2003年真题及解析
2003年全国攻读硕士学位研究生入学考试英语试题Section I Use of EnglishDirections：Read the following text. Choose the best word(s) for each numbered blank and mark A, B, C OR D on ANSWER SHEET 1. (10 points)Teachers need to be aware of the emotional, intellectual, and physical changes that young adults experience. And they also need to give serious 1 to how they can best 2 such changes. Growing bodies need movement and 3 , but not just in ways that emphasize competition. 4 they are adjusting to their new bodies and a whole host of new intellectual and emotional challenges, teenagers are especially self-conscious and need the 5 that comes from achieving success and knowing that their accomplishments are 6 by others. However, the typical teenage lifestyle is already filled with so much competition that it would be 7 to plan activities in which there are more winners than losers, 8 ，publishing newsletters with many student-written book reviews, 9 student artwork, and sponsoring book discussion clubs. A variety of small clubs can provide 10 opportunities for leadership, as well as for practice in successful 11 dynamics. Making friends is extremely important to teenagers, and many shy students need the 12 of some kind of organization with a supportive adult 13 visible in the background.In these activities, it is important to remember that the young teens have 14 attention spans. A variety of activities should be organized 15 participants can remain active as long as they want and then go on to 16 else without feeling guilty and without letting the other participants 17 . This does not mean that adults must accept irresponsibility. 18 they can help students acquire a sense of commitment by 19 for roles that are within their 20 and their attention spans and by having clearly stated rules.1. [A] thought [B] idea [C] opinion [D] advice2. [A] strengthen [B] accommodate [C] stimulate [D] enhance3. [A] care [B] nutrition [C] exercise [D] leisure4. [A] If [B] Although [C] Whereas [D] Because5. [A] assistance [B] guidance [C] confidence [D] tolerance6. [A] claimed [B] admired [C] ignored [D] surpassed7. [A] improper [B] risky [C] fair [D] wise8. [A] in effect [B] as a result [C] for example [D] in a sense9. [A] displaying [B] describing [C] creating [D] exchanging10. [A] durable [B] excessive [C] surplus [D] multiple11. [A] group [B] individual [C] personnel [D] corporation12. [A] consent [B] insurance [C] admission [D] security13. [A] particularly [B] barely [C] definitely [D] rarely14. [A] similar [B] long [C] different [D] short15. [A] if only [B] now that [C] so that [D] even if16. [A] everything [B] anything [C] nothing [D] something17. [A] off [B] down [C] out [D] alone18. [A] On the contrary [B] On the average [C] On the whole [D] On the other hand19. [A] making [B] standing [C] planning [D] taking20. [A] capability [B] responsibility [C] proficiency [D] efficiencySection II Reading ComprehensionPart ADirections:Read the following four texts. Answer the questions below each text by choosing [A], [B], [C] or [D]. Mark your answers on ANSWER SHEET 1. (40 points)Text 1Wild Bill Donovan would have loved the Inter net. The American spymaster who built the Office of Strategic Services in the World War Ⅱand later laid the roots for the CIA was fascinated with information. Donovan believed in using whatever tools came to hand in the “great game” of espionage—spying as a “profession.” These days the Net, which has already re-made such everyday pastimes as buying books and sending mail, is reshaping Donovan’s vocation as well.The latest revolution isn’t simply a matter of gentlemen reading other gentlemen’s e-mail. That kind of electronic spying has been going on for decades. In the past three or four years, the World Wide Web has given birth to a whole industry of point-and-click spying. The spooks call it “open source intelligence,” and as the Net grows, it is becoming increasingly influential. In 1995 the CIA held a contest to see who could compile the most data about Burundi. The winner, by a large margin, was a tiny Virginia company called Open-Source Solutions，whose clear advantage was its mastery of the electronic world.Among the firms making the biggest splash in the new world is Straitford, Inc., a private intelligence-analysis firm based in Austin, Texas. Straitford makes money by selling the results of spying (covering nations from Chile to Russia) to corporations like energy-services firm McDermott International. Many of its predictions are available online at .Straiford president George Friedman says he sees the online world as a kind of mutually reinforcing tool for both information collection and distribution, a spymaster’s dream. Last week his firm was busy vacuuming up data bits from the far corners of the world and predicting a crisis in U kraine. “As soon as that report runs, we’ll suddenly get 500 new internet sign-ups from Ukraine,” says Friedman, a former political science professor. “And we’ll hear back from some of them.” Open-source spying does have its risks, of course, since it can be difficult to tell good information from bad. That’s where Straitford earns its keep.Friedman relies on a lean staff of 20 in Austin. Several of his staff members have military-intelligence backgrounds. He sees the firm’s outsider status as the key to its success. Straitford’s briefs don’t sound like the usual Washington back-and-forthing, whereby agencies avoid dramatic declarations on the chance they might be wrong. Straitford, says Friedman, takes pride in its independent voice.21. The emergence of the Net has .