安徽省2012年普通高等学校专升本招生考试
高等数学
注意事项：
1．试卷共8页，请用签字笔答题，答案按要求写在指定的位置。 2．答题前将密封线内的项目填写完整。
一、选择题（下列每小题的选项中，只有一项是符合题意的，请将表示该选项的字母填在题后的括号内。共10小题，每小题3分，共30分）
1.若函数???
??>+≤=0,sin 0,3)(x a x
x x e x f x 在0=x 在处连续，则=a （ C ）
A. 0
B. 1
C. 2
D. 3
解：由)0()00()00(f f f =-=+得231=?=+a a ,故选C. 2.当0→x 时，与函数2
)(x x f =是等价无穷小的是（ A ） A. )1ln(2
x + B. x sin C.
x tan D. x cos 1-
解：由11ln(lim 1ln()(lim )
22
0)20=+=+→→x x x x f x x ,故选A.
3.设)(x f y =可导，则'-)]([x e f =（ D ）
A. )(x
e
f -' B. )(x e f -'- C. )(x x e f e --' D. )(x x e f e --'-
解：)()()()]([x
x x x x
e f e e e f e f -----'-='?'='
,故选D. 4.设
x 1是)(x f
的一个原函数，则?=dx x f x )(3
（ B ） A.
C x +2
2
1 B. C x +-221 C. C x +331 D. C x x +ln 414
解：因x 1是)(x f
的一个原函数,所以211)(x x x f -='
???
??=,所以
C x xdx dx x f x +-=-=??23
2
1)(故选B. 5.下列级数中收敛的是（ C ）
A. ∑∞
=-1
374n n
n n B. ∑
∞
=-1
2
31
n n C.
∑∞
=13
2
n n
n D. ∑∞
=1
21
sin n n
解：因121
)1(lim 212
2)1(lim 33313
<=+=+∞→+∞→n n n n n n n n ,所以∑∞=132n n n 收敛,
故选C. 6.交换?
???+=
10
2121
1
21),(),(y y
y
dx y x f dy dx y x f dy I 的积分次序，则下列各项正确的是（ B ）
A. ??
1
22
),(x x dy y x f dx B.
??
10
22
),(x x
dy y x f dy C.
?
?2
1
22
),(x x dy y x f dx D.
?
?
2
1
22
),(x x dy y x f dx
解：由题意画出积分区域如图:故选B.
7.设向量21,αα是非齐次线性方程组AX =b 的两个解，则下列向量中仍为该方程组解的是（ D ） A. 21αα+ B. 21αα- C. 212αα+ D. 212αα- 解：因,2)(2121b b b A A A =+=+=+αααα同理得
,0)(21=-ααA ,3)2(21b A =+αα,)2(21b A =-αα
故选D.
8.已知向量)2,5,4,0(),0,,0,2(),1,1,2,1(321--==-=αααk 线性相关,则=k （ D ） A. -2 B. 2 C. -3 D. 3
解:????? ??+--+--→????? ??---+--→????? ?
?---=????? ??03002240112125402240112
12540002112132
1k k k k ααα 由于123,,ααα线性相关,所以123(,,)2r ααα≤,因此3=k
9.设B A ,为事件，且,2.0)(,4.0)(,6.0)(===AB P B P A P 则=)(B A P （ A ） A.0.2 B. 0. 4 C. 0.6 D. 0.8
解: 2.0)]()()([1)(1)()(=-+-=+-=+=AB P B P A P B A P B A P B A P
10.有两个口袋,甲袋中有3个白球和1个黑球,乙袋中有1个白球和3个黑球.现从甲袋中任取一个球放入乙袋，再从乙袋中任取一个球,则取出白球的概率是（ B ） A.
163 B. 207 C. 41 D. 2
1 解: 由全概率公式得20
7
51415243=?+?=
p
二、填空题(本题共10小题，每小题3分，共30分，把答案填在题中横线上。)
11．设函数2161
31arcsin
x
x y --
-=，则函数的定义域为)4,2[-. 解：424442016,131
12<≤-??
??<<-≤≤-?>-≤-≤-x x x x x .
