diff --git a/工具/latex界面修改题目内容.py b/工具/latex界面修改题目内容.py index 31bc3b97..f85465a6 100644 --- a/工具/latex界面修改题目内容.py +++ b/工具/latex界面修改题目内容.py @@ -1,6 +1,6 @@ import os,re,json """这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭""" -problems = "18063,18077" +problems = "18103" editor = "王伟叶" def generate_number_set(string,dict): diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index 879f2b32..2dcf78e1 100644 --- a/工具/批量收录题目.py +++ b/工具/批量收录题目.py @@ -1,8 +1,8 @@ #修改起始id,出处,文件名 -starting_id = 18082 +starting_id = 18104 raworigin = "" filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目14.tex" -editor = "20230610\t王伟叶" +editor = "20230611\t王伟叶" indexed = True IndexDescription = "试题" diff --git a/工具/新题相似相同比对.py b/工具/新题相似相同比对.py index 53275e1c..17326930 100644 --- a/工具/新题相似相同比对.py +++ b/工具/新题相似相同比对.py @@ -5,7 +5,7 @@ old_problems_range = "1:50000" threshold = 0.85 # 待比对的文件 -filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目13.tex" +filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目14.tex" #生成数码列表, 逗号分隔每个区块, 区块内部用:表示整数闭区间 def generate_number_set(string): diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index a4de1cde..a562ff82 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -464084,6 +464084,1326 @@ "space": "4em", "unrelated": [] }, + "018104": { + "id": "018104", + "content": "设集合$A=\\{x | x=3 k+1,\\ k \\in \\mathbf{Z}\\}$, $B=\\{x | x=3 k+2,\\ k \\in \\mathbf{Z}\\}$, 全集为整数集, 则$\\overline{A \\cup B}=$\\bracket{20}.\n\\fourch{$\\{x | x=3 k,\\ k \\in \\mathbf{Z}\\}$}{$\\{x | x=3 k-1,\\ k \\in \\mathbf{Z}\\}$}{$\\{x | x=3 k-2,\\ k \\in \\mathbf{Z}\\}$}{$\\varnothing$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题1", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018105": { + "id": "018105", + "content": "若复数$(a+\\mathrm{i})(1-a \\mathrm{i})=2$, 则$a=$\\bracket{20}.\n\\fourch{$-1$}{$0$}{$1$}{$2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题2", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018106": { + "id": "018106", + "content": "执行下面的程序框图, 输出的$B=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, node distance = 10pt]\n\\node [draw, rounded corners] (start) {开始};\n\\node [draw, below = of start] (init) {$n=1$, $A=1$, $B=2$};\n\\node [draw, diamond, below = of init, aspect = 2] (judge) {$n\\le 3$?};\n\\node [draw, below = of judge] (step1) {$A=A+B$};\n\\node [draw, below = of step1] (step2) {$B=A+B$};\n\\node [draw, below = of step2] (step3) {$n=n+1$};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of step3] (output) {输出$B$};\n\\node [draw, rounded corners, below = of output] (end) {结束};\n\\coordinate [right = 15pt of judge] (stepx);\n\\coordinate [left = 15pt of step3] (stepy);\n\\foreach \\i/\\j in {start/init,init/judge,step1/step2,step2/step3,output/end}\n{\\draw [->] (\\i)--(\\j);};\n\\draw [->] (judge) -- node [right] {是} (step1);\n\\draw (judge) -- (stepx) node [above, midway] {否};\n\\draw [->] (stepx) -- (stepx|-output) -> (output);\n\\draw [->] (step3) -- (stepy) -- (stepy|-judge) -> (judge);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$21$}{$34$}{$55$}{$89$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题3", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018107": { + "id": "018107", + "content": "向量$|\\overrightarrow {a}|=|\\overrightarrow {b}|=1$, $|\\overrightarrow {c}|=\\sqrt{2}$, 且$\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$, 则$\\cos \\langle\\overrightarrow {a}-\\overrightarrow {c}, \\overrightarrow {b}-\\overrightarrow {c}\\rangle=$\\bracket{20}.