1996年全国硕士研究生入学统一考试数学(一)试卷
一、填空题(本题共5小题,每小题3分,满分15分.把答案填在题中横线上)
(5)设A 是43?矩阵,且A 的秩()2,r =A 而102020,103??
??=??
??-??
B 则()r AB =_____________. 二、选择题(本题共5小题,每小题3分,满分15分.每小题给出的四个选项中,只有一个符合题目要求,把所选项前的字母填在题后的括号内)
(5)四阶行列式112233440000
0000a b a b a b b a 的值等于
(A )12341234a a a a b b b b -
(B )12341234a a a a b b b b + (C )12123434()()a a b b a a b b --
(D )23231414()()a a b b a a b b --
八、（本题满分6分）
设,T A =-I ξξ其中I 是n 阶单位矩阵,ξ是n 维非零列向量,T ξ是ξ的转置.证明 (1)2=A A 的充分条件是 1.T =ξξ (2)当1T =ξξ时,A 是不可逆矩阵. 九、（本题满分8分）
已知二次型222
123123121323(,,)55266f x x x x x cx x x x x x x =++-+-的秩为2, (1)求参数c 及此二次型对应矩阵的特征值. (2)指出方程123(,,)1f x x x =表示何种二次曲面.
1997年全国硕士研究生入学统一考试数学(一)试卷
一、填空题(本题共5小题,每小题3分,满分15分.把答案填在题中横线上)
(4)设12243,311t -??
??=??
??-??
A B 为三阶非零矩阵,且,=AB O 则t =_____________. 二、选择题(本题共5小题,每小题3分,满分15分.每小题给出的四个选项中,只有一个符合题目要求,把所选项前的字母填在题后的括号内)
(4)设111122232333,,,a b c a b c a b c ??????
??????===??????????????????
ααα则三条直线 1112223330,0,0a x b y c a x b y c a x b y c ++=++=++= (其中220,1,2,3i i a b i +≠=)交于一点的充要条件是 (A )123,,ααα线性相关
(B )123,,ααα线性无关
(C )秩123(,,)r =ααα秩12(,)r αα (D )123,,ααα线性相关12,,αα线性无关 七、(本题共2小题,第(1)小题5分,第(2)小题6分,满分11分)
(1)设B 是秩为2的54?矩阵123,[1,1,2,3],[1,1,4,1],[5,1,8,9]T T T ==--=--ααα是齐次线性方程组x =B 0的解向量,求x =B 0的解空间的一个标准正交基.
(2)已知111??
??=????-??ξ是矩阵2125312a b -????=????--??
A 的一个特征向量. 1)试确定,a b 参数及特征向量ξ所对应的特征值. 2)问A 能否相似于对角阵?说明理由.
八、（本题满分5分）
设A 是n 阶可逆方阵,将A 的第i 行和第j 行对换后得到的矩阵记为.B (1)证明B 可逆. (2)求1.-AB
1998年全国硕士研究生入学统一考试数学(一)试卷
一、填空题(本题共5小题,每小题3分,满分15分.把答案填在题中横线上)
(4)设A 为n 阶矩阵*,0,≠A A 为A 的伴随矩阵,E 为n 阶单位矩阵.若A 有特征值,λ则*2()+A E 必有特征值________.
二、选择题(本题共5小题,每小题3分,满分15分.每小题给出的四个选项中,只有一个符合题目要求,把所选项前的字母填在题后的括号内) (4)设矩阵 111222333a b c a b c a b c ??
????????
是满秩的,则直线333121212x a y b z c a a b b c c ---==---与直线111232323
x a y b z c a a b b c c ---==---
(A )相交于一点 (B )重合(C )平行但不重合（D )异面 十、（本题满分6分）
已知二次曲面方程2222224x ay z bxy xz yz +++++=可以经过正交变换x y z ξηζ????
????=????????????
P 化为椭圆柱面方程2244,ηξ+=求,a b 的值和正交矩阵.P 十一、（本题满分4分） 设A 是n 阶矩阵,若存在正整数,k 使线性方程组k x =A 0有解向量,α且1.k -≠A α0 证明:向量组1,,,k -αA αA αL 是线性无关的.
