llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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v:shapes="_x0000_i1030">两边对x求导，得
_{}_{}

**（3****）**答案_{}

**[****解]** 由行列式的定义知，含有_{}的项只有

_{}，故_{}应填-36.

**（4****）**答案2

**[****解]** 由题设应有矩阵B-E不可逆，于是_{}

(5) 答案 1

**[****解]** 由已知Ax=0有非零解,且A满足行列式_{},知A的三个特征值为_{}, 于是行列式_{}_{}

**(6)** 应填 _{}

**[****解]** 记_{}，则_{}

=1_{}=_{}

** **

**二、选择题**

**（1****）**答案C

**[****解]** 因为_{}，_{}，于是有

_{}，可见f(x)在x=0处连续. 而

_{}，

_{}

故f(x)在点x=0处不可导，应选(C).

**（2****）**答案** **A

**[****解]** 因为 _{}=_{}

故(A)为正确选项.

(3) 答案 D

[解] 函数f(x,y)在区域D内有二阶偏导数,推不出其二阶偏导数连续,又不能得到f(x,y)在D内可微,更不能得到一阶偏导数连续. 故应选D.

**(4)** 答案 B.

**[****解]** 矩阵_{}与_{}等价,但其列向量组不等价,排除(A); 若Ax=0与Bx=0同解,则r(A)=r(B),因此有A,B等价,故(B)正确. 注意(C),(D)均为必要而非充分条件,比如

_{}与_{}的特征值和特征多项式相同,但A,B不相似.

(5) 答案C

[解] 因为_{},而右端显然可以不为零,所以(C)不正确, 故选(C).

**（6****）**答案 A

**[****解]** 由题设知，aX-bY仍服从正态分布，且其数学期望为E[aX-bY]=aEX-bEY=a-b，可见当a-b=1时，有_{}, 对比四个选项知，只有(A)满足a-b=1，因此正确选项为(A).

**三、[****解]**

令u=xt, _{}, 原方程化为

_{}

方程两边关于x求导,得 _{}

即 _{}

**四**、[**解**]

由_{}，即 _{}，所以 _{}

又 _{}

对于极限 _{}，

（1）如果 0<_{}<1, 有 _{}，则 _{}，由洛必达法则 _{}_{}，这与已知相矛盾.

(2) 如果 _{}>1 ，有 _{}，同理有 _{}0，矛盾.

所以 _{}=1.

于是 _{}=_{}

=_{}

所以 _{}

** **

**五**、[**解**]

_{}

1）_{}

2）_{}

** **

**六、[****解] **

由_{}，有 _{}

点(x,f(x))处的切线方程是

_{}

令Y=0, 得切线在x轴上的截距为 _{}

注意到_{}，于是

_{}

故 _{}

=_{}

=_{}

**七、**[**解**]

由于 _{}，而D可分解为关于x和y轴对称的D1,D2，令
_{}于是

_{},

解得 _{} 故有 _{}，令x=1，得 _{}

**八、**[**证]**
(1) 令_{} (由分部积分得到)，则 F(0)=F(1), 由罗尔定理，知存在_{}，使 _{}_{}.

(2) 令 _{}，则

_{}，

即严格单调增加，所以_{}必唯一.

**注：**本题属隐含问题，形式上用介值定理，实际上用微分中值定理.

**九、[解] **

当r(A)=n时, _{}线性无关,设Bx=0,即

_{}

则 _{}

于是有 _{}

可见当n为偶数时, Bx=0有非零解: _{}; 当n为奇数时,Bx=0只有零解.

**十、[****解]**

(1) 设_{},_{}为A的特征值, 则有 _{},即_{}为_{}的特征值,但

_{},故_{},即A的特征值只能为0.

而(A+E)x=Ax+Ex=_{},可见A+E的特征值全为_{}, 故行列式

_{}

(2) 假设A能与对角阵相似.则有可逆阵P,使 _{},其中_{}是由A的特征值构成的对角阵,即_{}=0, 从而可推出A=0,这与A_{}矛盾, 所以, A不能与对角阵相似.

**十一、[****解]**

由X和Y独立，易知*P*{*Z*=0}=(1-*p*)^{2}+*p*^{2}, *P*{*Z*=1}=2*p*(1-*p*). 要使Z与X独立，必须

_{} , *i*=0,1; *j*=0,1

即

_{} 解得*p*=1/2

** **

**十二、[****解]**

根据连续复利公式，t年末售出总收入R的现值为_{}

于是 _{}，

其中 _{}

=_{}_{}

这里_{} 于是

_{}

令 _{}

当 r=0.06时，t=_{}.