12．设曲线22-+=x x y 在点M 处的切线斜率为3,则点M 的坐标是)0,1(. 解：12+='x y ，由1312=?=+='x x y ，从而0=y ，故填)0,1(.
13．设函数x x y arctan =，则=
''y 2
2)
1(2
x +. 解：2
1arctan x x x y ++=',222
2222)1(2
)1(2111x x x x x y +=+-+++=''. 14．
=+?dx x
x 2012)1(ln
C x ++2013)1(ln 2013
. 解：
C x x d x dx x x ++=++=+??
2013
)1(ln )1(ln )1(ln )1(ln 20132012
2012. 15．=?∞++-dx xe x 01= e . 解：
e dx xe e dx xe
x x ==?
?
+∞-∞++-0
1
.
16．幂级数∑∞
=-15
)2(n n
n
n x 的收敛域为)7,3[-.
解：由152215lim 5)2(15)2(lim )()(lim 11
1<-=-+=-+-=∞→++∞→+∞→x x n n n
x n x x u x u n n n n n n n
n n .
得73<<-x 级数收敛,
当3-=x 时,级数为∑∞
=-1
)1(n n
n 收敛; 当7=x 时,级数为∑∞
=11n n 发散;
故收敛域为)7,3[-.
17．设A 是n 阶矩阵，E 是n 阶单位矩阵，且，032=--E A A 则=--1
)2(E A E A +.
解：)()
2())(2(031
2
E A E A E E A E A E A A +=-?=+-?=---
18．设????
?
??-=10010
1110A ，记1
-A 表示A 的逆矩阵, *A 表示A 的伴随矩阵,则 =-*
1)(A ????
?
??----100101110.
19．设型随机变量),8,1(~N X 且),()(c X P c X P ≥=<则c = 1.
解：由正态分布的对称性得1==μc .
20．设型随机变量X 在区间]4,2[上服从均匀分布,则方差=
)(X D 3
1
.
解：直接由均匀分布得3
1
12)24()(2=-=X D .
三、计算题：本大题共8小题，其中第21-27题每题7分，第28题11分，共60分。
21．计算极限x x
x x 20tan sin lim
-→.
解：原式= 20sin lim x
x
x x -→ =x
x
x 2cos 1lim
0-→
=2
sin lim 0x
x →=0.
22．求由方程xy y x =确定的隐函数的导数dx
dy
. 解：两边取对数得y x y x ln ln ln +=, 两边求导得y y
x y y x y '+='+
11ln , 从而)
1()
ln 1(--=
x x y x y dx dy .
23．计算定积分
?
-22
2
2
1
1dx x x
解：令t x sec =,则,tan sec tdt t dx =当2=x 时, 4
π
=
t ;当2=x 时, 3
π
=
t .
所以原式= ?3
4
2
tan sec tan sec π
πdt t t t
t = ?34
cos π
πtdt = =|34
sin π
πt =
)23(2
1
-.
24．求微分方程02=--'x e y y 的通解.
解：原方程可整理为x
e y y =-'2
这是一阶线性微分方程,其中x
e x Q x P =-=)(,2)(. 所以原方程的通解为
??
????+??=?-C dx e x Q e y dx x P dx x P )()()( )(22C dx e e e dx
x dx
+??
=?-.
)(2C dx e e x x +=?-)(2C e e x x +-=-
x x Ce e 2+-=
25．计算二重积分
??D
yd x σ2,其中D 是由直线222===xy x y x 和、所围成的区域. 解：区域D 如图阴影部分所示.
故
??
D
yd x σ2
??=x
x
y y x x 22221
d d ?=
2
12222
d 2
1|
y y x x
x
?-=
214
)d 44(2
1x x |215
)252(x x -=5
210=.
26．设矩阵????
?
??---=320031
10
1A ，,231?????
??=B 且满足X B A B AX +=+2
,求矩阵X .
解：由X B A B AX +=+2可得B E A E A B E A X E A ))(()()(2+-=-=-
因024
2
041
100||≠-=---=-E A ,所以E A -可逆, 因此B E A X )(+=????? ???????
?