十二、（本题满分5分）
已知方程组(Ⅰ)
1111221,222112222,221122,220
00
n n n n n n n n n a x a x a x a x a x a x a x a x a x +++=+++=+++=L L M
L
的一个基础解析为11121,221222,212,2(,,,),(,,,),,(,,,).T T T n n n n n n b b b b b b b b b L L L L 试写出线性方程组(Ⅱ)
1111221,222112222,221122,220
00
n n n n n n n n n b y b y b y b y b y b y b y b y b y +++=+++=+++=L L M
L
的通解,并说明理由.
1999年全国硕士研究生入学统一考试数学(一)试卷
一、填空题(本题共5小题,每小题3分,满分15分.把答案填在题中横线上) (4)设n 阶矩阵A 的元素全为1,则A 的n 个特征值是 _____________.
二、选择题(本题共5小题,每小题3分,满分15分.每小题给出的四个选项中,只有一个符合题目要求,把所选项前的字母填在题后的括号内) (4)设A 是m n ?矩阵,B 是n m ?矩阵,则
(A )当m n >时,必有行列式||0≠AB (B )当m n >时,必有行列式||0=AB
(C )当n m >时,必有行列式||0≠AB
(D )当n m >时,必有行列式||0=AB
十、(本题满分8分)
设矩阵153,10a c b c a -??
??=????--??
A 其行列式||1,=-A 又A 的伴随矩阵*
A 有一个特征值0λ,属于0λ的一个特征向量为(1,1,1),T
=--α求,,a b c 和0λ的值. 十一、(本题满分6分)
设A 为m 阶实对称矩阵且正定,B 为m n ?实矩阵,T B 为B 的转置矩阵,试证T B AB 为正定矩阵的充分必要条件是B 的秩().r n =B
2000年全国硕士研究生入学统一考试数学(一)试卷
一、填空题(本题共5小题,每小题3分,满分15分.把答案填在题中横线上)
(4)已知方程组12312
112323120x a x a x ????????????+=????????????-??????无解,则a
= _____. 二、选择题(本题共5小题,每小题3分,满分15分.
(4)设n 维列向量组1,,()m m n <ααL 线性无关,则n 维列向量组1,,m ββL 线性无关的充分必要条件为
(A )向量组1,,m ααL 可由向量组1,,m ββL 线性表示 (B )向量组1,,m ββL 可由向量组1,,m ααL 线性表示
(C )向量组1,,m ααL 与向量组1,,m ββL 等价 (D )矩阵1(,,)m =A ααL 与矩阵1(,,)m =B ββL 等价 十、(本题满分6分)
设矩阵A 的伴随矩阵*10000100,10
1003
8?????
?=???
?-??
A 且11
3--=+ABA BA E ,其中E 为4阶单位矩阵,求矩阵B . 十一、(本题满分8分)
某适应性生产线每年1月份进行熟练工与非熟练工的人数统计,然后将16
熟练工支援其他生产部门,其缺额由招收新的非熟练工补齐.新、老非熟练
工经过培训及实践至年终考核有25成为熟练工.设第n 年1月份统计的熟练工与非熟练工所占百分比分别为n x 和,n y 记成向量.n n x y ?? ???
(1)求11n n x y ++?? ???与n n x y ??
???
的关系式并写成矩阵形式:11.n n n n x x y y ++????=
? ?????A (2)验证1241,11-????== ? ?????ηη是A 的两个线性无关的特征向量,并求出相应的特征值. (3)当111212x y ??
?
??= ? ? ?
?? ???
时,求11.
n n x y ++?? ???
一、填空题(本题共5小题,每小题3分,满分15分.把答案填在题中横线上)
(4)设24+-=A A E O ,则1(2)--A E = _____________.
二、选择题(本题共5小题,每小题3分,满分15分.每小题给出的四个选项中,只有一个符合题目要求,把所选项前的字母填在题后的括号内)
(4)设11114
0001
1110
00,111100001
1110
0????
