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:
- $\overbrace{U}(M \cup N)$
- $N \bigcup \partial_{U} M$
- $\partial_{U}(M \cap N)$
- $M \cup \triangleleft_{U} N$

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:
- $\frac{\pi}{10}$
- $\frac{\pi}{5}$
- $\frac{3 \pi}{10}$
- $\frac{2 \pi}{5}$

topic: High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


question: 若$z=1+i$,则$|iz+3\overline{z}|=\left(\ \ \right)$

options:
- $4\sqrt{5}$
- $4\sqrt{2}$
- $2\sqrt{5}$
- $2\sqrt{2}$

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:
- $a=1$，$b=-2$
- $a=-1$，$b=2$
- $a=1$，$b=2$
- $a=-1$，$b=-2$

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$45^{\circ}$


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
- $\mathrm{e}^{-1}$
- $\mathrm{e}^{-2}$

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:
- $a=1,b=-1$
- $a=1,b=1$
- $a=-1,b=1$
- $a=-1,b=-1$

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:
- $2 \sqrt{2}$
- $3 \sqrt{2}$
- $4 \sqrt{2}$
- $5 \sqrt{2}$


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}] $

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:
- $\frac{1}{6}$
- $\frac{1}{3}$
- $\frac{1}{2}$
- $\frac{2}{3}$

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
- $\frac{1}{2}$

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:
- $3\overrightarrow{m}-2\overrightarrow{n}$
- $-2\overrightarrow{m}+3\overrightarrow{n}$
- $3\overrightarrow{m}+2\overrightarrow{n}$
- $2\overrightarrow{m}+3\overrightarrow{n}$

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:
- $\{2,3,5\}$
- $\{1,3,4\}$
- $\{1,2,4,5\}$
- $\{2,3,4,5\}$

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

topic: High--Intermediate Numbers and Algebra--Sequences


options:
- $-2$
- $-1$
- $1$
- $2$

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

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:
- $\frac{{{x^2}}}{{18}}+\frac{{{y^2}}}{{16}}=1$
- $\frac{{{x^2}}}{9}+\frac{{{y^2}}}{8}=1$
- $\frac{{{x^2}}}{3}+\frac{{{y^2}}}{2}=1$
- $\frac{{{x^2}}}{2}+y^{2}=1$

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:
- $\{-1,2\}$
- $\{1,2\}$
- $\{1,4\}$
- $\{-1,4\}$

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:
- $-\frac{\sqrt{3}}{2}$
- $-\frac{1}{2}$
- $\frac{1}{2}$
- $\frac{\sqrt{3}}{2}$

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:
- $-\frac{1}{5}$
- $-\frac{2}{5}$
- $\frac{2}{5}$
- $\frac{4}{5}$

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:
- $1$
- $\frac{3}{2}$
- $\frac{5}{2}$
- $3$

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:
- $(1,1)$
- $(-1,2)$
- $(1,3)$
- $(-1,-4)$

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:
- $\frac{{\sqrt{3}}}{2}$
- $\frac{{\sqrt{2}}}{2}$
- $\frac{1}{2}$
- $\frac{1}{3}$

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:
- $\frac{1}{3}$
- $\frac{1}{2}$
- $\frac{{\sqrt{3}}}{3}$
- $\frac{{\sqrt{2}}}{2}$


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:
- $\frac{1}{5}$
- $\frac{\sqrt{5}}{5}$
- $\frac{2 \sqrt{5}}{5}$
- $\frac{4 \sqrt{5}}{5}$

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:
- $2$
- $2\sqrt{2}$
- $3$
- $3\sqrt{2}$

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

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:
- $2\in M$
- $3\in M$
- $4\notin M$
- $5\notin M$

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:
- $2$
- $3$
- $4$
- $5$

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 $


question: 若$z=-1+\sqrt{3}i$,则$\frac{z}{{z\overline{z}-1}}=\left(\ \ \right)$

options:
- $-1+\sqrt{3}i$
- $-1-\sqrt{3}i$
- $-\frac{1}{3}+\frac{{\sqrt{3}}}{3}i$
- $-\frac{1}{3}-\frac{{\sqrt{3}}}{3}i$

