南京信息工程大学试卷一标准答案

南京信息工程大学20XX ─ 20XX 学年 第 2 学期高等数学2课程试卷( Ｂ 卷) 及参考答案注意：1、本课程为 （表明必修或选修）， 学时为 ，学分为2、本试卷共 页；考试时间 分钟； 出卷时间： 年 月3、姓名、学号等必须写在指定地方； 考试时间： 年 月 日4、本考卷适用专业年级： 任课教师： XXX以上内容为教师填写）专业年级 班级学号 姓名一、填充题 (每小题 3 分，共 15 分)1．设L 是周长为a 的椭圆22143x y +=，则曲线积分22(234)L xy x y ds ++⎰=__12a ______． 2．已知:z ∑=zdS ∑=⎰⎰3R π. 3．设{(,,)|,01}D x y z a x b y =≤≤≤≤，且()1Dyf x d σ=⎰⎰，则()baf x dx =⎰_______2_____.4．将xoy 坐标面上的椭圆14922=+y x 绕x 轴旋转一周, 所生成的旋转曲面方程为 222194x y z ++= 5．微分方程230y y y '''--=的通解为 312x x y C e C e -=+，（12,C C 为常数）.二、选择题(每小题 3 分，共 15 分)1．级数11(1)n n n ∞=+∑ （ A ）（A ）发散 （B ） 收敛于1 （C ） 收敛于0 （D ）无法判断收敛性2． 22xydx ax dy +在xOy 面内是某一函数(,)u x y 的全微分，则a = （ C ）. (A) 1- (B) .2- (C) 1 (D) 23．2．设y x z =, 则zx∂=∂ ( A ) A. 1y yx - B. ln ||y x x C. (ln )y y x x x+ D. ln y x x 4．若区域222:1x y z Ω++≤取外侧，则积分222()xy z dv Ω++⎰⎰⎰等于 ( B )(A) 2120sin d d r dr ππθϕϕ⎰⎰⎰ (B)2140sin d d r dr ππθϕϕ⎰⎰⎰(C)211221()d d z dz πθρρρ-+⎰⎰⎰ (D)21d d πθρρ⎰⎰5．若级数1nn a∞=∑收敛 ，1nn b∞=∑发散，则级数1()nn n ab ∞=+∑ ( Ａ )(A) 一定发散 (B) 一定收敛 (C) 条件收敛 (D) 不能确定三、判别下列各级数是否收敛？若收敛，是绝对收敛还是条件收敛？(本题２０分) 1．1(1)sin3n nn π∞=-∑ 绝对收敛解 因为|(1)sin|sin333nnnnπππ-=≤，--------------------------------------------------------4分而级数13nn π∞=∑收敛，所以原级数1(1)sin3n nn π∞=-∑绝对收敛.------------------------------１０分2．132nnn n ∞=⋅∑ 解 因为1133(1)2lim1322n n n n nn n ++→∞+⋅=>⋅，--------------------------------------------------------------4分 由比值审敛法知，该级数发散.---------------------------------------------------------------１０分四．已知曲线方程:sin ,1cos ,4sin2t x t t y t z Γ=-=-=, 求对应于2t π=的点处的切线 及法平面方程 (本题１０分)解 (1,1,2)T = -------------------------------------------------------------------------------４分切线方程11211x y π-+-==-----------------------------------------------------８分 法平面方程402x y π+--= -----------------------------------------------------１０分五．求微分方程2x y y y e -'''++=的通解 (本题10分)解 对应的齐次方程的通解12()x y C x C e -=+， --------------------------------------4分 设非次方程的特解2x y Ax e *-=， ---------------------------------------------------------6分 则2(2)x y Ax Ax e -'=-，2(24)x y A Ax Ax e -''=-+代入解得12A =- ------------------------------------------------------------------------------8分 从而原方程的通解为2121()2xy C x C x e -=++ （12,C C 为常数） ------------10分六．求幂级数11n n n x n ∞=+∑的收敛域及和函数，并求1(1)2nn nn ∞=+∑ 的值(本题10分). 解 (2)lim1(1)(1)n n nR n n →∞+==++， --------------------------------------------------------------2分当1x =±时级数发散，故原级数的收敛域为11x -<<， -----------------------------4分又 111111n n n n n n x x n n ∞∞==+-=++∑∑1111n nn n x x n ∞∞===-+∑∑11ln(1),0||110,0x x x xx ⎧+-<<⎪=-⎨⎪=⎩， -------------------------------------------------------8分 令12x =，得12(1ln 2)(1)2nn n n ∞==-+∑. --------------------------------------------------10分 七．将1()arctan1xf x x+=-展为x 的幂级数 (本题10分). 解 221()(1),(11)1n n n f x x x x ∞='==--<<+∑-----------------------------------------------4分 0()(0)()xf x f f x dx '-=⎰221000(1)(1)21n x n nn n n x dx x n ∞∞+==-=-=+∑∑⎰------------------------------------------------------------8分 所以 2101(1)arctan ,(11)1421n n n x x x x n π∞+=+-=+-<<-+∑--------------------------------------10分八．计算曲面积分2(81)(1)4I y xdydz z y dzdx yzdxdy ∑=++-=⎰⎰，其中∑是由曲线13z y x ⎧=≤≤⎪⎨=⎪⎩ 绕y 轴旋转一周所成的曲面，它的法向量与y 轴正向的夹角恒大于2π.(本题10分) 解 曲面∑：221y x z -=+，设2212:3x z y ⎧+≤∑⎨=⎩，取右侧， ---------------------------2分则11I ∑+∑∑=-⎰⎰⎰⎰，由高斯公式212312dv d d dy πρθρπ+∑+∑Ω===⎰⎰⎰⎰⎰⎰⎰， --------------------------------------------6分而122(13)32zxD dzdx π∑=-=-⎰⎰⎰⎰- --------------------------------------------------------------8分从而 23234I πππ=+=. ----------------------------------------------------------------------10分。