[A] received support from fans like Donovan[B] remolded the intelligence services[C] restored many common pastimes[D] revived spying as a profession22. Donovan’s story is mentioned in the text to.[A] introduce the topic of online spying[B] show how he fought for the US[C] give an episode of the information war[D] honor his unique services to the CIA23. The phrase “making the biggest splash” (line 1,paragraph 3) most probably means.[A] causing the biggest trouble[B] exerting the greatest effort[C] achieving the greatest success[D] enjoying the widest popularity24. It can be learned from paragraph 4 that .[A] straitford’s prediction about Ukraine has proved true[B] straitford guarantees the truthfulness of its information[C] straitford’s business is characterized by unpredictability[D] straitford is able to provide fairly reliable information25. Straitford is most proud of its .[A] official status[B] nonconformist image[C] efficient staff[D] military backgroundText 2To paraphrase 18th-century statesman Edmund Burke, “all that is needed for the triumph of a misguided cause is that good people do nothing.” One such cause now seeks to end biomedical research because of the theo ry that animals have rights ruling out their use in research. Scientists need to respond forcefully to animal rights advocates, whose arguments are confusing the public and thereby threatening advances in health knowledge and care. Leaders of the animal rights movement target biomedical research because it depends on public funding, and few people understand the process of health care research. Hearing allegations of cruelty to animals in research settings, many are perplexed that anyone would deliberately harm an animal.For example, a grandmotherly woman staffing an animal rights booth at a recent street fair was distributing a brochure that encouraged readers not to use anything that comes from or is tested in animals—no meat, no fur, no medicines. Asked if she opposed immunizations, she wanted to know if vaccines come from animal research. When assured that they do, she replied, “Then I would have to say yes.” Asked what will happen when epidemics return, she said, “Don’t worry, scientists will find some way of using computers.” Such well-meaning people just don’t understand.Scientists must communicate their message to the public in a compassionate, understandable way—in human terms, not in the language of molecular biology. We need to make clear the connection between animal research and a grandmother’s hip replacement, a father’s bypass operation, a baby’s vaccinations, and even a pet’s shots. To those who are unaware that animal research was needed to produce these treatments, as well as new treatments and vaccines, animal research seems wasteful at best and cruel at worst.Much can be done. Scientists could “adopt” middle school classes and present their own research. They should be quick to respond to letters to the editor, lest animal rights misinformation go unchallenged and acquire adeceptive appearance of truth. Research institutions could be opened to tours, to show that laboratory animals receive humane care. Finally, because the ultimate stakeholders are patients, the health research community should actively recruit to its cause not only well-known personalities such as Stephen Cooper, who has made courageous statements about the value of animal research, but all who receive medical treatment. If good people do nothing, there is a real possibility that an uninformed citizenry will extinguish the precious embers of medical progress.26. The author begins his article with Edmund Burke’s words to.[A] call on scientists to take some actions[B] criticize the misguided cause of animal rights[C] warn of the doom of biomedical research[D] show the triumph of the animal rights movement27. Misled people tend to think that using an animal in research is .[A] cruel but natural[B] inhuman and unacceptable[C] inevitable but vicious[D] pointless and wasteful28. The example of the grandmotherly woman is used to show the public’s.[A] discontent with animal research[B] ignorance about medical science[C] indifference to epidemics[D] anxiety about animal rights29. The author believes that, in face of the challenge from animal rights advocates, scientists should .[A] communicate more with the public[B] employ hi-tech means in research[C] feel no shame for their cause[D] strive to develop new cures30. From the text we learn that Stephen Cooper is .[A] a well-known humanist[B] a medical practitioner[C] an enthusiast in animal rights[D] a supporter of animal researchText 3In recent years, railroads have been combining with each other, merging into supersystems, causing heightened concerns about monopoly. As recently as 1995, the top four railroads accounted for under 70 percent of the total ton-miles moved by rails. Next year, after a series of mergers is completed, just four railroads will control well over 90 percent of all the freight moved by major rail carriers.Supporters of the new supersystems argue that these mergers will allow for substantial cost reductions and better coordinated service. Any threat of monopoly, they argue, is removed by fierce competition from trucks. But many shippers complain that for heavy bulk commodities traveling long distances, such as coal, chemicals, and grain, trucking is too costly and the railroads therefore have them by the throat.The vast consolidation within the rail industry means that most shippers are served by only one rail company. Railroads typically charge such“captive”shippers 20 to 30 percent more than they do when another railroad iscompeting for the business. Shippers who feel they are being overcharged have the right to appeal to the federal government's Surface Transportation Board for rate relief, but the process is expensive, time consuming, and will work only in truly extreme cases.Railroads justify rate discrimination against captive shippers on the grounds that in the long run it reduces everyone's cost. If railroads charged all customers the same average rate, they argue, shippers who have the option of switching to trucks or other forms of transportation would do so, leaving remaining customers to shoulder the cost of keeping up the line. It's theory to which many economists subscribe, but in practice it often leaves railroads in the position of determining which companies will flourish and which will fail.