2000年试题分析、详解= 和评注

MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C3A47E.C4C31660" 此文档为“单个文件网页”，也称为“Web 档案”文件。如果您看到此消息，但是您的浏览器或编辑器不支持“Web 档案”文件。请下载支持“Web 档案”的浏览器，如 Microsoft Internet Explorer。 ------=_NextPart_01C3A47E.C4C31660 Content-Location: file:///C:/A718D1D4/2003-4.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" 2000年试题分析、详解= 和评注

2003年考研数学（四）İ= 95;题评注

一、填空题（本题共6小题，每&#= 23567;题4分，满分24分= . 把答案填在题中= 横线上）

（1）<= /b>极限_{=3D <=
u>_.}

【分析】本题属_{<=
!--[if gte vml 1]>}型未= 定式，化为指数函数= 7714;极限即可.

【详解】 _=3D

        =    =3D

【评注】对于型未= 定式的极= 限，也可直接用公式_{_=3D}进行= 计算，因此本题也可= 6825;样求解：

_=3D

【评注】完全类似例题见= 《数学复习指南》P.23【Ë= 63;1.30】和= 《文登数学全真模拟= 5797;卷》数学四P.29第一= 大题第（1）小题.

（2）_=3D  .

【分析】对称区间&#= 19978;的积分应注意利用ŝ= 87;积函数的对称性，这= 里有

【详解】 _=3D

        =         =3D

        =         =3D

           =      =3D

        =         =3D_.

【评注】本题属基本题型= ，主要考查对称区间= 9978;的积分性质和分布ఱ= 5;分法.

原题见《文登数= 学全真模拟试卷》数= 3398;二P.37第一= 题第（3）小题（&#= 23436;全是原题，答案也Ç= 68;样），完全类似题见= 《文登数学全真模拟= 5797;卷》数学三P.71第一= 大题第（2）小题.

（3）设a>0，<= span lang=3DEN-US>_{而<=
span
lang=3DEN-US>D表示全平&#=
38754;，则_{=3D   <=
u>_.}}

【分析】本题积分&#= 21306;域为全平面，但只ć= 77;当时，= 被积函数才不为零，= 2240;此实际上只需在满ห= 5;此不等式的区域内积&= #20998;即可.

【详解】 _=3D

        =    =3D

【评注】若被积函数只在= 某区域内不为零，则= 0108;重积分的计算只需ࢷ= 2;积分区域与被积函数&= #19981;为零的区域的公共%= 096;分上积分即可.

完全类似例题见= 《数学复习指南》P.191【Ë= 63;8.16-17】<= span lang=3DEN-US> .

（4）设A,B均为= 三阶矩阵，E是三阶单&#= 20301;矩阵. 已知AB=3D2A+B,B=3D_,则<= /p>
_{=3D .}

【分析】应先化简，从<= span lang=3DEN-US>AB=3D2A+B中确= 定_.

【详解】由AB=3D2A+B, 知

        = AB-B=3D2A-2E+2E,

即有      ，<= /p>
        = ，<= span lang=3DEN-US> _，<=
/p>
可见      _{=3D_{=3D_.}}

【评注】本题实质上是已= 知矩阵等式求逆的问= 9064;，应先分解出因式A-E，写= 成逆矩阵的定义形式= 5292;从而确定(A-E) 的逆= 矩阵.

完全类似例题见= 《数学最后冲刺》P.92【Ë= 63;7】<= span lang=3DEN-US>.

（5）设n维向量_{；<=
span
lang=3DEN-US>E为=
n阶单位矩阵，矩=
阵}

         ， = ，<= /p>
其中A的逆矩阵&#= 20026;B，则a=3D    -1    .

【分析】这里为<= span lang=3DEN-US>n阶矩阵，&#= 32780;为数= ，直接通过进行= 计算并注意利用乘法= 0340;结合律即可.

【详解】由题设，有

        =

        =       =3D

        =       =3D

        =       =3D

        =       =3D_,

于是有 _{，即=
，解=
得
由于A<0 ,故a=3D-1.}

【评注】完全类似例题见= 《数学复习指南》P.305第<= span lang=3DEN-US>2大题第（5）小题 .