南京信息工程大学-高等数学(上册)-试卷B(含答案)南京信息工程大学试卷学年 第 1学期 高等数学 课程试卷( B 卷)本试卷共 页；考试时间 120分钟；任课教师 课程组 ；一、单项选择题 (本大题有4小题, 每小题4分, 共16分) 1.)(0),sin (cos )(　处有则在设=+=x x x x x f .（A ）(0)2f '= （B ）(0)1f '=（C ）(0)0f '= （D ）()f x 不可导.2.　）时（　，则当，设133)(11)(3→-=+-=x x x x xx βα.（A ）()()x x αβ与是同阶无穷小，但不是等价无穷小； （B ）()()x x αβ与是等价无穷小；（C ）()x α是比()x β高阶的无穷小； （D ）()x β是比()x α高阶的无穷小.3. 若()()()02xF x t x f t dt=-⎰，其中()f x 在区间上(1,1)-二阶可导且'>()0f x ，则（ ）.（A ）函数()F x 必在0x =处取得极大值； （B ）函数()F x 必在0x =处取得极小值；（C ）函数()F x 在0x =处没有极值，但点(0,(0))F 为曲线()y F x =的拐点； （D ）函数()F x 在0x =处没有极值，点(0,(0))F 也不是曲线()y F x =的拐点。

4.)()( , )(2)( )(1=+=⎰x f dt t f x x f x f 则是连续函数，且设（A ）22x（B ）222x +（C ）1x - （D ）2x +.二、填空题（本大题有4小题，每小题4分，共16分） 5. =+→xx x sin 2)31(lim .6. ,)(cos 的一个原函数是已知x f x x =⋅⎰x x xx f d cos )(则 .7.lim(cos cos cos )→∞-+++=22221L n n nnn n ππππ .8.=-+⎰21212211arcsin －dx xx x .三、解答题（本大题有5小题，每小题8分，共40分）9. 设函数=()y y x 由方程sin()1x y e xy ++=确定，求'()y x 以及'(0)y .10. .d )1(177x x x x ⎰+-求11.．　求，， 设⎰--⎪⎩⎪⎨⎧≤<-≤=132)(1020)(dx x f x x x x xe x f x12. 设函数)(x f 连续，=⎰1()()g x f xt dt，且→=0()limx f x A x ，A 为常数.求'()g x 并讨论'()g x 在=0x 处的连续性.13. 求微分方程2ln xy y x x '+=满足=-1(1)9y 的解.四、 解答题（本大题10分）14. 已知上半平面内一曲线)0()(≥=x x y y ，过点(,)01，且曲线上任一点M x y (,)00处切线斜率数值上等于此曲线与x 轴、y 轴、直线x x =0所围成面积的2倍与该点纵坐标之和，求此曲线方程.五、解答题（本大题10分）15. 过坐标原点作曲线x y ln =的切线，该切线与曲线x y ln =及x 轴围成平面图形D.(1) 求D 的面积A ；(2) 求D 绕直线x = e 旋转一周所得旋转体的体积六、证明题（本大题有2小题，每小题4分，共8分）16. 设函数)(x f 在[]0,1上连续且单调递减，证明对任意的[,]∈01q ，1()()≥⎰⎰qf x d x q f x dx.17. 设函数)(x f 在[]π,0上连续，且0)(0=⎰πx d x f ，cos )(0=⎰πdx x x f .证明：在()π,0内至少存在两个不同的点21,ξξ，使.0)()(21==ξξf f （提示：设⎰=xdxx f x F 0)()(）解答一、单项选择题(本大题有4小题, 每小题4分, 共16分) 1、D 2、A 3、C 4、C二、填空题（本大题有4小题，每小题4分，共16分）5. 6e . 6.c x x +2)cos (21　.7. 2π. 8.3π.三、解答题（本大题有5小题，每小题8分，共40分） 9. 解：方程两边求导(1)cos()()0x y e y xy xy y +''+++= cos()()cos()x y x ye y xy y x e x xy +++'=-+0,0x y ==，(0)1y '=-10. 解：767u x x dx du == 1(1)112()7(1)71u du duu u u u -==-++⎰⎰原式 1(ln ||2ln |1|)7u u c =-++ 7712ln ||ln |1|77x x C =-++11. 解：1033()x f x dx xe dx ---=+⎰⎰⎰3()x xd e --=-+⎰⎰00232cos (1sin )x x xe e d x πθθθ----⎡⎤=--+-=⎣⎦⎰令3214e π=--12. 解：由(0)0f =，知(0)0g =。

南京信息工程大学2019-2020年大学英语试卷Sample Test Four for Book OnePart I Listening Comprehension (20 points)Section ADirections:You’re going to hear ten short conversations between two speakers. At the end of each conversation, a question will be aked about what was said. After you hear the converdation and the question about it, read the four possible ansers and decide which one would be the best answer to the question you’ve heard. Then mark your answer.DDABD DACDA1. A. She has done the exercise.B. She knows the man very well.C. She wants the man to do the exercise with her.D. She can help the man.2. A. At 7:15. B. At 7:35.C. At 7:20.D. At 8:05.3. A. Golf and tennis. B. Skiing and tennis.C. Golf and skating.D. Skiing and skating.4. A. The man's. B. John's.C. Cindy's.D. Cindy's sister's.5. A. The woman should leave at once.B. The woman shouldn't leave now.C. The woman must leave if she wants to catch the train.D. The woman will miss the train.6. A. To do the experiment with the woman.B. To finish the experiment