### 2014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(数一)

$\text{A.}$ $y=x+\sin x$ $\text{B.}$ $y=x^2+\sin x$ $\text{C.}$ $y=x+\sin \frac{1}{x}$ $\text{D.}$ $y=x^2+\sin \frac{1}{x}$

$\text{A.}$ 当 $f^{\prime}(x) \geq 0$ 时， $f(x) \geq g(x)$ $\text{B.}$ 当 $f^{\prime}(x) \geq 0$ 时， $f(x) \leq g(x)$ $\text{C.}$ 当 $f^{\prime \prime}(x) \geq 0$ 时， $f(x) \geq g(x)$ $\text{D.}$ 当 $f^{\prime \prime}(x) \geq 0$ 时， $f(x) \leq g(x)$

$\text{A.}$ $\int_0^1 \mathrm{~d} x \int_0^{x-1} f(x, y) \mathrm{d} y+\int_{-1}^0 \mathrm{~d} x \int_0^{\sqrt{1-x^2}} f(x, y) \mathrm{d} y$ $\text{B.}$ $\int_0^1 \mathrm{~d} x \int_0^{1-x} f(x, y) \mathrm{d} y+\int_{-1}^0 \mathrm{~d} x \int_{-\sqrt{1-x^2}}^0 f(x, y) \mathrm{d} y$ $\text{C.}$ $\int_0^{\frac{\pi}{2}} \mathrm{~d} \theta \int_0^{\frac{1}{\cos \theta+\sin \theta}} f(r \cos \theta, r \sin \theta) \mathrm{d} r$+ $\int_{\frac{\pi}{2}}^\pi \mathrm{d} \theta \int_0^1 f(r \cos \theta, r \sin \theta) \mathrm{d} r$ $\text{D.}$ $\int_0^{\frac{\pi}{2}} \mathrm{~d} \theta \int_0^{\frac{1}{\cos \theta+\sin \theta}} f(r \cos \theta, r \sin \theta) \mathrm{d} r$+ $\int_{\frac{\pi}{2}}^\pi \mathrm{d} \theta \int_0^1 f(r \cos \theta, r \sin \theta) r \mathrm{~d} r$

$$=\min _{a, b \in \mathrm{R}}\left\{\int_{-\pi}^\pi(x-a \cos x-b \sin x)^2 \mathrm{~d} x\right\} \text {, }$$

$\text{A.}$ $2 \sin x$ $\text{B.}$ $2 \cos x$ $\text{C.}$ $2 \pi \sin x$ $\text{D.}$ $2 \pi \cos x$

$\text{A.}$ $(a d-b c)^2$ $\text{B.}$ $-(a d-b c)^2$ $\text{C.}$ $a^2 d^2-b^2 c^2$ $\text{D.}$ $b^2 c^2-a^2 d^2$

$\text{A.}$ 必要非充分条件 $\text{B.}$ 充分非必要条件 $\text{C.}$ 充分必要条件 $\text{D.}$ 非充分非必要条件

$\text{A.}$ 0.1 $\text{B.}$ 0.2 $\text{C.}$ 0.3 $\text{D.}$ 0.4

$$Y_2=\frac{1}{2}\left(X_1+X_2\right)$$ 则
$\text{A.}$ $E\left(Y_1\right)>E\left(Y_2\right), D\left(Y_1\right)>D\left(Y_2\right)$ $\text{B.}$ $E\left(Y_1\right)=E\left(Y_2\right), D\left(Y_1\right)=D\left(Y_2\right)$ $\text{C.}$ $E\left(Y_1\right)=E\left(Y_2\right), D\left(Y_1\right) < D\left(Y_2\right)$ $\text{D.}$ $E\left(Y_1\right)=E\left(Y_2\right), D\left(Y_1\right)>D\left(Y_2\right)$

$$\oint_L z \mathrm{~d} x+y \mathrm{~d} z=$$

$$f(x ; \theta)=\left\{\begin{array}{cc} \frac{2 x}{3 \theta^2}, & < x < 2 \theta \\ 0 & \text {, 其他 } \end{array}\right.$$

$$\frac{\partial^2 z}{\partial x^2}+\frac{\partial^2 z}{\partial y^2}=\left(4 z+e^x \cos y\right) e^{2 x}$$

$$I=\iint_{\Sigma}(x-1)^3 \mathrm{~d} y \mathrm{~d} z+(y-1)^3 \mathrm{~d} z \mathrm{~d} x+(z-1) \mathrm{d} x \mathrm{~d} y$$

(1) 证明 $\lim _{n \rightarrow \infty} a_n=0$ ；
(2) 证明级数 $\sum_{n=1}^{\infty} \frac{a_n}{b_n}$ 收敛.

(1) 求方程组 $\boldsymbol{A x}=\mathbf{0}$ 的一个基础解系；
(2) 求满足 $\boldsymbol{A B}=\boldsymbol{E}$ 的所有矩阵 $B$.

$$A=\left(\begin{array}{cccc} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & & \vdots \\ 1 & 1 & \cdots & 1 \end{array}\right) \text { 与 } B=\left(\begin{array}{cccc} 0 & \cdots & 0 & 1 \\ 0 & \cdots & 0 & 2 \\ \vdots & & \vdots & \vdots \\ 0 & \cdots & 0 & n \end{array}\right)$$

$$P\{X=1\}=P\{X=2\}=\frac{1}{2}$$

(1) 求 $Y$ 的分布函数 $F_Y(y)$ ；
(2) 求期望 $\boldsymbol{E}(\boldsymbol{Y})$.

$$\boldsymbol{F}(x ; \theta)=\left\{\begin{array}{cc} 1-e^{-\frac{x^2}{\theta}}, & x \geq 0 \\ 0, & x < 0 \end{array}\right.$$

(1) 求 $E(X)$ 与 $E\left(X^2\right)$ ；
(2) 求 $\theta$ 的最大似然估计量 $\hat{\theta}_n$.
(3) 是否存在常数 $a$ ，使得对任意的 $\varepsilon>0$ ，都有
$$\lim _{n \rightarrow \infty} P\left\{\left|\hat{\theta}_n-a\right| \geq \varepsilon\right\}=0 \text { ? }$$

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