| {"question": "设集合 $U=\\mathbf{R}$, 集合 $M=\\{x \\mid x<1\\}, N=\\{x \\mid-1<x<2\\}$, 则 $\\{x \\mid x \\geq 2\\}=$ ", "options": ["$\\overbrace{U}(M \\cup N)$", "$N \\bigcup \\partial_{U} M$", "$\\partial_{U}(M \\cap N)$", "$M \\cup \\triangleleft_{U} N$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "在 $\\triangle A B C$ 中, 内角 $A, B, C$ 的对边分别是 $a, b, c$, 若 $a \\cos B-b \\cos A=c$, 且 $C=\\frac{\\pi}{5}$, 则 $\\angle B=(\\quad)$", "options": ["$\\frac{\\pi}{10}$", "$\\frac{\\pi}{5}$", "$\\frac{3 \\pi}{10}$", "$\\frac{2 \\pi}{5}$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "若$z=1+i$,则$|iz+3\\overline{z}|=\\left(\\ \\ \\right)$", "options": ["$4\\sqrt{5}$", "$4\\sqrt{2}$", "$2\\sqrt{5}$", "$2\\sqrt{2}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "已知$z=1-2i$,且$z+a\\overline{z}+b=0$,其中$a$,$b$为实数,则（ ）", "options": ["$a=1$，$b=-2$", "$a=-1$，$b=2$", "$a=1$，$b=2$", "$a=-1$，$b=-2$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "在长方体$ABCD-A_{1}B_{1}C_{1}D_{1}$中,已知$B_{1}D$与平面$ABCD$和平面$AA_{1}B_{1}B$所成的角均为$30^{\\circ}$,则（ ）", "options": ["$AB=2AD$", "$AB$与平面$AB_{1}C_{1}D$所成的角为$30^{\\circ}$", "$AC=CB_{1}$", "$B_{1}D$与平面$BB_{1}C_{1}C$所成的角为$45^{\\circ}$"], "answer": "D", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "6.已知函数 $f(x)=a \\mathrm{e}^{x}-\\ln x$ 在区间 $(1,2)$ 单调递增, 则 $a$ 的最小值为()", "options": ["$\\mathrm{e}^{2}$", "e", "$\\mathrm{e}^{-1}$", "$\\mathrm{e}^{-2}$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "设$（1+2i）a+b=2i$，其中a,b为实数，则（ ）", "options": ["$a=1,b=-1$", "$a=1,b=1$", "$a=-1,b=1$", "$a=-1,b=-1$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "在四棱雉 $P-A B C D$ 中, 底面 $A B C D$ 为正方形, $A B=4, P C=P D=3, \\angle P C A=45^{\\circ}$, 则 $\\triangle \\mathrm{PBC}$ 的面积为 ( )", "options": ["$2 \\sqrt{2}$", "$3 \\sqrt{2}$", "$4 \\sqrt{2}$", "$5 \\sqrt{2}$"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "设函数$f\\left(x\\right)=\\sin (\\omega x+\\frac{π}{3})$在区间$\\left(0,\\pi \\right)$恰有三个极值点、两个零点,则$\\omega $的取值范围是（ ）", "options": ["$[ \\dfrac {5}{3},\\dfrac {13}{6})$", "$[ \\dfrac {5}{3},\\dfrac {19}{6})$", "$(\\dfrac {13}{6},\\dfrac {8}{3}] $", "$(\\dfrac {13}{6},\\dfrac {19}{6}] $"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "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（ ）", "options": ["$\\frac{1}{6}$", "$\\frac{1}{3}$", "$\\frac{1}{2}$", "$\\frac{2}{3}$"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "10.已知等差数列 $\\left\\{a_{n}\\right\\}$ 的公差为 $\\frac{2 \\pi}{3}$, 集合 $S=\\left\\{\\cos a_n \\mid n \\in \\mathrm{N}^*\\right\\}$, 若 $S=\\{a, b\\}$, 则 $a b=($ ( )", "options": ["-1", "$-\\frac{1}{2}$", "0", "$\\frac{1}{2}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "在$\\triangle ABC$中,点$D$在边$AB$上,$BD=2DA$.记$\\overrightarrow{CA}=\\overrightarrow{m}$,$\\overrightarrow{CD}=\\overrightarrow{n}$,则$\\overrightarrow{CB}=\\left(\\ \\ \\right)$", "options": ["$3\\overrightarrow{m}-2\\overrightarrow{n}$", "$-2\\overrightarrow{m}+3\\overrightarrow{n}$", "$3\\overrightarrow{m}+2\\overrightarrow{n}$", "$2\\overrightarrow{m}+3\\overrightarrow{n}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设全集 $U=\\{1,2,3,4,5\\}$, 集合 $M=\\{1,4\\}, N=\\{2,5\\}$, 则 $\\overbrace{U}(M）=$ ", "options": ["$\\{2,3,5\\}$", "$\\{1,3,4\\}$", "$\\{1,2,4,5\\}$", "$\\{2,3,4,5\\}$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "记 $S_{n}$ 为等比数列 $\\left\\{a_{n}\\right\\}$ 的前 $n$ 项和, 若 $S_{4}=-5, S_{6}=21 S_{2}$, 则 $S_{8}=()$", "options": ["120", "85", "-85", "-120"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "已知向量$\\overrightarrow{a}$,$\\overrightarrow{b}$满足$|\\overrightarrow{a}|=1$,$|\\overrightarrow{b}|=\\sqrt{3}$,$|\\overrightarrow{a}-2\\overrightarrow{b}|=3$,则$\\overrightarrow{a}\\cdot \\overrightarrow{b}=\\left(\\ \\ \\right)$", "options": ["$-2$", "$-1$", "$1$", "$2$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "9.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 选择种数为 ( )", "options": ["120", "60", "40", "30"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "已知椭圆$C$:$\\frac{{{x^2}}}{{{a^2}}}+\\frac{{{y^2}}}{{{b^2}}}=1\\left(a \\gt b \\gt 0\\right)$的离心率为$\\frac{1}{3}$,$A_{1}$,$A_{2}$分别为$C$的左、右顶点,$B$为$C$的上顶点.若$\\overrightarrow{B{A_1}}\\cdot \\overrightarrow{B{A_2}}=-1$,则$C$的方程为（ ）", "options": ["$\\frac{{{x^2}}}{{18}}+\\frac{{{y^2}}}{{16}}=1$", "$\\frac{{{x^2}}}{9}+\\frac{{{y^2}}}{8}=1$", "$\\frac{{{x^2}}}{3}+\\frac{{{y^2}}}{2}=1$", "$\\frac{{{x^2}}}{2}+y^{2}=1$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "已知集合$A=\\{-1,1,2,4\\}$,$B=\\{x||x-1|\\leqslant 1\\}$,则$A\\cap B=\\left(\\ \\ \\right)$", "options": ["$\\{-1,2\\}$", "$\\{1,2\\}$", "$\\{1,4\\}$", "$\\{-1,4\\}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "已知函数 $f(x)=\\sin (\\omega x+\\varphi)$ 在区间 $\\left(\\frac{\\pi}{6}, \\frac{2 \\pi}{3}\\right)$ 单调递增, 直线 $x=\\frac{\\pi}{6}$ 和 $x=\\frac{2 \\pi}{3}$ 为函数 $y=f(x)$ 的图像 的两条对称轴, 则 $f\\left(-\\frac{5 \\pi}{12}\\right)=(\\quad)$", "options": ["$-\\frac{\\sqrt{3}}{2}$", "$-\\frac{1}{2}$", "$\\frac{1}{2}$", "$\\frac{\\sqrt{3}}{2}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "4.