“Do we really want railroads to be the arbiters of who wins and who loses in the marketplace?”asks Martin Bercovici, a Washington lawyer who frequently represents shipper.Many captive shippers also worry they will soon be hit with a round of huge rate increases. The railroad industry as a whole, despite its brightening fortuning fortunes, still does not earn enough to cover the cost of the capital it must invest to keep up with its surging traffic. Yet railroads continue to borrow billions to acquire one another, with Wall Street cheering them on. Consider the $10.2 billion bid by Norfolk Southern and CSX to acquire Conrail this year. Conrail's net railway operating income in 1996 was just $427 million, less than half of the carrying costs of the transaction. Who's going to pay for the rest of the bill? Many captive shippers fear that they will, as Norfolk Southern and CSX increase their grip on the market.31. According to those who support mergers, railway monopoly is unlikely because .[A] cost reduction is based on competition.[B] services call for cross-trade coordination.[C] outside competitors will continue to exist.[D] shippers will have the railway by the throat.32. What is many captive shippers' attitude towards the consolidation in the rail industry?[A] Indifferent.[B] Supportive.[C] Indignant.[D] Apprehensive.33. It can be inferred from paragraph 3 that .[A] shippers will be charged less without a rival railroad.[B] there will soon be only one railroad company nationwide.[C] overcharged shippers are unlikely to appeal for rate relief.[D] a government board ensures fair play in railway business.34. The word “arbiters”(line 7,paragraph 4)most probably refers to those .[A] who work as coordinators.[B] who function as judges.[C] who supervise transactions.[D] who determine the price.35. According to the text, the cost increase in the rail industry is mainly caused by .[A] the continuing acquisition.[B] the growing traffic.[C] the cheering Wall Street.[D] the shrinking market.Text 4It is said that in England death is pressing, in Canada inevitable and in California optional. Small wonder. Americans’ life expectancy has nearly doubled over the past century. Failing hips can be replaced, clinical depression controlled, cataracts removed in a 30-minute surgical procedure. Such advances offer the aging population a quality of life that was unimaginable when I entered medicine 50 years ago. But not even a great health-care system can cure death—and our failure to confront that reality now threatens this greatness of ours.Death is normal; we are genetically programmed to disintegrate and perish, even under ideal conditions. We all understand that at some level, yet as medical consumers we treat death as a problem to be solved. Shielded by third-party payers from the co st of our care, we demand everything that can possibly be done for us, even if it’s useless. The most obvious example is late-stage cancer care. Physicians—frustrated by their inability to cure the disease and fearing loss of hope in the patient—too often offer aggressive treatment far beyond what is scientifically justified.In 1950, the US spent $12.7 billion on health care. In 2002, the cost will be $1,540 billion. Anyone can see this trend is unsustainable. Yet few seem willing to try to reverse it. Some scholars conclude that a government with finite resources should simply stop paying for medical care that sustains life beyond a certain age—say 83 or so. Former Colorado governor Richard Lamm has been quoted as saying that the old and infirm “have a dut y to die and get out of the way”, so that younger, healthier people can realize their potential.I would not go that far. Energetic people now routinely work through their 60s and beyond, and remain dazzlingly productive. At 78, Viacom chairman Sumner Redstone jokingly claims to be 53. Supreme Court Justice Sandra Day O’Connor is in her 70s, and former surgeon general C. Everett Koop chairs an Internet start-up in his 80s.These leaders are living proof that prevention works and that we can manage the health problems that come naturally with age. As a mere 68-year-old, I wish to age as productively as they have.Yet there are limits to what a society can spend in this pursuit. As a physician, I know the most costly and dramatic measures may be ineffective and painful. I also know that people in Japan and Sweden, countries that spend far less on medical care, have achieved longer, healthier lives than we have. As a nation, we may be overfunding the quest for unlikely cures while underfunding research on humbler therapies that could improve people’s lives.36. What is implied in the first sentence?[A] Americans are better prepared for death than other people.[B] Americans enjoy a higher life quality than ever before.