?
?
?
?== ? ? ?
?????
A B ,则A 与B (A )合同且相似 (B )合同但不相似 (C )不合同但相似
(D )不合同且不相似
九、(本题满分6分)
设12,,,s αααL 为线性方程组=AX O 的一个基础解系,1112221223121,,,s s t t t t t t =+=+=+βααβααβααL ,其中21,t t 为实常数,试问21,t t 满足什么条件时12,,,s βββL 也为=AX O 的一个基础解系?
十、(本题满分8分)
已知三阶矩阵A 和三维向量x ,使得2,,A A x x x 线性无关,且满足3232=-A A A x x x . (1)记2(,,),=P A A x x x 求B 使1-=A PBP .
(2)计算行列式+A E .
一、填空题(本题共5小题,每小题3分,满分15分.把答案填在题中横线上)
(4)已知实二次型3231212
32221321444)(),,(x x x x x x x x x a x x x f +++++=经正交变换可化为标准型216y f =,则a =_____________.
二、选择题(本题共5小题,每小题3分,满分15分.每小题给出的四个选项中,只有一个符合题目要求,把所选项前的字母填在题后的括号内)
(4)设有三张不同平面,其方程为i i i i d z c y b x a =++(3,2,1=i )它们所组成的线性方程组的系数矩阵与增广矩阵的秩都为2,则这三张平面可能的位置关系为
九、(本题满分6分)
已知四阶方阵1234(,,,)=A αααα, 1234,,,αααα均为四维列向量,其中234,,ααα线性无关,1232=-ααα.若1234=+++βαααα,求线性方程组x =A β的通解.
十、(本题满分8分)
设,A B 为同阶方阵,
(1)若,A B 相似,证明,A B 的特征多项式相等.
(2)举一个二阶方阵的例子说明(1)的逆命题不成立. (3)当,A B 为实对称矩阵时,证明(1)的逆命题成立.
2003年全国硕士研究生入学统一考试数学(一)试卷
一、填空题(本题共6小题,每小题4分,满分24分.把答案填在题中横线上)
(4)从2R 的基1211,01????== ? ?-????αα到基1211,12????
== ? ?????
ββ的过渡矩阵为 .
二、选择题(本题共6小题,每小题4分,满分24分.每小题给出的四个选项中,只有一个符合题目要求,把所选项前的字母填在题后的括号内)
(4)设向量组I :12,,,r αααL 可由向量组II :12,,,s βββL 线性表示,则
(A )当s r <时,向量组II 必线性相关 (B )当s r >时,向量组II 必线性相关 (C )当s r <时,向量组I 必线性相关 (D )当s r >时,向量组I 必线性相关 (5)设有齐次线性方程组0x =A 和0x =B ,其中,A B 均为n m ?矩阵,现有4个命题:
① 若0x =A 的解均是0x =B 的解,则秩()≥A 秩()B ② 若秩()≥A 秩()B ,则0x =A 的解均是0x =B 的解 ③ 若0x =A 与0x =B 同解,则秩()=A 秩()B ④ 若秩()=A 秩()B , 则0x =A 与0x =B 同解 以上命题中正确的是
(A )①② (B )①③ (C )②④ (D )③④ 九 、(本题满分10分)
设矩阵322232223????=??????A ,010101001??
??=??????P ,1*-=B P A P ,求2+B E 的特征值与特征向量,其中*
A 为A 的伴随矩阵,E 为3阶单位矩阵.
十 、(本题满分8分)
已知平面上三条不同直线的方程分别为:1l 032=++c by ax ,:2l 032=++a cy bx ,:3l 032=++b ay cx .试证这三条直线交于一点的充分必要条件为.0=++c b a
2004年全国硕士研究生入学统一考试数学(一)试卷
一、填空题(本题共6小题,每小题4分,满分24分.把答案填在题中横线上)
(5)设矩阵210120001??
??=??
????