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:
- $\frac{1}{8}$
- $\frac{1}{6}$
- $\frac{1}{4}$
- $\frac{1}{2}$

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

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

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

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:
- $\frac{2}{5}$
- $\frac{\sqrt{30}}{2}$
- $\frac{3}{5}$
- $\frac{\sqrt{35}}{2}$

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:
- $\sqrt{5}$
- $2\sqrt{2}$
- $\sqrt{10}$
- $\frac{{5\sqrt{10}}}{4}$


question: 设集合$A=\{1,2\}$,$B=\{2,4,6\}$,则$A\cup B=\left(\ \ \right)$

options:
- $\{2\}$
- $\{1,2\}$
- $\{2,4,6\}$
- $\{1,2,4,6\}$

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:
- $\pi$
- $\sqrt{6} \pi$
- $3 \pi$
- $3 \sqrt{6} \pi$


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:
- $(-\infty,-2)$
- $(-\infty,-3)$
- $(-4,-1)$
- $(-3,0)$

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:
- $20$
- $18$
- $13$
- $6$

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:
- $-6$
- $-5$
- $5$
- $6$

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:
- $\{x\left|\right.0\leqslant x \lt 2\}$
- $\{x|\frac{1}{3}\leqslant x \lt 2\}$
- $\{x\left|\right.3\leqslant x \lt 16\}$
- $\{x|\frac{1}{3}\leqslant x \lt 16\}$

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:
- $\frac{5}{6}$
- $\frac{2}{3}$
- $\frac{1}{2}$
- $\frac{1}{3}$

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:
- $\frac{1}{3}$
- $\frac{1}{2}$
- $\frac{{\sqrt{3}}}{3}$
- $\frac{{\sqrt{2}}}{2}$


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:
- $\frac{1+\sqrt{2}}{2}$
- $\frac{1+2 \sqrt{2}}{2}$
- $1+\sqrt{2}$
- $2+\sqrt{2}$

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$45^{\circ}$


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

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:
- $\frac{1}{2}$
- $-\frac{1}{2}$
- $1$
- $-1$

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

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

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:
- $1$
- $5$
- $7$
- $25$

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

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:
- $a \lt b \lt c$
- $c \lt b \lt a$
- $c \lt a \lt b$
- $a \lt c \lt b$

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:
- $-\frac{π}{2}$，$\frac{π}{2}$
- $-\frac{{3π}}{2}$，$\frac{π}{2}$
- $-\frac{π}{2}$，$\frac{π}{2}+2$
- $-\frac{{3π}}{2}$，$\frac{π}{2}+2$

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:
- $(1,1)$
- $(-1,2)$
- $(1,3)$
- $(-1,-4)$

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:
- $1.0\times 10^{9}m^{3}$
- $1.2\times 10^{9}m^{3}$
- $1.4\times 10^{9}m^{3}$
- $1.6\times 10^{9}m^{3}$


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 $\frac{{\sqrt{13}}}{2}$
- $\frac{{\sqrt{13}}}{2}$
- $\frac{{\sqrt{17}}}{2}$

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:
- $\frac{1}{6}$
- $\frac{1}{4}$
- $\frac{1}{3}$
- $\frac{1}{2}$

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

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

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:
- $a=1$，$b=-3$
- $a=-1$，$b=3$
- $a=-1$，$b=-3$
- $a=1$，$b=3$

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:
- $(-2,1]$
- $(-3,-2)\cup \left[1,3\right)$
- $\left[-2,1\right)$
- $(-3,-2]\cup \left(1,3\right)$

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:
- $\frac{1}{6}$
- $\frac{1}{3}$
- $\frac{1}{2}$
- $\frac{2}{3}$

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

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

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:
- $\{0,1,2\}$
- $\{-2,-1,0\}$
- $\{0,1\}$
- $\{1,2\}$

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:
- $\frac{2}{3}$
- $\frac{\sqrt{2}}{3}$
- $-\frac{\sqrt{2}}{3}$
- $-\frac{2}{3}$

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
- $1-\mathrm{i}$
- $1+\mathrm{i}$

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

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:
- $\tan \left(\alpha -\beta \right)=1$
- $\tan \left(\alpha +\beta \right)=1$
- $\tan \left(\alpha -\beta \right)=-1$
- $\tan \left(\alpha +\beta \right)=-1$

