55) -- (0,56) -- (0.2,57) -- (-0.2,58) -- (0,59) -- (0,105);\n\\draw (0,55) node [below left] {$O$};\n\\foreach \\i in {1,2,...,10}\n{\\draw (\\i,55.5) -- (\\i,55) node [below] {$\\i$};};\n\\foreach \\i in {60,65,...,100}\n{\\draw [dotted] (10.5,\\i) -- (0,\\i) node [left] {$\\i\\%$};};\n\\draw (5.5,45) node {居民编号};\n\\draw (-2,80) node [rotate = 90] {正确率};\n\\filldraw (12,70) circle (0.05 and {7/16}) ++ (1,0) node {讲座后};\n\\filldraw (12,80) node {\\tiny$\\times$} node {\\tiny$+$} ++ (1,0) node {讲座前};\n\\foreach \\i/\\j/\\k in {1/65/90,2/60/85,3/70/80,4/60/90,5/65/85,6/75/85,7/90/95,8/85/100,9/80/85,10/95/100}\n{\\filldraw (\\i,\\j) node {\\tiny$\\times$} node {\\tiny$+$} (\\i,\\k) circle (0.05 and {7/16});};\n\\end{tikzpicture}\n\\end{center}\n\\onech{讲座前问卷答题的正确率的中位数小于$70 \\%$}{讲座后问卷答题的正确率的平均数大于$85 \\%$}{讲座前问卷答题的正确率的标准差小于讲座后正确率的标准差}{讲座后问卷答题的正确率的极差大于讲座前正确率的极差}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题2", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017356": { + "id": "017356", + "content": "若$z=1+\\mathrm{i}$. 则$|\\mathrm{i} z+3 \\overline {z}|=$\\bracket{20}.\n\\fourch{$4 \\sqrt{5}$}{$4 \\sqrt{2}$}{$2 \\sqrt{5}$}{$2 \\sqrt{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题3", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017357": { + "id": "017357", + "content": "如图, 网格纸上绘制的是一个多面体的三视图, 网格小正方形的边长为$1$, 则该多面体的体积为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\foreach \\i in {0,1,2,3,4,5,6}\n{\\draw [dashed,thin] (\\i,0) -- (\\i,4) (\\i+8,0) -- (\\i+8,4) (\\i,-6) -- (\\i,-2);};\n\\foreach \\i in {0,1,2,3,4}\n{\\draw [dashed,thin] (0,\\i) -- (6,\\i) (8,\\i) -- (14,\\i) (0,\\i-6) -- (6,\\i-6);};\n\\draw [ultra thick] (1,1) -- (5,1) -- (3,3) -- (1,3) -- cycle;\n\\draw [ultra thick] (10,1) rectangle (12,3);\n\\draw [ultra thick] (1,-5) rectangle (5,-3) (3,-5) -- (3,-3);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$8$}{$12$}{$16$}{$20$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题4", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017358": { + "id": "017358", + "content": "将函数$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$($\\omega>0$)的图像向左平移$\\dfrac{\\pi}{2}$个单位长度后得到曲线$C$, 若$C$关于$y$轴对称, 则$\\omega$的最小值是\\bracket{20}.\n\\fourch{$\\dfrac{1}{6}$}{$\\dfrac{1}{4}$}{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题5", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017359": { + "id": "017359", + "content": "从分别写有$1,2,3,4,5,6$的$6$张卡片中无放回随机抽取$2$张, 则抽到的$2$张卡片上的数字之积是$4$的倍数的概率为\\bracket{20}\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{1}{3}$}{$\\dfrac{2}{5}$}{$\\dfrac{2}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题6", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017360": { + "id": "017360", + "content": "函数$y=(3^x-3^{-x}) \\cos x$在区间$[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}]$的图像大致为\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{(pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{abs((pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180))});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{-(pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi/2,0) node [below] {$-\\dfrac{\\pi}{2}$};\n\\draw (pi/2,0) node [below] {$\\dfrac{\\pi}{2}$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi/2:pi/2,samples = 100] plot (\\x,{-abs((pow(3,\\x)-pow(3,-\\x))*cos(\\x/pi*180))});\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题7", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017361": { + "id": "017361", + "content": "当$x=1$时, 函数$f(x)=a \\ln x+\\dfrac{b}{x}$取得最大值$-2$, 则$f'(2)=$\\bracket{20}.