录入2023届全国高考甲乙文科卷
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#修改起始id,出处,文件名
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starting_id = 18104
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raworigin = ""
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filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目14.tex"
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editor = "20230611\t王伟叶"
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starting_id = 18170
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raworigin = "2023届全国高考"
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filename = r"C:\Users\wangweiye\Documents\wwy sync\临时工作区\23届高考文科.tex"
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editor = "20230612\t王伟叶"
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indexed = True
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IndexDescription = "试题"
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@ -465404,6 +465404,926 @@
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"space": "4em",
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"unrelated": []
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},
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"018170": {
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"id": "018170",
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"content": "设全集$U=\\{1,2,3,4,5\\}$, 集合$M=\\{1,4\\}$, $N=\\{2,5\\}$, 则$N \\cup \\overline{M}=$\\bracket{20}.\n\\fourch{$\\{2,3,5\\}$}{$\\{1,3,4\\}$}{$\\{1,2,4,5\\}$}{$\\{2,3,4,5\\}$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题1",
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"edit": [
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"20230612\t王伟叶"
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"space": "",
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},
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"018171": {
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"id": "018171",
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"content": "$\\dfrac{5(1+\\mathrm{i}^3)}{(2+\\mathrm{i})(2-\\mathrm{i})}=$\\bracket{20}.\n\\fourch{$-1$}{$1$}{$1-\\mathrm{i}$}{$1+\\mathrm{i}$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题2",
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"remark": "",
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"unrelated": []
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},
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"018172": {
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"id": "018172",
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"content": "已知向量$\\overrightarrow{a}=(3,1)$, $\\overrightarrow{b}=(2,2)$, 则$\\cos \\langle\\overrightarrow{a}+\\overrightarrow{b}, \\overrightarrow{a}-\\overrightarrow{b}\\rangle=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{17}$}{$\\dfrac{\\sqrt{17}}{17}$}{$\\dfrac{\\sqrt{15}}{15}$}{$\\dfrac{2 \\sqrt{5}}{5}$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题3",
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"edit": [
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"20230612\t王伟叶"
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018173": {
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"id": "018173",
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"content": "某校文艺部有$4$名学生, 其中高一、高二年级各$2$名. 从这$4$名学生中随机选$2$名组织校文艺汇演, 则这$2$名学生来自不同年级的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{6}$}{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{2}{3}$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题4",
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"20230612\t王伟叶"
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"remark": "",
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"space": "",
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"unrelated": []
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"018174": {
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"id": "018174",
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"content": "已知向量$S_n$为等差数列$\\{a_n\\}$的前$n$项和若$a_2+a_6=10$, $a_4 a_8=45$, 则$S_5=$\\bracket{20}.\n\\fourch{$25$}{$22$}{$20$}{$15$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题5",
