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$$1 . \text { 已知集合 } A = \left\{ x \mid - 5 < x ^ { 3 } < 5 \right\} , B = \left\{ - 3 , - 1 , 0 , 2 , 3 \right\} , \text { 则 } A \cap B = \text { 【答案】 } A$$
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一、单项选择题:本题共 8 小题,每小题 5 分,共 40 分。在每小题给出的四个选项中,只有一项是符合题目要求的。
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1. 已知集合 \(A=\left\{x \mid -5 < x^{3} < 5\right\}, B=\left\{-3,-1,0,2,3\right\}\),则 \(A \cap B=\) 【答案】A
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A. \(\{-1,0\}\) B. \(\{2,3\}\) C. \(\{-3,-1,0\}\) D. \(\{-1,0,2\}\)
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【解析】\(A \cap B=\{-1,0\}\),选 A。
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2. 若 \(\frac{2}{z-1}=1+i\),则 \(z=\) 【答案】C
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A. \(-1-i\) B. \(-1+i\) C. \(1-i\) D. \(1+i\)
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3. 已知向量 \(\vec{a}=(0,1)\),\(\vec{b}=(2,x)\),若 \(\vec{b} \perp (\vec{b}-4\vec{a})\),则 \(x=\) 【答案】D
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A. \(-2\) B. \(-1\) C. \(1\) D. \(2\)
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【解析】\(\vec{b}-4\vec{a}=(2,x-4)\),\(\vec{b} \perp (\vec{b}-4\vec{a})\),\(\therefore \vec{b}(\vec{b}-4\vec{a})=0\),
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\(\therefore 4+x(x-4)=0\),\(\therefore x=2\),选 D。
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4. 已知 \(\cos(\alpha+\beta)=m\),\(\tan \alpha \tan \beta=2\),则 \(\cos(\alpha-\beta)=\) 【答案】A
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A. \(-3m\) B. \(-\frac{m}{3}\) C. \(\frac{m}{3}\) D. \(3m\)
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【解析】\(\left\{\begin{array}{l}\cos \alpha \cos \beta-\sin \alpha \sin \beta=m \\\frac{\sin \alpha \sin \beta}{\cos \alpha \cos \beta}=2\end{array}\right.\),\(\therefore \left\{\begin{array}{l}\sin \alpha \sin \beta=-2m \\\cos \alpha \cos \beta=-m\end{array}\right.\)
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\(\cos(\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta=-m-2m=-3m\),选 A。
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5. 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 【答案】B
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A. \(2\sqrt{3}\pi\) B. \(3\sqrt{3}\pi\) C. \(6\sqrt{3}\pi\) D. \(9\sqrt{3}\pi\)
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【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 \(\frac{1}{3}\pi r^{2}h\)。
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