\n\\fourch{$-1$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}{$1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题8", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017362": { + "id": "017362", + "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 已知$B_1D$与平面$ABCD$和平面$AA_1B_1B$所成的角均为$30^{\\circ}$, 则\\bracket{20}.\n\\fourch{$AB=2AD$}{$AB$与平面$AB_1C_1D$所成的角为$30^{\\circ}$}{$AC=CB_1$}{$B_1D$与平面$BB_1C_1C$所成的角为$45^{\\circ}$}\n\\fourch{$\\sqrt{5}$}{$2 \\sqrt{2}$}{$\\sqrt{10}$}{$\\dfrac{5 \\sqrt{10}}{4}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题9", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017363": { + "id": "017363", + "content": "甲、乙两个圆锥的母线长相等, 侧面展开图的圆心角之和为$2\\pi$, 侧面积分别为$S_{\\text{甲}}$和$S_{\\text{乙}}$, 体积分别为$V_{\\text{甲}}$和$V_{\\text{乙}}$, 若$\\dfrac{S_{\\text{甲}}}{{S_{\\text{乙}}}}=2$, 则$\\dfrac{V_{\\text{甲}}}{{V_{\\text{乙}}}}=$\\bracket{20}.\n\\fourch{$\\sqrt{5}$}{$2 \\sqrt{2}$}{$\\sqrt{10}$}{$\\dfrac{5 \\sqrt{10}}{4}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题10", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017364": { + "id": "017364", + "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的离心率为$\\dfrac{1}{3}$, $A_1, A_2$分别为$C$的左、右顶点, $B$为$C$的上顶点. 若$\\overrightarrow{BA_1} \\cdot \\overrightarrow{BA_2}=-1$, 则$C$的方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{18}+\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}{9}+\\dfrac{y^2}{8}=1$}{$\\dfrac{x^2}{3}+\\dfrac{y^2}{2}=1$}{$\\dfrac{x^2}{2}+y^2=1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题11", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017365": { + "id": "017365", + "content": "已知$9^m=10$, $a=10^m-11$, $b=8^m-9$, 则\\bracket{20}.\n\\fourch{$a>0>b$}{$a>b>0$}{$b>a>0$}{$b>0>a$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题12", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017366": { + "id": "017366", + "content": "已知向量$\\overrightarrow {a}=(m, 3)$, $\\overrightarrow {b}=(1, m+1)$. 若$\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题13", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017367": { + "id": "017367", + "content": "设点$M$在直线$2 x+y-1=0$上, 点$(3,0)$和$(0,1)$均在圆$M$上, 则圆$M$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题14", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017368": { + "id": "017368", + "content": "记双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的离心率为$e$, 写出满足条件``直线$y=2 x$与$C$无公共点''的$e$的一个值\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题15", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017369": { + "id": "017369", + "content": "已知$\\triangle ABC$中, 点$D$在边$BC$上, $\\angle ADB=120^{\\circ}$, $AD=2$, $CD=2BD$. 