### 2023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)

$\text{A.}$ $\left.\frac{\partial f}{\partial x}\right|_{(0,1)}$ 不存在, $\left.\frac{\partial f}{\partial y}\right|_{(0,1)}$ 存在 $\text{B.}$ $\left.\frac{\partial f}{\partial x}\right|_{(0,1)}$ 存在, $\left.\frac{\partial f}{\partial y}\right|_{(0,1)}$ 不存在 $\text{C.}$ $\left.\frac{\partial f}{\partial x}\right|_{(0,1)},\left.\frac{\partial f}{\partial y}\right|_{(0,1)}$ 均存在 $\text{D.}$ $\left.\frac{\partial f}{\partial x}\right|_{(0,1)},\left.\frac{\partial f}{\partial y}\right|_{(0,1)}$ 均不存在

$\text{A.}$ $F(x)= \begin{cases}\ln \left(\sqrt{1+x^2}-\bar{x}\right), & x \leq 0 \\ (x+1) \cos x-\sin x, & x>0\end{cases}$ $\text{B.}$ $F(x)=\left\{\begin{array}{l}\ln \left(\sqrt{1+x^2}-x\right)+1, x \leq 0 \\ (x+1) \cos x-\sin x, x>0\end{array}\right.$ $\text{C.}$ $\boldsymbol{F}(x)= \begin{cases}\ln \left(\sqrt{1+x^2}+x\right), & x \leq 0 \\ (x+1) \sin x+\cos x, & x>0\end{cases}$ $\text{D.}$ $\boldsymbol{F}(x)=\left\{\begin{array}{l}\ln \left(\sqrt{1+x^2}+x\right)+1, x \leq 0 \\ (x+1) \sin x+\cos x, x>0\end{array}\right.$

$\text{A.}$ $a < 0, b>0$ $\text{B.}$ $a>0, b>0$ $\text{C.}$ $a=0, b>0$ $\text{D.}$ $a=0, b < 0$

$\text{A.}$ 充分必要条件 $\text{B.}$ 充分不必要条件 $\text{C.}$ 必要不充分条件 $\text{D.}$ 既不充分也不必要条件

$\text{A.}$ $\left(\begin{array}{cc}|\boldsymbol{A}| B^* & -B^* A^* \\ O & |B| A^*\end{array}\right)$ $\text{B.}$ $\left(\begin{array}{cc}|A| B^* & -A^* B^* \\ O & |B| A^*\end{array}\right)$ $\text{C.}$ $\left(\begin{array}{cc}|B| A^* & -B^* A^* \\ O & |A| B^*\end{array}\right)$ $\text{D.}$ $\left(\begin{array}{cc}|\boldsymbol{B}| \boldsymbol{A}^* & -\boldsymbol{A}^* B^* \\ -\boldsymbol{O} & |\boldsymbol{A}| \boldsymbol{B}^*\end{array}\right)$

$\text{A.}$ $y_1^2+y_2^2$ $\text{B.}$ $y_1^2-y_2^2$ $\text{C.}$ $y_1^2+y_2^2-4 y_3^2$ $\text{D.}$ $y_1^2+y_2^2-y_3^2$

$\text{A.}$ $k\left(\begin{array}{l}3 \\ 3 \\ 4\end{array}\right), k \in \mathbf{R}$ $\text{B.}$ $k\left(\begin{array}{c}3 \\ 5 \\ 10\end{array}\right), k \in \mathbf{R}$ $\text{C.}$ $k\left(\begin{array}{c}-1 \\ 1 \\ 2\end{array}\right), k \in \mathbf{R}$ $\text{D.}$ $k\left(\begin{array}{l}1 \\ 5 \\ 8\end{array}\right), k \in \mathbf{R}$

$\text{A.}$ $\frac{1}{\mathrm{e}}$ $\text{B.}$ $\frac{1}{2}$ $\text{C.}$ $\frac{2}{\mathrm{e}}$ $\text{D.}$ 1

$$S_1^2=\frac{1}{n-1} \sum_{i=1}^n\left(X_i-\bar{X}\right)^2, S_2^2=\frac{1}{m-1} \sum_{i=1}^m\left(Y_i-\bar{Y}\right)^2$$

$\text{A.}$ $\frac{S_1^2}{S_2^2} \sim F(n, m)$ $\text{B.}$ $\frac{S_1^2}{S_2^2} \sim F(n-1, m-1)$ $\text{C.}$ $\frac{2 S_1^2}{S_2^2} \sim F(n, m)$ $\text{D.}$ $\frac{2 S_1^2}{S_2^2} \sim F(n-1, m-1)$

$\text{A.}$ $\frac{\sqrt{\pi}}{2}$ $\text{B.}$ $\frac{\sqrt{2 \pi}}{2}$ $\text{C.}$ $\sqrt{\pi}$ $\text{D.}$ $\sqrt{2 \pi}$

$\lim _{x \rightarrow \infty} x^2\left(2-x \sin \frac{1}{x}-\cos \frac{1}{x}\right)=$

$\mathrm{d} f(x, y)=\frac{x \mathrm{~d} y-y \mathrm{~d} x}{x^2+y^2}, f(1,1)=\frac{\pi}{4} ，$

$\sum_{n=0}^{\infty} \frac{x^{2 n}}{(2 n)!}=$

$$X \sim B(1, p), Y \sim B(2, p), p \in(0,1)$$

$$a e^x+y^2+y-\ln (1+x) \cos y+b=0$$

(1) 求 $a, b$ 的值.
(2) 判断 $x=0$ 是否为 $y(x)$ 的极值点.

$$D=\left\{(x, y) \left\lvert\, 0 \leq y \leq \frac{1}{x \sqrt{1+x^2}}\right., x \geq 1\right\}$$

(1) 求 $D$ 的面积;
(2) 求 $D$ 绕 $x$ 轴旋转所成旋转体的体积.

(1) 若 $f(0)=0$ ，则存在 $\xi \in(-a, a)$ ，使得

$$f^{\prime \prime}(\xi)=\frac{1}{a^2}[f(a)+f(-a)]$$

(2) 若 $f(x)$ 在 $(-a, a)$ 内取得极值，则存在 $\eta \in(-a, a)$ ，使得

$$\left|f^{\prime \prime}(\eta)\right| \geq \frac{1}{2 a^2}|f(a)-f(-a)|$$

$$A\left(\begin{array}{l} x_1 \\ x_2 \\ x_3 \end{array}\right)=\left(\begin{array}{c} x_1+x_2+x_3 \\ 2 x_1-x_2+x_3 \\ x_2-x_3 \end{array}\right)$$

(1) 求矩阵 $\boldsymbol{A}$.
(2) 求可迕矩阵 $P$ 与对角矩阵 $\boldsymbol{\Lambda}$ ，使得 $\bar{P}^{-1} \boldsymbol{A} \boldsymbol{P}=\boldsymbol{\Lambda}$.

\begin{aligned} & \quad f(x)=\frac{e^x}{\left(1+e^x\right)^2},-\infty < x < +\infty \\ & \text { 令 } Y=e^X \text {. } \end{aligned}

(1) 求 $X$ 的分布函数；
(2) 求 $Y$ 的概率密度;
（3） $\boldsymbol{Y}$ 的期望是否存在?

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