向量 $|\\vec{a}|=|\\vec{b}|=1,|\\vec{c}|=\\sqrt{2}$, 且 $\\vec{a}+\\vec{b}+\\vec{c}=\\overrightarrow{0}$, 则 $\\cos \\langle\\vec{a}-\\vec{c}, \\vec{b}-\\vec{c}\\rangle=(\\quad)$", "options": ["$-\\frac{1}{5}$", "$-\\frac{2}{5}$", "$\\frac{2}{5}$", "$\\frac{4}{5}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "记函数$f\\left(x\\right)=\\sin (\\omega x+\\frac{π}{4})+b\\left(\\omega\\ \\ \\gt 0\\right)$的最小正周期为$T$.若$\\$2pi$ \\lt T$ ,且$y=f\\left(x\\right)$的图像关于点（$\\frac{3π}{2}$,$2)$中心对称,则$f(\\frac{π}{2})=\\left(\\ \\ \\right)$", "options": ["$1$", "$\\frac{3}{2}$", "$\\frac{5}{2}$", "$3$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "设 $A, B$ 为双曲线 $x^{2}-\\frac{y^{2}}{9}=1$ 上两点, 下列四个点中, 可为线段 $A B$ 中点的是 $()$", "options": ["$(1,1)$", "$(-1,2)$", "$(1,3)$", "$(-1,-4)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "椭圆$C$:$\\frac{{{x^2}}}{{{a^2}}}+\\frac{{{y^2}}}{{{b^2}}}=1\\left(a \\gt b \\gt 0\\right)$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$\\frac{1}{4}$,则$C$的离心率为（ ）", "options": ["$\\frac{{\\sqrt{3}}}{2}$", "$\\frac{{\\sqrt{2}}}{2}$", "$\\frac{1}{2}$", "$\\frac{1}{3}$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "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（ ）", "options": ["$\\frac{1}{3}$", "$\\frac{1}{2}$", "$\\frac{{\\sqrt{3}}}{3}$", "$\\frac{{\\sqrt{2}}}{2}$"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "8.已知双曲线 $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ 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 $|A B|=(\\quad)$", "options": ["$\\frac{1}{5}$", "$\\frac{\\sqrt{5}}{5}$", "$\\frac{2 \\sqrt{5}}{5}$", "$\\frac{4 \\sqrt{5}}{5}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "设$F$为抛物线$C:y^{2}=4x$的焦点,点$A$在$C$上,点$B\\left(3,0\\right)$,若$|AF|=|BF|$,则$|AB|=\\left(\\ \\ \\right)$", "options": ["$2$", "$2\\sqrt{2}$", "$3$", "$3\\sqrt{2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "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 ( )", "options": ["$1+\\frac{3 \\sqrt{2}}{2}$", "4", "$1+3 \\sqrt{2}$", "7"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "设全集$U=\\{1,2,3,4,5\\}$,集合$M$满足$\\complement _{U}M=\\{1,3\\}$,则（ ）", "options": ["$2\\in M$", "$3\\in M$", "$4\\notin M$", "$5\\notin M$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "已知向量$\\overrightarrow{a}=\\left(2,1\\right)$,$\\overrightarrow{b}=\\left(-2,4\\right)$,则$|\\overrightarrow{a}-\\overrightarrow{b}|=\\left(\\ \\ \\right)$", "options": ["$2$", "$3$", "$4$", "$5$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "已知正三棱台的高为$1$,上下底面的边长分别为$3\\sqrt{3}$和$4\\sqrt{3}$,其顶点都在同一球面上,则该球的表面积为（ ）", "options": ["$100\\pi $", "$128\\pi $", "$144\\pi $", "$192\\pi $"], "answer": "A", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "若$z=-1+\\sqrt{3}i$,则$\\frac{z}{{z\\overline{z}-1}}=\\left(\\ \\ \\right)$", "options": ["$-1+\\sqrt{3}i$", "$-1-\\sqrt{3}i$", "$-\\frac{1}{3}+\\frac{{\\sqrt{3}}}{3}i$", "$-\\frac{1}{3}-\\frac{{\\sqrt{3}}}{3}i$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "5.设 $O$ 为平面坐标系的坐标原点, 在区域 $\\left\\{(x, y) \\mid 1 \\leq x^{2}+y^{2} \\leq 4\\right\\}$ 内随机取一点, 记该点为 $A$, 则直线 $O A$ 的倾斜角不大于 $\\frac{\\pi}{4}$ 的概率为 ( )", "options": ["$\\frac{1}{8}$", "$\\frac{1}{6}$", "$\\frac{1}{4}$", "$\\frac{1}{2}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "$\\left|2+\\mathrm{i}^{2}+2 \\mathrm{i}^{3}\\right|=(\\quad\\quad)$", "options": ["1", "2", "$\\sqrt{5}$", "5"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "5.已知正项等比数列 $\\left\\{a_{n}\\right\\}$ 中, $a_{1}=1, S_{n}$ 为 $\\left\\{a_{n}\\right\\}$ 前 $n$ 项和, $S_{5}=5 S_{3}-4$, 则 $S_{4}=(\\quad)$", "options": ["7", "9", "15", "30"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "已知函数$f\\left(x\\right)=\\cos ^{2}x-\\sin ^{2}x$,则（ ）", "options": ["$f\\left(x\\right)$在$(-\\frac{π}{2}$，$-\\frac{π}{6})$上单调递减", "$f\\left(x\\right)$在$(-\\frac{π}{4}$，$\\frac{π}{{12}})$上单调递增", "$f\\left(x\\right)$在$(0$，$\\frac{π}{3})$上单调递减", "$f\\left(x\\right)$在（$\\frac{π}{4}$，$\\frac{{7π}}{{12}})$上单调递增"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "己知椭圆 $\\frac{x^{2}}{9}+\\frac{y^{2}}{6}=1, F_{1}, F_{2}$ 两个焦点, $O$ 为原点, $P$ 为椭圆上一点, $\\cos \\angle F_{1} P F_{2}=\\frac{3}{5}$, 则 $|P O|=(\\quad)$", "options": ["$\\frac{2}{5}$", "$\\frac{\\sqrt{30}}{2}$", "$\\frac{3}{5}$", "$\\frac{\\sqrt{35}}{2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "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_{甲}$和$S_{乙}$,体积分别为$V_{甲}$和$V_{乙}$.