当$\\dfrac{AC}{AB}$取得最小值时, $BD=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题16", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017370": { + "id": "017370", + "content": "甲、乙两城之间的长途客车均由$A$和$B$两家公司运营, 为了解这两家公司长途客车的运行情况, 随机调查了甲、乙两城之间的$500$个班次, 得到下面列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& 准点班次数 & 未准点班次数 \\\\\\hline\n$A$& 240 & 20 \\\\\\hline\n$B$& 210 & 30 \\\\\\hline\n\\end{tabular}\n\\end{center}\n(1) 根据上表, 分别估计这两家公司甲、乙两城之间的长途客车准点的概率;\\\\\n(2) 能否有$90 \\%$的把握认为甲、乙两城之间的长途客车是否准点与客车所属公司有关?\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$,\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$P(\\chi^2 \\geq k)$& 0.100 & 0.050 & 0.010 \\\\\\hline\n$k$& 2.706 & 3.841 & 6.635 \\\\\\hline\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题17", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017371": { + "id": "017371", + "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和. 已知$\\dfrac{2S_n}{n}+n=2 a_n+1$.\\\\\n(1) 证明: $\\{a_n\\}$是等差数列;\\\\\n(2) 若$a_4, a_7, a_9$成等比数列, 求$S_n$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题18", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017372": { + "id": "017372", + "content": "小明同学参加综合实践活动, 设计了一个封闭的包装盒, 包装盒如图所示: 底面$ABCD$是边长为$8$(单位: $\\text{cm}$)的正方形, $\\triangle EAB$, $\\triangle FBC$, $\\triangle GCD$, $\\triangle HDA$均为正三角形, 且它们所在的平面都与平面$ABCD$垂直.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,-2) node [right] {$C$} coordinate (C);\n\\draw (0,0,-2) node [below] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) ++ (0,{sqrt(3)},0) node [above] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) ++ (0,{sqrt(3)},0) node [right] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(D)$) ++ (0,{sqrt(3)},0) node [above] {$G$} coordinate (G);\n\\draw ($(D)!0.5!(A)$) ++ (0,{sqrt(3)},0) node [left] {$H$} coordinate (H);\n\\draw (A)--(B)--(C)(E)--(F)--(G)--(H)--cycle(H)--(A)--(E)--(B)--(F)--(C);\n\\draw [dashed] (A)--(D)--(C)(H)--(D)--(G)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $EF\\parallel$平面$ABCD$;\\\\\n(2) 求该包装盒的容积 (不计包装盒材料的厚度).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题19", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017373": { + "id": "017373", + "content": "已知函数$f(x)=x^3-x$, $g(x)=x^2+a$, 曲线$y=f(x)$在点$(x_1, f(x_1))$处的切线也是曲线$y=g(x)$的切线.\\\\\n(1) 若$x_1=-1$, 求$a$;\\\\\n(2) 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题20", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017374": { + "id": "017374", + "content": "设抛物线$C: y^2=2 p x$($p>0$)的焦点为$F$, 点$D(p, 0)$, 过$F$的直线交$C$于$M, N$两点. 当直线$MD$垂直于$x$轴时, $|MF|=3$.\\\\\n(1) 求$C$的方程;\\\\\n(2) 设直线$MD, ND$与$C$的另一个交点分别为$A, B$, 记直线$MN, AB$的倾斜角分别为$\\alpha$, $\\beta$. 当$\\alpha-\\beta$取得最大值时, 求直线$AB$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题21", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017375": { + "id": "017375", + "content": "在直角坐标系$xOy$中, 曲线$C_1$的参数方程为$\\begin{cases}x=\\dfrac{2+t}{6},\\\\ y=\\sqrt{t}\\end{cases}$($t$为参数), 曲线$C_2$的参数方程为$\\begin{cases}x=-\\dfrac{2+s}{6},\\\\ y=-\\sqrt{s}\\end{cases}$($s$为参数).\\\\\n(1) 写出$C_1$的普通方程;\\\\\n(2) 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 曲线$C_3$的极坐标方程为$2 \\cos \\theta-\\sin \\theta=0$, 求$C_3$与$C_1$交点的直角坐标, 及$C_3$与$C_2$交点的直角坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题22", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017376": { + "id": "017376", + "content": "已知$a, b, c$均为正数, 且$a^2+b^2+4 c^2=3$, 证明:\\\\\n(1) $a+b+2 c \\leq 3$;\\\\\n(2) 若$b=2 c$, 则$\\dfrac{1}{a}+\\dfrac{1}{c} \\geq 3$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考甲卷文科试题23", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017377": { + "id": "017377", + "content": "设全集$U=\\{1,2,3,4,5\\}$, 集合$M$满足$\\overline{M}=\\{1,3\\}$, 则\\bracket{20}.