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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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ShapeID="_x0000_i1064" DrawAspect="Content" ObjectID="_1111259747"> D表示全平面,则=    .

分析 本题积分区域为全平面,但只有当时,被积函数才不为零,因此实际上只需在满足此不等式的区域内积分即可.

详解  =

           =

评注  若被积函数只在某区域内不为零,则二重积分的计算只需在积分区域与被积函数不为零的区域的公共部分上积分即可.

完全类似例题见《数学复习指南》P.1918.16-17 .

4A,B均为三阶矩阵,E是三阶单位矩阵. 已知AB=2A+B,B=,

=   .

分析 应先化简,从AB=2A+B中确定.

详解  AB=2A+B,

          AB-B=2A-2E+2E,

即有     

           

可见      ==.

评注 本题实质上是已知矩阵等式求逆的问题,应先分解出因式A-E,写成逆矩阵的定义形式,从而确定(A-E) 的逆矩阵.

完全类似例题见《数学最后冲刺》P.927.

 

5n维向量En阶单位矩阵,矩阵

        

其中A的逆矩阵为B,则a=    -1     .

分析 这里n阶矩阵,而为数,直接通过进行计算并注意利用乘法的结合律即可.

详解  由题设,有

         

              =

              =

              =

              =,

于是有  ,即  ,解得  由于A<0 ,a=-1.

评注 完全类似例题见《数学复习指南》P.3052大题第(5)小题 .

 

6设随机变量X Y的相关系数为0.5, EX=EY=0,, =      6       .

分析 利用期望与相关系数的公式进行计算即可.

详解  因为

       =

                 =4+

                 =4+2

评注 本题的核心是逆向思维,利用公式,而这种分析方法是文登辅导班上重点介绍过的.

 

二、选择题(本题共6小题,每小题4分,满分24. 每小题给出的四个选项中,只有一项符合题目要求,把所选项前的字母填在题后的括号内)

1曲线

(A) 仅有水平渐近线.                (B) 仅有铅直渐近线.

(C) 既有铅直又有水平渐近线.        (D) 既有铅直又有斜渐近线.       [  D  ]

分析 先考虑是否有水平渐近线,若无水平渐近线应进一步考虑是否存在斜渐近线,而是否存在铅直渐近线,应看函数是否存在无定义点.

详解  时,极限均不存在,故不存在水平渐近线;

          又因为 ,所以有斜渐近线y=x.

另外,在 x=0 无定义,且,可见 x=0为铅直渐近线.

故曲线既有铅直又有斜渐近线,应选(D).

评注 本题为常规题型,完全类似例题见《数学复习指南》P.153 6.30-31.

2设函数,其中x=1处连续,则f(x)x=1处可导的

 (A) 充分必要条件.       B)必要但非充分条件.

(C) 充分但非必要条件 .   (D) 既非充分也非必要条件.                 [  A  ]                                                     

分析  被积函数含有绝对值,应当作分段函数看待,利用f(x)x=1处左右导数定义讨论即可.

详解  因为

        

        

可见,f(x)x=1处可导的充分必要条件是  故应选(A).

评注 函数表达式中含有绝对值、取极值符号(max,min)等,均应当作分段函数处理.一般地,函数在点处可导的充要条件是

完全类似例题见《数学题型集粹与练习题集》P.28 2.6】和《考研数学大串讲》P.19的公式.

3设可微函数f(x,y)在点取得极小值,则下列结论正确的是

 (A) 处的导数等于零.  B处的导数大于零.

(C) 

详解 

           

所以   

                   =

评注 本题考查半抽象复合函数求二阶偏导.

完全类似例题见《文登数学全真模拟试卷》数学四P.67第六题和《数学复习指南》P.1717.20,7.22.

 

、(本题满分8分)

计算二重积分

       

其中积分区域D=

分析 从被积函数与积分区域可以看出,应该利用极坐标进行计算.

详解 作极坐标变换:,有

        

           =

,则

         .