[C] Americans are over-confident of their medical technology.[D] Americans take a vain pride in their long life expectancy.37. The author uses the example of caner patients to show that .[A] medical resources are often wasted[B] doctors are helpless against fatal diseases[C] some treatments are too aggressive[D] medical costs are becoming unaffordable38. The author’s attitude toward Richard Lamm’s remark is one of.[A] strong disapproval [B] reserved consent[C] slight contempt [D] enthusiastic support39. In contras to the US, Japan and Sweden are funding their medical care.[A] more flexibly [B] more extravagantly[C] more cautiously [D] more reasonably40. The text intends to express the idea that.[A]medicine will further prolong pe ople’s lives[B]life beyond a certain limit is not worth living[C] death should be accepted as a fact of life[D] excessive demands increase the cost of health carePart BDirections:Read the following text carefully and then translate the underlined segments into Chinese. Your translation should be written clearly on ANSWER SHEET 2. (10 points)Human beings in all times and places think about their world and wonder at their place in it. Humans are thoughtful and creative, possessed of insatiable curiosity.（41）Furthermore, humans have the ability to modify the environment in which they live, thus subjecting all other life forms to their own peculiar ideas and fancies. Therefore, it is important to study humans in all their richness and diversity in a calm and systematic manner, with the hope that the knowledge resulting from such studies can lead humans to a more harmonious way of living with themselves and with all other life forms on this planet Earth.“Anthropology” derives from the Greek words anthropos “human” and logos“the study of.” By its very name, anthropology encompasses the study of all humankind.Anthropology is one of the social sciences.（42）Social science is that branch of intellectual enquiry which seeks to study humans and their endeavors in the same reasoned, orderly, systematic, and dispassioned manner that natural scientists use for the study of natural phenomena.Social science disciplines include geography, economics, political, science, psychology, and sociology. Each of these social sciences has a subfield or specialization which lies particularly close to anthropology.All the social sciences focus upon the study of humanity. Anthropology is a field-study oriented discipline which makes extensive use of the comparative method in analysis.（43）The emphasis on data gathered first-hand, combined with a cross-cultural perspective brought to the analysis of cultures past and present, makes this study a unique and distinctly important social science.Anthropological analyses rest heavily upon th e concept of culture. Sir Edward Tylor’s formulation of the concept of culture was one of the great intellectual achievements of 19th century science.（44）Tylor defined culture as “…that complex whole which includes belief, art, morals, law, custom, and any other capabilities and habits acquired by man as a member of society.” This insight, so profound in its simplicity, opened up an entirely new way of perceiving and understanding human life. Implicit within Tylor’s definition is the concept that culture is learned. shared, and patterned behavior.（45）Thus, the anthropological concept of “culture,” like the concept of “set” in mathematics, is an abstract concept which makes possible immense amounts of concrete research and understanding.Section III Writing46. Directions:Study the following set of drawings carefully and write an essay entitled in which you should1）describe the set of drawings, interpret its meaning, and2）point out its implications in our life.You should write about 200 words neatly on ANSWER SHEET 2. (20 points)第一部分英语知识运用试题解析一、文章总体分析文章主要论述了教师们应该关注青少年在成长时期所经历的情感、心智和生理上的变化，并采取方法帮助他们适应这些变化，健康成长。
2003年英语真题+答案解析
河南省2003年普通高等学校选拔优秀专科毕业生进入本科阶段学习考试公共英语Part Ⅰ Vocabulary and Structure (1×30 points)Directions:There are 30 incomplete sentences in this part. For each sentence there are 4 choices marked A,B,C and D. Choose the one that best completes the sentence. Then mark the corresponding letter on the Answer Sheet with a single line through the center.1._______that the trade between the two countries reached its highest point.A. It was the 1990sB. That it was in the 1990sC. It was in the 1990sD. During the 1990s2. Mary said that Hemingway was her_______writer and that she had read nearly all of his novels.A. FamousB. popularC. CommonD. favorite3. She_______English since she was a child.A. LearningB. had learnedC. is learningD. has been learning4. Not until ten o’clock _______it was too late to return.A. they realizedB. they did realizeC. did they realizeD. realized they5. Once a decision is made，it must be_______ firmly.A. carried onB. carried outC. carried awayD. carried off6. _______born in Chicago，the writer is most famous for his stories about New York City.A. AlthoughB. SinceC. WhenD. As7. The pen I am writing my letter with is different from______.A. that oneB. oneC. the oneD. the ones8. Without his wise mother，he______such a successful scientist.A. would not becomeB. should not have becomeC. may not have becomeD. could not have become9. ______，we shall go out for a picnic on Sunday.A. Weather permitsB. Weather permittingC. With weather permitsD. With weather permitting10. The color______ from yellow through green to black.A. RangesB. constitutesC. ComposesD. consists11. Experiments ______that accurate measurements be made.A. ShowB. proveC. DemandD. head12. I