\n\\fourch{$-\\dfrac{1}{5}$}{$-\\dfrac{2}{5}$}{$\\dfrac{2}{5}$}{$\\dfrac{4}{5}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题4", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018108": { + "id": "018108", + "content": "已知正项等比数列$\\{a_n\\}$中, $a_1=1$, $S_n$为$\\{a_n\\}$前$n$项和, $S_5=5S_3-4$, 则$S_4=$\\bracket{20}.\n\\fourch{$7$}{$9$}{$15$}{$30$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题5", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018109": { + "id": "018109", + "content": "有$50$加人报名足球倶乐部, $60$人报名乒乓球倶乐部, $70$人报名足球或乒乓球倶乐部, 若已知某人报足球倶乐部, 则其报乒乓球倶乐部的概率为\\bracket{20}.\n\\fourch{$0.8$}{$0.4$}{$0.2$}{$0.1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题6", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018110": { + "id": "018110", + "content": "``$\\sin ^2 \\alpha+\\sin ^2 \\beta=1$''是``$\\sin \\alpha+\\cos \\beta=0$''的\\bracket{20}.\n\\twoch{充分条件但不是必要条件}{必要条件但不是充分条件}{充要条件}{既不是充分条件也不是必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题7", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018111": { + "id": "018111", + "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的离心率为$\\sqrt{5}$, 其中一条渐近线与圆$(x-2)^2+(y-3)^2=1$交于$A, B$两点, 则$|AB|=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{\\sqrt{5}}{5}$}{$\\dfrac{2 \\sqrt{5}}{5}$}{$\\dfrac{4 \\sqrt{5}}{5}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题8", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018112": { + "id": "018112", + "content": "有五名志愿者参加社区服务, 共服务星期六、星期天两天, 每天从中任选两人参加服务, 则恰有$1$人连续参加两天服务的选择种数为\\bracket{20}.\n\\fourch{$120$}{$60$}{$40$}{$30$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题9", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018113": { + "id": "018113", + "content": "已知$f(x)$为函数$y=\\cos (2 x+\\dfrac{\\pi}{6})$向左平移$\\dfrac{\\pi}{6}$个单位所得函数, 则$y=f(x)$与$y=\\dfrac{1}{2} x-\\dfrac{1}{2}$的交点个数为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题10", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018114": { + "id": "018114", + "content": "在四棱锥$P-ABCD$中, 底面$ABCD$为正方形, $AB=4$, $PC=PD=3$, $\\angle PCA=45^{\\circ}$, 则$\\triangle PBC$的面积为\\bracket{20}.\n\\fourch{$2 \\sqrt{2}$}{$3 \\sqrt{2}$}{$4 \\sqrt{2}$}{$5 \\sqrt{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题11", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018115": { + "id": "018115", + "content": "已知椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{6}=1$, $F_1$、$F_2$为两个焦点, $O$为原点, $P$为椭圆上一点, $\\cos \\angle F_1PF_2=\\dfrac{3}{5}$, 则$|PO|=$\\bracket{20}.\n\\fourch{$\\dfrac{2}{5}$}{$\\dfrac{\\sqrt{30}}{2}$}{$\\dfrac{3}{5}$}{$\\dfrac{\\sqrt{35}}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题12", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018116": { + "id": "018116", + "content": "若$y=(x-1)^2+a x+\\sin (x+\\dfrac{\\pi}{2})$为偶函数, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题13", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018117": { + "id": "018117", + "content": "设$x, y$满足约束条件$\\begin{cases}-2x+3y\\le 3, \\\\ 3x-2y\\le 3,\\\\ x+y\\ge 1,\\end{cases}$ $z=3 x+2 y$, 