   ,则

     

        =

        =

        =

       =

因此   

       

评注 本题属常规题型,明显地应该选用极坐标进行计算,在将二重积分化为定积分后,再通过换元与分步积分(均为最基础的要求),即可得出结果,综合考查了二重积分、换元积分与分步积分等多个基础知识点.

 

六、(本题满分9分)

a>1内的驻点为 a为何值时,t(a)最小?并求出最小值.

分析 先由f(t)的导数为零确定驻点t(a),它是关于a的函数,再把此函数对a求导,然后令此导数为零,得到可能极值点,进一步判定此极值为最小值即可.

详解  ,得唯一驻点

             

考察函数a>1时的最小值.

     

得唯一驻点

         

    时,;当时,,因此为极小值,从而是最小值.

评注 本题属基本题型,只是函数表达式由驻点给出,求极值与最值的要求均是最基本的.

类似例题见《数学复习指南》P.1446.11-12.

 

七、(本题满分9分)

y=f(x) 是第一象限内连接点A(0,1),B(1,0)的一段连续曲线,M(x,y)为该曲线上任意一点,点CMx轴上的投影,O为坐标原点. 若梯形OCMA的面积与曲边三角形CBM的面积之和为,求f(x)的表达式.

分析 梯形OCMA的面积可直接用梯形面积公式计算得到,曲边三角形CBM的面积可用定积分计算,再由题设,可得一含有变限积分的等式,两边求导后可转化为一阶线性微分方程,然后用通解公式计算即可.

详解  根据题意,有

   .

两边关于x求导,得

       

时,得

       

此为标准的一阶线性非齐次微分方程,其通解为            y

                    A

=                     M

=                    O    C        B     x

=

x=0时,f(0)=1.

由于x=1时,f(1)=0 ,故有2+C=0,从而C=-2. 所以

    

评注 本题一阶线性微分方程的求解比较简单,一般教材中都可找到标准的求解方法,完全类似例题见《数学题型集粹与练习题集》P.290第七题.

 

八、(本题满分8分)

设某商品从时刻0到时刻t的销售量为 欲在T 时将数量为A的该商品销售完,试求

(1)       t时的商品剩余量,并确定k的值;

(2)       在时间段[0T]上的平均剩余量.

分析 在时刻t的剩余量y(t)可用总量A减去销量x(t)得到; 由于y(t)随时间连续变化,因此在时间段[0,T] 上的平均剩余量,即函数平均值可用积分表示.

详解 (1) 在时刻t商品的剩余量为

           

                = 

 =0,得

            

因此

                     

(2) 依题意,[0T]上的平均值为

       

          =

          =

因此在时间段[0T] 上的平均剩余量为

评注 函数f(x)[a,b] 上的平均值记为

本题考查了函数平均值的概念,但大纲中只对数学一、二明确提出要求,而数学三、四的考试大纲中没有相应的要求,因此本题有超纲的嫌疑.

 

九、(本题满分13分)

设有向量组(I):和向量组(II): 试问:当a为何值时,向量组(I)与(II)等价?当a为何值时,向量组(I)与(II)不等价?

分析 两个向量组等价也即两个向量组可以相互线性表示,而两个向量组不等价,只需其中一组有一个向量不能由另一组线性表示即可. 而线性表示问题又可转化为对应非齐次线性方程组是否有解的问题,这可通过化增广矩阵为阶梯形来判断. 一个向量是否可由线性表示,只需用初等行变换化增广矩阵()为阶梯形讨论,而一组向量是否可由线性表示,则可结合起来对矩阵()同时作初等行变换化阶梯形,然后类似地进行讨论即可.

详解 作初等行变换,有

   =

                       .

(1) 时,有行列式,秩(,故线性方程组均有唯一解. 所以,可由向量组(I)线性表示.

同样,行列式,秩(,故可由向量组(II)线性表示. 因此向量组(I)与(II)等价.

(2) a=-1时,有

     .

由于秩(秩(,线性方程组无解,故向量不能由线性表示. 因此,向量组(I)与(II)不等价.

评注1 涉及到参数讨论时,一般联想到利用行列式判断,因此,本题也可这样分析:

因为行列式,可见

(1) 时,秩,因此三维列向量组等价,即向量组(I)与(II)等价.