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"edit": [
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"20230612\t王伟叶"
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],
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"remark": "",
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"space": "",
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},
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"018175": {
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"id": "018175",
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"content": "执行如图的程序框图, 则输出的$B=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, node distance = 10pt]\n\\node [draw, rounded corners] (start) {开始};\n\\node [draw, trapezium, trapezium left angle = 60, trapezium right angle = 120, below = of start] (input) {输入$n=3$, $A=1$, $B=2$, $k=1$};\n\\node [draw, diamond, aspect = 2, below = of input] (switch) {$k\\le n$};\n\\node [draw, below = of switch] (step1) {$A=A+B$};\n\\node [draw, below = of step1] (step2) {$B=A+B$};\n\\node [draw, below = of step2] (step3) {$k=k+1$};\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\\foreach \\i/\\j in {start/input,input/switch,step1/step2,step2/step3,output/end}\n{\\draw [->] (\\i) -- (\\j);};\n\\draw [->] (switch) -- (step1) node [midway, right] {是};\n\\coordinate [right = 15pt of switch] (stepx);\n\\coordinate [left = 15pt of step3] (stepy);\n\\draw [->] (switch) -- (stepx) node [midway, above] {否} -- (stepx|-output) -- (output);\n\\draw [->] (step3) -- (stepy) -- (stepy|-switch) -- (switch);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$21$}{$34$}{$55$}{$89$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题6",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018176": {
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"id": "018176",
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"content": "设$F_1, F_2$为椭圆$C: \\dfrac{x^2}{5}+y^2=1$的两个焦点, 点$P$在$C$上, 若$\\overrightarrow{PF_1} \\cdot \\overrightarrow{PF_2}=0$, 则$|PF_1| \\cdot|PF_2|=$\\bracket{20}.\n\\fourch{$1$}{$2$}{$4$}{$5$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题7",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018177": {
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"id": "018177",
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"content": "曲线$y=\\dfrac{\\mathrm{e}^x}{x+1}$在点$(1, \\dfrac{\\mathrm{e}}{2})$处的切线方程为\\bracket{20}.\n\\fourch{$y=\\dfrac{\\mathrm{e}}{4} x$}{$y=\\dfrac{\\mathrm{e}}{2} x$}{$y=\\dfrac{\\mathrm{e}}{4} x+\\dfrac{\\mathrm{e}}{4}$}{$y=\\dfrac{\\mathrm{e}}{2} x+\\dfrac{3 \\mathrm{e}}{4}$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题8",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018178": {
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"id": "018178",
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"content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的离心率为$\\sqrt{5}$, $C$的一条渐近线与圆$(x-2)^2+(y-3)^2=1$交于$A, B$两点, 则$|AB|=$\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{5}}{5}$}{$\\dfrac{2 \\sqrt{5}}{5}$}{$\\dfrac{3 \\sqrt{5}}{5}$}{$\\dfrac{4 \\sqrt{5}}{5}$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题9",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018179": {
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"id": "018179",
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"content": "在三棱锥$P-ABC$中, $\\triangle ABC$是边长为$2$的等边三角形, $PA=PB=2$, $PC=\\sqrt{6}$, 则该棱锥的体积为\\bracket{20}.\n\\fourch{$1$}{$\\sqrt{3}$}{$2$}{$3$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题10",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018180": {
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"id": "018180",
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"content": "已知函数$f(x)=\\mathrm{e}^{-(x-1)^2}$. 记$a=f(\\dfrac{\\sqrt{2}}{2})$, $b=f(\\dfrac{\\sqrt{3}}{2})$, $c=f(\\dfrac{\\sqrt{6}}{2})$, 则\\bracket{20}.\n\\fourch{$b>c>a$}{$b>a>c$}{$c>b>a$}{$c>a>b$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题11",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018181": {
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"id": "018181",