for the woman.C. To take care of the woman.D. To take the woman home.7. A. Stay inside. B. Sit under the sun.C. Go to the park by herself.D. Join her friend.8. A. Go to bed.B. Learn to work on the computer.C. Finish her paper.D. Stay up all night.9. A. He doesn't like the weather in New York.B. He doesn't think the woman agrees with him.C. He is not thinking of a trip to New York.D. He wants to stay where he is.10.A. She failed the exam because she didn't spend enough time in the lab.B. She passed the exam because she spent a lot of time in the lab after class.C. She won't do well on the exam if she doesn't spend more time in the lab.D. She's going to spend a lot of time in the lab after class so that she can't pass theexam.Section B DCACDDirections:Listen to the dialogue and choose the best answer to each question you hear.1/11 A. A formal talk between the boss and his secretary.B. An informal talk between two friends.C. A formal talk between a professor and a student.D. An interview between the interviewer and the job applicant.2/12 A. General manager.B. An official worker.C. Head of the Personnel Department.D. A secretary.3/13 A. Office worker.B. Interpreter.C. Manager of Beijing Branch.D. Computer operator.4/14 A. She types well.B. She is English.C. She is good at languages.D. She used to be a school teacher.5/15 A. The man is not satisfied with the speed Miss Jenkins types.B. The man is hesitating whether to accept Miss Jenkins or not.C. The man does not think Miss Jenkins is qualified.D. The man will probably recommend Miss Jenkins to the general manager. Section C BABBADirections:Listen to the dialogue and choose the best answer to each question you hear.1/16 A. She has come to have an interview with Mrs Davis.B. She has come to work in the office.C. She has come to apply for a place in the finance section.D. She has come to get a reply to her job application.2/17 A. Director of the finance section.B. General manager of the company.C. Secretary of the finance section.D. Typist of the finance section.3/18 A. At 8:30.B. At 8:45.C. At 9:00.D. At 9:30.4/19 A. She’s just lost her job.B. She’s just got a promotion.C. She’s just been appointed secretary to the general manager.D. She’s just been transferred to another company.5/20 A. Because she is starting a new job today.B. Because she finds the job very difficult.C. Because she doesn’t know how to deal with her new boss.D. Because she doesn’t know how to find a good job.Part II Use of English (10 points)Directions:In this part there are 10 incomplete dialogues. For each dialogue there are four choices marked A, B, C and D. Choose the ONE answer that best completes the dialogue. Then mark the corresponding letter on the Answer Sheet with a single line through the center.CBBAB BDACB21.--- Many thanks.---____________.A. It doesn’t matterB. Never mindC. It’s my pleasureD. The same to you22.--- I’m awfully sorry.---____________.A. Don’t mention itB. That’s rightC. All rightD. That’s all right23.--- How do you do?---____________./?A. Very wellB. How do you doC. I’m a doctorD. Nice to have known you24.--- Sorry to trouble you.---____________.A. It’s a pleasureB. I don’t think soC. No, I don’t mind at allD. Never mind25.--- Do you mind if I smoke here?