则$z$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题14", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018118": { + "id": "018118", + "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E, F$分别为棱$CD, A_1B_1$的中点, 则以$EF$为直径的球面与正方体的所有棱的交点总数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题15", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018119": { + "id": "018119", + "content": "$\\triangle ABC$中, $\\angle BAC=60^{\\circ}$, $AB=2$, $BC=\\sqrt{6}$, $AD$平分$\\angle BAC$与$BC$交于点$D$, 则$AD=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题16", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018120": { + "id": "018120", + "content": "已知数列$\\{a_n\\}$中, $a_2=1$, 设$S_n$为$\\{a_n\\}$的前$n$项和, $2S_n=n a_n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求数列$\\{\\dfrac{a_{n+1}}{2^n}\\}$的前$n$项和$T_n$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题17", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018121": { + "id": "018121", + "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, $AA_1=2$, $A_1C\\perp$底面$ABC$, $\\angle ACB=90^{\\circ}$, 点$A$到平面$BCC_1B_1$的距离为$1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,0,1) node [below] {$A$} coordinate (A);\n\\draw ({sqrt(3)/sqrt(2)},0,{sqrt(3)/sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)},0) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)+(B)-(A)$) node [right] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)+(C)-(A)$) node [above] {$C_1$} coordinate (C_1);\n\\draw (A)--(B)--(B_1)--(A_1)--cycle(A_1)--(C_1)--(B_1);\n\\draw (A)--(B_1);\n\\draw [dashed] (A)--(C)--(B)(C)--(C_1)(C)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1C=AC$;\\\\\n(2) 若$A_1A$到$B_1B$的距离为$2$, 求$AB_1$与平面$BCC_1B_1$所成角的正弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题18", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018122": { + "id": "018122", + "content": "为研究某药物对小鼠的生长抑制作用, 将$40$只小鼠均分为两组, 分别为对照组 (不加药物) 和实验组 (加药物).\n(1) 从$40$只小鼠中任取两只小鼠, 设对照组小鼠数目为$X$, 求$X$的分布列和期望;\\\\\n(2) 测得$40$只小鼠的体重如下(单位: $g$)(已经按照从小到大排好).\\\\\n对照组:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline 17.3 & 18.4 & 20.1 & 20.4 & 21.5 & 23.2 & 24.6 & 24.8 & 25.0 & 25.4 \\\\\n\\hline 26.1 & 26.3 & 26.4 & 26.5 & 26.8 & 27.0 & 27.4 & 27.5 & 27.6 & 28.3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n实验组:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline 5.4 & 6.6 & 6.8 & 6.9 & 7.8 & 8.2 & 9.4 & 10.0 & 10.4 & 11.2 \\\\\n\\hline 14.4 & 17.3 & 19.2 & 20.2 & 23.6 & 23.8 & 24.5 & 25.1 & 25.2 & 26.0 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(I) 求$40$只小鼠体重的中位数$m$, 并完成下面$2 \\times 2$列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline &$0$)交于$A, B$两点, 且$|AB|=4 \\sqrt{15}$.\\\\\n(1) 求$p$的值;\\\\\n(2) $F$为$y^2=2 p x$的焦点, $M$、$N$为抛物线上两点且$\\overrightarrow{MF} \\cdot \\overrightarrow{NF}=0$, 求$\\triangle MNF$面积的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题20", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018124": { + "id": "018124", + "content": "已知函数$f(x)=a x-\\dfrac{\\sin x}{\\cos ^3 x}$, $x \\in(0, \\dfrac{\\pi}{2})$.\\\\\n(1) 若$a=8$, 讨论函数$f(x)$的单调性;\\\\\n(2) 若$f(x)<\\sin 2 x$恒成立, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题21", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018125": { + "id": "018125", + "content": "已知$P(2,1)$, 直线$l: \\begin{cases}x=2+t \\cos \\alpha, \\\\ y=1+t \\sin \\alpha\\end{cases}$($t$为参数), $l$与$x$轴, $y$轴正半轴交于$A, B$两点, $|PA| \\cdot|PB|=4$.