### 2006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)

$\text{A.}$ $0 < \mathrm{d} y < \Delta y$ $\text{B.}$ $0 < \Delta y < \mathrm{d} y$ $\text{C.}$ $\Delta y < \mathbf{d} y < 0$ $\text{D.}$ $\mathrm{d} y < \Delta y < 0$

$$\int_0^{\frac{\pi}{4}} \mathrm{~d} \theta \int_0^1 f(r \cos \theta, r \sin \theta) r \mathrm{~d} r \text { 等于 }$$
$\text{A.}$ $\int_0^{\frac{\sqrt{2}}{2}} \mathrm{~d} x \int_x^{\sqrt{1-x^2}} f(x, y) \mathrm{d} y$ $\text{B.}$ $\int_0^{\frac{\sqrt{2}}{2}} \mathrm{~d} x \int_0^{\sqrt{1-x^2}} f(x, y) \mathrm{d} y$ $\text{C.}$ $\int_0^{\frac{\sqrt{2}}{2}} \mathrm{~d} y \int_y^{\sqrt{1-y^2}} f(x, y) \mathrm{d} x$ $\text{D.}$ $\int_0^{\frac{\sqrt{2}}{2}} \mathrm{~d} y \int_0^{\sqrt{1-y^2}} f(x, y) \mathrm{d} x$

$\text{A.}$ $\sum_{n=1}^{\infty}\left|a_n\right|$ 收敛. $\text{B.}$ $\sum_{n=1}^{\infty}(-1)^n a_n$ 收敛. $\text{C.}$ $\sum_{n=1}^{\infty} a_n a_{n+1}$ 收敛. $\text{D.}$ $\sum_{n=1}^{\infty} \frac{a_n+a_{n+1}}{2}$ 收敛.

$\text{A.}$ 若 $f_x^{\prime}\left(x_0, y_0\right)=0$ ，则 $f_y^{\prime}\left(x_0, y_0\right)=0$ $\text{B.}$ 若 $f_x^{\prime}\left(x_0, y_0\right)=0$ ，则 $f_y^{\prime}\left(x_0, y_0\right) \neq 0$ $\text{C.}$ 若 $f_x^{\prime}\left(x_0, y_0\right) \neq 0$ ，则 $f_y^{\prime}\left(x_0, y_0\right)=0$ $\text{D.}$ 若 $f_x^{\prime}\left(x_0, y_0\right) \neq 0$ ，则 $f_y^{\prime}\left(x_0, y_0\right) \neq 0$

$\text{A.}$ 若 $\alpha_1, \alpha_2, \cdots, \alpha_s$ 线性相关，则 $A \alpha_1, A \alpha_2, \cdots, A \alpha_s$ 线性相关. $\text{B.}$ 若 $\alpha_1, \alpha_2, \cdots, \alpha_s$ 线性相关，则 $A \alpha_1, A \alpha_2, \cdots, A \alpha_s$ 线性无关. $\text{C.}$ 若 $\alpha_1, \alpha_2, \cdots, \alpha_s$ 线性无关，则 $A \alpha_1, A \alpha_2, \cdots, A \alpha_s$ 线性相关. $\text{D.}$ 若 $\alpha_1, \alpha_2, \cdots, \alpha_s$ 线性无关，则 $A \alpha_1, A \alpha_2, \cdots, A \alpha_s$ 转性无关.

$\text{A.}$ $C=P^{-1} A P$ $\text{B.}$ $C=P A P^{-1}$ $\text{C.}$ ${C}={P}^T {A} {P}$ $\text{D.}$ ${C}={P A} {P}^T$

$\text{A.}$ $P(A \cup B)>P(A)$ $\text{B.}$ $P(A \cup B)>P(B)$ $\text{C.}$ $P(A \cup B)=P(A)$ $\text{D.}$ $P(A \cup B)=P(B)$

$\text{A.}$ $\sigma_1 < \sigma_2$ $\text{B.}$ $\sigma_1>\sigma_2$ $\text{C.}$ $\mu_1 < \mu_2$ $\text{D.}$ $\mu_1>\mu_2$

$\lim _{x \rightarrow 0} \frac{x \ln (1+x)}{1-\cos x}=$

(1) 证明 $\lim _{n \rightarrow \infty} x_n$ 存在，并求之.
(2) 计算 $\lim _{n \rightarrow \infty}\left(\frac{x_{n+1}}{x_n}\right)^{\frac{1}{x_n^2}}$.

(1) 验证 $f^{\prime \prime}(u)+\frac{f^{\prime}(u)}{u}=0$
（2）若 $f(1)=0, f^{\prime}(1)=1$ ，求函数 $f(u)$ 的表达式.

$$\oint_L y f(x, y) \mathrm{d} x-x f(x, y) \mathrm{d} y=0$$

(1) 证明方程组系数矩阵 $A$ 的秩 $r(A)=2$ ；
(2) 求 $a, b$ 的值及方程组的通解.

(1) 求 $A$ 的特征值与特征向量
(2) 求正交矩阵 $Q$ 和对角矩阵 $\Lambda$ ，使得 $Q^T A Q=\Lambda$.

$$f_X(x)=\left\{\begin{array}{l} 1 / 2,-1 < x < 0 \\ 1 / 4,0 \leq x < 2 \\ 0 \quad, \text { 其他 } \end{array}\right.$$

(1) 求 $Y$ 的概率密度 $f_Y(y)$;
(2) $F\left(-\frac{1}{2}, 4\right)$.

$$f(x ; \theta)=\left\{\begin{array}{lc} \theta, & 0 < x < 1 \\ 1-\theta, 1 \leq x < 2 \\ 0, & \text { 其他 } \end{array}\right.$$

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