---____________.A. That’s all rightB. Of course notC. OKD. Yes, please26.--- How’s your family?---____________.A. Thanks all the sameB. Thanks for callingC. Not too badD. Don’t mention it27.--- How about going for a drive?---____________.A. It’s OKB. Yes, we shallC. Yes, pleaseD. Good idea28.--- Would you like a coffee?---____________.A. Yes, pleaseB. Yes, I prefer teaC. I like coffee very muchD. Thank you very much29.--- What’s the matter with you?---____________.A. It doesn’t matterB. Not too badC. I don’t feel wellD. That’s all right30.--- Can you help me?---____________.A. No, I don’t knowB. Sure. What is it?C. Don’t mention itD. That’s itPart III Reading Comprehension (30 points)Directions:There are three passages in this part. Each passage is followed by five questions. For each questions there are four choices marked A, B, C and D. You should decide on the best choice and mark the corresponding letter on the Answer Sheet with a single line through the center.Passage 1The first true piece of sports equipment that man invented was the ball.In ancient Egypt, as everywhere, pitching stones was a favorite children’s game. But a badly thrown rock could hurt a child. Looking for something less dangerous to throw, the Egyptians made what were probably the first balls.At first, balls were made of grass of leaves held together by vines. Later they were made of pieces of animal skin sewed together and stuffed with feathers or hay.Even though the Egyptians were warlike, they found time for peaceful games. Before long they had developed a number of ball games, each with it s own set of rules. Perhaps they played ball more for instruction than for fun. Ball playing was thought of mainly as a way to teach young men the speed and skill they would need for war.ADADA31. The ball was probably invented because ______.A. throwing stones often caused injuriesB. throwing stones was not funC. games with stones did not have rulesD. rocks were too heavy to throw32. The first balls were probably made of ______.A. animal skins stuffed with rocksB. twists of hayC. hides stuffed with hay or feathersD. grass and leaves tied with vines33. This selection says that the Egyptians played ______.A. many different games with ballsB. many different kinds of gamesC. only one ball gameD. different games with similar rules34. The Egyptians thought that ball playing was ______.A. childishB. difficultC. not enjoyableD. worthwhile35. The best title for this selection is ______.A. The First Ball GamesB. How Egyptian Children Played GamesC. Egyptian SportsD. The Beginning of SportsPassage 2Although weddings vary greatly from country to country, they are happy events in every culture. In most American weddings, the bride wears a traditional style of dress. There is also an old saying that tells women what they should wear or carry when they marry:"Something old, something new, something borrowed, something blue."This "good luck" saying dates back to Victorian times in the 1800s, and many American brides take it very seriously, following the tradition of the wedding dress."Something old" symbolizes the connection with the bride's family and the past. Many brides wear a piece of family jewelry or their mother' or grandmother's wedding dress."Something new" symbolizes good luck and success in the bride's