\\\\\n(1) 求$\\alpha$的值;\\\\\n(2) 以原点为极点, $x$轴的正半轴为极轴建立极坐标系, 求$l$的极坐标方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考甲卷理科试题22", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018126": { + "id": "018126", + "content": "已知$a>0$, $f(x)=2|x-a|-a$.\\\\\n(1) 求不等式$f(x)=latex, scale = 0.5]\n\\foreach \\i in {0,1,...,6}\n{\\draw [gray] (\\i,0) --++ (0,5);\n\\draw [gray] ({\\i+7},0) --++ (0,5);\n\\draw [gray] (\\i,-6) --++ (0,5);};\n\\foreach \\i in {0,1,...,5}\n{\\draw [gray] (0,\\i) --++ (6,0);\n\\draw [gray] (7,\\i) --++ (6,0);\n\\draw [gray] (0,{\\i-6}) --++ (6,0);};\n\\draw [ultra thick] (2,1) rectangle (4,4) (2,3) -- (4,3);\n\\draw [ultra thick] (9,1) --++ (2,0) --++ (0,2) --++ (-1,0) --++ (0,1) --++ (-1,0) --cycle;\n\\draw [ultra thick] (2,-2) rectangle (4,-4) (2,-3) --++ (2,0);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$24$}{$26$}{$28$}{$30$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题3", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018130": { + "id": "018130", + "content": "已知$f(x)=\\dfrac{x \\mathrm{e}^x}{\\mathrm{e}^{a x}-1}$足偶函数, 则$a=$\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$1$}{$2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题4", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018131": { + "id": "018131", + "content": "设$O$为平面坐标系的坐标原点, 在区域$\\{(x, y) | 1 \\leq x^2+y^2 \\leq 4\\}$内随机取一点, 记该点为$A$, 则直线$OA$的倾斜角不大于$\\dfrac{\\pi}{4}$的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{8}$}{$\\dfrac{1}{6}$}{$\\dfrac{1}{4}$}{$\\dfrac{1}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题5", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018132": { + "id": "018132", + "content": "已知函数$f(x)=\\sin (\\omega x+\\varphi)$在区问$(\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3})$单调递增, 直线$x=\\dfrac{\\pi}{6}$和$x=\\dfrac{2 \\pi}{3}$为函数$y=f(x)$的图像的两条对称轴, 则$f(-\\dfrac{5 \\pi}{12})=$\\bracket{20}.\n\\fourch{$-\\dfrac{\\sqrt{3}}{2}$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题6", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018133": { + "id": "018133", + "content": "甲乙两位同学从$6$种课外读物中各自选读$2$种, 则这两人选读的课外读物中恰有$1$种相同的选法共有\\bracket{20}.\n\\fourch{$30$ 种}{$60$ 种}{$120$ 种}{$240$ 种}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题7", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018134": { + "id": "018134", + "content": "已知圆锥$PO$的底面半径为$\\sqrt{3}$, $O$为底面圆心, $PA, PB$为圆锥的母线, $\\angle AOB=120^{\\circ}$, 若$\\triangle PAB$的面积等于$\\dfrac{9 \\sqrt{3}}{4}$, 则该圆锥的体积为\\bracket{20}\n\\fourch{$\\pi$}{$\\sqrt{6} \\pi$}{$3 \\pi$}{$3 \\sqrt{6} \\pi$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题8", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018135": { + "id": "018135", + "content": "已知$\\triangle ABC$为等腰直角三角形, $AB$为斜边, $\\triangle ABD$为等边三角形, 若二面角$C-AB-D$为$150^{\\circ}$, 则直线$CD$与平面$ABC$所成角的正切值为\\bracket{20}.