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"content": "函数$y=f(x)$的图像由$y=\\cos (2 x+\\dfrac{\\pi}{6})$的图像向左平移$\\dfrac{\\pi}{6}$个单位长度得到, 则$y=f(x)$的图像与直线$y=\\dfrac{1}{2} x-\\dfrac{1}{2}$的交点个数为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题12",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018182": {
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"id": "018182",
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"content": "记$S_n$为等比数列$\\{a_n\\}$的前$n$项和. 若$8S_6=7S_3$, 则$\\{a_n\\}$的公比为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题13",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018183": {
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"id": "018183",
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"content": "若$f(x)=(x-1)^2+a x+\\sin (x+\\dfrac{\\pi}{2})$为偶函数, 则$a=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题14",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018184": {
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"id": "018184",
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"content": "若$x, y$满足约束条件$\\begin{cases}3 x-2 y \\leq 3, \\\\ -2 x+3 y \\leq 3,\\\\ x+y \\geq 1,\\end{cases}$ 则$z=3 x+2 y$的最大值为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题15",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018185": {
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"id": "018185",
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"content": "在正方体$ABCD-A_1B_1C_1D_1$中, $AB=4$, $O$为$AC_1$的中点, 若该正方体的棱与球$O$的球面有公共点, 则球$O$的半径的取值范围是\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题16",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018186": {
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"id": "018186",
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"content": "记$\\triangle ABC$的内角$A, B, C$的对边分别为$a, b, c$, 已知$\\dfrac{b^2+c^2-a^2}{\\cos A}=2$.\\\\\n(1) 求$b c$;\\\\\n(2) 若$\\dfrac{a \\cos B-b \\cos A}{a \\cos B+b \\cos A}-\\dfrac{b}{c}=1$, 求$\\triangle ABC$的面积.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2023届全国高考甲卷文科试题17",
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"edit": [
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"20230612\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018187": {
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"id": "018187",
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"content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, $A_1C \\perp$平面$ABC$, $\\angle ACB=90^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (0,0,{sqrt(2)}) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)},0) node [above] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)+(C)-(A)$) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)+(B)-(A)$) node [right] {$B_1$} coordinate (B_1);\n\\draw (A)--(B)--(B_1)--(A_1)--cycle(B)--(A_1)--(C_1)--(B_1);\n\\draw [dashed] (A)--(C)--(B)(C)--(A_1)(C)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$ACC_1A_1 \\perp$平面$BB_1C_1C$;\\\\\n(2) 设$AB=A_1B$, $AA_1=2$, 求四棱锥$A_1-BB_1C_1C$的高.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考甲卷文科试题18",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018188": {
|
||||
"id": "018188",
|
||||
"content": "一项试验旨在研究臭氧效应, 试验方案如下: 选$40$只小白鼠, 随机地将其中$20$只分配到试验组, 另外$20$只分配到对照组, 试验组的小白鼠饲养在高浓度臭氧环境, 对照组的小白鼠饲养在正常环境, 一段时间后统计每只小白鼠体重的增加量(单位: $\\text{g}$). 试验结果如下\\\\\n对照组的小白鼠体重的增加量从小到大排序为:\n\\begin{center}\n\\begin{tabular}{cccccccccc}\n15.2 & 18.8 & 20.2 & 21.3 & 22.5 & 23.2 & 25.8 & 26.5 & 27.5 & 30.1 \\\\\n32.6 & 34.3 & 34.8 & 35.6 & 35.6 & 35.8 & 36.2 & 37.3 & 40.5 & 43.2\n\\end{tabular}\n\\end{center}\n试验组的小白鼠体重的增加量从小到大排序为:\n\\begin{center}\n\\begin{tabular}{cccccccccc}\n7.8 & 9.2 & 11.4 & 12.4 & 13.2 & 15.5 & 16.5 & 18.0 & 18.8 & 19.2 \\\\ 19.8 & 20.2 & 21.6 & 22.8 & 23.6 & 23.9 & 25.1 & 28.2 & 32.3 & 36.5\n\\end{tabular}\n\\end{center}\n(1) 计算试验组的样本平均数;\\\\\n(2) (I) 求$40$只小白鼠体重的增加量的中位数$m$, 再分别统计两样本中小于$m$与不小于$m$的数据的个数, 完成如下列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline &$<m$&$\\geq m$\\\\\n\\hline 对照组 & & \\\\\n\\hline 试验组 & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(II) 根据(I)中的列联表, 能否有$95 \\%$的把握认为小白鼠在高浓度臭氧环境中与在正常环境中体重的增加量有差异?、、\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(b+c)(a+c)(b+d)}$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$k_0$& 0.10 & 0.05 & 0.010 & 0.005 & 0.001 \\\\\n\\hline$p(\\chi^2 \\geq k_0)$& 2.706 & 3.841 & 6.635 & 7.879 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考甲卷文科试题19",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018189": {