new life. The wedding dress is often the new item. However, the bride may wear new jewelry or new shoes."Something borrowed" reminds the bride that friends and family will help her when she needs them. The borrowed object might be something such as a beautiful handkerchief, or a best friend's wedding veil."Something blue" is the symbol of faithfulness and loyalty. Often the blue item is the garter worn on the bride's leg. It might also be a ribbon or a piece of jewelry.The bride carries flowers as a symbol of happiness. In the past, both the bride and groom wore flowers. At the end of the wedding reception, the bride throws her flowers to all the single women who attend the party. The single woman who catches the flowers is believed to be the next to marry.Throwing the garter began in France because pieces of the bride's clothing were considered lucky. The bride would throw the garter to the guests at the wedding and whoever caught it could expect good luck. In the United States, the groom traditionally removes the garter from the bride and throws it to the unmarried men. The man who catches it is thought to be the next to marry. At some weddings, the man who catches the garter will place it on the leg of the woman who caught the flowers, or they may start the next dance. It is also common for those who have received the flowers and the garter to have their photo taken with the bride and groom.Whatever the bride wears, it is also traditional that she be the best-dressed and most important woman at the wedding party. Therefore, it is considered very rude for other women at the wedding to dress more beautifully than the bride.BACDB36.What do weddings have in common in every culture?bA.They are serious things.B.They are happy events.C.They are traditional.D.They are beautiful.37.The bride should wear "something old" because this means that ___a__.A.she will not forget her family and the pastB.she will not have enough money to buy new clothesC.she will be a good wife after marriageD.she will be lucky38.In France, _____ symbolizes good luck. cA.catching the flowers from the brideB.wearing the veil of the brideC.getting any piece of the bride's clothingD.dressing beautifully39.Which of the following is NOT true?dA.The bride usually carries flowers.B.The groom throws the bride's garter to the unmarried men at the wedding.C.The bride's garter is considered something that will bring good luck.D.The bride will throw the flowers to all the women at her wedding party.40.What if other women at the wedding dress more beautifully than the bride?bA.It is considered very nice.B.It is considered rather impolite.C.It is considered lucky.D.It is considered quite traditional.Passage 3Imagine you are Alice, stepping through the looking glass. Suddenly everything is reversed. Doorknobs are on the wrong side of doors. The gearshift in your car is in the wrong place.Twenty-five million Americans wake up every day in just such an awkward situation. They are the one in ten of us who are left-handed and must face the world designed for the right-handed majority.Why we are left-or right-handed remains one of the great unsolved mysteries of science. We know that nearly two out of three lefties are male and that left-handedness runs in families. According to