\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{\\sqrt{2}}{5}$}{$\\dfrac{\\sqrt{3}}{5}$}{$\\dfrac{2}{5}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题9", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018136": { + "id": "018136", + "content": "已知$\\{a_n\\}$是公差为$\\dfrac{2 \\pi}{3}$的等差数列, 若$\\{\\cos a_n | n \\in \\mathbf{N}^*\\}=\\{a, b\\}$, 则$a b=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$-\\dfrac{1}{2}$}{$0$}{$1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题10", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018137": { + "id": "018137", + "content": "已知双曲线$C: x^2-\\dfrac{y^2}{9}=1$, 则以下可能为双曲线$C$的弦的中点的是\\bracket{20}.\n\\fourch{$(1,1)$}{$(-1,2)$}{$(1,3)$}{$(-1,-4)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题11", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018138": { + "id": "018138", + "content": "已知圆$O$半径为$1, PA$与圆$O$相切, 切点为$A,|OP|=\\sqrt{2}$, 过点$P$的直线与圆$O$交于$B, C$两点, $D$为$BC$中点, 则$\\overrightarrow{PD} \\cdot \\overrightarrow{PA}$的最大值为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}+\\dfrac{\\sqrt{2}}{2}$}{$1+\\dfrac{\\sqrt{2}}{2}$}{$1+\\sqrt{2}$}{$2+\\sqrt{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题12", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018139": { + "id": "018139", + "content": "已知点$A(1, \\sqrt{5})$在抛物线$C: y^2=2 p x$上, 则$A$到$C$的准线的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题13", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018140": { + "id": "018140", + "content": "若$x, y$满足约束条件$\\begin{cases}x-3 y \\leq-1, \\\\ x+2 y \\leq 9, \\\\ 3 x+y \\geq 7,\\end{cases}$ 则$z=2 x-y$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题14", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018141": { + "id": "018141", + "content": "已知$\\{a_n\\}$为等比数列, $a_2 a_4 a_5=a_3 a_6$, $a_9 a_{10}=-8$, 则$a_7=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题15", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018142": { + "id": "018142", + "content": "已知$a \\in(0,1)$, 函数$f(x)=a^x+(1+a)^x$是$(0,+\\infty)$上的增函数, $a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题16", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018143": { + "id": "018143", + "content": "某厂为比较甲乙两种工艺对橡胶产品伸缩率的处理效应, 进行$10$次配对试验, 每次配对试验选用材质相同的两个橡胶产品, 随机地选其中一个用甲工艺处理, 另一个用乙工艺处理, 测量处理后的橡胶产品的伸缩率, 甲.乙两种工艺处理后的橡胶产品的伸缩率分别记为$x_i$, $y_i$($i=1,2, \\cdots,10$), 试验结果如下表\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 试验序号$i$& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 伸缩率$x_i$& 545 & 533 & 551 & 522 & 575 & 544 & 541 & 568 & 596 & 548 \\\\\n\\hline 伸缩率$y_i$& 536 & 527 & 543 & 530 & 560 & 533 & 522 & 550 & 576 & 536 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n$z_i=x_i-y_i$($i=1,2, \\cdots, 10$), 记$z_1, z_2, \\cdots, z_{10}$的样本平均数为$\\overline {z}$, 样本方差为$s^2$.\\\\\n(1) 求$\\overline {z}$, $s^2$;\\\\\n(2) 判断甲工艺处理后的橡胶产品的伸缩率较乙工艺处理后的橡胶产品的伸缩率是否有显著提高(如果$\\overline {z} \\geq 2 \\sqrt{\\dfrac{s^2}{10}}$, 则认为甲工艺处理后的橡胶产品的伸缩率较乙工艺处理后的橡胶产品的伸缩率有显著提高, 否则不认为有显著提高).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题17", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018144": { + "id": "018144", + "content": "$\\triangle ABC$中, $\\angle A=120^{\\circ}$, $AB=2$, $AC=1$.\\\\\n(1) 求$\\sin \\angle ABC$;\\\\\n(2) 若$D$为$BC$上的一点, $\\angle BAD=90^{\\circ}$, 求$\\triangle BAD$的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题18", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018145": { + "id": "018145", + "content": "如图, 三棱锥$P-ABC$中, $\\angle ABC=90^{\\circ}$, $AB=2$, $BC=2 \\sqrt{2}$, $PB=PC=\\sqrt{6}$, $AD=\\sqrt{5} OD$, $BF \\perp AO$, $O, D, E$分别为$BC, PB, AP$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-45:0.5cm)}, scale = 1.5]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw ({sqrt(2)},{sqrt(3)},-1) node [above] {$P$} coordinate (P);\n\\draw (A)--(C)--(P)--(B)--cycle;\n\\draw [dashed] (B)--(C);\n\\draw (A)--(P);\n\\draw ($(B)!0.5!