若$\\frac{{S}_{甲}}{{S}_{乙}}=2$,则$\\frac{{{V_甲}}}{{{V_乙}}}=\\left(\\ \\ \\right)$", "options": ["$\\sqrt{5}$", "$2\\sqrt{2}$", "$\\sqrt{10}$", "$\\frac{{5\\sqrt{10}}}{4}$"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "设集合$A=\\{1,2\\}$,$B=\\{2,4,6\\}$,则$A\\cup B=\\left(\\ \\ \\right)$", "options": ["$\\{2\\}$", "$\\{1,2\\}$", "$\\{2,4,6\\}$", "$\\{1,2,4,6\\}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "已知圆雉 $P O$ 的底面半径为 $\\sqrt{3}, O$ 为底面圆心, $P A, P B$ 为圆雉的母线, $\\angle A O B=120^{\\circ}$, 若 $\\triangle P A B$ 的面 积等于 $\\frac{9 \\sqrt{3}}{4}$, 则该圆雉的体积为 $($ ( )", "options": ["$\\pi$", "$\\sqrt{6} \\pi$", "$3 \\pi$", "$3 \\sqrt{6} \\pi$"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "函数 $f(x)=x^{3}+a x+2$ 存在 3 个零点, 则 $a$ 的取值范围是（）", "options": ["$(-\\infty,-2)$", "$(-\\infty,-3)$", "$(-4,-1)$", "$(-3,0)$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "若实数$x$,$y$满足约束条件$\\left\\{\\begin{array}{l}x-2≥0,\\\\ 2x+y-7≤0,\\\\ x-y-2≤0,\\end{array}\\right.$则$z=3x+4y$的最大值是______", "options": ["$20$", "$18$", "$13$", "$6$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "已知向量$\\overrightarrow{a}=\\left(3,4\\right)$,$\\overrightarrow{b}=\\left(1,0\\right)$,$\\overrightarrow{c}=\\overrightarrow{a}+t\\overrightarrow{b}$,若$ \\lt \\overrightarrow{a}$,$\\overrightarrow{c} \\gt = \\lt \\overrightarrow{b}$,$\\overrightarrow{c} \\gt $,则实数$t=\\left(\\ \\ \\right)$", "options": ["$-6$", "$-5$", "$5$", "$6$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "若集合$M=\\{x|\\sqrt{x} \\lt 4\\}$,$N=\\{x\\left|\\right.3x\\geqslant 1\\}$,则$M\\cap N=\\left(\\ \\ \\right)$", "options": ["$\\{x\\left|\\right.0\\leqslant x \\lt 2\\}$", "$\\{x|\\frac{1}{3}\\leqslant x \\lt 2\\}$", "$\\{x\\left|\\right.3\\leqslant x \\lt 16\\}$", "$\\{x|\\frac{1}{3}\\leqslant x \\lt 16\\}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "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", "options": ["$\\frac{5}{6}$", "$\\frac{2}{3}$", "$\\frac{1}{2}$", "$\\frac{1}{3}$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "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（ ）", "options": ["$\\frac{1}{3}$", "$\\frac{1}{2}$", "$\\frac{{\\sqrt{3}}}{3}$", "$\\frac{{\\sqrt{2}}}{2}$"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "已知 $\\mathrm{e} O$ 的半径为 1 , 直线 $P A$ 与 $\\mathrm{e} O$ 相切于点 $A$, 直线 $P B$ 与 $\\mathrm{e} O$ 交于 $B, C$ 两点, $D$ 为 $B C$ 的中点, 若 $|P O|=\\sqrt{2}$, 则 $\\overrightarrow{P A} \\cdot \\overrightarrow{P D}$ 的最大值为 $(\\quad)$", "options": ["$\\frac{1+\\sqrt{2}}{2}$", "$\\frac{1+2 \\sqrt{2}}{2}$", "$1+\\sqrt{2}$", "$2+\\sqrt{2}$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "在长方体$ABCD-A_{1}B_{1}C_{1}D_{1}$中,已知$B_{1}D$与平面$ABCD$和平面$AA_{1}B_{1}B$所成的角均为$30^{\\circ}$,则（ ）", "options": ["$AB=3AD$", "$AB$与平面$AB_{1}C_{1}D$所成的角为$30^{\\circ}$", "$AC=CB_{1}$", "$B_{1}D$与平面$BB_{1}C_{1}C$所成的角为$45^{\\circ}$"], "answer": "D", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "12.函数 $y=f(x)$ 的图象由 $y=\\cos \\left(2 x+\\frac{\\pi}{6}\\right)$ 的图象向左平移 $\\frac{\\pi}{6}$ 个单位长度得到, 则 $y=f(x)$ 的图象与直线 $y=\\frac{1}{2} x-\\frac{1}{2}$ 的交点个数为 $(\\quad)$", "options": ["1", "2", "3", "4"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "若直线$2x+y-1=0$是圆$\\left(x-a\\right)^{2}+y^{2}=1$的一条对称轴,则$a=\\left(\\ \\ \\right)$", "options": ["$\\frac{1}{2}$", "$-\\frac{1}{2}$", "$1$", "$-1$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "10.已知 $f(x)$ 为函数 $y=\\cos \\left(2 x+\\frac{\\pi}{6}\\right)$ 向左平移 $\\frac{\\pi}{6}$ 个单位所得函数, 则 $y=f(x)$ 与 $y=\\frac{1}{2} x-\\frac{1}{2}$ 的交点个数为 $(\\quad)$", "options": ["1", "2", "3", "4"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "若函数$f\\left(x\\right)$的定义域为$R$,且$f\\left(x+y\\right)+f\\left(x-y\\right)=f\\left(x\\right)f\\left(y\\right)$,$f\\left(1\\right)=1$,则$\\sum_{k=1}^{22}f\\left(k\\right)=\\left(\\ \\ \\right)$", "options": ["$-3$", "$-2$", "0", "1"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "若复数$z$满足$i\\cdot z=3-4i$,则$|z|=\\left(\\ \\ \\right)$", "options": ["$1$", "$5$", "$7$", "$25$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "已知 $f(x)=\\frac{x \\mathrm{e}^{x}}{\\mathrm{e}^{a x}-1}$ 是偶函数, 则 $a=(\\quad)$", "options": ["-2", "-1", "1", "2"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "设$a=0.1e^{0.1}$,$b=\\frac{1}{9}$,$c=-\\ln 0.9$,则（ ）", "options": ["$a \\lt b \\lt c$", "$c \\lt b \\lt a$", "$c \\lt a \\lt b$", "$a \\lt c \\lt b$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "函数$f\\left(x\\right)=\\cos x+\\left(x+1\\right)\\sin x+1$在区间$\\left[0,2\\pi \\right]$的最小值、最大值分别为（ ）", "options": ["$-\\frac{π}{2}$，$\\frac{π}{2}$", "$-\\frac{{3π}}{2}$，$\\frac{π}{2}$", "$-\\frac{π}{2}$，$\\frac{π}{2}+2$", "$-\\frac{{3π}}{2}$，$\\frac{π}{2}+2$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "设 $A, B$ 为双曲线 $x^{2}-\\frac{y^{2}}{9}=1$ 上两点, 下列四个点中, 可为线段 $A B$ 中点的是 $(\\quad)$", "options": ["$(1,1)$", "$(-1,2)$", "$(1,3)$", "$(-1,-4)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "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$148.5m$时,相应水面的面积为$140.0km^{2}$；水位为海拔$157.5m$时,相应水面的面积为$180.0km^{2}$.