A ,矩阵
B 满足**2=+ABA BA E ,其中*A 为A 的伴随矩阵,E 是单位矩阵,则B =__________ . 二、选择题(本题共8小题,每小题4分,满分32分.每小题给出的四个选项中,只有一个符合题目要求,把所选项前的字母填在题后的括号内)
(11)设A 是3阶方阵,将A 的第1列与第2列交换得B ,再把B 的第2列加到第3列得C ,则满足=AQ C 的可逆矩阵Q 为
(A )??????????101001010 (B )??????????100101010 (C )??????????110001010 (D )??
???
?????100001110 (12)设,A B 为满足=AB O 的任意两个非零矩阵,则必有
(A )A 的列向量组线性相关,B 的行向量组线性相关 (B )A 的列向量组线性相关,B 的列向量组线性相关 (C )A 的行向量组线性相关,B 的行向量组线性相关 (D )A 的行向量组线性相关,B 的列向量组线性相关 (20)(本题满分9分)
设有齐次线性方程组121212(1)0,2(2)20,(2),()0,
n n
n a x x x x a x x n nx nx n a x ++++=??++++=?≥??
?++++=?L L L L L L L L L
试问a 取何值时,该方程组有非零解,并求出其通解.
(21)(本题满分9分)
设矩阵12314315a -??
??=--??
????
A 的特征方程有一个二重根,求a 的值,并讨论A 是否可相似对角化.
2005年全国硕士研究生入学统一考试数学(一)试卷
一、填空题(本题共6小题,每小题4分,满分24分.把答案填在题中横线上)
(5)设123,,ααα均为3维列向量,记矩阵
123(,,)=A ααα,123123123(,24,39)=++++++B ααααααααα, 如果1=A ,那么=B .
二、选择题(本题共8小题,每小题4分,满分32分.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)
(11)设21,λλ是矩阵A 的两个不同的特征值,对应的特征向量分别为12,αα,则1α,12()+A αα线性无关的充分必要条件是 (A )01≠λ (B )02≠λ (C )01=λ (D )02=λ
(12)设A 为(2)n n ≥阶可逆矩阵,交换A 的第1行与第2行得矩阵**.,B A B 分别为,A B 的伴随矩阵,则 (A )交换*A 的第1列与第2列得*B (B )交换*A 的第1行与第2行得*B
(C )交换*A 的第1列与第2列得*-B (D )交换*A 的第1行与第2行得*-B
(20)(本题满分9分)
已知二次型212322
21321)1(22)1()1(),,(x x a x x a x a x x x f +++-+-=的秩为2. (1)求a 的值；
(2)求正交变换x y =Q ,把),,(321x x x f 化成标准形. (3)求方程),,(321x x x f =0的解.
(21)(本题满分9分)
已知3阶矩阵A 的第一行是c b a c b a ,,),,,(不全为零,矩阵12324636k ??
??=??
????
B (k 为常数),且=AB O ,求线性方程组0x =A 的通解.
2006年全国硕士研究生入学统一考试数学(一)试卷
一、填空题(本题共6小题,每小题4分,满分24分.把答案填在题中横线上)
(5)设矩阵2112??
= ?-??
A ,E 为2阶单位矩阵,矩阵
B 满足2=+BA B E ,则B = .
(6)设随机变量X 与Y 相互独立,且均服从区间[0,3]上的均匀分布,则{}max{,}1P X Y ≤= .
二、选择题(本题共8小题,每小题4分,满分32分. 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)
(11)设12,,,,s αααL 均为n 维列向量,A 是m n ?矩阵,下列选项正确的是 (A )若12,,,,s αααL 线性相关,则12,,,,s A αA αA αL 线性相关
(B )若12,,,,s αααL 线性相关,则12,,,,s A αA αA αL 线性无关
(C )若12,,,,s αααL 线性无关,则12,,,,s A αA αA αL 线性相关
(D )若12,,,,s αααL 线性无关,则12,,,,s A αA αA αL 线性无关. (12)设A 为3阶矩阵,将A 的第2行加到第1行得B ,再将B 的第1列的-1倍加到第2列得C ,记1100
10001??
?
= ? ??