(2) a=-1时,,秩,而行列式,可见r=3, 因此线性方程组无解,故向量不能由线性表示. 即向量组(I)与(II)不等价.

评注2 向量组(I)与(II)等价,相当于均为整个向量组的一个极大线性无关组,问题转化为求向量组的极大线性无关组,这可通过初等行变换化阶梯形进行讨论.

本题完全类似分析思路的例题见《数学复习指南》P.3474.13】和《数学最后冲刺》P.9414.

 

十、(本题满分13分)

设矩阵可逆,向量是矩阵的一个特征向量,对应的特征值,其中是矩阵A的伴随矩阵. 试求a,b的值.

分析 题设已知特征向量,应想到利用定义:,又与伴随矩阵相关的问题,应利用进行化简.

详解 矩阵属于特征值的特征向量为,由于矩阵A可逆,故可逆.于是,且

          .

两边同时左乘矩阵A,得

        

        

由此,得方程组

                         

由式(1),(2)解得

              ;

由式(1),(3)解得

               a=2.

由于

        

根据(1)式知,特征向量所对应的特征值

所以,当时,

      时,

评注 本题若先求出,再按特征值、特征向量的定义进行分析,则计算过程将非常复杂. 一般来说,见到,首先应想到利用公式进行化简.

本题类似的例题见《数学复习指南》P.3655.7】和《数学最后冲刺》P.903.

 

十一、(本题满分13分)

设随机变量X的概率密度为

   

F(x)X的分布函数. 求随机变量Y=F(X)的分布函数.

分析 先求出分布函数F(x) 的具体形式,从而可确定Y=F(X) ,然后按定义求Y 的分布函数即可。注意应先确定Y=F(X)的值域范围,再对y分段讨论.

详解 易见,当x<1时,F(x)=0; x>8 时,F(x)=1.

对于,有

    

G(y)是随机变量Y=F(X)的分布函数. 显然,当时,G(y)=0;当时,G(y)=1.

 对于,有

       

              =

              =

于是,Y=F(X)的分布函数为

              

评注 事实上,本题X为任意连续型随机变量均可,此时Y=F(X)仍服从均匀分布:

y<0时,G(y)=0;

时,G(y)=1;

0时,

                     =

                     =

评注 本题是《数学复习指南》P.4312.23】原题(实际上还是此题的特殊情形).

 

十二、(本题满分13分)

对于任意二事件A B

           

称做事件AB的相关系数.

(1)    证明事件AB独立的充分必要条件是其相关系数等于零;

(2)    利用随机变量相关系数的基本性质,证明

分析 (1) 利用事件AB独立的定义P(AB)=P(A)P(B)即可;(2) 随机变量XY的相关系数为,而需将转化为用随机变量表示,显然,若有以及即可,这只需定义

  

详解  (1) 的定义,可见当且仅当

             P(AB)-P(A)P(B)=0,

而这恰好是二事件AB独立的定义,即AB独立的充分必要条件.

(2) 考虑随机变量XY:

       

由条件知,XY都服从01分布:

易见

,    ;

因此,事件AB的相关系数就是随机变量XY的相关系数.

    于是由二随机变量相关系数的基本性质,可见

评注 如上01分布与随机事件之间的关系值得注意,较好地将两者联系起来了,为借助相互的性质提供了便利.

完全类似例题见《数学题型集粹与练习题集》P.298第十一题和《考研数学大串讲》P.24925.

 

注: 1.《数学复习指南》 (2003版,经济类)世界图书出版公司

主编: 陈文灯、黄先开

2.《数学题型集粹与练习题集》(2003版,经济类)世界图书出版公司

主编: 陈文灯、黄先开

3.《文登数学全真模拟试卷》(2003版,经济类)世界图书出版公司

主编: 陈文灯、黄先开

4.《数学最后冲刺》(2003版,经济类)世界图书出版公司

主编: 陈文灯、黄先开

5.《考研数学大串讲》(2002版,经济类)世界图书出版公司

主编: 黄先开、曹显兵、施明存

 

(文登学校供稿)