（6）设随机变量X 和= Y的相关系数为<= span lang=3DEN-US>0.5, EX=3DEY=3D0,_{, 则<=
span
lang=3DEN-US>_{=3D      6      .}}

【分析】利用期望与相关= 系数的公式进行计算= 1363;可.

【详解】因为

     = ; _=3D

        =          =3D4+₌

        =          =3D4+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= 6741;导班上重点介绍过௚= 0;.

二、= 选择题（本题共6小题，每小题<= span lang=3DEN-US>4分，满分24分= . 每小题给出的四= 个选项中，只有一项= 1526;合题目要求，把所๧= 3;项前的字母填在题后&= #30340;括号内）

（1）<= /b>曲线

(A) <= /span>仅有水平渐近线= .    &nbs= p;         =   (B) 仅有铅直&#= 28176;近线.

(C) <= /span>既有铅直又有水= 平渐近线.        (D) 既有铅直又有斜= 渐近线.    &nbs= p; [ D ]

【分析】先考虑是否有水= 平渐近线，若无水平= 8176;近线应进一步考虑਷= 9;否存在斜渐近线，而&= #26159;否存在铅直渐近线A= 292;应看函数是否存在无= ;定义点.

【详解】当时，= 极限均不= 存在，故不存在水平= 8176;近线；

        = 又因为 _{，<=
span
lang=3DEN-US>_{，所=
以有斜渐近线y=3Dx.}}

另外，在 x=3D0 处= 无定= 义，且，可= 见 x=3D0为铅= 直渐近线.

故曲线_{既有=
铅直又有斜渐近线，=
4212;选(D).}

【评注】本题为常规题型= ，完全类似例题见《= 5968;学复习指南》P.153 【Ë= 63;6.30-31】<= span lang=3DEN-US>.

（2）设函数，其= 中在<= span lang=3DEN-US>x=3D1处连= 续，则是<= span lang=3DEN-US>f(x)在<= span lang=3DEN-US>x=3D1处可= 导的

(A) 充分必要条件<= span lang=3DEN-US>.    &nbs= p; （= 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 ]        =             &nb= sp;            =             &nb= sp;

【分析】被积函数含有绝= 对值，应当作分段函= 5968;看待，利用f(x)在<= span lang=3DEN-US>x=3D1处左= 右导数定义讨论即可.

【详解】因为

         ，<= /p>
         ，<= /p>
可见，f(x)在x=3D1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= 1495;(max,min)等，= 均应当作分段函数处= 9702;.一般地，&#= 20989;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= 0064;题集》P.28 【Ë= 63;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= 7491;确的是

(A) 在<= span lang=3DEN-US>_{处的=
导数等于零. （B）在<=
span
lang=3DEN-US>_{处的=
导数大于零.}}

(C) 在<= span lang=3DEN-US>_{处的=
导数小于零.   (D) 在<=
span
lang=3DEN-US>_{处的=
导数不存在.}}

        =             &nb= sp;            =             &nb= sp;            =           [ A ]

【分析】可微必有偏导数= 存在，再根据取极值= 0340;必要条件即可得结෹= 0;.

【详解】可微函数f(x,y)在点取得= 极小值，根据取极值= 0340;必要条件知，即= 在<= span lang=3DEN-US>_{处的=
导数等于零, 故应选(A).}

【评注1】本题= 考查了偏导数的定义= 5292;在<= span lang=3DEN-US>_{处的=
导数即；而=
在<=
span
lang=3DEN-US>_{处的=
导数即}}

【评注2】本题= 也可用排除法分析，= 1462;，在= (0,0)处可= 微且取得极小值，并= 9988;有，可= 排除(B),(C),(D), 故正= 确选项为(A).

（4）设矩阵

    .

已知矩阵A相似于B，则秩(A-2E)= 与秩(A-E)之和等于

(A) 2.        (B) 3.         (C) 4.        =    (D) 5.        =          [ C ]

【分析】利用相似矩阵有= 相同的秩计算，秩(A-2E)与秩= (A-E)之和= 等于秩(B-2E)与秩= (B-E)之和= .