?
P ,则 (A )1-=C P AP (B )1-=C PAP (C )T =C P AP (D )T =C PAP
(20)(本题满分9分)
已知非齐次线性方程组1234123412
341435131
x x x x x x x x ax x x bx +++=-??
++-=-??++-=?有3个线性无关的解,
(1)证明方程组系数矩阵A 的秩()2r =A . (2)求,a b 的值及方程组的通解.
(21)(本题满分9分)
设3阶实对称矩阵A 的各行元素之和均为3,向量()()121,2,1,0,1,1T
T
=--=-αα是线性方程组0x =A 的两个解. (1)求A 的特征值与特征向量.
(2)求正交矩阵Q 和对角矩阵A ,使得T =Q AQ A .
2019考研英语二真题及答案Word版
Section I Use of English Directions: Read the following text. Choose the best word(s) for each numbered blank and mark A, B, C or D on the ANSWER SHEET. (10 points) Weighing yourself regularly is a wonderful way to stay aware of any significant weight fluctuations. 1 , when done too often, this habit can sometimes hurt more than it 2 . As for me, weighing myself every day caused me to shift my focus from being generally healthy and physically active to focusing 3 on the scale. That was bad to my overall fitness goals. I had gained weight in the form of muscle mass, but thinking only of 4 the number on the scale, I altered my training program. That conflicted with how I needed to train to 5 my goals. I also found that weighing myself daily did not provide an accurate 6 of the hard work and progress I was making in the gym. It takes about three weeks to a month to notice any significant changes in your weight 7 altering your training program. The most 8 changes will be observed in skill level, strength and inches lost. For these 9 , I stopped weighing myself every day and switched to a bimonthly weighing schedule 10 . Since weight loss is not my goal, it is less important for me to 11 my weight each week. Weighing every other week allows me to observe and 12 any significant weight changes. That tells me whether I need to 13 my training program. I use my bimonthly weigh-in 14 to get information about my nutrition as well. If my training intensity remains the same, but I’m constantly 15 and dropping weight, this is a 16 that I need to increase my daily caloric intake. The 17 to stop weighing myself every day has done wonders for my overall health, fitness and well-being. I’m experiencing increased zeal for working out since I no longer carry the burden of a 18 morning weigh-in. I’ve also experienced greater success in achieving my specific fitness goals, 19 I’m trai ning according to those goals, not the numbers on a scale. Rather than 20 over the scale, turn your focus to how you look,
自考英语二历年真题及答案(2005-2014)史上最全
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2002 Directions：Translate the following passage into Chinese and put your translation on the ANSWER SHEET． Since 1981，farmers in Holland have been encouraged to adopt“green”farming techniques that were thought to benefit plant and bird life．Farmers who have voluntarily adopted these measures are compensated by the European Union．The goal of the program is to work against the negative effects of modem fanning，such as declines in species diversity and the disturbance of local nesting grounds．The“green”methods of farming cost the European Union about 1.7 billion Euros annually．This is about 4 percent of the budget for“Common Agricultural Policy,”and the compensation is expected to rise to 10 percent within the next few years． Various forms of“green farming”employed around the world have proved successful, and all new methods thought to be environmentally sensitive should be subject to sound scientific evaluation to determine whether they are actually meeting the intended goals． Part V Writing(30 minutes，15 points) Directions：You are to write in no less than 120 words about the title “What I Consider Important in Life”．Your composition should be based on the Chinese outline given below．
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2017全国考研英语二真题和答案解析.doc
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(完整word版)2018年10月自考英语二真题及答案
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第二部分：阅渎选择(第11～15题，每题2分，共l0分) 阅渎下面短文，请从短文后所给各题的4个选项(A、B、C、D)中选出1个最佳 选项，并在答题卡相应位置上将该项涂黑。
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第四部分：填句补文(第26～30题，每题2分，共10分) 下面的短文有5处空自，短文后有6个句子，其中5个取自短文，请根据短文内容 将其分别放回原有位置，以恢复文章原貌，请将正确选项的字母写在答题卡上。
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