|
||||
"id": "018189",
|
||||
"content": "已知函数$f(x)=a x-\\dfrac{\\sin x}{\\cos ^2 x}, x \\in(0, \\dfrac{\\pi}{2})$.\\\\\n(1) 当$a=1$时, 讨论$f(x)$的单调性;\\\\\n(2) 若$f(x)+\\sin x<0$, 求$a$的取值范围.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考甲卷文科试题20",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018190": {
|
||||
"id": "018190",
|
||||
"content": "已知直线$x-2 y+1=0$与抛物线$C: y^2=2 p x$($p>0$) 交于$A, B$两点, $|AB|=4 \\sqrt{15}$.\\\\\n(1) 求$p$;\\\\\n(2) 设$F$为$C$的焦点, $M, N$为$C$上两点, 且$\\overrightarrow{FM} \\cdot \\overrightarrow{FN}=0$, 求$\\triangle MFN$面积的最小值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考甲卷文科试题21",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018191": {
|
||||
"id": "018191",
|
||||
"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届全国高考甲卷文科试题22",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018192": {
|
||||
"id": "018192",
|
||||
"content": "已知$a>0$, $f(x)=2|x-a|-a$.\\\\\n(1) 求不等式$f(x)<x$的解集;\\\\\n(2) 若函数$y=f(x)$的图像与$x$轴围城的三角形面积为$2$, 求$a$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考甲卷文科试题23",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018193": {
|
||||
"id": "018193",
|
||||
"content": "$|2+\\mathrm{i}^2+2 \\mathrm{i}^3|=()$\\bracket{20}.\n\\fourch{$1$}{$2$}{$\\sqrt{5}$}{$5$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题1",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018194": {
|
||||
"id": "018194",
|
||||
"content": "设全集$U=\\{0,1,2,4,6,8\\}$, 集合$M=\\{0,4,6\\}$, $N=\\{0,1,6\\}$, 则$M \\cup \\overline{N}=$\\bracket{20}.\n\\fourch{$\\{0,2,4,6,8\\}$}{$\\{0,1,4,6,8\\}$}{$\\{1,2,4,6,8\\}$}{$U$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题2",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018195": {
|
||||
"id": "018195",
|
||||
"content": "如图, 网格纸上绘制的是个零件的三视图, 网格小正方形的边长为$1$, 则该零件的表面积为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=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届全国高考乙卷文科试题3",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018196": {
|
||||
"id": "018196",
|
||||
"content": "在$\\triangle ABC$中, 内角$A, B, C$的对边分别是$a, b, c$, 若$a \\cos B-b \\cos A=c$, 且$C=\\dfrac{\\pi}{5}$, 则$\\angle B=$\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{10}$}{$\\cdot \\dfrac{\\pi}{5}$}{$\\dfrac{3 \\pi}{10}$}{$\\dfrac{2 \\pi}{5}$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题4",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018197": {
|
||||
"id": "018197",
|
||||
"content": "已知函数$f(x)=\\dfrac{x \\mathrm{e}^x}{\\mathrm{e}^{a x}-1}$是偶函数, 则实数$a=$\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$1$}{$2$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题5",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018198": {
|
||||
"id": "018198",
|
||||
"content": "正方形$ABCD$的边长是$2$, $E$是$AB$的中点, 则$\\overrightarrow{EC} \\cdot \\overrightarrow{ED}=$\\bracket{20}.\n\\fourch{$\\sqrt{5}$}{$3$}{$2 \\sqrt{5}$}{$5$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题6",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018199": {
|
||||
"id": "018199",
|
||||
"content": "已知$O$是平面直角坐标系的原点, 在区域$\\{(x, y) | 1 \\leq x^2+y^2 \\leq 4\\}$内随机取一点$A$, 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届全国高考乙卷文科试题7",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018200": {
|
||||
"id": "018200",
|
||||
"content": "函数$f(x)=x^3+a x+2$存在$3$个零点, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty,-2)$}{$(-\\infty,-3)$}{$(-4,-1)$}{$(-3,0)$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题8",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018201": {
|
||||
"id": "018201",
|
||||
"content": "某学校举办作文比赛, 共$6$个主题, 每位参赛同学从中随机抽取一个主题准备作文, 则甲、乙两位参赛同学抽到不同主题概率为\\bracket{20}.\n\\fourch{$\\dfrac{5}{6}$}{$\\dfrac{2}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{1}{3}$}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题9",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018202": {
|
||||
"id": "018202",
|
||||
