### 李永乐武忠祥宋浩陈默等著2023考研数学最后三套过线急救版（数学三）

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

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

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

$\text{A.}$ 绝对收敛. $\text{B.}$ 条件收敛. $\text{C.}$ 发散. $\text{D.}$ 收敛性与 $k$ 的取值有关.

$\text{A.}$ $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right)$. $\text{B.}$ $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 2 & 6 \\ 0 & 0 & 3\end{array}\right)$. $\text{C.}$ $\left(\begin{array}{lll}0 & 1 & 0 \\ 2 & 0 & 0 \\ 0 & 0 & 3\end{array}\right)$. $\text{D.}$ $\left(\begin{array}{lll}0 & 1 & 0 \\ 2 & 0 & 6 \\ 0 & 0 & 3\end{array}\right)$.

(1) $\boldsymbol{A}+\boldsymbol{B}$ 可逆; (2) $\boldsymbol{A B}=\boldsymbol{B A}$; (3) $\boldsymbol{A}-\boldsymbol{E}$ 可逆; (4) $(\boldsymbol{B}-\boldsymbol{E}) \boldsymbol{x}=0$ 有非零解.

$\text{A.}$ 1个 $\text{B.}$ 2个 $\text{C.}$ 3个 $\text{D.}$ 4个

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

$\text{A.}$ $A, B, C$ 两两独立. $\text{B.}$ $P(A+B+C)=1-P(\bar{A}) P(\bar{B}) P(\bar{C})$. $\text{C.}$ $P(A B C)=P(A) P(B) P(C)$. $\text{D.}$ $P(B-A)=1$.

$\text{A.}$ $N(-1,1)$. $\text{B.}$ 与 $Y$ 同分布. $\text{C.}$ $N(0,1)$. $\text{D.}$ $N\left(\frac{1}{3}, \frac{2}{3}\right)$.

$\text{A.}$ $F(1,1)$. $\text{B.}$ $\chi^2(1)$. $\text{C.}$ $N(0,1)$. $\text{D.}$ $t(1)$.

$\lim _{\substack{x \rightarrow 2 \\ y \rightarrow+\infty}}\left(\cos \frac{x^2}{y}\right)^{\frac{y^2+x}{x^3}}=$

$$\int_0^1 f(x) \mathrm{d} x=3, \int_0^1 x f(x) \mathrm{d} x=3,$$

$$\frac{a}{f^{\prime}\left(\xi_1\right)}+\frac{b}{f^{\prime}\left(\xi_2\right)}+\frac{c}{f^{\prime}\left(\xi_3\right)}=1 .$$

$$f(x, x)=(x-2)^2+(x-2) \ln x,$$

$$\frac{\partial^2 z}{\partial x^2}+\frac{\partial^2 z}{\partial y^2}=\mathrm{e}^{2 x}\left(z+\mathrm{e}^x \cos y\right) .$$
(I) 验证: $f^{\prime \prime}(u)-f(u)=u$;
(II) 若 $f(0)=f^{\prime}(0)=1$, 求出函数 $f(u)$ 的表达式.

(I) 求常数 $a$ 的值;
（II) 求所作可逆线性变换的矩阵 $\boldsymbol{P}$.

(I) 求条件密度函数 $f_{X \mid Y}(x \mid y)$ 与 $f_{Y \mid X}(y \mid x)$;
(II) $F(x, y)$ 是 $(X, Y)$ 的分布函数,求 $F\left(\frac{\mathrm{e}}{2}, \ln \frac{\mathrm{e}}{2}\right)$;
(III) 设 $\left(Y_1, Y_2, \cdots, Y_n\right)$ 是取自 $Y$ 的样本, $S^2=\frac{1}{n-1} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)^2$ 为样本方差, 求 $E\left(S^2\right)$.

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