one study, almost half the offspring of two left-handed parents will be left-handed. The Scots-Irish family Kerr (from the Gaelic word for “left”) produced so many left-handers that in 1470 the family built its castle’s spiral stairways with a reverse twist to favor left-handed swordsmen.On the other hand, heredity (遗传) alone cannot explain lefties. At least 84 percent of them are born of two right-handed parents. And in 12 percent of genetically identical twins, one will be right-handed, the other left.Perhaps the greatest puzzle of all is not why some people are left-handed, butrather why so few are. So, scientists are trying to discover the truth of the matter, and they are beginning to gain insight into many ways left-handed differ from right-handed by considering how their brains work.BBDDD41. How many of the Americans are left-handers?A. 10%B. 12%C. 50%D. 80%42. Among the left-handers ______.A. few are menB. most of them are womenC. half of them are menD. two thirds are men43. According to the passage, which of the following is NOT true?A.Almost half of the children of two left-handed parents will be left-handed.B.At least 84 percent of left-handers are born of two right-handed parents.C.One in ten of Americans are left-handed.D.In the majority of genetically identical twins, one will be right-handed, theother left.44. In the third paragraph, the word “offspring” means ______.A. relativesB. sonsC. daughtersD. children45. The reason why so few are left-handed ______.A. has nothing to do with heredityB. remains one of the great unsolved mysteries of scienceC. has little to do with the working of their brainsD. is that left-handers are mainly twinsPart IV Vocabulary and Structure (25 points)Section ADirections:In this section there are 15 incomplete sentences. For each sentence there are four choices marked A, B, C and D. Choose the ONE answer that best completes the sentence. Then mark the corresponding letter on the Answer Sheet with a single line through the center.BCBAB BACBA CACDC46.The hostess silently _____ the cup from the table without any guest's knowledge.A. remindedB. reducedC. refusedD. removed47.In some cultures, a knife _____ the highest power.A.looksB. setsC. symbolizesD. seeks48.The forming of this custom _____ the last century, when Chinese women had low social position.A. comes back toB. dates back toC. is the day back toD. adjusts back to49.Although we are of the same age, we _____ in height and weight.A. varyB. variousC. differentD. very50.Some soldiers are very _____ to their military officer and follow his orders strictly (严格地).A. loyaltyB. loyalC. faithfulnessD. faith51._____ is responsible will have to pay for the damage (损坏).A. WhatB. WhateverC. WhoD. Whoever52.All of those who _____ the peace conference (会议) were asked to sign their names on a piece of cloth.A. attendedB. presentedC. wentD. was53.The air hostess (空中小姐) _____ the passengers to fasten their seat belts when the plane was about to land.A. rememberedB. repliedC. remindedD. questioned54.Born deaf (聋的), she could not use _____ language to communicate with us.A. nonverbalB. verbalC. foreignD. body55.Being a _____ of America, the famous movie star decided to run for US president.A. nativeB. maleC. originalD. female56.One cannot ____ across his