\n\\fourch{$2 \\in M$}{$3 \\in M$}{$4 \\notin M$}{$5 \\notin M$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题1", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017378": { + "id": "017378", + "content": "已知$z=1-2 \\mathrm{i}$, 且$z+a \\cdot \\overline {z}+b=0$, 其中$a, b$为实数, 则\\bracket{20}.\n\\fourch{$a=1$, $b=-2$}{$a=-1$, $b=2$}{$a=1$, $b=2$}{$a=-1$, $b=-2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题2", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017379": { + "id": "017379", + "content": "已知向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=\\sqrt{3}$, $|\\overrightarrow {a}-2 \\overrightarrow {b}|=3$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}=$\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$1$}{$2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题3", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017380": { + "id": "017380", + "content": "嫦娥二号卫星在完成探月任务后, 继续进行深空探测, 成为我国第一颗环绕太阳飞行的人造卫星. 为研究嫦娥二号绕日周期与地球绕日周期的比值, 用到数列$\\{b_n\\}$: $b_1=1+\\dfrac{1}{a_1}$, $b_2=1+\\dfrac{1}{a_1+\\dfrac{1}{a_2}}$, $b_3=1+\\dfrac{1}{a_1+\\dfrac{1}{a_2+\\dfrac{1}{a_3}}}$, $\\cdots$, 以此类推, 其中$a_k$($k=1,2, \\cdots$)均为正整数, 则\\bracket{20}.\n\\fourch{$b_1=latex, node distance = 10pt]\n\\node [draw, rounded corners] (start) {开始};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of start] (step1) {输入$a=1$, $b=1$, $n=1$};\n\\node [draw, below = of step1] (step2) {$b=b+2a$};\n\\node [draw, below = of step2] (step3) {$a=b-a$, $n=n+1$};\n\\node [draw, diamond, aspect = 2, below = of step3] (switch) {$|\\dfrac{b^2}{a^2}-2|<0.01$};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of switch] (step4) {输出$n$};\n\\node [draw, rounded corners, below = of step4] (end) {结束};\n\\coordinate [left = 15pt of switch] (stepx);\n\\foreach \\i/\\j in {start/step1,step1/step2,step2/step3,step3/switch,step4/end}\n{\\draw [->] (\\i)--(\\j);};\n\\draw [->] (switch) -- node [right] {是} (step4);\n\\draw (switch) -- (stepx) node[below, midway] {否};\n\\draw [->] (stepx) -- (stepx|-step1) -> (step1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$3$}{$4$}{$5$}{$6$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题6", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017383": { + "id": "017383", + "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E, F$分别为$AB, BC$的中点, 则\\bracket{20}.\n\\twoch{平面$B_1EF \\perp$平面$BDD_1$}{平面$B_1EF \\perp$平面$A_1BD$}{平面$B_1EF\\parallel$平面$A_1AC$}{平面$B_1EF\\parallel$平面$A_1C_1D$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题7", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017384": { + "id": "017384", + "content": "已知等比数列$\\{a_n\\}$的前$3$项和为$168$, $a_2-a_5=42$, 则$a_6=$\\bracket{20}.\n\\fourch{$14$}{$12$}{$6$}{$3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题8", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017385": { + "id": "017385", + "content": "已知球$O$的半径为$1$, 四棱锥的顶点为$O$, 底面的四个顶点均在球$O$的球面上, 则当该四棱锥的体积最大时, 其高为\\bracket{20}.