【详解】因为矩阵A相似于B，于是有矩阵<= span lang=3DEN-US>A-2E与矩= 阵B-2E相似= ，矩阵A-E与矩= 阵B-E 相似= ，且相似矩阵有相同= 0340;秩，而

秩(B-2E)=3D秩，秩= (B-E)=3D秩<= span lang=3DEN-US>_，<=
/p>
可见有秩(A-2E)+秩(A-E)=3D 秩(B-2E)+秩(B-E)=3D4，故应选(C).

【评注】若，则= ，且= 相似矩阵有相同的行= 1015;式、相同的秩和相ࡧ= 6;的特征值等性质. 见《数学&#= 22797;习指南》P.360相似= 矩阵及其性质.

（5）对于任意二事件A和= B

(A) 若，则= A,B一定= 独立.     (B) 若，则= A,B有可= 能独立.

(C) 若，则= A,B一定= 独立.     (D) 若，则= A,B一定= 不独立.

        =           [ B ]

【分析】本题考查独立与= 互斥事件之间的关系= 5292;事实上，独立与互ਪ= 1;事件之间没有必然的&= #20114;推关系.

【详解】推不= 出P(AB)=3DP(A)P(B), 因此= 推不出A,B一定= 独立，排除(A); 若<= span lang=3DEN-US>_{，则=
P(AB)=3D0，但=
P(A)P(B)是否=
为零不确定，因此(C),(D) 也不=
成立，故正确选项为(B).}

【评注】当P(A)_,P(B)时，= 若A,B相互= 独立，则一定有，从= 而有_.可见= ，当A,B相互= 独立时，往往A,B并不= 是互斥的.

完全类似例题见= 《数学复习指南》P.415第二= 大题第（7）小题.

（6）设随机变量X和= Y都服从正态分布= ，且它们不相关，则

(A) X与Y一定独立.         (B) (X,Y)服从= 二维正态分布.

(C) X与Y未必独立.         (D) X+Y服从= 一维正态分布.        =      [ C ]

【分析】本题考查正态分= 布的性质以及二维正= 4577;分布与一维正态分०= 7;之间的关系.只有(X,Y) 服从= 二维正态分布时，不= 0456;关与独立才是等价௚= 0;.

【详解】只有当(X,Y) = 服从二维正态分= 布时，X与= Y不相关_{<=
!--[if gte vml 1]>

X}与<= span lang=3DEN-US>Y独立，本&#= 39064;仅仅已知X和= Y服从正态分布，= 因此，由它们不相关= 5512;不出X与= Y一定独立，排除= (A); 若<= span lang=3DEN-US>X和= Y都服从正态分布= 且相互独立，则(X,Y)服从= 二维正态分布，但题= 5774;并不知道X,Y是否= 独立，可排除(B); 同样= 要求X与= Y相互独立时，才= 能推出X+Y服从= 一维正态分布，可排= 8500;(D).故正= 确选项为(C).

【评注】 ① 若X与= Y均服从正态分布= 且相互独立，则(X,Y)服从= 二维正态分布.

② 若X与= Y均服从正态分布= 且相互独立，则服从= 一维正态分布.

③ 若(X,Y)服从= 二维正态分布，则X与= Y相互独立_X与<= span lang=3DEN-US>Y不相关.

完全类似结论见《ă= 68;学复习指南》P.458的<= span lang=3DEN-US>[注= ].

三 、（本题满分8分）=

设



试补充定&#= 20041;f(0),使得= f(x)在<= span lang=3DEN-US>_上连=
续.

【详解】

        = _{=3D -₊}

        =           =3D -

        =           =3D -_&=
nbsp;

        =          =3D -

        =          =3D -

由于f(x)在<= span lang=3DEN-US>_{上连=
续，因此定义}

        ，<= /p>
使= f(x)在上连= 续.

【评注】本题实质上是一= 求极限问题，但以这= 1181;形式表现出来，还್= 1;查了连续的概念.完全类似&#= 20363;题在一般教科书上ų= 17;可找到，或参见《文= 登数学全真模拟试卷= 2299;数学三P.24第三= 题.