tried to catch the ball but it was ______my reach.A. BeyondB. besidesC. in addition toD. as well as13. Those scientists were conducting an experiment and expected a good______ .A. EffectB. resultC. ConsequenceD. affect14. Under heavy gunfire，those cameramen risked their lives to give______battle field reports.A. word-for -wordB. face- to- faceC.down -to- earthD. on- the- spot15. I appreciate ______to your birthday party.A. to be invitedB. to have invitedC. having invitedD. being invited16. The rain was heavy and ______the house was damaged.A. ConsequentlyB. constantlyC. ConsistentlyD. continuously17. When it ______school work，the professor will treat his students equally.A. comes to evaluateB. comes to evaluatingC. will come to evaluateD. will come to evaluating18. _____, you will never convince him.A. However long you argueB. However you argue longC. How you argue longD. How long you argue19. ______new products have been successfully trial produced.A. A great dealB. A large amount ofC. A plenty ofD. A large number of20. He has made a discovery，______of great importance to the progress of science and technology.A. I think which isB. which I think isC. that I think isD. of which I think it is21. Smoking is so harmful to personal health that it kills people ______each year than automobile accidents.A. six more timesB. six times moreC. over six timesD. six times22. What they have done for us can’t be measured in ______of money.A. WayB. meansC. TermsD. place23. Go and get some water，______?A. don’t youB. shall youC. can’t youD. won’t you24. You must pack plenty of food for the journey，______，you’ll need warm clothes，so pack them too.A. on the wholeB. otherwiseC. HoweverD. likewise25. Meat ______easily in hot weather.A. turns badlyB. becomes badlyC. goes badD. changes worse26. I couldn’t find ______，and so I took this one.A. a large coat enoughB. a large enough coatC. an enough large coatD. a coat enough large27. What_____ would happen if he knew you felt that way?A. will you thinkB. do you thinkC. you thinkD. you would think28. It is _____impossible to find a good educational computer program in this city.A. BarelyB. merelyC. HardlyD. nearly29. _____him do this job by himself?A. Why not letB. Why not to letC. Why don’t letD. Why you not to let30. Almost _____of the population of this country is literate.A. Ninth tenthsB. nine tenthC. Nine tenthsD. ninth tenPart Ⅱ Cloze (1×20 points)Directions：In this section，there are 20 blanks in the following. For each blank there are four choices marked A，B，C and D. You should choose the right answer that best fits into the passage.Scientists used to explore 31 the surface of the ocean. Now they 32 below the surface，too. They want to know about the ocean water and the 33 and animal life 34 in the ocean.In 1934 the scientist William Beebe dived 3，000 feet below the surface in a hollow steel ball. In 1935 Auguste Picard dived 10，330 feet. In 1960 his son Jean dived to a 35 of 35，800 feet.All these early dives were deep. But the divers could not stay down for very long. They had to 36 the surface after a few seconds. Scientists needed to stay down longer to study life below the surface. 37 ，they succeeded. Cousteau，a Frenchman，was able to 38 down to a depth of 36 feet for one month and to a depth of 90 feet for a week.Now scientists are developing even 39 equipment. With this new equipment，men can stay 40 the surface for days or 41 weeks. In 1962 Consteau 42 a research station 35 feet below the surface. Then in 1964 he set up another station on the ocean floor of the Red Sea.This was the first undersea station to operate 43 help from the surface.Many countries are now studying undersea 44 . The Soviet Union has an undersea laboratory in the Crimean Sea. The United States has a laboratory 50 feet 45 on the ocean floor 46 the Virgin Islands. In 1970 five men lived there for two weeks. Then a 47 of five women scientists stayed in the laboratory. 48 came other teams of men. All were there to explore the ocean depths and 49 plans for the use of its resources. Scientists hope to find enough 50 ，vegetable，and animal wealth there to provide food for the whole world.31. A. in B. to C. On D. under32. A. explore B. are exploring C. explored D. will explore33. A. plant B. situation C.surroundings D. circumstance34. A. deeply B. normally C. commonly D. deep35. A. depth B. range C. length D. rate36. A. come to B. come over C. come across D. come back up to37. A. However B. Gradually C. moreover D. Though38. A. stay B. remain C. keep D. take39. A. good B. well C. better D. more40. A. below B. on C. in D. to41. A. even B. more C. several D. some42. A. put up B. came up C. made up D. set up43. A. with B. under C. for D. without44. A. living B. situation C. structure D. environment45. A. below B. down C. under D. deep46. A. away B. on C. off D. to47. A. group B. gang C. crowd D. team48. A. after B. next C. later D. then49. A. make B. to make C. making D. made50. A. mineral B. mine C. minefield D. mine layerPart Ⅲ Reading Comprehension(3×20 points)Directions: There are four passages in this part. Each passage is followed by some questions orunfinished statements. For each of them there are four choices marked A, B, C and D, you should decide the best choice.Passage 1With a tremendous roar from its rocket engine，the satellite is sent up into the sky. Minutes later，at an altitude of 300 miles，this tiny electronic moon begins to orbit about the earth. Its radio begins to transmit a staggering amount of information about the satellite’s orbital path，the amount of radiation it detects，and the presence of