?---=23122
0021
102????
?
??-=250
27．设行列式132
1
31213
2
113211)(++++=
x x x x x D ,求)(x D 在0=x 处的导数.
解：132731273
21732171321
31213
2113211)(+++++++=
++++=
x x x x x x x x x x x x D
2
11
10
1110
10001)
7(1
32
1
31213
2
113211)
7(--+=++++=x x x x x x x x
)23)(7()2)(1)(7(22+-+=--+=x x x x x x x x .
故)32)(7()23)(72()(22-+++-+='x x x x x x x D . 从而14)0(='D .
28．已知离散型随机变量X 的密度函数为?????
????≥<≤<≤<=.2,1,21,
21,10,,0,0)(x
x x a x x F 且数学期望34)(=X E .
求: (1) a 的值; (2) X 的分布列；(3)方差D (X )．
解：(1) 由分布函数的性质知,随机变量X 的可能取值为0、1、2，且
2
1)2(,21)1(,)0(==-=
===X P a X P a X P
图5-7
因3
423212)21(10)(=-=?+-?+?=a a a X E 所以6
1
=
a . (2) 由（1）即得
(3) 3
223160)(2
222=?+?+?=X E ,
四、证明题与应用题：本大题
共3小题，每小题10分，共30分。
29．设)(2y
x f xy u =，其中)(t f 可微，u y
z
y x z x
3:=??+??证明. 证明：因为
y
y x f xy y x f y x u 1
)()(22?'+=?? ),()(2
y
x f xy y x
f y '+=
???
? ??-?'+=??22)()(2y x y x f xy y x xyf y u )()(22y
x f x y x xyf '-=,
故)()(2)()(2222y
x
f y x y x f xy y x f y x y x f xy y u y x u x
'-+'+=??+?? )(32
y
x
f xy =u 3=. ????(9分)
30．设D 是由曲线e x x y ==,ln 及x 轴所围成的的平面区域
求: (1) 平面区域D 的面积S ; (2) D 绕y 轴旋转一周所成的旋转体的体积V ． 解:区域D 如图阴影部分所示。曲线x y ln =与x 轴及 e x =的交点坐标分别为)1,(),0,1(e （1）平面区域D 的面积
1)ln (d ln |1
1
=-==?e
e x x x x x S .
（2）D 绕y 轴旋转一周所成的旋转体的体积V
).
1(2
2
d d )(121
221
221
22
|+=
-
==-=-??=??e e e y e e y
e e V y y y π
π
πππππ
31．证明不等式：当e b a >>时，
)71828.2(ln ln ≈< a a b a b . 证明: 设),(,ln )(+∞∈=e x x x x f ,则),(,0ln 1)(+∞∈>+='e x x x f , 所以),(ln )(+∞∈=e x x x x f 在上单调递增,从而当当e b a >>时,有 )()(b f a f >,即b b a a ln ln >,即 b a a b （D）would make 4 If you had told me earlier, I______to meet you at the hotel. （A）had come （B）will have come （C）would come （D）would have come 5 Kim went to visit the teachers in the primary school______he worked three years ago. （A）when （B）where （C）which （D）that 6 The language school started a new _____ to help young learners with reading and writing. （A）program （B）design （C）event （D）progress 英语 注意事项： 1．试卷共8页，用钢笔或圆珠笔将答案写在答题纸上，写在试卷上一律无效。 2. 答题前将答题纸上密封线内的项目填写清楚。 Part I Vocabulary and Structure (1 point each. 30 points in all) Directions: There are 30 incomplete sentences. You arc required to complete each one by deciding on the most appropriate word or words from the four choices marked A,B,C and D,and then write the corresponding letter on the Answer Sheet. l. Jane hadn't been to London before. had her husband. A. Neither B. Either C. So D. Or 2. No sooner had he arrived in Rome he heard of the good news. A. when B. than C. then D.until 3. Mr. Cook has not commented these reports. A. in B. of C. with D.on 4．It is to stare directly at the sun. A. harmful B.harmless https://www.doczj.com/doc/dc7593114.html,fortable D.safe 5. It's no use with a drunken man. A. argue B.to argue C.arguing D.being argued 6. The program on education is watching. A. worthy B.worthily C.worthiness D.worth 7. the garden, the old man went to have a rest. A. Having been watered B. Watering C. Having watered D. Being watered 8．