四 、（本题满分8分）=

设f(u,v)具有二阶连续偏= 导数，且满足，又= _,求<= span lang=3DEN-US>

【分析】本题是典型的复= 合函数求偏导问题：，<= span lang=3DEN-US>，直= 接利用复合函数求偏= 3548;公式即可，注意利ஷ= 2;

【详解】，<= /p>

故=         =     ，<= /p>
_{<=
!--[if gte vml 1]>}

所以

        =            =3D

【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= 5797;卷》数学四P.67第六= 题和《数学复习指南= 2299;P.171【Ë= 63;7.20,7.22】<= span lang=3DEN-US>.

五 、（本题满分8分）=

计算二重积分<= /p>


其中积分&#= 21306;域D=3D

【分析】从被积函数与积= 分区域可以看出，应= 5813;利用极坐标进行计౛= 9;.

【详解】作极坐标变换：= ，有=



        =    =3D

令，则=

         _.

记   ，则=



        =3D

        =3D

        =3D

     = ; =3D

因此    ，<= /p>


【评注】本题属常规题型= ，明显地应该选用极= 2352;标进行计算，在将ߚ= 8;重积分化为定积分后&= #65292;再通过换元与分步= 215;分（均为最基础的要= ;求），即可得出结果&#= 65292;综合考查了二重积Ñ= 98;、换元积分与分步积= 分等多个基础知识点.

六、（本题满分= 9分）=

设a>1，在<= span lang=3DEN-US>_{内的=
驻点为问<=
span
lang=3DEN-US>a为何值时&#=
65292;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，&#= 20877;把此函数对a求导，然&#= 21518;令此导数为零，得Ò= 40;可能极值点，进一步= 判定此极值为最小值= 1363;可.

【详解】由，得= 唯一驻点

        =

考察函数在<= span lang=3DEN-US>a>1时的= 最小值. 令

      ，<= /p>
得唯一驻&#= 28857;

        =

    当时，= ；当= 时，= ，因= 此为极= 小值，从而是最小值.

【评注】本题属基&#= 26412;题型，只是函数表ů= 98;式由驻点给出，求极= 值与最值的要求均是= 6368;基本的.

类似例题见《数= 学复习指南》P.144【Ë= 63;6.11-12】<= span lang=3DEN-US>.

七、（本题满分= 9分）=

设y=3Df(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C为= M在x轴上的投影，<= span lang=3DEN-US>O为坐标原&#= 28857;. 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= 1040;，曲边三角形CBM的面= 积可用定积分计算，= 0877;由题设，可得一含੍= 7;变限积分的等式，两&= #36793;求导后可转化为一&= 454;线性微分方程，然后= ;用通解公式计算即可.

【详解】根据题意，有<= /p>
   _.

两边关于x求导，得



当= 时，= 得



此为标准&#= 30340;一阶线性非齐次微Ñ= 98;方程，其通解为        =     y

<= ![endif]>        = _=
  A

=3D_{=

M}

=3D_{=

O    C
B     x<=
/span>}

=3D

当= x=3D0时，f(0)=3D1.

由于x=3D1时，= f(1)=3D0 ,故有= 2+C=3D0，从= 而C=3D-2. 所以=



【评注】本题一阶线性微= 分方程的求解比较简= 1333;，一般教材中都可৙= 4;到标准的求解方法，&= #23436;全类似例题见《数= 398;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&#= 20026;，<= span lang=3DEN-US>_{欲在=
T 时将数量&#=
20026;A的该商品&#=
38144;售完，试求}

(1)      = t时的商品&#= 21097;余量，并确定k的值；

(2)      = 在时间段[0，T]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= 4179;均值可用积分表示= .

【详解】 (1) 在时= 刻t商品的剩&#= 20313;量为

        =

        =       =3D，<= span lang=3DEN-US>

由_=3D0，得=

        =      ，<= /p>
因此

        =

(2) <= /span>依题意，在<= span lang=3DEN-US>[0，= T]上的平均值为<= /p>


        = =3D

        = =3D

因此在时间段[0，= T] 上的平均剩余量= 为

【评注】函数f(x)在[a,b] 上的平均值记为=

本题考查了函数平ß= 43;值的概念，但大纲中= 只对数学一、二明确= 5552;出要求，而数学三ӌ= 9;四的考试大纲中没有&= #30456;应的要求，因此本'= 064;有超纲的嫌疑.