meteorites. Information of all kinds races back to the earth. No human being could possibly copy down all these facts，much less remember and organize them. But an electronic computer can.The marvel of the machine age，the electronic computer has been in use only since 1946. It can do simple computations—add，subtract，multiply，and divide—with lightning speed and perfect accuracy. It can multiply two 10digit numbers in 1／1，000 second，a problem that would take an average person five minutes to do with pencil and paper. Some computers can work 500，000 times faster than any person can.Once it is given a program，that is，a carefully worked out set of instructions devised by a technician trained in computer language，a computer can gather a wide range of information for many purposes. For the scientist it can get information from outer space or from the depths of the ocean. In business and industry the computer prepares factory inventories，keeps track of sales trends and production needs，mails dividend checks，and makes out company payrolls. It can keep bank accounts up to date and make out electric bills. If you are planning a trip by plane，the computer will find out what route to take and what space is available.51. Why does the author regard the electronic computer as the marvel of the machine age?A. Because electronic computers are rare.B. Because people know little about electronic computers.C. Because electronic computers can do much more kinds of work that human beings can’t.D. Because electronic computers have been widely suspected.52. Electronic computers were given the name in the beginning just for the simple reason that_______ .A. they could do computationsB. they could work much faster than human beingsC. they looked like an abacusD. they could remember data quickly53. According to the passage，which of the following is NOT true?A. Computers began to be used in 1946.B. Computers can add，subtract，multiply and divide.C. Computers were widely used during World War Ⅱ.D. No human being can work as fast as a computer.54. A computer can calculate_______.A. with lightning speedB. accuratelyC. with a speed 500，000 times faster than a personD. all above55. In the first sentence of last paragraph，“program” would probably mean_______.A. a performanceB. a showC. a worked out set of instructionsD. a TV playPassage 2On the old time farm in America there were chickens and turkeys. There were also cows，pigs，and other livestock. But there were very few machines. Most of the work was done by the entire farm family with the h elp of a “hired” man. Sometimes extra labourers were needed in busy seasons. Horses provided 79 percent of the power used，human labor 15 percent，and machines only 6 percent.Today all that has changed. On many modern farms machines now supply 96 percent of the power，human labor 3 percent，and horses 1 percent. Modern farms are enterprising businesses which keep only the livestock that can pay its way. The children go to school by bus every morning，the parents work on the farm or in the house，and hired help is seldom needed. Their work has been replaced by a whole army of farm machines.Farmers in the machine age also use the new fertilizers，new sprays，new feeds，new hybrid seeds，and other helps developed by farm sciences. As a result the farmers are able to produce more food with less labour. This means fewer but larger farms and fewer but moreprosperous farmers.56. In old days，most of the work on a farm was done by_______.A. all the farm familyB. livestockC. farm machinesD. both A and B57. From the first paragraph，we learn that in old days the most important farming power was____.A. LivestockB. farmersC. MachinesD. extra laborers58. On today’s farms，the chief reason why livestock is still raised would probably be____.A. that farmers conventions are difficult to be given upB. that farmers want to gain more profitC. that farmers have to do soD. that farmers work has been replaced by machines59. On modern farms，farmers need mostly all the followings EXCEPT_______.A. farm scienceB. seedsC. FertilizersD. hard work60. In the last sentence of Paragraph 3，“fewer but larger farms and fewer but more prosperous farmers” means_______.A. hired help is seldom neededB. productivity on the farms has been raised greatlyC. productivity on the farms has fallen sharplyD. more food is never neededPassage 3City traffic jam—one of the least wanted effects of the motor vehicle—is something with which we’re all familiar and for which most of us have an answer. But which solution is best?Some people suggest for better roads，others for cars to be banned(禁止) from city centers and yet others say better public transport would attract drivers from their lonely and boring journeys.But the important question is what natural power creates a big city center. We are，after all，in an age of electronic communication; our big shopping areas have moved out of city centers，and our living areas moved out of them long ago.Yet some force causes offices and service industries related to them to gather in London or New York or Tokyo. This suggests that far from the problems of a crowded environment forcing companies and people to move out，there is a critical (重大的) size beyond which more companies are attracted to move in. Nobody seems to know why，yet the answer is important to the way traffic jam is dealt with.61. Which of the following is NOT mentioned in the passage as the cause of the traffic problem?A. The poor public transport.B. The over concentration of shopping centers.C. The great number of cars in the city centers.D. The bad road conditions.62. According