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$148.5m$上升到$157.5m$时,增加的水量约为（$\\sqrt{7}\\approx 2.65)\\left(\\ \\ \\right)$", "options": ["$1.0\\times 10^{9}m^{3}$", "$1.2\\times 10^{9}m^{3}$", "$1.4\\times 10^{9}m^{3}$", "$1.6\\times 10^{9}m^{3}$"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "双曲线$C$的两个焦点为$F_{1}$,$F_{2}$,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$\\cos \\angle F_{1}NF_{2}=\\frac{3}{5}$,则$C$的离心率为（ ）", "options": ["$\\frac{{\\sqrt{5}}}{2}$", "$\\frac{{\\sqrt{5}}}{2}$ 或者 $\\frac{{\\sqrt{13}}}{2}$", "$\\frac{{\\sqrt{13}}}{2}$", "$\\frac{{\\sqrt{17}}}{2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "将函数$f\\left(x\\right)=\\sin (\\omega x+\\frac{π}{3})\\left(\\omega\\ \\ \\gt 0\\right)$的图像向左平移$\\frac{π}{2}$个单位长度后得到曲线$C$,若$C$关于$y$轴对称,则$\\omega $的最小值是（ ）", "options": ["$\\frac{1}{6}$", "$\\frac{1}{4}$", "$\\frac{1}{3}$", "$\\frac{1}{2}$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "2.若复数 $(a+\\mathrm{i})(1-a \\mathrm{i})=2, a \\in \\mathrm{R}$, 则 $a=(\\quad)$", "options": ["-1", "0", "1", "2"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "1.在复平面内, $(1+3 \\mathrm{i})(3-\\mathrm{i})$ 对应的点位于（ ）", "options": ["第一象限", "第二象限", "第三象限", "第四象限"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "已知$a$,$b\\in \\bm{R}$,$a+3i=\\left(b+i\\right)i(i$为虚数单位),则______", "options": ["$a=1$，$b=-3$", "$a=-1$，$b=3$", "$a=-1$，$b=-3$", "$a=1$，$b=3$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "已知全集$U=\\{x\\left|\\right.-3 \\lt x \\lt 3\\}$,集合$A=\\{x\\left|\\right.-2 \\lt x\\leqslant 1\\}$,则$\\complement _{U}A=\\left(\\ \\ \\right)$", "options": ["$(-2,1]$", "$(-3,-2)\\cup \\left[1,3\\right)$", "$\\left[-2,1\\right)$", "$(-3,-2]\\cup \\left(1,3\\right)$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "4.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 ( )", "options": ["$\\frac{1}{6}$", "$\\frac{1}{3}$", "$\\frac{1}{2}$", "$\\frac{2}{3}$"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "6.有 60 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（）", "options": ["0.8", "0.4", "0.2", "0.1"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "设$x\\in \\bm{R}$,则\"$\\sin x=1$\"是\"$\\cos x=0$\"的______", "options": ["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"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "设集合$A=\\{-2,-1,0,1,2\\}$,$B=\\{x|0\\leqslant x \\lt \\frac{5}{2}\\}$,则$A\\cap B=\\left(\\ \\ \\right)$", "options": ["$\\{0,1,2\\}$", "$\\{-2,-1,0\\}$", "$\\{0,1\\}$", "$\\{1,2\\}$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "5.已知椭圆 $C: \\frac{x^{2}}{3}+y^{2}=1$ 的左、右焦点分别为 $F_{1}, F_{2}$, 直线 $y=x+m$ 与 $C$ 交 于 $A, B$ 两点, 若 $\\triangle F_{1} A B$ 面积是 $\\triangle F_{2} A B$ 面积的 2 倍, 则 $m=()$", "options": ["$\\frac{2}{3}$", "$\\frac{\\sqrt{2}}{3}$", "$-\\frac{\\sqrt{2}}{3}$", "$-\\frac{2}{3}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "2.$\\frac{5\\left(1+i^{3}\\right)}{(2+i)(2-i)}=(\\quad)$", "options": ["-1", "1", "$1-\\mathrm{i}$", "$1+\\mathrm{i}$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "设$\\{a_{n}\\}$是公差不为$0$的无穷等差数列,则\"$\\{a_{n}\\}$为递增数列\"是\"存在正整数$N_{0}$,当$n \\gt N_{0}$时,$a_{n} \\gt 0$\"的（ ）", "options": ["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"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "若$\\sin \\left(\\alpha +\\beta \\right)+\\cos \\left(\\alpha +\\beta \\right)=2\\sqrt{2}\\cos (\\alpha +\\frac{π}{4})\\sin \\beta $,则（ ）", "options": ["$\\tan \\left(\\alpha -\\beta \\right)=1$", "$\\tan \\left(\\alpha +\\beta \\right)=1$", "$\\tan \\left(\\alpha -\\beta \\right)=-1$", "$\\tan \\left(\\alpha +\\beta \\right)=-1$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "在$\\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{p A}\\cdot\\overrightarrow{p B}$的取值范围是（）", "options": ["[-5,3]", "[-3,5]", "[-6,4]", "[-4,6]"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "为了得到函数$y=2\\sin 3x$的图象,只要把函数$y=2\\sin (3x+\\dfrac{π}{5})$图象上所有的点______", "options": ["向左平移$\\dfrac{π}{5}$个单位长度", "向右平移$\\dfrac{π}{5}$个单位长度", "向左平移$\\dfrac{π}{{15}}$个单位长度", "向右平移$\\dfrac{π}{{15}}$个单位长度"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "设全集$U=\\{-2,-1,0,1,2,3\\}$,集合$A=\\{-1,2\\}$,$B=\\{x|x^{2}-4x+3=0\\}$,则$\\complement _{U}(A\\cup B)=\\left(\\ \\ \\right)$", "options": ["$\\{1,3\\}$", "$\\{0,3\\}$", "$\\{-2,1\\}$", "$\\{-2,0\\}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "已知数列$\\{a_{n}\\}$满足$a_{1}=1$,$a_{n+1}=a_{n}-\\dfrac{1}{3}a_{n}^{2}(n\\in \\bm{N}^{*})$,则______", "options": ["$2 \\lt 100a_{100} \\lt \\dfrac{5}{2}$", "$\\dfrac{5}{2} \\lt 100a_{100} \\lt 3$", "$3 \\lt 100a_{100} \\lt \\dfrac{7}{2}$", "$\\dfrac{7}{2} \\lt 100a_{100} \\lt 4$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "8.