thoughts to others well if he doesn't master enough words.A. passB. handC. transportD. put57.Could you give me more of his _____ information before I agree to interview him?A. backgroundB. underground\C. backD. behind58.His _____ expression (表情) shows that he is unhappy about that.A. bodyB. handC. facialD. verbal59.Mum always tells me to work more _____ so that I can have more time to play and relax.A. efficientB. easilyC. easeD. efficiently60.Though defeated, the guest team _____ us to another match this Saturday.A. changedB. wantedC. challengedD. madeSection BDirections:There are 10 blanks in the following passage. For each blank there arefour choices marked A, B, C and D. You should choose the ONE answer that best completes the passage. Then mark the corresponding letter on the Answer Sheet with a single line through the center.If you visit a big city anywhere in the world, you will probably find a restaurant which serves the food of your native country. Most large cities 61 the United States offer an international sample of 62 . Many people enjoy eating the food 63 other nations. This is probably one reason 64 there are so many different kinds of restaurants in the United States. 65 second reason is that many Americans come from other parts of the world. They enjoy tasting the foods of their 66 lands. In the city of Detroit, for example, there are many people from Western Europe, Latin America 67 the Far East. There are many restaurants in Detroit 68 serve the foods of these areas. There are many other international restaurants, 69 . Americans enjoy the foods in these restaurants as well as the chance to 70 understand the people and their way of life.ACBDC BABABa61. A. in B. at C. for D. toc62. A. things B. cities C. foods D. nationsa63. A. about B. of C. to D. withd64. A. for B. of C. which D. whyc65. A. Same B. Then C. A D. Ford66. A. national B. country C. state D. nativea67. A. and B. with C. of D. thenb68. A. for B. which C. what D. therea69. A. too B. either C. then D. thoughb70. A. do B. better C. serve D. morePart V Writing (15 points)Directions:For this part, you are allowed thirty minutes to complete the following letter by using the given information and translate the Chinese into English. Recipient of the letter: Mr. AndersonSender of the letter: Gao KemingDate: (Use today’s date)1August 30th, 20042 Dear Mr. Anderson ,I was very pleased to hear that you were 3promoted to Manager (提升经理) of the Overseas Department. Please 4accept my warm congratulations (接受我的祝贺).For many years, you 5 have worked very hard for the company (为公司努力工作). Your winning of the position shows that your good service hasbeen recognized by all. I am sure that the department will make greater progress under your guidance. I 6wish you success (祝你成功) in your future career.Sincerely yours,Gao Keming7ANSWER SHEETPart I Listening Comprehension (20 points)01.[A][B][C][D]02.[A][B][C][D]03.[A][B][C][D]04.[A][B][C][D]05.[A][B][C][D]06.[A][B][C][D]07.[A][B][C][D]08.[A][B][C][D]09.[A][B][C][D]10.[A][B][C][D]11.[A][B][C][D]12.[A][B][C][D]13.[A][B][C][D]14.[A][B][C][D]15.[A][B][C][D]16.[A][B][C][D]17.[A][B][C][D]18.[A][B][C][D]19.[A][B][C][D]20.[A][B][C][D]Part II Use of English (10 points)21.[A][B][C][D]22.[A][B][C][D]23.[A][B][C][D]24.[A][B][C][D]25.