\n\\fourch{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{3}$}{$\\dfrac{\\sqrt{2}}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题9", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017386": { + "id": "017386", + "content": "某棋手与甲、乙、丙三位棋手各比赛一盘, 各盘比赛结果相互独立. 已知该棋手与甲、乙、丙比赛获胜的概率分别为$p_1, p_2, p_3$且$p_3>p_2>p_1>0$. 记该棋手连胜两盘的概率为$p$, 则\\bracket{20}.\n\\twoch{$p$与该棋手和甲, 乙, 丙的比赛次序无关}{该棋手在第二盘与甲比赛, $p$最大}{该棋手在第二盘与乙比赛, $p$最大}{该棋手在第二盘与丙比赛, $p$最大}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题10", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017387": { + "id": "017387", + "content": "双曲线$C$的两个焦点$F_1, F_2$, 以$C$的实轴为直径的圆记为$D$, 过$F_1$作$D$的切线与$C$交于$M, N$两点, 且$\\cos \\angle F_1NF_2=\\dfrac{3}{5}$, 则$C$的离心率为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{5}}{2}$}{$\\dfrac{3}{2}$}{$\\dfrac{\\sqrt{13}}{2}$}{$\\dfrac{\\sqrt{17}}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题11", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017388": { + "id": "017388", + "content": "已知函数$f(x), g(x)$的定义域均为$\\mathbf{R}$, 且$f(x)+g(2-x)=5$, $g(x)-f(x-4)=7$. 若$y=g(x)$的图像关于直线$x=2$对称, $g(2)=4$, 则$\\displaystyle\\sum_{k=1}^{22} f(k)=$\\bracket{20}.\n\\fourch{$-21$}{$-22$}{$-23$}{$-24$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题12", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017389": { + "id": "017389", + "content": "从甲、乙等$5$名同学中随机选$3$名参加社区服务工作, 则甲、乙都入选的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题13", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017390": { + "id": "017390", + "content": "过四点$(0,0)$, $(4,0)$, $(-1,1)$, $(4,2)$中的三点的一个圆的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题14", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017391": { + "id": "017391", + "content": "记函数$f(x)=\\cos (\\omega x+\\varphi)$($\\omega>0$, $0<\\varphi<\\pi$)的最小正周期为$T$, 若$f(T)=\\dfrac{\\sqrt{3}}{2}$, $x=\\dfrac{\\pi}{9}$为$f(x)$的零点, 则$\\omega$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题15", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017392": { + "id": "017392", + "content": "已知$x=x_1$和$x=x_2$分别是函数$f(x)=2 a^x-\\mathrm{e} x^2$($a>0$且$a \\neq 1$)的极小值点和极大值点, 若$x_1=latex,scale = 2]\n\\draw (0,0,0) node [below] {$E$} coordinate (E);\n\\draw ({sqrt(3)},0,0) node [right] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (0,0,-1) node [below] {$C$} coordinate (C);\n\\draw ($(D)!0.7!(B)$) node [above] {$F$} coordinate (F);\n\\draw (A)--(B)--(D)--cycle(A)--(F);\n\\draw [dashed] (A)--(D)(D)--(C)--(B)(F)--(C)--(A)(D)--(E)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$BED \\perp$平面$ACD$;\\\\\n(2) 设$AB=BD=2$, $\\angle ACB=60^{\\circ}$, 点$F$在$BD$上, 当$\\triangle AFC$的面积最小时, 求$CF$与平面$ABD$所成角的正弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题18", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017395": { + "id": "017395", + "content": "某地经过多年的环境治理, 已将荒山改造成了绿水青山, 为估计一林区某种树木的总材积量, 随机选取了$10$棵这种树木, 测量每棵树的根部横截面积 (单位: $m^2$) 和材积量 (单位: $m^3$), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n样本号$i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 总和 \\\\\\hline\n根部横截面积$x_i$ & 0.04 & 0.06 & 0.04 & 0.08 & 0.08 & 0.05 & 0.05 & 0.07 & 0.07 & 0.06 & 0.6 \\\\\\hline\n材积量$y_i$& 0.25 & 0.40 & 0.22 & 0.54 & 0.51 & 0.34 & 0.36 & 0.46 & 0.42 & 0.40 & 3.9 \\\\\\hline\n\\end{tabular}\n\\end{center}\n并计算得$\\displaystyle\\sum_{i=1}^{10} x_i^2=0.038$, $\\displaystyle\\sum_{i=1}^{10} y_i^2=1.6158$, $\\displaystyle\\sum_{i=1}^{10} x_i y_i=0.2474$.