### 2021年全国硕士研究生招生统一考试数学试题及详细参考解答(数三)

$\text{A.}$ 低阶无穷小 $\text{B.}$ 等价无穷小 $\text{C.}$ 高阶无穷小 $\text{D.}$ 同阶但非等价无穷小

$\text{A.}$ 连续且取极大值 $\text{B.}$ 连续且取极小值 $\text{C.}$ 可导且导数等于 0 $\text{D.}$ 可导且导数不为 0

$\text{A.}$ $(e,+\infty)$ $\text{B.}$ $(0, e)$ $\text{C.}$ $\left(0, \frac{1}{e}\right)$ $\text{D.}$ $\left(\frac{1}{e},+\infty\right)$

$$f\left(x+1, \mathrm{e}^x\right)=x(x+1)^2, f\left(x, x^2\right)=2 x^2 \ln x$$

$\text{A.}$ $\mathrm{d} x+\mathrm{d} y$ $\text{B.}$ $\mathrm{d} x-\mathrm{d} y$ $\text{C.}$ $\mathrm{d} y$ $\text{D.}$ $-\mathrm{d} y$

$$f\left(x_1, x_2, x_3\right)=\left(x_1+x_2\right)^2+\left(x_2+x_3\right)^2-\left(x_3-x_1\right)^2$$

$\text{A.}$ 2,0 $\text{B.}$ 1,1 $\text{C.}$ 2,1 $\text{D.}$ 1,2

$B=\left(\begin{array}{l}\alpha_1^T \\ \alpha_2^T \\ \alpha_3^T\end{array}\right), \beta=\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right), k$ 为任意常数，

$\text{A.}$ $\alpha_2+\alpha_3+\alpha_4+k \alpha_1$ $\text{B.}$ $\alpha_1+\alpha_3+\alpha_4+k \alpha_2$ $\text{C.}$ $\alpha_1+\alpha_2+\alpha_4+k \alpha_3$ $\text{D.}$ $\alpha_1+\alpha_2+\alpha_3+k \alpha_4$

$\text{A.}$ $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right),\left(\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right)$ $\text{B.}$ $\left(\begin{array}{ccc}1 & 0 & 0 \\ 2 & -1 & 0 \\ -3 & 2 & 1\end{array}\right),\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$ $\text{C.}$ $\left(\begin{array}{ccc}1 & 0 & 0 \\ 2 & -1 & 0 \\ -3 & 2 & 1\end{array}\right),\left(\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right)$ $\text{D.}$ $\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 3 & 1\end{array}\right),\left(\begin{array}{ccc}1 & 2 & -3 \\ 0 & -1 & 2 \\ 0 & 0 & 1\end{array}\right)$

$\text{A.}$ 若 $P(A \mid B)=P(A)$ ，则 $P(A \mid \bar{B})=P(A)$ $\text{B.}$ 若 $P(A \mid B)>P(A)$ ，则 $P(\bar{A} \mid \bar{B})>P(\bar{A})$ $\text{C.}$ 若 $P(A \mid B)>P(A \mid \bar{B})$ ，则 $P(A \mid B)>P(A)$ $\text{D.}$ 若 $P(A \mid A \cup B)>P(\bar{A} \mid A \cup B)$ ，则 $P(A)>P(B)$

$\text{A.}$ $E(\hat{\theta})=\theta, D(\hat{\theta})=\frac{\sigma_1^2+\sigma_2^2}{n}$ $\text{B.}$ $E(\hat{\theta})=\theta, D(\hat{\theta})=\frac{\sigma_1^2+\sigma_2^2-2 \rho \sigma_1 \sigma_2}{n}$ $\text{C.}$ $E(\hat{\theta}) \neq \theta, D(\hat{\theta})=\frac{\sigma_1^2+\sigma_2^2}{n}$ $\text{D.}$ $E(\hat{\theta}) \neq \theta, \quad D(\hat{\theta})=\frac{\sigma_1^2+\sigma_2^2-2 \rho \sigma_1 \sigma_2}{n}$

$\text{A.}$ $\frac{1}{4}$ $\text{B.}$ $\frac{3}{8}$ $\text{C.}$ $\frac{1}{2}$ $\text{D.}$ $\frac{5}{8}$

$\int_{\sqrt{5}}^5 \frac{x}{\sqrt{\left|x^2-9\right|}} \mathrm{d} x=$

$$I=\iint_D e^{(x+y)^2}\left(x^2-y^2\right) \mathrm{d} x \mathrm{~d} y$$

(1) 求 $y_n(x)$ ；
(2) 求级数 $\sum_{n=1}^{\infty} y_n(x)$ 的收敛域及和函数.

（1）求 $X$ 的概率密度；（2）求 $Z$ 的概率密度；(3) 求 $E\left(\frac{X}{Y}\right)$.

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