[A][B][C][D]26.[A][B][C][D]27.[A][B][C][D]28.[A][B][C][D]29.[A][B][C][D]30.[A][B][C][D]Part III Reading Comprehension (30 points)31.[A][B][C][D]32.[A][B][C][D]33.[A][B][C][D]34.[A][B][C][D]35.[A][B][C][D]36.[A][B][C][D]37.[A][B][C][D]38.[A][B][C][D]39.[A][B][C][D]40.[A][B][C][D]41.[A][B][C][D]42.[A][B][C][D]43.[A][B][C][D]44.[A][B][C][D]45.[A][B][C][D]Part IV Vocabulary and Structure (25 points)46.[A][B][C][D]47.[A][B][C][D]48.[A][B][C][D]49.[A][B][C][D] 50.[A][B][C][D]51.[A][B][C][D]52.[A][B][C][D]53.[A][B][C][D] 54.[A][B][C][D]55.[A][B][C][D]56.[A][B][C][D]57.[A][B][C][D] 58.[A][B][C][D]59.[A][B][C][D]60.[A][B][C][D]61.[A][B][C][D] 62.[A][B][C][D]63.[A][B][C][D]64.[A][B][C][D]65.[A][B][C][D] 66.[A][B][C][D]67.[A][B][C][D]68.[A][B][C][D]69.[A][B][C][D] 70.[A][B][C][D]Part V Writing (15 points)1_______________________________________________________ 2_______________________________________________________ 3_______________________________________________________ 4_______________________________________________________ 5_______________________________________________________ 6_______________________________________________________ 7_______________________________________________________。

南京信息工程大学试卷20XX －20XX 学年 第 2 学期 高等代数（下） 课程试卷( A 卷)本试卷共 2 页；考试时间 120分钟；任课教师 杨兴东 昝立博 ；出卷时间20XX 年 6月专业 学号 姓名 得分一、填空题（15分）：1．设A 是正交矩阵，则A = ；1-A = ；A 的特征值为 . 2．设3阶方阵A 的特征值为1,0,1-，22B A A E =--,则B = .3．设线性变换ϕ在线性空间V 的一组基321,,εεε下的矩阵⎪⎪⎪⎭⎫⎝⎛=333231232221131211a a a a a a a a a A ， 则ϕ在基1ε，25ε，3ε下的矩阵为 .4. 若3112x -⎛⎫ ⎪-⎝⎭与204y ⎛⎫⎪⎝⎭相似，则x = ，y = . 5．欧氏空间2R 中基1ε=⎪⎪⎭⎫ ⎝⎛31，2ε=⎪⎪⎭⎫ ⎝⎛-12的度量矩阵为 . 二、选择题（15分）：1．如果B A ,都是n 阶正定阵，则下列结论错误的是 （ ） （A ）B A +是正定阵； （B ）B A -是正定阵； （C ）1-A 是正定阵； （D ）B A +-1是正定阵.2．设ϕ是数域P 上线性空间V 的线性变换，α和β是ϕ的分别属于特征值λ和μ 的特征向量，那么 （ ）（A ）若α和β线性无关，则λ≠μ； （B ）若α和β线性相关，则λ≠μ； （C ）若λ=μ，则α和β线性相关； （D ）若λ≠μ，则α和β线性无关． 3. 设B A ,为n 阶矩阵，E 为n 阶单位矩阵，且A 与B 相似，则 （ ） （A ）A 与B 有相同的特征矩阵； （B ） A 与B 有相同的特征值和特征向量； （C ）对任意常数λ, A E -λ与B E -λ相似；（D ）A 与B 相似于同一个对角阵.4. 设n 维线性空间V 的线性变换ϕ在V 的一组基下的矩阵是A ,且A 的秩为r ，则ϕ的值域()V ϕ与核()V ker 的维数分别为 （ ）（A ） ,r r (B) ,r n r - (C) ,n r r - (D) ,0n5. 下列命题中正确的是 （ ） （A ）线性变换在不同基下的矩阵是合同的； （B ）欧式空间中不同基的度量矩阵是相似的； （C ）设B A ,是两个n 阶正定阵，则A 与B 合同； （D ）任意一个复方阵都相似于一个对角阵. 三、（10分） 问t 取何值时，实二次型222123123121323(,,)5224f x x x x x x tx x x x x x =+++-+正定?四、（12分） 设121211212111,,,11030117ααββ-⎛⎫⎛⎫⎛⎫⎛⎫⎪ ⎪ ⎪ ⎪-- ⎪ ⎪ ⎪ ⎪==== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭, 1W =()21,ααL ，2W =()21,ββL ，求21W W ⋂和21W W +的基及维数．五、（12分）求复系数矩阵1332613148A -⎛⎫⎪=-- ⎪ ⎪--⎝⎭的Jordan 标准形．六、（12分）设P 是数域，{}A A P A V T n n =∈=⨯1，{}A A P A V T n n -=∈=⨯2， 证明：(1) 1V 与2V 都是n n P ⨯的子空间；(2) =⨯n n P 21V V ⊕. 七、（16分）设实二次型123121323(,,)222f x x x x x x x x x =++ (1) 写出二次型()321,,x x x f 的矩阵A ；(2) 求正交线性替换Py x =化二次型()321,,x x x f 为标准形；(3) 写出二次型()321,,x x x f 的规范形，指出它的秩，正、负惯性指数和符号差，并判别()321,,x x x f 的正定性.八、（8分）设A 为实反对称矩阵，即A A T -=，证明： （1）A 的特征值只能是0或纯虚数； （2）2A E -是正定阵．20xx-20xx 学年《高等代数》（下）期末试卷（A ）参考答案与评分标准一、填空题（本题满分为15分）：1)-4,-6,-12; 2) -24,-25; 3)1112132321223132332222a a a a aa a a a ⎛⎫⎪ ⎪⎪ ⎪⎝⎭; 4) 011101110⎛⎫⎪⎪ ⎪⎝⎭; 1,-1; 5)51110⎛⎫ ⎪⎝⎭.二、选择题（本题满分为15分）：1） D ； 2） C ； 3) C ； 4) B ； 5） B ．三、解：二次型的矩阵为1112125t A t -⎛⎫ ⎪= ⎪ ⎪-⎝⎭----------------2分当A 的顺序主子式都大于0时，即22123111|1|10,10;12=-5401125t t P P t P A tt t t -==>==->==->-时， 原二次型正定；联立方程2210540t t t ⎧->⎨-->⎩,解得405t -<< ----------------8分因此当405t -<<时，原二次型正定。