\\\\\n(1) 估计该林区这种树木平均一棵的根部横截面积与平均一棵的材积量;\\\\\n(2) 求该林区这种树木的根部横截面积与材积量的样本相关系数 (精确到 0.01);\\\\\n(3) 现测量了该林区所有这种树木的根部横截面积, 并得到所有这种树木的根部横截面积总和为$186 \\mathrm{m}^2$. 已知树木的材积量与其根部横截面积近似成正比. 利用以上数据给出该林区这种树木的总材积量的估计值.\\\\\n附: 相关系数$r=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})(y_i-\\overline {y})}{\\sqrt{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})^2 \\sum_{i=1}^n(y_i-\\overline {y})^2}}$, $\\sqrt{1.896} \\approx 1.377$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题19", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017396": { + "id": "017396", + "content": "已知椭圆$E$的中心为坐标原点, 对称轴为$x$轴, $y$轴, 且过$A(0,-2)$, $B(\\dfrac{3}{2},-1)$两点.\\\\\n(1) 求$E$的方程;\\\\\n(2) 设过点$P(1,-2)$的直线交$E$于$M, N$两点, 过$M$且平行于$x$的直线与线段$AB$交于点$T$, 点$H$满足$\\overrightarrow{MT}=\\overrightarrow{TH}$, 证明: 直线$HN$过定点.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题20", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017397": { + "id": "017397", + "content": "已知函数$f(x)=\\ln (1+x)+a x e^{-x}$.\\\\\n(1) 当$a=1$时, 求曲线$f(x)$在点$(0, f(0))$处的切线方程;\\\\\n(2) 若$f(x)$在区间$(-1,0)$, $(0,+\\infty)$各恰有一个零点, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题21", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017398": { + "id": "017398", + "content": "在直角坐标系$xOy$中, 曲线$C$的方程为$\\begin{cases}x=\\sqrt{3} \\cos 2 t \\\\ y=2 \\sin t\\end{cases}$($t$为参数). 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 已知直线$l$的极坐标方程为$\\rho \\sin (\\theta+\\dfrac{\\pi}{3})+m=0$.\\\\\n(1) 写出$l$的直角坐标方程;\\\\\n(2) 若$l$与$C$有公共点, 求$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题22", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017399": { + "id": "017399", + "content": "已知$a, b, c$为正数, 且$a^{\\frac{3}{2}}+b^{\\frac{3}{2}}+c^{\\frac{3}{2}}=1$, 证明:\\\\\n(1) $a b c \\leq \\dfrac{1}{9}$;\\\\\n(2) $\\dfrac{a}{b+c}+\\dfrac{b}{a+c}+\\dfrac{c}{a+b} \\leq \\dfrac{1}{2 \\sqrt{abc}}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷理科试题23", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017400": { + "id": "017400", + "content": "集合$M=\\{2,4,6,8,10\\}$, $N=\\{x |-1=latex, node distance = 10pt]\n\\node [draw, rounded corners] (start) {开始};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of start] (step1) {输入$a=1$, $b=1$, $n=1$};\n\\node [draw, below = of step1] (step2) {$b=b+2a$};\n\\node [draw, below = of step2] (step3) {$a=b-a$, $n=n+1$};\n\\node [draw, diamond, aspect = 2, below = of step3] (switch) {$|\\dfrac{b^2}{a^2}-2|<0.01$};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of switch] (step4) {输出$n$};\n\\node [draw, rounded corners, below = of step4] (end) {结束};\n\\coordinate [left = 15pt of switch] (stepx);\n\\foreach \\i/\\j in {start/step1,step1/step2,step2/step3,step3/switch,step4/end}\n{\\draw [->] (\\i)--(\\j);};\n\\draw [->] (switch) -- node [right] {是} (step4);\n\\draw (switch) -- (stepx) node[below, midway] {否};\n\\draw [->] (stepx) -- (stepx|-step1) -> (step1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$3$}{$4$}{$5$}{$6$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题7", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017407": { + "id": "017407", + "content": "右图是下列四个函数中的某个函数在区间$[-3,3]$的大致图像, 