曲线 $y=\\frac{\\mathrm{e}^{x}}{x+1}$ 在点 $\\left(1, \\frac{\\mathrm{e}}{2}\\right)$ 处的切线方程为 $(\\quad)$", "options": ["$y=\\frac{\\mathrm{e}}{4} x$", "$y=\\frac{\\mathrm{e}}{2} x$", "$y=\\frac{\\mathrm{e}}{4} x+\\frac{\\mathrm{e}}{4}$", "$y=\\frac{\\mathrm{e}}{2} x+\\frac{3 \\mathrm{e}}{4}$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "已知 $\\alpha$ 为锐角, $\\cos \\alpha=\\frac{1+\\sqrt{5}}{4}$, 则 $\\sin \\frac{\\alpha}{2}=$ ( )", "options": ["$\\frac{3-\\sqrt{5}}{8}$", "$\\frac{-1+\\sqrt{5}}{8}$", "$\\frac{3-\\sqrt{5}}{4}$", "$\\frac{-1+\\sqrt{5}}{4}$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "已知等比数列$\\{a_{n}\\}$的前$3$项和为$168$,$a_{2}-a_{5}=42$,则$a_{6}=\\left(\\ \\ \\right)$", "options": ["$4$", "$2$", "$6$", "$3$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "当$x=1$时,函数$f\\left(x\\right)=a\\ln x+\\frac{b}{x}$取得最大值$-2$,则${f'}\\left(2\\right)=\\left(\\ \\ \\right)$", "options": ["$-1$", "$-\\frac{1}{2}$", "$\\frac{1}{2}$", "$1$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "9.已知双曲线 $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ 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 $|A B|=(\\quad)$", "options": ["$\\frac{\\sqrt{5}}{5}$", "$\\frac{2 \\sqrt{5}}{5}$", "$\\frac{3 \\sqrt{5}}{5}$", "$\\frac{4 \\sqrt{5}}{5}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "“ $\\sin ^{2} \\alpha+\\sin ^{2} \\beta=1$ ” 是“ $\\sin \\alpha+\\cos \\beta=0$ ”的 ( )", "options": ["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"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "2.设集合 $A=\\{0,-a\\}, B=\\{1, a-2,2 a-2\\}$, 若 $A \\subseteq B$, 则 $a=()$", "options": ["2", "1", "$\\frac{2}{3}$", "-1"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "设 $F_{1}, F_{2}$ 为椭圆 $C: \\frac{x^{2}}{5}+y^{2}=1$ 的两个焦点, 点 $P$ 在 $C$ 上, 若 $\\overrightarrow{P F_{1}} \\cdot \\overrightarrow{P F_{2}}=0$, 则 $\\left|P F_{1}\\right| \\cdot\\left|P F_{2}\\right|=(\\quad)$", "options": ["1", "2", "4", "5"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "已知函数$f\\left(x\\right)$,$g\\left(x\\right)$的定义域均为$R$,且$f\\left(x\\right)+g\\left(2-x\\right)=5$,$g\\left(x\\right)-f\\left(x-4\\right)=7$.若$y=g\\left(x\\right)$的图像关于直线$x=2$对称,$g\\left(2\\right)=4$,则$\\sum _{k=1}^{22}f\\left(k\\right)=\\left(\\ \\ \\right)$", "options": ["$-21$", "$-22$", "$-23$", "$-24$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "正方形 $A B C D$ 的边长是 $2, E$ 是 $A B$ 的中点, 则 $\\overrightarrow{E C} \\cdot \\overrightarrow{E D}=(\\quad)$", "options": ["$\\sqrt{5}$", "3", "$2 \\sqrt{5}$", "5"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "设全集 $U=\\{0,1,2,4,6,8\\}$, 集合 $M=\\{0,4,6\\}, N=\\{0,1,6\\}$, 则 $M \\cup \\partial_{U} N=(\\quad)$", "options": ["$\\{0,2,4,6,8\\}$", "$\\{0,1,4,6,8\\}$", "$\\{1,2,4,6,8\\}$", "$U$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "1.设 $z=\\frac{2+\\mathrm{i}}{1+\\mathrm{i}^{2}+\\mathrm{i}^{5}}$, 则 $\\bar{z}=(\\quad)$", "options": ["$1-2 \\mathrm{i}$", "$1+2 i$", "$2-\\mathrm{i}$", "$2+\\mathrm{i}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "已知正四棱锥的侧棱长为$l$,其各顶点都在同一球面上.若该球的体积为$36\\pi $,且$3\\leqslant l\\leqslant 3\\sqrt{3}$,则该正四棱锥体积的取值范围是（ ）", "options": ["$[18$，$\\frac{{81}}{4}]$", "$[\\frac{{27}}{4}$，$\\frac{{81}}{4}]$", "$[\\frac{{27}}{4}$，$\\frac{{64}}{3}]$", "$\\left[18,27\\right]$"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "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", "options": ["30 种", "60 种", "120 种", "240 种"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "已知$a=\\frac{{31}}{{32}}$,$b=\\cos \\frac{1}{4}$,$c=4\\sin \\frac{1}{4}$,则（ ）", "options": ["$c \\gt b \\gt a$", "$b \\gt a \\gt c$", "$a \\gt b \\gt c$", "$a \\gt c \\gt b$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "3.某学校为了解学生参加体育运动的情况, 用比例分配的分层随机抽样方法作抽样调查, 拟从初中部和高中部两层共抽取60名学生, 已知该校初中部和高中部分别有400名和200名学生, 则不同的抽样结果共有 ()", "options": ["$\\mathrm{C}_{400}^{45} \\cdot \\mathrm{C}_{200}^{15}$ 种", "$\\mathrm{C}_{400}^{20} \\cdot \\mathrm{C}_{200}^{40}$ 种", "$\\mathrm{C}_{400}^{30} \\cdot \\mathrm{C}_{200}^{30}$ 种", "$\\mathrm{C}_{400}^{40} \\cdot \\mathrm{C}_{200}^{20}$ 种"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "4.