(C)$) node [above left] {$O$} coordinate (O);\n\\draw ($(A)!0.5!(C)$) node [below right] {$F$} coordinate (F);\n\\draw ($(P)!0.5!(B)$) node [left] {$D$} coordinate (D);\n\\draw ($(P)!0.5!(A)$) node [right] {$E$} coordinate (E);\n\\draw (A)--(D)--(E)--(F)(B)--(E);\n\\draw [dashed] (A)--(O)(B)--(F)--(O)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $EF\\parallel$平面$ADO$;\\\\\n(2) 平面$ADO \\perp$平面$BEF$;\\\\\n(3) 求二面角$D-AO-C$的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题19", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018146": { + "id": "018146", + "content": "已知曲线$C$的方程为$\\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1$($a>b>0$), 离心率为$\\dfrac{\\sqrt{5}}{3}$, 曲线过点$A(-2,0)$.\\\\\n(1) 求曲线$C$的方程;\\\\\n(2) 过点$(-2,3)$的直线交曲线$C$于$P, Q$两点, 直线$AP, AQ$与$y$轴交于$M, N$两点, 证明: 线段$MN$的中点是定点.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题20", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018147": { + "id": "018147", + "content": "已知函数$f(x)=(\\dfrac{1}{x}+a) \\ln (x+1)$.\\\\\n(1) 若$a=-1$, 求$f(x)$在$(1, f(1))$处的切线方程;\\\\\n(2) 是否存在$a, b$使$y=f(\\dfrac{1}{x})$图像关于直线$x=b$轴对称?\\\\\n(3) 若$f(x)$在$(0,+\\infty)$上存在极值点, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题21", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018148": { + "id": "018148", + "content": "在直角坐标系$x O y$中, 以坐标原点$O$为极点, $x$轴正半轴为极轴建立极坐标系, 曲线$C_1$的极坐标方程为$\\rho=2 \\sin \\theta(\\dfrac{\\pi}{4} \\leq \\theta \\leq \\dfrac{\\pi}{2})$, 曲线$C_2: \\begin{cases}x=2 \\cos \\alpha, \\\\ y=2 \\sin \\alpha\\end{cases}$($\\alpha$为参数, $\\dfrac{\\pi}{2}<\\alpha<\\pi$).\\\\\n(1) 写出$C_1$的直角坐标方程;\\\\\n(2) 若直线$y=x+m$既与$C_1$没有公共点, 也与$C_2$没有公共点, 求$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题22", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018149": { + "id": "018149", + "content": "已知$f(x)=2|x|+|x-2|$.\\\\\n(1) 求不等式$f(x) \\leq 6-x$的解集;\\\\\n(2) 在直角坐标系$x O y$中, 求不等式组$\\begin{cases}f(x) \\leq y, \\\\ x+y-6 \\leq 0\\end{cases}$所确定的平面区域的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考乙卷理科试题23", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018150": { + "id": "018150", + "content": "已知集合$U=\\{1,2,3,4,5\\}$, $A=\\{1,3\\}$, $B=\\{1,2,4\\}$, 则$A \\cup(\\complement_UB)=$\\bracket{20}.\n\\fourch{$\\{1,3,5\\}$}{$\\{1,3\\}$}{$\\{1,2,4\\}$}{$\\{1,2,4,5\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题1", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018151": { + "id": "018151", + "content": "``$a^2=b^2$''是``$a^2+b^2=2 a b$''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}{既不充分又不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题2", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018152": { + "id": "018152", + "content": "若$a=1.01^{0.5}$, $b=1.01^{0.6}$, $c=0.6^{0.5}$, 则$a, b, c$的大小关系为\\bracket{20}.