Her to the job left her with very little free time. A. devotion B.faith C.trust D.interest 9. The children have been since their father left. A.out of style B.under control C.out of control D.in style 10. It is reported great changes have been made. A.which B. that C. while D.what 11. The manager promised to keep me of how the project was going on. A. be informed B. informed C. inform https://www.doczj.com/doc/dc7593114.html,rming 12. He announced that the company had been transferred to city. A. another B. others C. the other D.one another 13. A great cheer went up from the crowd as they caught of the hero. A. vision B. view C. sight D.glance 14. Physics my favorite subject when I studied in the university. A. were B.was C.is D.are 15. we can succeed or not depends on how well we cooperate. A. Then B. Thus C. What D. Whether 16. I will lend you the book you return it to me in time. A. on condition that B. in case C. in order that D. so that 17. Mr.Brown is supposed to for Italy last month. 安徽省普通高等学校专升本考试2008年真题 ParｔI V oｃaｂｕｌａrｙaｎd Sｔｒucture(1 ｐoinｔeaｃh，3０ｐoinｔs ｉｎａll ) Direcｔioｎs: Ｔhere aｒe ３0 incompleｔe senｔｅnces.Yｏｕａrｅrequirｅd ｔｏcomｐletｅｅach onｅby decｉｄingｏn theｍosｔapprｏpriate worｄor wordｓfr ｏm the 4 choices maｒkｅd A，B, C ａｎｄＤ，ｔhen wrｉte thｅｃoｒｒespondiｎg let ｔer on the Ａnｓｗer Sheｅt． 1.Ｉｔwas thｅdocｔor's carelessnｅss thａｔ＿_＿____tｈe ｓｅrious acciｄｅnｔ. A．ｒｅsuｌteｄfrom B. resulted ｉn Ｃ.brouｇht up D. brouｇht in 2.Ｈｅｒeｇretted ＿_＿___＿toｏmuch time oｎｃomputer ｇａｍes. A．to cost B. costｉng C. tｏspend Ｄ. spenｄinｇ 3.Cｈinese ｃｕlture and Japanesｅculturｅhave ａlot__＿__＿_＿_ Ａ.iｎｆacｔ B. iｎsight Ｃ．in comｍｏnＤ．in touｃh 4.Ｔhe smａlｌvｉｌlage ｉs ｓａid_____iｎ a battlｅｄuring Ｗｏrｌd Wａr ＩI． A．to ｈａｖe beｅn destｒoyed Ｂ.ｔo be ｄeｓｔroyed Ｃ．tｏdestrｏy D. to have destroｙed 5.Ｔhe ｇoverｎment has taｋｅn ｅfｆective mｅasures to ＿_＿___ｉtseｌf to ｔｈe ｃｈangiｎgｗorｌｄ. Ａ．settle B. adｏpt Ｃ.cｈaｎge D.adjust 6.＿____＿a woｒd，tｈis mｏviｅｃaｎbe ｒａteｄａs a ｓｕｃcessfulｄomes ｔicｆilm． A．Of Ｂ．At C.In D. By 7.Ｍany a cｕstomer______aｂｏut ｔhｅpoｏｒquality oｆthe waｓhingｍａchines. https://www.doczj.com/doc/dc7593114.html,plaｉns Ｂ.ｃomplain C.are compｌainｉnｇD．hａve cｏｍplaｉneｄ 8.Iｔis rｅpｏrted thatｓoｍe scientｉstｓin tｈe Ｕnitｅd Statｅs ｈavｅfｏuｎｄa ｎew ____＿__foｒcanceｒ. A．wａy B. anｓｗerＣ.apｐｒoａch D. cure 9.Heｍade sｔudenｔs think for ｔhｅｍｓelｖes,_______ｔeｌlｉngｔhem wｈa ｔto think. 安徽省普通高等学校专升本考试2008年真题 Part I Vocabulary and Structure(1 point each, 30 points in all ) Directions: There are 30 incomplete sentences. You are required to complete each one by deciding on the most appropriate word or words from the 4 choices marked A, B, C and D, then write the corresponding letter on the Answer Sheet. 1.It was the doctor's carelessness that _______the serious accident. A. resulted from B. resulted in C .brought up D. brought in 2.He regretted _______too much time on computer games. A. to cost B. costing C. to spend D. spending 3.Chinese culture and Japanese culture have a lot_________ A. in fact B. in sight C. in common D. in touch 4.The small village is said _____in a battle during World War II. A.to have been destroyed B.to be destroyed C. to destroy D. to have destroyed 5.The government has taken effective measures to ______itself to the changing world. A.settle B. adopt C. change D. adjust 6.______a word, this movie can be rated as a successful domestic film. A.Of B. At C. In D. By 7.Many a customer______about the poor quality of the washing machines. https://www.doczj.com/doc/dc7593114.html,plains https://www.doczj.com/doc/dc7593114.html,plain C. are complaining D. have complained 8.It is reported that some scientists in the United States have found a new _______ for cancer. A. way B. answer C. approach D. cure 9.He made students think for themselves, _______telling them what to think. A. more than B. other than C. rather than D. better than 10.Much_________she likes him, she would never consider marrying him. A. while B. as C. although D. despite 11.It is said that ten percent of the local children would leave school ________about 14 to learn a trade. A. at B. of C. in D. to 12.The government regards _______ as its duty to provide the best service for the public. A. this B. it C. what D. that 13.My watch is nowhere to be found. I ________when I was on the bus. A.must drop it B．