若 $f(x)=(x+a) \\ln \\frac{2 x-1}{2 x+1}$ 为偶函数, 则 $a=()$", "options": ["-1", "0", "$\\frac{1}{2}$", "1"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "设$F$为抛物线$C:y^{2}=4x$的焦点,点$A$在$C$上,点$B\\left(3,0\\right)$,若$|AF|=|BF|$,则$|AB|=\\left(\\ \\ \\right)$", "options": ["$sqrt{2}$", "$2\\sqrt{2}$", "$3sqrt{3}$", "$3\\sqrt{2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "已知等比数列$\\{a_{n}\\}$的前$3$项和为$168$,$a_{2}-a_{5}=42$,则$a_{6}=\\left(\\ \\ \\right)$", "options": ["$14$", "$12$", "$6$", "$3$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "从8名女生，4名男生中选出6名学生组成课外小组，如果按性别比例分层抽样，则不同的抽取方法种数为____", "options": ["$\\mathbf{C}^4_8\\cdot\\mathbf{C}^2_4$", "$\\mathbf{C}^4_8\\cdot\\mathbf{C}^3_4$", "$\\mathbf{C}^4_12$", "$\\mathsf{A}_{8}^{4}\\cdot\\mathsf{A}_{4}^{2}$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "若$\\alpha,\\beta\\in(0,\\pi)$则“$\\alpha>\\beta$”是 “$\\cos\\alpha>\\cos\\beta$”的什么条件____", "options": ["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"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "动点P在直线$x+y-4=0$，$O$为原点，则$|OP|$的最小值为____", "options": ["$\\sqrt{10}$", "$2\\sqrt{2}$", "$\\sqrt{6}$", "2"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "已知力F与水平方向的夹角为$30^{\\circ}$（斜向上），大小为50N，一个质量为8kg的木块受力F的作用在动摩擦因数$\\mu=0.02$的水平平面上运动了20m，则力F和摩擦力f所做的功分别为$\\left(g=10m/s^2\\right)$____", "options": ["$500\\sqrt{3}J,-22J$", "$500\\sqrt{3}J,22J$", "$500J,10\\sqrt{3}J$", "$500J,-10\\sqrt{3}J$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "若a,b>0,则\"$a>b$\"是\"$a^{3}+b^{3}>a^{2}b+a b^{2}$\"的____", "options": ["充分非必要条件", "必要非充分条件", "充分且必要条件", "既非充分也非必要条件"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "《吕氏春秋·音律篇》记载了利用“三分损益”制定关于“宫、商、角、徵、羽”五音的方法，以一段均匀的发声管为基数“宫”,然后将此发声管均分成三段,舍弃其中的一段保留二段,这就是“三分损一\"，余下来的三分之二长度的发声管所发出的声音就是“徵”;将“徵”管均分成三份，再加上一份，即“徵”管长度的三分之四,这就是“三分益一”，于是就产生了“商\"；“商”管保留分之二，“三分损一”，于是得出“羽”;羽管“三分益一”，即羽管的三分之四的长度,就是角”。如果按照三分损益律,基数“宫”发声管长度为1，则“羽”管的长度为____", "options": ["$\\frac{16}{27}$", "$\\frac{27}{16}$", "$\\frac{64}{81}$", "$\\frac{81}{64}$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "正方体$ABCD—A_1B_1C_1D_1$中，线段$BB_1$与线段$AD_1$所成角的余弦值为____", "options": ["$\\frac{\\sqrt{2}}{3}$", "$\\frac{\\sqrt{3}}{2}$", "$\\frac{1}{2}$", "$\\frac{\\sqrt{2}}{2}$"], "answer": "D", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "已知$\\sin x+\\cos x=\\frac{\\sqrt{2}}{2}$,则$\\sin^{4}x+\\cos^{4}x=$____", "options": ["$\\frac{9}{8}$", "$\\frac{7}{8}$", "$\\frac{5}{4}$", "$\\frac{3}{4}$"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "已知函数$f(x)=x^{2}+ax+b(a<0,b>0)$有两个不同零点x1,x2。-2和x1,x2三个数适当排序后既可成为等差数列,也可成为等比数列,则函数f(x)的解析式为____", "options": ["$f(x)=x^2-5x-4$", "$f(x)=x^{2}+5x+4$", "$f(x)=x^{2}-5x+4$", "$f(x)=x^{2}+5x-4$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "已知直线$a x+b y-1=0$(a,b不全为0)，与圆$x^{2}+y^{2}=50$有公共点，且公共点的横、纵坐标均为整数，那么这样的直线有____", "options": ["66条", "72条", "74条", "78条"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "已知$\\cos\\left(\\alpha-{\\frac{\\pi}{2}}\\right)=-{\\frac{2{\\sqrt{5}}}{5}}$， $\\alpha\\in\\left(\\pi,\\frac{3\\pi}{2}\\right)$，则$\\tan\\alpha$____", "options": ["2", "$\\frac{3}{2}$", "1", "$\\frac{1}{2}$"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "有5把外形一样的钥匙，其中3把能开锁，2把不能开锁．现准备通过一一试开将其区分出来，每次随机抽出一把进行试开，试开后不放回，则恰好试开3次就将能开锁的和不能开锁的钥匙区分出来的概率为____", "options": ["$\\frac{3}{10}$", "$\\frac{1}{10}$", "$\\frac{7}{15}$", "$\\frac{4}{15}$"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "圆$(x+2)+y=4$与圆$(x-2)+(y-1)=9$的位置关系为____", "options": ["内切", "相交", "外切", "相离"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "阿基米德不仅是著名的物理学家，也是著名的数学家，他利用“通近法”得到椭圆的面积除以圆周率等于椭圆的长半轴长与短半轴长的乘积．若椭圆C的对称轴为坐标轴，焦点在y轴上，且C椭圆的离心率为$\\frac{\\sqrt{7}}{4}$，面积为$12 \\pi$，则椭圆C的方程为____", "options": ["$\\frac{x^{2}}{4}+\\frac{y^{2}}{36}=1$", "$\\frac{x^{2}}{3}+\\frac{y^{2}}{4}=1$", "$\\frac{x^{2}}{18}+\\frac{y^{2}}{32}=1$", "$\\frac{x^{2}}{9}+\\frac{y^{2}}{16}=1$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "过点$P(2{\\sqrt{6}},2{\\sqrt{6}})$的直线l与曲线$y={\\sqrt{13-x^{2}}}$交于A,B两点，若$2\\overrightarrow{P A}=5\\overrightarrow{A B}$,则直线l的斜率为____", "options": ["$2-\\sqrt{3}$", "$2+\\sqrt{3}$", "$2-\\sqrt{3}$或者$2+\\sqrt{3}$", "$2-\\sqrt{3}$或者$\\sqrt{3}-1$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "在直三棱柱$ABC-A_{1}B_{1}C_{1}$的棱所在直线中，与直线$BC_1$异面的直线条数为____", "options": ["1", "2", "3", "4"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "$a=1+\\sin0.1$，$b=e^{0.1}$，$c=1.01^{10}$，$d=\\frac{17}{16}$，a，b，c，d间的大小关系为____", "options": ["$b>a>d>c$", "$b>c>a>d$", "$b>c>d>a$", "$b>a>c>d$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "已知实数$a,b\\in(1,+\\infty)$，且$2\\left(a + b \\right)=e^{2a}+2\\ln b+1$，e为自然对数的底数，则____", "options": ["$1<b<a$", "$a<b<2a$", "$2a<b<e^{a}$", "$e^a<b<e^{2a}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "下列事件中，是随机事件的是____", "options": ["守株待兔", "瓮中捉鳖", "水中捞月", "水滴石穿"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "某班级要从6名男生、3名女生中选派6人参加社区宣传活动，如果要求至少有2名女生参加，那么不同的选派方案种数为____", "options": ["19", "38", "55", "65"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "已知函数f(x)的定义域是$\\left( 0,+\\infty \\right )$，且满足$2x^{2}f(x)+x^{3}f(x)=\\ln x$，$f\\left(\\sqrt{e}\\right)=\\frac{1}{4e}$（其中e为自然常数，$e \\approx 2.718$），则下列说法正确的是____", "options": ["$f\\left(x\\right)$在$\\left( 0,+ \\infty \\right)$上单调递增", "$f\\left(x\\right)$在$\\left( 0,+ \\infty \\right)$上单调递减", "$f\\left(x\\right)$在$\\left( 0,+ \\infty \\right)$上有极大值", "$f\\left(x\\right)$在$\\left( 0,+ \\infty \\right)$上有极小值"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "已知i是虚数单位，z为复数，$2+\\frac{1}{i}=z(3+i)$，则在复平面内z对应的点位于____", "options": ["第一象限", "第二象限", "第三象限", "第四象限"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "已知集合$A=\\left\\{x\\in Z \\mid -x^{2}+2x+3 \\geq 0 \\right\\}$,$B=\\{y\\mid y<2\\}$，则$A \\cap B$____", "options": ["$\\emptyset$", "$\\left[-1,2\\right)$", "$\\left\\{0,1\\right\\}$", "$\\left\\{-1，0,1\\right\\}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "等差数列$\\left\\{a_{n}\\right\\}$中，若$a_{4}+a_{6}+a_{8}+a_{10}+a_{12}=120$，则$a_{8}$的值为", "options": ["20", "24", "36", "72"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "将《红楼梦》《水浒传》《西游记》《三国演义》四本书随机地分发给甲、乙、丙三人，每人至少分得一本，则下列说法正确的是____", "options": ["事件“甲分得一本”与事件“丙分得两本”为互斥事件", "事件“乙分得《三国演义》”与事件“丙分得《水浒传》”为对立事件", "事件“甲分得两本”与事件“乙分得两本”为对立事件", "事件“甲分得《红楼梦》”与事件“乙分得《红楼梦》”为互斥事件"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} |
| {"question": "半径为4的球面上有A、B、C、D四点，且AB，AC，AD两两互相垂直，则$\\Delta A B C,\\Delta A C D,\\Delta A D B$面积之和最大值为____", "options": ["8", "16", "32", "64"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "与$2022^{\\circ}$终边相同的角是____", "options": ["$-488^{\\circ}$", "$-148^{\\circ}$", "$142^{\\circ}$", "$222^{\\circ}$"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "已知集合$A=\\left\\{x\\mid2<x<4\\right\\},B=\\left\\{x\\mid\\left(x-1\\right)\\left(x-3\\right)<0\\right\\}$，则$A\\cap B=$？____", "options": ["$(1,3)$", "$(1,4)$", "$(2,3)$", "$(2,4)$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "已知点$P(x,y)$在圆$x^2+y^2-4x+3=0$上运动，则$\\frac{y}{x}$的最大值是____", "options": ["$\\frac{\\sqrt{2}}{4}$", "$\\frac{1}{2}$", "$\\frac{\\sqrt{2}}{3}$", "$\\frac{\\sqrt{3}}{3}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "已知$S_n$为等差数列$\\left\\{a_n\\right\\}$的前n项和，若$S_1=1$，$\\frac{S_{4}}{S_{2}}=4$，则$\\frac{S_{6}}{S_{4}}$的值为____", "options": ["$\\frac{9}{4}$", "$\\frac{3}{2}$", "$\\frac{5}{4}$", "4"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "若复数$z=1-\\mathrm{i}$，则$\\left|\\frac{z}{1-z}\\right|$的值为____", "options": ["1", "$\\sqrt{2}$", "$2\\sqrt{2}$", "4"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "已知x,y满足不等式组$\\begin{cases}2x+y-2\\le0\\\\ x-2y-1\\le0\\\\ x\\ge0\\end{cases}$，则点P(x,y)所在区域的面积是____", "options": ["1", "2", "$\\frac{5}{4}$", "$\\frac{4}{5}$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "平面上有两组互不重合的点,$A_1,A_2,......A_m$与$B_1,B_2,......B_n$,$(m,n \\in N^+, n\\geq2)$,$\\forall t\\in[1,n]$，$t \\in N^{+}$， $\\left|\\sum_{i=1}^{m}\\overrightarrow{A_{i}B_{t}}\\right|=t$，则$\\sum_{i=1}^{n-1}B_{i}B_{i+1}$的范围是____", "options": ["$\\left[ \\frac{n}{m},\\frac{2n}{m} \\right]$", "$\\left[ \\frac{n}{m},\\frac{n^2}{m} \\right]$", "$\\left[ \\frac{n}{m},\\frac{n+n^2}{2m} \\right]$", "$\\left[ \\frac{n-1}{m},\\frac{n^2-1}{m} \\right]$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "已知命题P:$\\exists n{\\in}N,n^{2}>2^{n}$，则$\\neg p$为____", "options": ["$\\forall n \\in N,n^2>2^n$", "$\\exists n \\in N,n^2\\leq2^n$", "$\\forall n \\in N,n^{2} \\leq 2^{n}$", "$\\forall n \\notin N,n^2 \\leq 2^n$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "函数$f(x)=\\left|x-2\\right|- \\ln x$在定义域内零点可能落在下列哪个区间内____", "options": ["$\\left( 0,1\\right)$", "$\\left( 2,3\\right)$", "$\\left( 3,4\\right)$", "$\\left( 4,5\\right)$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "圆$(x-2)^{2}+y^{2}=4$与圆$(x+2)^2+(y+3)^{2}=9$的位置关系为____", "options": ["内切", "外切", "相交", "相离"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "复数$z=a+bi(a,b\\neq0)$，若$\\left|{\\frac{1}{z}}-{1}\\right|=2$，则下面哪一个选项的值和a,b无关____", "options": ["$\\left|z+\\frac{1}{3}\\right|$", "$\\left|z+\\frac{1}{2}\\right|$", "$\\left|z-\\frac{1}{2} \\right|$", "$\\left|z-\\frac{1}{4}\\right|$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "复数$({\\frac{3+i}{1-i}})^{2}$的值为____", "options": ["$-3-4i$", "$-3+4i$", "$3-4i$", "$3+4i$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "已知a，b为非零实数，则\"$a<b$\"是\"$\\frac{a}{\\left|b\\right|}<\\frac{b}{\\left|a\\right|}$\"的____", "options": ["充分而不必要条件", "必要而不充分条件", "充要条件", "既不充分也不必要条件"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "已知数列$\\left\\{a_{n}\\right\\}$，$\\left\\{b_{n}\\right\\}$,$a_{n+1}=\\left [ \\frac{a_n}{2} \\right ] $，$b_{n+1}=\\frac{\\left [ b_n\\right ]}{2} $,$n \\in N^+$，其中$\\left[x \\right]$为不大于x的最大整数。