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]

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

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:
- $\{1,3\}$
- $\{0,3\}$
- $\{-2,1\}$
- $\{-2,0\}$

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:
- $2 \lt 100a_{100} \lt \dfrac{5}{2}$
- $\dfrac{5}{2} \lt 100a_{100} \lt 3$
- $3 \lt 100a_{100} \lt \dfrac{7}{2}$
- $\dfrac{7}{2} \lt 100a_{100} \lt 4$

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:
- $y=\frac{\mathrm{e}}{4} x$
- $y=\frac{\mathrm{e}}{2} x$
- $y=\frac{\mathrm{e}}{4} x+\frac{\mathrm{e}}{4}$
- $y=\frac{\mathrm{e}}{2} x+\frac{3 \mathrm{e}}{4}$

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:
- $\frac{3-\sqrt{5}}{8}$
- $\frac{-1+\sqrt{5}}{8}$
- $\frac{3-\sqrt{5}}{4}$
- $\frac{-1+\sqrt{5}}{4}$

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:
- $4$
- $2$
- $6$
- $3$

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:
- $-1$
- $-\frac{1}{2}$
- $\frac{1}{2}$
- $1$

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:
- $\frac{\sqrt{5}}{5}$
- $\frac{2 \sqrt{5}}{5}$
- $\frac{3 \sqrt{5}}{5}$
- $\frac{4 \sqrt{5}}{5}$

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

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

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

topic: High--Intermediate Geometry--Equations of Conic Sections


options:
- $-21$
- $-22$
- $-23$
- $-24$

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

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:
- $\{0,2,4,6,8\}$
- $\{0,1,4,6,8\}$
- $\{1,2,4,6,8\}$
- $U$

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:
- $1-2 \mathrm{i}$
- $1+2 i$
- $2-\mathrm{i}$
- $2+\mathrm{i}$

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:
- $[18$，$\frac{{81}}{4}]$
- $[\frac{{27}}{4}$，$\frac{{81}}{4}]$
- $[\frac{{27}}{4}$，$\frac{{64}}{3}]$
- $\left[18,27\right]$


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

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:
- $c \gt b \gt a$
- $b \gt a \gt c$
- $a \gt b \gt c$
- $a \gt c \gt b$

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

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

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:
- $sqrt{2}$
- $2\sqrt{2}$
- $3sqrt{3}$
- $3\sqrt{2}$

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:
- $14$
- $12$
- $6$
- $3$

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:
- $\mathbf{C}^4_8\cdot\mathbf{C}^2_4$
- $\mathbf{C}^4_8\cdot\mathbf{C}^3_4$
- $\mathbf{C}^4_12$
- $\mathsf{A}_{8}^{4}\cdot\mathsf{A}_{4}^{2}$

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

topic: High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


options:
- $\sqrt{10}$
- $2\sqrt{2}$
- $\sqrt{6}$
- 2

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:
- $500\sqrt{3}J,-22J$
- $500\sqrt{3}J,22J$
- $500J,10\sqrt{3}J$
- $500J,-10\sqrt{3}J$

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

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:
- $\frac{16}{27}$
- $\frac{27}{16}$
- $\frac{64}{81}$
- $\frac{81}{64}$

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:
- $\frac{\sqrt{2}}{3}$
- $\frac{\sqrt{3}}{2}$
- $\frac{1}{2}$
- $\frac{\sqrt{2}}{2}$


question: 已知$\sin x+\cos x=\frac{\sqrt{2}}{2}$,则$\sin^{4}x+\cos^{4}x=$____

options:
- $\frac{9}{8}$
- $\frac{7}{8}$
- $\frac{5}{4}$
- $\frac{3}{4}$

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:
- $f(x)=x^2-5x-4$
- $f(x)=x^{2}+5x+4$
- $f(x)=x^{2}-5x+4$
- $f(x)=x^{2}+5x-4$

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

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
- $\frac{1}{2}$

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:
- $\frac{3}{10}$
- $\frac{1}{10}$
- $\frac{7}{15}$
- $\frac{4}{15}$

topic: High--Intermediate Probability and Statistics--Foundations of Probability Theory


options:
- 内切
- 相交
- 外切
- 相离


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:
- $\frac{x^{2}}{4}+\frac{y^{2}}{36}=1$
- $\frac{x^{2}}{3}+\frac{y^{2}}{4}=1$
- $\frac{x^{2}}{18}+\frac{y^{2}}{32}=1$
- $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$