则函数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (-3,0.2) -- (-3,0) node [below] {$-3$};\n\\draw [domain = -3:3, samples = 100] plot (\\x,{(-\\x*\\x*\\x+3*\\x)/(\\x*\\x+1)});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$y=\\dfrac{-x^3+3 x}{x^2+1}$}{$y=\\dfrac{x^3-x}{x^2+1}$}{$y=\\dfrac{2 x \\cos x}{x^2+1}$}{$y=\\dfrac{2 \\sin x}{x^2+1}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题8", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017408": { + "id": "017408", + "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E, F$分别为$AB, BC$的中点, 则\\bracket{20}.\n\\twoch{平面$B_1EF \\perp BDD_1$}{平面$B_1EF \\perp A_1BD$}{平面$B_1EF\\parallel A_1AC$}{平面$B_1EF\\parallel A_1C_1D$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题9", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017409": { + "id": "017409", + "content": "已知等比数列$\\{a_n\\}$的前$3$项和为$168$, $a_2-a_5=42$, 则$a_6=$\\bracket{20}.\n\\fourch{$14$}{$12$}{$6$}{$3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题10", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017410": { + "id": "017410", + "content": "函数$f(x)=\\cos x+(x+1) \\sin x+1$在区间$[0,2 \\pi]$的最小值、最大值分别为\\bracket{20}.\n\\fourch{$-\\dfrac{\\pi}{2}$, $\\dfrac{\\pi}{2}$}{$-\\dfrac{3 \\pi}{2}$, $\\dfrac{\\pi}{2}$}{$-\\dfrac{\\pi}{2}$, $\\dfrac{\\pi}{2}+2$}{$-\\dfrac{3 \\pi}{2}$, $\\dfrac{\\pi}{2}+2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题11", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017411": { + "id": "017411", + "content": "已知球$O$的半径为$1$, 四棱锥的顶点为$O$, 底面的四个顶点均在球$O$的的球面上, 当该四棱锥的体积最大时, 其高为\\bracket{20}.\n\\fourch{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{3}$}{$\\dfrac{\\sqrt{2}}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题12", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017412": { + "id": "017412", + "content": "记$S_n$为等差数列$\\{a_n\\}$的前$n$项和. 若$2S_3=3S_2+6$, 则公差$d=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题13", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017413": { + "id": "017413", + "content": "从甲、乙等$5$名同学中随机选$3$名参加社区服务工作, 则甲、乙都入选的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题14", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017414": { + "id": "017414", + "content": "过四点$(0,0)$, $(4,0)$, $(-1,1)$, $(4,2)$中的三点的一个圆的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题15", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017415": { + "id": "017415", + "content": "若$f(x)=\\ln |a+\\dfrac{1}{1-x}|+b$是奇函数, 则$a=$\\blank{50}, $b=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题16", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017416": { + "id": "017416", + "content": "记$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$, 已知$\\sin C \\sin (A-B)=\\sin B \\sin (C-A)$.\\\\\n(1) 若$A=2B$, 求$C$;\\\\\n(2) 证明: $2 a^2=b^2+c^2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题17", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017417": { + "id": "017417", + "content": "如图, 四面体$ABCD$中, $AD \\perp CD$, $AD=CD$, $\\angle ADB=\\angle BDC$, $E$为$AC$中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [below] {$E$} coordinate (E);\n\\draw ({sqrt(3)},0,0) node [right] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (0,0,-1) node [below] {$C$} coordinate (C);\n\\draw ($(D)!0.7!