若$a_1=b_1=m$，$m\\leq1000$，$m \\in N^+$，有且仅有4个不同的，使得$a_{t}\\neq b_t$，则m一共有多少种不同的取值____", "options": ["120", "126", "210", "252"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "已知非零向量$\\vec{a}$，$\\vec{b}$满足$\\left|\\vec{a}\\right|=\\left|\\vec{b}\\right|$，且满足$\\left|\\vec{a}+\\vec{b}\\right|=\\left|2\\vec{a}-\\vec{b}\\right|$，则$\\vec{a}$与$\\vec{b}$的夹角为____", "options": ["$\\frac{2\\pi}{3}$", "$\\frac{\\pi}{2}$", "$\\frac{\\pi}{3}$", "$\\frac{\\pi}{6}$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} |
| {"question": "设$f(x)$是定义在$\\mathbb{R}$上的奇函数，当$x\\leq0$，$f(x)=2x^2-x$，则$f(1)$等于____", "options": ["-3", "-1", "1", "3"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "已知函数$f(x)=\\left\\{\\begin{matrix}\r\ne^x+2,x\\le 1 \\\\ \\log _2(x^2-1), x> 1\r\n\\end{matrix}\\right.$，则$f\\left[f\\left(0\\right)\\right]$____", "options": ["3", "-3", "-2", "2"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "若关于x的方程$x^{2}+a x+a^{2}-1=0$,有一个正根和一个负根，则实数a的取值范围为____", "options": ["$-\\frac{2\\sqrt{3}}{3}\\leq a \\leq\\frac{2\\sqrt{3}}{3}$", "$-\\frac{2\\sqrt{3}}{3}<\\alpha<\\frac{2\\sqrt{3}}{3}$", "$-1\\leq a\\leq1$", "$-1<a<1$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} |
| {"question": "设i为虚数单位，复数z满足$\\left(1-i \\right)\\cdot z=2i$,那么$\\left | z \\right | $的值为____", "options": ["1", "$\\sqrt{2}$", "$\\sqrt{3}$", "2"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "已知O为坐标原点，双曲线C:$\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$，$(a>0,b>0)$的右焦点为F(C,0),离心率为$e=\\frac{2\\sqrt{3}}{3}$，过点F的直线与C的的两条渐近线的交点分别为A，B。$\\Delta OA B$为直角三角形，$\\left|AB\\right|=3$，则C的方程为____", "options": ["$\\frac{x^2}{4}-\\frac{y^2}{2}=1$", "$\\frac{x^2}{3}-y^2=1$", "$\\frac{x^2}{6}-\\frac{y^2}{3}=1$", "$\\frac{x^2}{8}-\\frac{y^2}{4}=1$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} |
| {"question": "要得到函数$y={\\sqrt{2}} \\cos x$的图象，只需将函数$y=\\sqrt{2}\\sin\\left(x+\\frac{\\pi}{4}\\right)$的图像____", "options": ["向上平移$\\frac{\\pi}{4}$个单位", "向下平移$\\frac{\\pi}{4}$个单位", "向左平移$\\frac{\\pi}{4}$个单位", "向右平移$\\frac{\\pi}{4}$个单位"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "设集合$A=\\left\\{ x\\mid y=1n\\frac{3+x}{3-x} \\right\\}$，集合$B=\\{y|y=2^{|X|}+1\\}$，则$A \\cap B$的值为____", "options": ["$(-\\infty,-2]\\cup(3,+\\infty)$", "$(-\\infty,-2]$", "$[2,3)$", "$(3,+\\infty)$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} |
| {"question": "已知圆C1:$x^2+y^2-4x-6y+9=0$,圆C2:$x^2+y^2+12x+6y-19=0$,则两圆位置关系是____", "options": ["相交", "内切", "外切", "相离"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} |
| {"question": "将函数$y=\\tan(\\omega x-1)(\\omega>0)$的图像向左平移2个单位长度后,与函数$y=\\tan(\\omega x+3)$的图象重合,则的最小值等于____", "options": ["$2-\\frac{\\pi}{2}$", "1", "$\\pi - 2$", "2"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} |
| {"question": "圆锥的底面半径为2，高为4.一个圆柱的下底面在圆锥的底面上，上底面的圆周在圆锥的侧面上，当圆柱侧面积为$4 \\pi$时，该圆柱的体积为____", "options": ["$\\pi$", "$2\\pi$", "$3\\pi$", "$4\\pi$"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} |
| {"question": "如果数列$\\left\\{\\alpha_{n}\\right\\}$是等差数列,$a_1>0,d\\neq0$,则____", "options": ["$a_{1}a_{8}>a_{4}a_{5}$", "$a_{1}a_{8}<a_{4}a_{5}$", "$a_{1}+a_{8}>a_{4}+a_{5}$", "$a_{1}a_{8}=a_{4}a_{5}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "令i表示虚数单位，设复数$z=\\left(a+i\\right)^{2}$在复平面上的对应点在虚轴负半轴上，则实数a的值是____", "options": ["-1", "1", "$\\sqrt{2}$", "$-\\sqrt{3}$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} |
| {"question": "二项式$\\left(x^{2}+{\\frac{2}{\\sqrt{x}}}\\right)^{10}$的展开式中的常数项是____", "options": ["第10项", "第9项", "第8项", "第7项"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} |
| {"question": "设函数$f(x)=e^{x}+x-2$，$g(x)=\\ln x+x^2-3$，若实数a,b满足$f(a)=0$,$g(b)=0$,则____", "options": ["$g(a)<0<f(b)$", "$f(b)<0<g(a)$", "$0<g(a)<f(b)$", "$f(b)<g(a)<0$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
| {"question": "已知$f(x)=x^3$，则$\\lim\\limits_{\\Delta x\\to0}\\frac{f(-1+\\Delta x)-f(-1)}{\\Delta x}=$____", "options": ["0", "-3", "2", "3"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} |
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