might drop C. must have dropped it D. should have dropped it. 14. The price of pork has been very______these days. A. costly B. high C. expensive D.cheap 15. You must______the rubbish before you leave the office. 安徽省专升本英语-6 (总分：150.00，做题时间：90分钟) 一、Part Ⅰ Vocabulary and Structure(总题数：30，分数：30.00) 1.The bicycle you referred to isn"t ______. It belongs to ______. （分数：1.00） A.me; you B.mine; hers C.hers; his D.his; her √ 解析： 2.—I thought you were going to the mountains last week. —I"d want to, but I was ______ busy to leave. （分数：1.00） A.too much B.so much C.much more D.much too √ 解析： 3.His health is ______. （分数：1.00） A.as poor, if not poor than his sister B.poor as his sister"s if not poor C.as poor as if not poorer than his sister"s √ D.as poor, if not poorer than sister"s 解析： 4.The kite flew ______ in the sky and everyone spoke ______ of it. （分数：1.00） A.high; high B.highly; highly C.high; highly √ D.highly; high 解析： 5.That young man has made so much noise that he ______ not have been allowed to attend the concert. （分数：1.00） A.need B.must C.would D.should √ 解析： 6.______ do it myself than try to persuade such a silly fellow like him. （分数：1.00） A.I"d like B.I"d rather √ C.I"d better D.I"d like to 解析： 2019年安徽专升本招生考试（公共英语）模拟试题 Part I Vocabulary and Structure(30points) Directions:In this part, there are 30 incomplete sentence. For each sentence there are four choices marked A,B,C and D. Choose the ONE that best completes the sentence. 1.He was speaking so fast at the conference ________we could hardly follow him. A. what B. as C. but D. that 2.People ________in summer in the lakes around the city, but now there is no lake available. A. used to swimming B. was used to swimming C. were used to swim D. used to swim 3.--What time is the football game on? --I have ________,but I can check it for you in the newspaper. A. not idea B. no idea C. no ideas D. not ideas 4.We’ve got two TV sets, but we still can ’t watch anything because________ works properly. A. each B. either C. neither D. every 5.Jack worked late into the night, ________he had not finished even half of the job. A. only finding B. only found C. only find D. only to find 6.There is no doubt ________he is a good employee, for he has all the qualities that the company needs. A. as B. Who C. that D. what 7.It suddenly ________me that we could turn to the police foe help. A. happened to B. occurred to C. took place D. turned up 8.The young man cheated his friend of much money, ________was really disgraceful. A. it B. who C. that D. which 9.“John is learning German in his free time. ”“ ________. ” A. So his sister does B. So is his sister C. So does his sister D. So his sister is 安徽省普通高等学校专升本招生考试英语真题试题注意事项： 1. 试卷共8页，用钢笔或圆珠笔直接答在试题卷上。 2. 答卷前将密封线内的项目填写清楚。Part I Vocabulary and Structure (1 point each, 30 points in all) Directions: There are 30 incomplete sentences. You are required to complete each one by deciding on the most appropriate word or words from the 4 choices marked A,B,C and D, then write the corresponding letter on the Answer Sheet. 