\n\\fourch{$c>a>b$}{$c>b>a$}{$a>b>c$}{$b>a>c$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题3", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018153": { + "id": "018153", + "content": "函数$f(x)$的图象如下图所示, 则$f(x)$的解析式可能为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -4:4, samples = 100] plot (\\x,{2*cos(\\x/pi*180)/(\\x*\\x+1)});\n\\draw (2,0) node [above] {$2$} (-2,0) node [above] {$-2$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{2(\\mathrm{e}^x-\\mathrm{e}^{-x})}{x^2+2}$}{$\\dfrac{2 \\sin x}{x^2+1}$}{$\\dfrac{2(\\mathrm{e}^x+\\mathrm{e}^{-x})}{x^2+2}$}{$\\dfrac{2 \\cos x}{x^2+1}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题4", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018154": { + "id": "018154", + "content": "已知函数$f(x)$的一条对称轴为直线$x=2$, 一个周期为$4$, 则$f(x)$的解析式可能为\\bracket{20}.\n\\fourch{$\\sin (\\dfrac{\\pi}{2} x)$}{$\\cos (\\dfrac{\\pi}{2} x)$}{$\\sin (\\dfrac{\\pi}{4} x)$}{$\\cos (\\dfrac{\\pi}{4} x)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题5", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018155": { + "id": "018155", + "content": "已知$\\{a_n\\}$为等比数列, $S_n$为数列$\\{a_n\\}$的前$n$项和, $a_{n+1}=2S_n+2$, 则$a_4$的值为\\bracket{20}.\n\\fourch{$3$}{$18$}{$54$}{$152$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题6", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018156": { + "id": "018156", + "content": "调查某种花萝长度和花瓣长度, 所得数据如图所示, 其中相关系数$r=0.8245$, 下列说法正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\draw [->] (0,0) -- (1.2,0) node [below] {花瓣长度};\n\\draw [->] (0,0) -- (0,1.2) node [left] {花萼长度};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0.37/0.23,0.35/0.34,0.54/0.63,0.42/0.44,1.00/0.90,0.77/0.88,0.75/0.75,0.96/0.82,0.65/0.56,0.71/0.61,0.18/0.07,0.42/0.58,0.50/0.80,0.43/0.36,0.50/0.57,0.56/0.50,0.75/0.79,0.55/0.33,0.07/0.21,0.14/0.21,0.43/0.25,0.75/0.52,0.64/0.82,0.72/0.67,0.33/0.18,0.83/0.69,0.56/0.69,0.39/0.21,0.34/0.49,0.93/0.68}\n{\\filldraw (\\i,\\j) circle (0.01);};\n\\draw [dashed, domain = 0.1:1.1] plot (\\x,{0.83*\\x+0.067});\n\\end{tikzpicture}\n\\end{center}\n\\onech{花瓣长度和花萼长度没有相关性}{花瓣长度和花萼长度呈负相关}{花瓣长度和花萼长度呈正相关}{若从样本中抽取一部分, 则这部分的相关系数一定是$0.8245$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题7", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018157": { + "id": "018157", + "content": "在三棱锥$P-ABC$中, 线段$PC$上的点$M$满足$PM=\\dfrac{1}{3} PC$, 线段$PB$上的点$N$满足$PN=\\dfrac{2}{3} PB$, 则三棱锥$P-AMN$和三棱锥$P-ABC$的体积之比为\\bracket{20}.\n\\fourch{$\\dfrac{1}{9}$}{$\\dfrac{2}{9}$}{$\\dfrac{1}{3}$}{$\\dfrac{4}{9}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题8", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018158": { + "id": "018158", + "content": "双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左右焦点分别为$F_1$和$F_2$, 过$F_2$作其中一条渐近线的垂线, 垂足为$P$. 已知$PF_2=2$, 直线$PF_1$的斜率为$\\dfrac{\\sqrt{2}}{4}$, 则双曲线的方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{8}-\\dfrac{y^2}{4}=1$}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{8}=1$}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{2}=1$}{$\\dfrac{x^2}{2}-\\dfrac{y^2}{4}=1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题9", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018159": { + "id": "018159", + "content": "已知$\\mathrm{i}$是虚数单位, 化简$\\dfrac{5+14 \\mathrm{i}}{2+3 \\mathrm{i}}$的结果为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题10", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018160": { + "id": "018160", + "content": "在$(2 