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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。
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
解析:
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。
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。
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 \(\frac{1}{3}\pi r^{2}h\)。