=

九、（本题满分= 13分）=

设有向量组（I）：，<= span lang=3DEN-US>_{，<=
span
lang=3DEN-US>_{和向=
量组（II）：，<=
span
lang=3DEN-US>_{，<=
span
lang=3DEN-US>_{试问=
：当a为何值时&#=
65292;向量组（I）与（II）等价？当a为何值时&#=
65292;向量组（I）与（II）不等价？}}}}

【分析】两个向量&#= 32452;等价也即两个向量ń= 52;可以相互线性表示，= 而两个向量组不等价= 5292;只需其中一组有一ߑ= 0;向量不能由另一组线&= #24615;表示即可. 而线性表&#= 31034;问题又可转化为对ò= 12;非齐次线性方程组是= 否有解的问题，这可= 6890;过化增广矩阵为阶੷= 9;形来判断. 一个向量是否= 可由线性= 表示，只需用初等行= 1464;换化增广矩阵（）为= 阶梯形讨论，而一组= 1521;量是否= 可由线性= 表示，则可结合起来= 3545;矩阵（）同= 时作初等行变换化阶= 6799;形，然后类似地进ඡ= 2;讨论即可.

【Ŧ= 14;解】作初等行&#= 21464;换，有

   _=3D

        =             &nb= sp; _.

(1) 当时，= 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&#= 32452;（I）与（II）等价.

(2) 当a=3D-1时，= 有

     _{_.}

由于= 秩（）<= span lang=3DEN-US>_{秩（=
，线=
性方程组无解=
，故向量不能=
由线性=
表示. 因此，向&#=
37327;组（I）与（II）不等价.}

【评注1】涉及= 到参数讨论时，一般= 2852;想到利用行列式判ਪ= 9;，因此，本题也可这&= #26679;分析：

因为行列&#= 24335;，<= span lang=3DEN-US>_，可=
见

(1) <= /span>当时，= 秩，因= 此三维列向量组与<= span lang=3DEN-US>等价= ，即向量组（I）与（II）等价.

(2) <= /span>当a=3D-1时，，秩，而= 行列式，可= 见_{_r}（<= span lang=3DEN-US>_=3D3,因此= 线性方程组无解= ，故向量不能= 由线性= 表示. 即向量组&#= 65288;I）与（II）不等价.

【评注2】 = 向量组（I）与（II）等价，相当于= 与<= span lang=3DEN-US>_{均为=
整个向量组的一=
个极大线性无关组，=
8382;题转化为求向量组的极=
大线性无关组，这可=
6890;过初等行变换化阶੷=
9;形进行讨论.}

本题完全类似分= 析思路的例题见《数= 3398;复习指南》P.347【Ë= 63;4.13】和= 《数学最后冲刺》P.94【Ë= 63;14】<= span lang=3DEN-US>.

=

十、（本题满分= 13分）=

设矩阵可逆= ，向量是矩= 阵的一= 个特征向量，是<= span lang=3DEN-US>_{对应=
的特征值，其中是矩=
阵A的伴随矩&#=
38453;. 试求a,b和<=
span
lang=3DEN-US>_的值=
.}

【分析】题设已知&#= 29305;征向量，应想到利ī= 92;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，<= span lang=3DEN-US>，且=

        = _.

两边同时左乘矩= 阵A，得

         ，<= /p>
         ，<= /p>
即

₌，<= /p>
由此，得&#= 26041;程组

_{=
        =
            &nb=
sp;}

由式(1),(2)解得=

         ₌或_;

由式(1),(3)= 解得

        =        a=3D2.

由于

         _，<=
/p>
根据(1)式知= ，特征向量所对= 应的特征值

所以，当时，= ；<= /p>
      当= 时，=

【评注】本题若先求出<= span lang=3DEN-US>_{，再=
按特征值、特征向量=
0340;定义进行分析，则෶=
5;算过程将非常复杂. 一般来说&#=
65292;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.365【Ë= 63;5.7】和= 《数学最后冲刺》P.90【Ë= 63;3】<= span lang=3DEN-US>.