1.Will you speak louder so as make people _____ you? A.to hear B.heard C.hearing D.hear 2.The car industry can’t survive ______ the government help. A.without B.with C.besides D.except 3.I had considerable difficulty ________ her to go out for a drink with me. A.to persuade B.to have persuaded C.persuade D.persuading 4.Mr. Black, our new English teacher, is strict _________ kind. A.or B.but C.with D.as well 5._______ China Today is a good way of improving our English. A.Having read B.Have read C.Reading D.Read 6.If I ________ you, I wouldn’t miss the chance tomorrow morning. A.be B.will be C.am D.were 7.Those who want to attend the meeting should _______ the form. A.fill in B.fill up C.fill with D.fill of 8.After ______ for the job, the interviewees will be required to take a health check. A.to interview B.being interviewed C.interviewing D.having interviewed https://www.doczj.com/doc/dc7593114.html,pared ________ our small flat, Bill’s house seemed like a palace. A.in B.for C.with D.as 10.——Each of the hard-working students ________ to go to college. ——So do we. A.hope B.hoping C.hoped D.hopes 11.He says ______ clearly ________ beginners understand most of his words. A.such…that B.so…that C.such a …that D.so a …that https://www.doczj.com/doc/dc7593114.html,nguage 安徽省普通高等学校专升本考试2012年真题 注意事项： 1.试卷共8页，请用黑色签字笔答题，答案写在答题纸上，写在试卷上无效。 2.答题前请将答题纸上密封线内的项目填写完整。 PartI Vocabulary and Structure(1 point each;30 point in all) Directions:There are 30 incomplete sentences. You are required to complete each one by deciding on the most appropriate word or words from the four choices marked A,B,C and D,and then write the write the corresponding letter on the Answer sheet. 1.—Hello, ____ Kate？ —Yes, speaking. A. is this B. is that C. are you D. is it 2. By the time I got home, my mother ____ to bed. A. went B. was getting C. had gone D. has gone 3 ____ everything ____ ，it was not a bad holiday. A. brining--into effects B. learning--by heart C. petting--into practice D. taking--into account 4. When waiting at a bus stop for a long time, most people often look ____ and asleep. A. bored B. surprised C. excited D. amused 5. ____ the effective method , I answered all the questions correctly. A. Thanks to B. In case of C. As well as D.Apart from 6. If you are like most people , your intelligence ____ from season to season. A. turns B. change C. varies D. ranges 7. No cream hot me, thanks. I am ____ a diet. A. in B. on C. from D. off 8. ____ really worries George is that his parents expect too much of him. A. What B. This C. Whichever D. It 9.Mary loves Jazz, and she has played ____ for years. A. piano B. a piano C. one piano D. the piano 10.Ted made up has mind soon and accepted the invitation ____ . A. without hesitation B. without doubt C.without understanding D. without exception 11.It is good manners ____ help to others when they are in need. A. to obtain B. obtaining C. to offer D. offering 12. ____ you've grown your favorite flowers, the following job is to take care. A. Although B. Before C. While D. 13. Huangshan Mountain is really a magnificent place for an enjoyable and ____ vacation. A.dull B. annual C. relaxing D. noisy https://www.doczj.com/doc/dc7593114.html,ura ____ her parents travel abroad once a year. A. and B. together with C. as well as D. accompanied by 15.—Who broke the window ? —Not me—It must have been ____ else.安徽专升本英语真题以及答案.doc
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