"content": "函数$f(x)=\\sin (\\omega x+\\varphi)$在区间$(\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3})$上单调递增, 直线$x=\\dfrac{\\pi}{6}$和$x=\\dfrac{2 \\pi}{3}$是函数$y=f(x)$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届全国高考乙卷文科试题10",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018203": {
|
||||
"id": "018203",
|
||||
"content": "已知实数$x, y$满足$x^2+y^2-4 x-2 y-4=0$, 则$x-y$的最大值是\\bracket{20}.\n\\fourch{$1+\\dfrac{3 \\sqrt{2}}{2}$}{4}{$1+3 \\sqrt{2}$}{7}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "选择题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题11",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018204": {
|
||||
"id": "018204",
|
||||
"content": "设$A, B$为双曲线$x^2-\\dfrac{y^2}{9}=1$上两点, 下列四个点中, 可以为线段$AB$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届全国高考乙卷文科试题12",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018205": {
|
||||
"id": "018205",
|
||||
"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届全国高考乙卷文科试题13",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018206": {
|
||||
"id": "018206",
|
||||
"content": "若$\\theta \\in(0, \\dfrac{\\pi}{2}), \\tan \\theta=\\dfrac{1}{2}$, 则$\\sin \\theta-\\cos \\theta=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题14",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018207": {
|
||||
"id": "018207",
|
||||
"content": "若$x, y$满足约束条件$\\begin{cases}x-3 y \\leq-1 \\\\x+2 y \\leq 9, \\\\\n3 x+y \\geq 7,\\end{cases}$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届全国高考乙卷文科试题15",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018208": {
|
||||
"id": "018208",
|
||||
"content": "已知点$S, A, B, C$均在半径为$2$的球面上, $\\triangle ABC$是边长为$3$的等边三角形, $SA \\perp$平面$ABC$, 则$SA=$\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题16",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018209": {
|
||||
"id": "018209",
|
||||
"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:\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记$z_i=x_i-y_i$($i=1,2, \\cdots, 10$), 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届全国高考乙卷文科试题17",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018210": {
|
||||
"id": "018210",
|
||||
"content": "记$S_n$为等差数列$\\{a_n\\}$的前$n$项和, 已知$a_2=11$, $S_{10}=40$.\n(1) 求$\\{a_n\\}$的通项公式;\\\\\n(2) 求数列$\\{|a_n|\\}$的前$n$项和$T_n$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题18",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018211": {
|
||||
"id": "018211",
|
||||
"content": "如图, 在三棱锥$P-ABC$中, $AB \\perp BC$, $AB=2$, $BC=2 \\sqrt{2}$, $PB=PC=\\sqrt{6}$, $BP$、$AP$、$BC$的中点分别为$D$、$E$、$O$, 点$F$在$AC$上, $BF \\perp AO$.\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)(O)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $EF\\parallel$平面$ADO$;\\\\\n(2) 若$\\angle POF=120^{\\circ}$, 求三棱锥$P-ABC$的体积.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题19",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018212": {
|
||||
"id": "018212",
|
||||
"content": "已知函数$f(x)=(\\dfrac{1}{x}+a) \\ln (x+1)$.\\\\\n(1) 当$a=-1$时, 求曲线$y=f(x)$在$(1, f(1))$处的切线方程;\\\\\n(2) 若函数$f(x)$在$(0,+\\infty)$单调递增, 求$a$的取值范围.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题20",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018213": {
|
||||
"id": "018213",
|
||||
"content": "已知椭圆$C: \\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1$($a>b>0$)的离心率是$\\dfrac{\\sqrt{5}}{3}$, 点$A(-2,0)$在$C$上.\\\\\n(1) 求$C$的方程;\\\\\n(2) 过点$(2,3)$的直线交$C$于$P, Q$两点, 直线$AP, AQ$与$y$轴的交点分别为$M, N$, 证明: 线段$MN$的中点为定点.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "2023届全国高考乙卷文科试题21",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018214": {
|
||||
"id": "018214",
|
||||
"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$\\rho=2 \\sin \\theta$($\\dfrac{\\pi}{4} \\leq \\theta \\leq \\dfrac{\\pi}{2}$), $C_2: \\begin{cases}x=2 \\cos \\alpha, \\\\ y=2 \\sin \\alpha\\end{cases}$($\\alpha$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届全国高考乙卷文科试题22",
|
||||
"edit": [
|
||||
"20230612\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018215": {
|
||||
"id": "018215",
|
||||
"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届全国高考乙卷文科试题23",
|
||||
"edit": [
|
||||
"20230612\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.",
|
||||
|
|
|
|||
Reference in New Issue