十一、（本题满= 分13分）=

设随机变量X的概率密&#= 24230;为



F(x)是X的分布函&#= 25968;. 求随机变&#= 37327;Y=3DF(X)的分= 布函数.

【分析】先求出分&#= 24067;函数F(x) 的具= 体形式，从而可确定Y=3DF(X) ,然后= 按定义求Y 的分布函&#= 25968;即可。注意应先确ê= 50;Y=3DF(X)的值= 域范围，再= 对y分段讨论.

【详解】易见，当x<1时，F(x)=3D0; 当x>8 时，F(x)=3D1.

对于，有=



设G(y)是随机变量Y=3DF(X)的分= 布函数. 显然，当时，= G(y)=3D0；当= 时，= G(y)=3D1.

对于，有=



        =       =3D

        =       =3D

于是，Y=3DF(= X)的分布函数为<= /p>
        =

【评注】事实上，本题<= span lang=3DEN-US>X为任意连&#= 32493;型随机变量均可，Ē= 92;时Y=3DF(X)仍服= 从均匀分布:

当y<0时，G(y)=3D0;

当时，= G(y)=3D1;

当 0时，=

        =              =3D

        =              =3D

【评注】本题是《数学复= 习指南》P.431【Ë= 63;2.23】原= 题（实际上还是此题= 0340;特殊情形）.

十二、（本题满= 分13分）=

对于任意二事件A 和= B，，<= /p>
        =

称做事件A和= B的相关系数.

(1)    证明事件A和B独立的充分必要= 条件是其相关系数等= 0110;零；

(2)    利用随机变量相= 关系数的基本性质，= 5777;明

【分析】 (1) 利用= 事件A和= B独立的定义P(AB)=3DP(A)P(B)即可= ；(2) 随机= 变量X和= Y的相关系数为<= span lang=3DEN-US>_{，而=
需将转化=
为用随机变量表示，=
6174;然，若有以及=
，<=
span
lang=3DEN-US>_{即可=
，这只需定义}}

₌

【详解】 (1) 由的定= 义，可见当且= 仅当

        =      P(AB)-P(A)P(B)=3D0,

而这恰好&#= 26159;二事件A和= B独立的定义，即= 是<= span lang=3DEN-US>A和= B独立的充分必要= 条件.

(2) <= /span>考虑随机变量<= span lang=3DEN-US>X和= Y:



由条件知&#= 65292;X和= Y都服从0—1分布：

_{<=
!--[if gte vml 1]>}，<= span lang=3DEN-US>

易见

₌， = ；<= /p>
_{=
,    _;}

₌

因此，事&#= 20214;A和= B的相关系数就是= 随机变量X和= Y的相关系数.

    于是由二随机变= 量相关系数的基本性= 6136;，可见

【评注】如上0—1分布与随机事件= 之间的关系值得注意= 5292;较好地将两者联系ล= 5;来了，为借助相互的&= #24615;质提供了便利.

完全类似例题见= 《数学题型集粹与练= 0064;题集》P.298第十= 一题和《考研数学大= 0018;讲》P.249【Ë= 63;25】<= span lang=3DEN-US>.

注&#= 65306; 1.《数学复习指南= 2299; （2003版，经济类）世&= #30028;图书出版公司

主编: = 陈文灯、黄先开<= /span>

2.《数学题= 2411;集粹与练习题集》ᦀ= 8;2003版，经济类）世界Þ= 70;书出版公司

主编: = 陈文灯、黄先开<= /span>

3.《文登数= 3398;全真模拟试卷》（2003&#= 29256;，经济类）世界图È= 70;出版公司

主编: = 陈文灯、黄先开<= /span>

4.《数学最= 1518;冲刺》（2003版，经济&#= 31867;）世界图书出版公Ö= 96;

主编: = 陈文灯、黄先开<= /span>

5.《考研数= 3398;大串讲》（2002版，经&#= 27982;类）世界图书出版Ð= 44;司

主编: = 黄先开、曹显兵、施= 6126;存

(文登学校= 379;稿)

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