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

       

3答案

[] 由行列式的定义知,含有的项只有

,故应填-36.

4答案2

[]  由题设应有矩阵B-E不可逆,于是

(5)  答案 1

[]  由已知Ax=0有非零解,A满足行列式,A的三个特征值为, 于是行列式

(6)  应填

[] ,则

        =1=

 

二、选择题

1答案C

 [] 因为,于是有

,可见f(x)x=0处连续.

         

          

f(x)在点x=0处不可导,应选(C).

2答案 A

[] 因为 =

(A)为正确选项.

(3) 答案 D

[] 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)D内可微,更不能得到一阶偏导数连续. 故应选D.

(4)  答案 B.

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

的特征值和特征多项式相同,A,B不相似.

(5) 答案C

[] 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

[] 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、[]

u=xt, , 原方程化为

         

方程两边关于x求导, 

       

[]

,即 ,所以

  

   对于极限

   1)如果 0<<1, ,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

 

[]

1

2

 

六、[] 

,有

(x,f(x))处的切线方程是

         

Y=0, 得切线在x轴上的截距为

注意到,于是

        

 

 

 

      =

      =

七、[] 

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、[]  (1) (由分部积分得到),则 F(0)=F(1), 由罗尔定理,知存在,使 .

(2) ,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] 

r(A)=n, 线性无关,设Bx=0,即

  

于是有

可见当n为偶数时, Bx=0有非零解: ; n为奇数时,Bx=0只有零解.

十、[]

(1) ,A的特征值, 则有 ,的特征值,

,,A的特征值只能为0.

(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,=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、[]

 XY独立,易知P{Z=0}=(1-p)2+p2, P{Z=1}=2p(1-p). 要使ZX独立,必须

          , i=0,1; j=0,1

             解得p=1/2

 

十二、[] 

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

      

 

    r=0.06时,t=.