x^3-\\dfrac{1}{x})^6$的展开式中, $x^2$的系数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题11", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018161": { + "id": "018161", + "content": "过原点的一条直线与圆$C: (x+2)^2+y^2=3$相切, 交曲线$y^2=2 p x$($p>0$)于点$P$, 若$OP=8$, 则$p$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题12", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018162": { + "id": "018162", + "content": "甲乙丙三个盒子中装有一定量的黑球和白球, 其总数之比为$5: 4: 6$, 这三个盒子中黑球占总数得比例分别为$40 \\%, 25 \\%, 50 \\%$, 现从三个盒子中各取一个球, 取到的三个球都是黑球的概率为\\blank{50}; 将三个盒子混合在一起后任取一个球, 是白球的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题13", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018163": { + "id": "018163", + "content": "在$\\triangle ABC$中, $\\angle A=60^{\\circ}$, $BC=1$, 点$D$为$BC$的中点, 点$E$为$CD$的中点, 若设$\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{AC}=\\overrightarrow {b}$, 则$\\overrightarrow{AE}$可用$\\overrightarrow {a}, \\overrightarrow {b}$表示为\\blank{50}; 若$\\overrightarrow{BF}=\\dfrac{1}{3} \\overrightarrow{BC}$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{AF}$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题14", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018164": { + "id": "018164", + "content": "若函数$f(x)=a x^2-2 x-|x^2-a x+1|$有且仅有两个零点, 则$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题15", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018165": { + "id": "018165", + "content": "在$\\triangle ABC$中, 角$A, B, C$所对的边分別是$a, b, c$. 已知$a=\\sqrt{39}$, $b=2$, $\\angle A=120^{\\circ}$.\n(1) 求$\\sin B$的值;\\\\\n(2) 求$c$的值;\\\\\n(3) 求$\\sin (B-C)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题16", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018166": { + "id": "018166", + "content": "三棱台$ABC-A_1B_1C_1$中, 已知$A_1A \\perp$平面$ABC$, $AB \\perp AC$, $AB=AC=AA_1=2$, $A_1C_1=1$, $N$为线段$AB$的中点, $M$为线段$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (0,0,1) node [left] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (1,0,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw ($(A)!0.5!(B)$) node [left] {$N$} coordinate (N);\n\\draw (B)--(C)--(C_1)--(B_1)--cycle(B_1)--(A_1)--(C_1);\n\\draw [dashed] (B)--(A)--(C)(A)--(A_1);\n\\draw [dashed] (A_1)--(N)(C_1)--(A)--(M);\n\\draw (C_1)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1N\\parallel$平面$C_1MA$;\\\\\n(2) 求平面$C_1MA$与平面$ACC_1A_1$所成角的余弦值;\\\\\n(3) 求点$C$到平面$C_1MA$的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题17", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018167": { + "id": "018167", + "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右顶点分别为$A_1, A_2$, 右焦点为$F$, 且$|A_1F|=3$, $|A_2F|=1$.\\\\\n(1) 求椭圆$C$的方程及离心率;\\\\\n(2) 设点$P$是椭圆$C$上一动点 (不与顶点重合), 直线$A_2P$交$y$轴于点$Q$, 若三角形$A_1PQ$的面积是三角形$A_2FP$面积的二倍, 求直线$A_2P$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题18", + "edit": [ + "20230611\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018168": { + "id": "018168", + "content": "已知$\\{a_n\\}$为等差数列, $a_3+a_5=16$, $a_5-a_3=4$.\\\\\n(1) 求$a_n$和$\\displaystyle\\sum_{i=2^{n-1}}^{2^n-1} a_i$;\\\\\n(2) 设$\\{b_n\\}$为等比数列, 当$2^{k-1} \\leq n \\leq 2^k-1$时, $b_k0$时, 证明: $f(x)>1$;\\\\\n(3) 证明: $\\dfrac{5}{6} \\leq \\ln (n !)-(n+\\dfrac{1}{2}) \\ln (n)+n \\leq 1$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届全国高考天津卷试题20", + "edit": [ + "20230611\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.",