(B)$) node [above] {$F$} coordinate (F);\n\\draw (A)--(B)--(D)--cycle(A)--(F);\n\\draw [dashed] (A)--(D)(D)--(C)--(B)(F)--(C)--(A)(D)--(E)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$BED \\perp$平面$ACD$;\\\\\n(2) 设$AB=BD=2$, $\\angle ACB=60^{\\circ}$, 点$F$在$BD$上, 当$\\triangle AFC$面积最小时, 求三棱锥$F-ABC$的体积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题18", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017418": { + "id": "017418", + "content": "某地经过多年的环境治理, 已将荒山改造成了绿水青山, 为估计一林区某种树木的总材积量, 随机选取了$10$棵这种树木, 测量每棵树的根部横截面积 (单位: $m^2$) 和材积量 (单位: $m^3$), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n样本号$i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 总和 \\\\ \\hline\n根部横截面积$x_i$ & 0.04 & 0.06 & 0.04 & 0.08 & 0.08 & 0.05 & 0.05 & 0.07 & 0.07 & 0.06 & 0.6 \\\\ \\hline\n材积量$y_i$& 0.25 & 0.40 & 0.22 & 0.54 & 0.51 & 0.34 & 0.36 & 0.46 & 0.42 & 0.40 & 3.9 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n并计算得$\\displaystyle\\sum_{i=1}^{10} x_i^2=0.038$, $\\displaystyle\\sum_{i=1}^{10} y_i^2=1.6158$, $\\displaystyle\\sum_{i=1}^{10} x_i y_i=0.2474$.\\\\\n(1) 估计该林区这种树木平均一棵的根部横截面积与平均一棵的材积量;\\\\\n(2) 求该林区这种树木的根部横截面积与材积量的样本相关系数 (精确到 0.01);\\\\\n(3) 现测量了该林区所有这种树木的根部横截面积, 并得到所有这种树木的根部横截面积总和为$186 \\mathrm{m}^2$. 已知树木的材积量与其根部横截面积近似成正比. 利用以上数据给出该林区这种树木的总材积量的估计值.\\\\\n附: 相关系数$r=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})(y_i-\\overline {y})}{\\sqrt{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})^2 \\sum_{i=1}^n(y_i-\\overline {y})^2}}$, $\\sqrt{1.896} \\approx 1.377$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题19", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017419": { + "id": "017419", + "content": "已知函数$f(x)=a x-\\dfrac{1}{x}-(a+1) \\ln x$.\\\\\n(1) 当$a=0$时, 求$f(x)$的最大值;\\\\\n(2) 若$f(x)$恰有一个零点, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题20", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017420": { + "id": "017420", + "content": "已知椭圆$E$的中心为坐标原点, 对称轴为$x$轴, $y$轴, 且过$A(0,-2), B(\\dfrac{3}{2},-1)$两点.\\\\\n(1) 求$E$的方程;\\\\\n(2) 设过点$P(1,-2)$的直线交$E$于$M, N$两点, 过$M$且平行于$x$轴的直线与线段$AB$交于点$T$, 点$H$满足$\\overrightarrow{MT}=\\overrightarrow{TH}$, 证明: 直线$HN$过定点.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题21", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017421": { + "id": "017421", + "content": "在直角坐标系$x O y$中, 曲线$C$的方程为$\\begin{cases}x=\\sqrt{3} \\cos 2 t, \\\\ y=2 \\sin t\\end{cases}$($t$为参数). 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 已知直线$l$的极坐标方程为$\\rho \\sin (\\theta+\\dfrac{\\pi}{3})+m=0$.\\\\\n(1) 写出$l$的直角坐标方程;\\\\\n(2) 若$l$与$C$有公共点, 求$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题22", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017422": { + "id": "017422", + "content": "已知$a, b, c$为正数, 且$a^{\\frac{3}{2}}+b^{\\frac{3}{2}}+c^{\\frac{3}{2}}=1$, 证明:\\\\\n(1) $a b c \\leq \\dfrac{1}{9}$;\\\\\n(2) $\\dfrac{a}{b+c}+\\dfrac{b}{a+c}+\\dfrac{c}{a+b} \\leq \\dfrac{1}{2 \\sqrt{abc}}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届全国高考乙卷文科试题23", + "edit": [ + "20230531\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "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.",