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$\sqrt{3}-1$

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


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:
- $b>a>d>c$
- $b>c>a>d$
- $b>c>d>a$
- $b>a>c>d$

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:
- $1<b<a$
- $a<b<2a$
- $2a<b<e^{a}$
- $e^a<b<e^{2a}$

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

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

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

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


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:
- $\emptyset$
- $\left[-1,2\right)$
- $\left\{0,1\right\}$
- $\left\{-1，0,1\right\}$

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


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

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


options:
- $-488^{\circ}$
- $-148^{\circ}$
- $142^{\circ}$
- $222^{\circ}$

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:
- $(1,3)$
- $(1,4)$
- $(2,3)$
- $(2,4)$

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:
- $\frac{\sqrt{2}}{4}$
- $\frac{1}{2}$
- $\frac{\sqrt{2}}{3}$
- $\frac{\sqrt{3}}{3}$


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


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


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
- $\frac{5}{4}$
- $\frac{4}{5}$

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:
- $\left[ \frac{n}{m},\frac{2n}{m} \right]$
- $\left[ \frac{n}{m},\frac{n^2}{m} \right]$
- $\left[ \frac{n}{m},\frac{n+n^2}{2m} \right]$
- $\left[ \frac{n-1}{m},\frac{n^2-1}{m} \right]$

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:
- $\forall n \in N,n^2>2^n$
- $\exists n \in N,n^2\leq2^n$
- $\forall n \in N,n^{2} \leq 2^{n}$
- $\forall n \notin  N,n^2 \leq 2^n$

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:
- $\left( 0,1\right)$
- $\left( 2,3\right)$
- $\left( 3,4\right)$
- $\left( 4,5\right)$

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


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:
- $\left|z+\frac{1}{3}\right|$
- $\left|z+\frac{1}{2}\right|$
- $\left|z-\frac{1}{2} \right|$
- $\left|z-\frac{1}{4}\right|$


options:
- $-3-4i$
- $-3+4i$
- $3-4i$
- $3+4i$


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

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

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:
- $\frac{2\pi}{3}$
- $\frac{\pi}{2}$
- $\frac{\pi}{3}$
- $\frac{\pi}{6}$

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


question:
已知函数$f(x)=\left\{\begin{matrix}
e^x+2,x\le 1 \\ \log _2(x^2-1), x> 1
\end{matrix}\right.$，则$f\left[f\left(0\right)\right]$____

options:
- 3
- -3
- -2
- 2

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:
- $-\frac{2\sqrt{3}}{3}\leq a \leq\frac{2\sqrt{3}}{3}$
- $-\frac{2\sqrt{3}}{3}<\alpha<\frac{2\sqrt{3}}{3}$
- $-1\leq a\leq1$
- $-1<a<1$

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

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:
- $\frac{x^2}{4}-\frac{y^2}{2}=1$
- $\frac{x^2}{3}-y^2=1$
- $\frac{x^2}{6}-\frac{y^2}{3}=1$
- $\frac{x^2}{8}-\frac{y^2}{4}=1$

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

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:
- $(-\infty,-2]\cup(3,+\infty)$
- $(-\infty,-2]$
- $[2,3)$
- $(3,+\infty)$

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


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

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:
- $\pi$
- $2\pi$
- $3\pi$
- $4\pi$


question: 如果数列$\left\{\alpha_{n}\right\}$是等差数列,$a_1>0,d\neq0$,则____

options:
- $a_{1}a_{8}>a_{4}a_{5}$
- $a_{1}a_{8}<a_{4}a_{5}$
- $a_{1}+a_{8}>a_{4}+a_{5}$
- $a_{1}a_{8}=a_{4}a_{5}$


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
- $\sqrt{2}$
- $-\sqrt{3}$

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


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:
- $g(a)<0<f(b)$
- $f(b)<0<g(a)$
- $0<g(a)<f(b)$
- $f(b)<g(a)<0$

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

topic: High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives