61 lines
1.9 KiB
Markdown
61 lines
1.9 KiB
Markdown
### 题目1
|
||
**题目序号**: 1
|
||
**题目内容**: 已知集合 $A=\left\{x \mid -5 < x^{3} < 5\right\}, B=\left\{-3,-1,0,2,3\right\}$，则 $A \cap B=$
|
||
**选项**:
|
||
A. $\{-1,0\}$
|
||
B. $\{2,3\}$
|
||
C. $\{-3,-1,0\}$
|
||
D. $\{-1,0,2\}$
|
||
**答案**: A
|
||
**解析**: $A \cap B=\{-1,0\}$，选 A。
|
||
|
||
---
|
||
|
||
### 题目2
|
||
**题目序号**: 2
|
||
**题目内容**: 若 $\frac{2}{z-1}=1+i$，则 $z=$
|
||
**选项**:
|
||
A. $-1-i$
|
||
B. $-1+i$
|
||
C. $1-i$
|
||
D. $1+i$
|
||
**答案**: C
|
||
|
||
---
|
||
|
||
### 题目3
|
||
**题目序号**: 3
|
||
**题目内容**: 已知向量 $\vec{a}=(0,1)$，$\vec{b}=(2,x)$，若 $\vec{b} \perp (\vec{b}-4\vec{a})$，则 $x=$
|
||
**选项**:
|
||
A. $-2$
|
||
B. $-1$
|
||
C. $1$
|
||
D. $2$
|
||
**答案**: D
|
||
**解析**: $\vec{b}-4\vec{a}=(2,x-4)$，$\vec{b} \perp (\vec{b}-4\vec{a})$，$\therefore \vec{b}(\vec{b}-4\vec{a})=0$，$\therefore 4+x(x-4)=0$，$\therefore x=2$，选 D。
|
||
|
||
---
|
||
|
||
### 题目4
|
||
**题目序号**: 4
|
||
**题目内容**: 已知 $\cos(\alpha+\beta)=m$，$\tan \alpha \tan \beta=2$，则 $\cos(\alpha-\beta)=$
|
||
**选项**:
|
||
A. $-3m$
|
||
B. $-\frac{m}{3}$
|
||
C. $\frac{m}{3}$
|
||
D. $3m$
|
||
**答案**: A
|
||
**解析**: $\left\{\begin{array}{l}\cos \alpha \cos \beta-\sin \alpha \sin \beta=m \\\frac{\sin \alpha \sin \beta}{\cos \alpha \cos \beta}=2\end{array}\right.$，$\therefore \left\{\begin{array}{l}\sin \alpha \sin \beta=-2m \\\cos \alpha \cos \beta=-m\end{array}\right.$ $\cos(\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta=-m-2m=-3m$，选 A。
|
||
|
||
---
|
||
|
||
### 题目5
|
||
**题目序号**: 5
|
||
**题目内容**: 已知圆柱和圆锥的底面半径相等，侧面积相等，且它们的高均为 $\sqrt{3}$，则圆锥的体积为
|
||
**选项**:
|
||
A. $2\sqrt{3}\pi$
|
||
B. $3\sqrt{3}\pi$
|
||
C. $6\sqrt{3}\pi$
|
||
D. $9\sqrt{3}\pi$
|
||
**答案**: B
|
||
**解析**: 设它们底面半径为 $r$，圆锥母线 $l$，$\therefore 2\pi r\sqrt{3}=\pi rl$，$\therefore l=\sqrt{3}$，则圆锥的体积为 $\frac{1}{3}\pi r^{2}h$。 |