to the passage，offices and their associated service industries move to big cities because of_______.A. the convenience of communicationB. the gathering of companiesC. some unknown forceD. the convenience of shopping63. The word “This”(Line 2，Para.4) refers to_______.A. some forceB. traffic jamC. the force that causes offices to move in or outD. the fact that companies gather in big cities64. This passage is most likely taken from_______.A. a novelB. a TV playC. an instructionD. a news report65. The topic of this passage would probably be_______.A. Best Answer to Traffic ProblemB. The Banning of Cars in the City CenterC. The Formation of City CenterD. Searching for Ways to Solve Traffic ProblemPassage 4Mrs. Cox teaches English in a large high school located in the inner area of a big city on the West Coast. Ever since she was a young girl，Mrs. Cox had wanted to become a teacher. She has taught eight years now and hasn’t changed her mind.After she graduated from high school，Mrs. Cox went on to college. Four years later，she received her bachelor’s degree (B.A.) in English and her teaching certificate. Then she was qualified to teach in the secondary schools of her state. In the summers，Mrs. Cox takes more classes. Someday she hopes to get a master’s degree (M.A.).With an M.A.，she will receive a higher salary.The school day at Mrs. Cox’s high school，like that in many high schools in the United States，is divided into six periods of one hour each. Mrs. Cox must teach five of these six periods. During her free period，which for her is from 2 to 3 p.m.，Mrs. Cox must meet with parents，order supplies，make out examinations，check assignments，and take care of many other things. In short，her free period isn’t really free at all. Mrs. Cox works steadily from the time she arrives at school in the morning until the time she leaves for home late in the afternoon.66. Mrs. Cox wants to be a teacher because_______.A. she likes teachingB. she is a young girlC. she has many problems to deal withD. she doesn’t mind what she is doing67. From paragraph two，we can infer_______.A. Mrs. Cox has received a teaching certificateB. Mrs. Cox likes teaching very muchC. Mrs. Cox takes more classesD. Mrs. Cox is a qualified teacher68. She will receive a higher salary if_______.A. she gets a master’s degreeB. she takes more classesC. she has studies normal teachingD. she gets a bachelor’s degree69. According to the passage, all the followings are true EXCEPT_______.A. she graduated from high schoolB. she graduated from collegeC. she has got her bachelor’s degreeD. she teaches five periods for a school day70. The best title for the passage would probably be_______.A. A Typical School DayB. Teacher Mrs. CoxC. Mrs. Cox Likes TeachingD. Mrs. Cox Has Got Her Teaching CertificatePart Ⅳ Translation (2.5×8 points)Section ADirections：There are 4 sentences in this section. You are required to translate them from Chinese into English.71.这口大钟大约有三个人那么高。
2003-数一真题、标准答案及解析
(3)2003年全国硕士研究生入学统一考试数学一真题、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)的平均值为40 (cm)，贝U 的置信度为0.95的置信区间是(注：标准正态分布函数值(1.96) 0.975, (1.645) 0.95.)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，只有一项符合题目要求, 把所选项前的字母填在题后的括号内)一个极小值点和两个极大值点 两个极小值点和一个极大值点 两个极小值点和两个极大值点已知函数f(x,y)在点(0,0)的某个邻域内连续，且lim f (x, y)―xyx 0, y 0(1)lim (cos x)x 0(2)曲面z x 2(3) 设x 21ln(1 x 2)2x 4y z 0平行的切平面的方程是a n cosnx()，则 a2 =(4) 从R 2的基到基1 的过渡矩阵为(5) 设二维随机变量 (X,Y)的概率密度为 f (x, y)6x, 0 0,x y 其他,1,则 P{X Y 1}(6) 已知一批零件的长度 X (单位：cm)服从正态分布 N(,1)，从中随机地抽取16个零件，得到长度(1) 设函数f(x)在()内连续，其导函数的图形如图所示，则f(x)有(A) (B) (C) 0, lim b nn1 ,lim c nn，则必有(A) a nb n 对任意n 成立.(B) b n C n 对任意(C) 极限lim a n C n 不存在. n(D)极限lim b n C n 不存在.n1，则2 2 2(x y )n y 2与平面 0n(A) 点(0,0)不是f(x,y)的极值点.(B) 点(0,0)是f(x,y)的极大值点.(C) 点(0,0)是f(x,y)的极小值点.(D)根据所给条件无法判断点(0,0)是否为f(x,y)的极值点.(4)设向量组I:1, 2, , r可由向量组II:1? 2 ,,s线性表示，则(A)当r s时, 向量组II必线性相关•(B)当(r s 时，向量组II必线性相关(C)当r s时，向量组I必线性相关•(D)当j r s 时，向量组1必线性相关[](5)设有齐次线性方程组Ax=0和Bx=0,其中A,B均为m n矩阵，现有4个命题:①若Ax=0的解均是Bx=0的解，则秩(A)秩(B)；②若秩(A)秩(B)，则Ax=0的解均是Bx=0的解；③若Ax=0与Bx=0同解，则秩(A)=秩(B)；④若秩(A)=秩(B)，则Ax=0与Bx=0同解.以上命题中正确的是(A)①②•(B)①③•(C)②④•(D)③④•[ ](6) 设随机变量X~t(n)(n1),Y 1X2，则(A)2Y~ (n )•(B)Y〜2(n 1).(C)Y ~ F(n,1)・(D)Y〜F(1, n).[]三、(本题满分10分)过坐标原点作曲线y=lnx的切线，该切线与曲线y=lnx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V.四、(本题满分12分)1 2x ( i)n将函数f (x) arctan 展开成x的幕级数，并求级数的和•1 2x n 02n 1五、(本题满分10分)已知平面区域D {(x, y) 0 x ,0 y }，L为D的正向边界•试证：sin y . sin x . sin y . sin x .(1) ;xe dy ye dx xe dy ye dx；sin y . sin x . - 2(2) ；xe dy ye dx 2 .六、(本题满分10分)某建筑工程打地基时，需用汽锤将桩打进土层•汽锤每次击打，都将克服土层对桩的阻力而作功层对桩的阻力的大小与桩被打进地下的深度成正比(比例系数为k,k>0)•汽锤第一次击打将桩打进地下根据设计方案，要求汽锤每次击打桩时所作的功与前一次击打时所作的功之比为常数r(0<r<1).问(1) 汽锤击打桩3次后，可将桩打进地下多深？(2) 若击打次数不限，汽锤至多能将桩打进地下多深？.设土a m.(注：m 表示长度单位米.) 七、(本题满分12分)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1 : ax2by 3c 0， 2 : bx2cy 3a 0， 3: cx2ay 3b 0.试证这三条直线交于一点的充分必要条件为 十一、(本题满分10分) 已知甲、乙两箱中装有同种产品，其中甲箱中装有3件合格品和3件次品，乙箱中仅装有3件合格品.从 甲箱中任取3件产品放入乙箱后，求：(1) 乙箱中次品件数的数学期望； (2) 从乙箱中任取一件产品是次品的概率 .十二、(本题满分8分) 设总体X 的概率密度为设函数y=y(x)在()内具有二阶导数，且y 0, x x(y)是y=y(x)的反函数.(1) 试将x=x(y)所满足的微分方程第(ydydxsin x)(-dx)3 0变换为y=y(x)满足的微分方程; dy求变换后的微分方程满足初始条件y(0) 0, y (0)、(本题满分12分) 设函数f(x)连续且恒大于零，2 2 2f (x y z )dv(t)y 2)dF(t))d，G(t)f(x 2D(t)t 2，1f(x )dx(t) {( x, y, z) x 2 2 y 2 .2-1z t }，D(t){(x, y) x 22 .2,yt}(1)讨论F(t)在区间 (0, )内的单调性(2)证明当t>0时， F(t)-G(t).九、(本题满分10分)3 2 2 0 1 0设矩阵A 2 3 2， P 1 0 1， B P 1A P ，求 B+2E 2 2 3 0 0 1a b c 0.其中的特征值与特征向量，其中A *为D(t)0是未知参数•从总体X 中抽取简单随机样本 X 1,X 2, ,X n ，记? min (X^X ?, ,X n ).(1) 求总体 X 的分布函数 F(x); (2) 求统计量 ?的分布函数 F ?(x)；(3) 如果用 ?作为 的估计量，讨论它是否具有无偏性f(x)2e 2(x ),x0, x其中2003年考研数学一真题评注、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)1ln(1 x 2)lim 1_ ln cosx【详解 1】lim(cosx)ln(1 x )= e x 0ln(1 x)x 0sin x1所以原式=e 2故所求的切平面方程为2(x 1) 4(y 2) (z 5) 0,即卩 2x 4y z 5._ 1= ,e .(1) i|m (cos x)【分析】1型未定式，化为指数函数或利用公式lim f (x)g(x) (1 ) = e lim(f(x) 1)g(x)进行计算求极限均而lim x 0 ln(1In cos xx 2) lim 竺空x 0x 2limcosxx 02x，故原式=e1 e"【详解2】 因为lim (cos x 1)x 0ln(1 x 2)1 2xlim 2—x 0 x 2(2) 曲面z2y 与平面2x4yz 0平行的切平面的方程是 2x 4y z 5. 【分析】 待求平面的法矢量为 n {2,4, 1}，因此只需确定切点坐标即可求出平面方程 ，而切点坐标可根据曲面z 2 2x y 切平面的法矢量与 n{2,4, 1}平行确定•【详解】22令 F (x, y, z) z x y ,则F x 2x ， F y 2y ， F z 1 .设切点坐标为(x 0,y °,Z 0)，则切平面的法矢量为 { 2x 。
2003年高考全国卷文科数学真题及答案
2003年高考全国卷文科数学真题及答案一、选择题（共12小题，每小题5分，满分60分）1．（5分）直线y＝2x关于x轴对称的直线方程为（ ）A．B．C．y＝﹣2x D．y＝2x2．（5分）已知x∈（，0），cos x，则tan2x等于（ ）A．B．C．D．3．（5分）抛物线y＝ax2的准线方程是y＝2，则a的值为（ ）A．B．C．8 D．﹣84．（5分）等差数列{a n}中，已知a1，a2+a5＝4，a n＝33，则n为（ ）A．48 B．49 C．50 D．515．（5分）双曲线虚轴的一个端点为M，两个焦点为F1、F2，∠F1MF2＝120°，则双曲线的离心率为（ ）A．B．C．D．6．（5分）设函数若f（x0）＞1，则x0的取值范围是（ ）A．（﹣1，1）B．（﹣1，+∞）C．（﹣∞，﹣2）∪（0，+∞）D．（﹣∞，﹣1）∪（1，+∞）7．（5分）已知f（x5）＝lgx，则f（2）＝（ ）A．lg2 B．lg32 C．D．8．（5分）函数y＝sin（x+φ）（0≤φ≤π）是R上的偶函数，则φ＝（ ）A．0 B．C．D．π9．（5分）已知点（a，2）（a＞0）到直线l：x﹣y+3＝0的距离为1，则a＝（ ）A．B．C．D．10．（5分）已知圆锥的底面半径为R，高为3R，它的内接圆柱的底面半径为，该圆柱的全面积为（ ）A．2πR2B．C．D．11．（5分）已知长方形的四个顶点A（0，0），B（2，0），C（2，1）和D（0，1），一质点从AB的中点P0沿与AB夹角为θ的方向射到BC上的点P1后，依次反射到CD、DA和AB 上的点P2、P3和P4（入射角等于反射角）若P4与P0重合，则tgθ＝（ ）A．B．C．D．112．（5分）棱长都为的四面体的四个顶点在同一球面上，则此球的表面积为（ ）A．3πB．4πC．3D．6π二、填空题（共4小题，每小题4分，满分16分）13．（4分）不等式的解集是 ．14．（4分）在的展开式中，x3的系数是 （用数字作答）15．（4分）在平面几何里，有勾股定理“设△ABC的两边AB，AC互相垂直，则AB2+AC2＝BC2”，拓展到空间，类比平面几何的勾股定理，研究三棱锥的侧面面积与底面面积间的关系，可以得出正确的结论是：“设三棱锥A﹣BCD的三个侧面ABC、ACD、ADB两两互相垂直，则 ．”16．（4分）如图，一个地区分为5个行政区域，现给地图着色，要求相邻区域不得使用同一颜色．现有4种颜色可供选择，则不同的着色方法共有 种．（以数字作答）三、解答题（共6小题，满分74分）17．（12分）已知正四棱柱ABCD﹣A1B1C1D1．AB＝1，AA1＝2，点E为CC1中点，点F为BD1中点．（1）证明EF为BD1与CC1的公垂线；（2）求点D1到面BDE的距离．18．（12分）已知复数z的辐角为60°，且|z﹣1|是|z|和|z﹣2|的等比中项．求|z|．19．（12分）已知数列{a n}满足a1＝1，a n＝3n﹣1+a n﹣1（n≥2）．（Ⅰ）求a2，a3；（Ⅱ）证明．20．（12分）已知函数f（x）＝2sin x（sin x+cos x）．（1）求函数f（x）的最小正周期和最大值；（2）在给出的直角坐标系中，画出函数y＝f（x）在区间上的图象．21．（12分）在某海滨城市附近海面有一台风，据监测，当前台风中心位于城市O（如图）的东偏南方向300km的海面P处，并以20km/h的速度向西偏北45°方向移动，台风侵袭的范围为圆形区域，当前半径为60km，并以10km/h的速度不断增大，问几小时后该城市开始受到台风的侵袭？22．（14分）已知常数a＞0，在矩形ABCD中，AB＝4，BC＝4a，O为AB的中点，点E、F、G分别在BC、CD、DA上移动，且，P为GE与OF的交点（如图），问是否存在两个定点，使P到这两点的距离的和为定值？若存在，求出这两点的坐标及此定值；若不存在，请说明理由．2003年全国统一高考数学试卷（文科）参考答案与试题解析一、选择题（共12小题，每小题5分，满分60分）1．（5分）直线y＝2x关于x轴对称的直线方程为（ ）A．B．C．y＝﹣2x D．y＝2x 【解答】解：∵直线y＝f（x）关于x对称的直线方程为y＝﹣f（x），∴直线y＝2x关于x对称的直线方程为：y＝﹣2x．故选：C．2．（5分）已知x∈（，0），cos x，则tan2x等于（ ）A．B．C．D．【解答】解：∵cos x，x∈（，0），∴sin x．∴tan x．∴tan2x．故选：D．3．（5分）抛物线y＝ax2的准线方程是y＝2，则a的值为（ ）A．B．C．8 D．﹣8 【解答】解：抛物线y＝ax2的标准方程是x2y，则其准线方程为y2，所以a．故选：B．4．（5分）等差数列{a n}中，已知a1，a2+a5＝4，a n＝33，则n为（ ）A．48 B．49 C．50 D．51【解答】解：设{a n}的公差为d，∵，a2+a5＝4，∴d4d＝4，即5d＝4，解得d．∴an（n﹣1），令a n＝33，即33，解得n＝50．故选：C．5．（5分）双曲线虚轴的一个端点为M，两个焦点为F1、F2，∠F1MF2＝120°，则双曲线的离心率为（ ）A．B．C．D．【解答】解：根据双曲线对称性可知∠OMF2＝60°，∴tan∠OMF2，即c b，∴a b，∴e．故选：B．6．（5分）设函数若f（x0）＞1，则x0的取值范围是（ ）A．（﹣1，1）B．（﹣1，+∞）C．（﹣∞，﹣2）∪（0，+∞）D．（﹣∞，﹣1）∪（1，+∞）【解答】解：当x0≤0时，，则x0＜﹣1，当x0＞0时，则x0＞1，故x0的取值范围是（﹣∞，﹣1）∪（1，+∞），故选：D．7．（5分）已知f（x5）＝lgx，则f（2）＝（ ）A．lg2 B．lg32 C．D．【解答】解：令x5＝2，∴得x，∵f（x5）＝lgx，∴f（2）＝lg lg2．故选：D．8．（5分）函数y＝sin（x+φ）（0≤φ≤π）是R上的偶函数，则φ＝（ ）A．0 B．C．D．π【解答】解：当φ＝0时，y＝sin（x+φ）＝sin x为奇函数不满足题意，排除A；当φ时，y＝sin（x+φ）＝sin（x）为非奇非偶函数，排除B；当φ时，y＝sin（x+φ）＝cos x，为偶函数，满足条件．当φ＝π时，y＝sin（x+φ）＝﹣sin x，为奇函数，故选：C．9．（5分）已知点（a，2）（a＞0）到直线l：x﹣y+3＝0的距离为1，则a＝（ ）A．B．C．D．【解答】解：由点到直线的距离公式得：，∵a＞0，∴a．故选：C．10．（5分）已知圆锥的底面半径为R，高为3R，它的内接圆柱的底面半径为，该圆柱的全面积为（ ）A．2πR2B．C．D．【解答】解：设圆锥内接圆柱的高为h，则，解得，所以圆柱的全面积为：s＝2．故选：B．11．（5分）已知长方形的四个顶点A（0，0），B（2，0），C（2，1）和D（0，1），一质点从AB的中点P0沿与AB夹角为θ的方向射到BC上的点P1后，依次反射到CD、DA和AB 上的点P2、P3和P4（入射角等于反射角）若P4与P0重合，则tgθ＝（ ）A．B．C．D．1【解答】解：由于若P4与P0重合，故P2、P3也都是所在边的中点，因为ABCD是长方形，根据对称性可知P0P1的斜率是，则tgθ．故选：C．12．（5分）棱长都为的四面体的四个顶点在同一球面上，则此球的表面积为（ ）A．3πB．4πC．3D．6π【解答】解：借助立体几何的两个熟知的结论：（1）一个正方体可以内接一个正四面体；（2）若正方体的顶点都在一个球面上，则正方体的体对角线就是球的直径．则球的半径R，∴球的表面积为3π，故选：A．二、填空题（共4小题，每小题4分，满分16分）13．（4分）不等式的解集是 （2，4] ．【解答】解：∵x0，∴x＞0，∵不等式，两边平方得，4x﹣x2＜x2，∴2x2﹣4x＞0，解得，x＞2，x＜0（舍去），∵4x﹣x2≥0，∴0≤x≤4，∴综上得：不等式的解集为：（2，4]，故答案为（2，4]．14．（4分）在的展开式中，x3的系数是 （用数字作答）【解答】解：根据题意，对于，有T r+1＝C99﹣r•x9﹣r•（）r＝（）r•C99﹣r•x9﹣2r，令9﹣2r＝3，可得r＝3，当r＝3时，有T4x3，故答案．15．（4分）在平面几何里，有勾股定理“设△ABC的两边AB，AC互相垂直，则AB2+AC2＝BC2”，拓展到空间，类比平面几何的勾股定理，研究三棱锥的侧面面积与底面面积间的关系，可以得出正确的结论是：“设三棱锥A﹣BCD的三个侧面ABC、ACD、ADB两两互相垂直，则 S△ABC2+S△ACD2+S△ADB2＝S△BCD2 ．”【解答】解：建立从平面图形到空间图形的类比，于是作出猜想：S△ABC2+S△ACD2+S△ADB2＝S△BCD2．故答案为：S△ABC2+S△ACD2+S△ADB2＝S△BCD2．16．（4分）如图，一个地区分为5个行政区域，现给地图着色，要求相邻区域不得使用同一颜色．现有4种颜色可供选择，则不同的着色方法共有 72 种．（以数字作答）【解答】解：由题意，选用3种颜色时：涂色方法C43•A33＝24种4色全用时涂色方法：C21•A44＝48种所以不同的着色方法共有72种．故答案为：72三、解答题（共6小题，满分74分）17．（12分）已知正四棱柱ABCD﹣A1B1C1D1．AB＝1，AA1＝2，点E为CC1中点，点F为BD1中点．（1）证明EF为BD1与CC1的公垂线；（2）求点D1到面BDE的距离．【解答】解：（1）取BD中点M．连接MC，FM．∵F为BD1中点，∴FM∥D1D且FM D1D．又EC CC1且EC⊥MC，∴四边形EFMC是矩形∴EF⊥CC1．又FM⊥面DBD1．∴EF⊥面DBD1．∵BD1⊂面DBD1．∴EF⊥BD1．故EF为BD1与CC1的公垂线．（Ⅱ）解：连接ED1，有V E﹣DBD1＝V D1﹣DBE．由（Ⅰ）知EF⊥面DBD1，设点D1到面BDE的距离为d．则．∵AA1＝2，AB＝1．∴，，∴．∴故点D1到平面DBE的距离为．18．（12分）已知复数z的辐角为60°，且|z﹣1|是|z|和|z﹣2|的等比中项．求|z|．【解答】解：设z＝（r cos60°+r sin60°i），则复数z的实部为．由题设|z﹣1|2＝|z|•|z﹣2|，即：（z﹣1）（1）＝|z|∴r2﹣r+1＝r，整理得r2+2r﹣1＝0．解得r1，r1（舍去）．即|z|1．19．（12分）已知数列{a n}满足a1＝1，a n＝3n﹣1+a n﹣1（n≥2）．（Ⅰ）求a2，a3；（Ⅱ）证明．【解答】解：（Ⅰ）∵a1＝1，∴a2＝3+1＝4，∴a3＝32+4＝13；（Ⅱ）证明：由已知a n﹣a n﹣1＝3n﹣1，n≥2故a n＝（a n﹣a n﹣1）+（a n﹣1﹣a n﹣2）+…+（a2﹣a1）+a1．n≥2当n＝1时，也满足上式．所以．20．（12分）已知函数f（x）＝2sin x（sin x+cos x）．（1）求函数f（x）的最小正周期和最大值；（2）在给出的直角坐标系中，画出函数y＝f（x）在区间上的图象．【解答】解：（1）f（x）＝2sin2x+2sin x cos x＝1﹣cos2x+sin2x所以函数的最小正周期为π，最大值为；（2）由（1）列表得：xy 11111故函数y＝f（x）在区间上的图象是：21．（12分）在某海滨城市附近海面有一台风，据监测，当前台风中心位于城市O（如图）的东偏南方向300km的海面P处，并以20km/h的速度向西偏北45°方向移动，台风侵袭的范围为圆形区域，当前半径为60km，并以10km/h的速度不断增大，问几小时后该城市开始受到台风的侵袭？【解答】解：如图建立坐标系：以O为原点，正东方向为x轴正向．在时刻：t（h）台风中心P（x，y）的坐标为令（x′，y′）是台风边缘线上一点，则此时台风侵袭的区域是（x′﹣x）2+（y′﹣y）2≤[r（t）]2，其中r（t）＝10t+60，若在t时，该城市受到台风的侵袭，则有（0﹣x）2+（0﹣y）2≤（10t+60）2，即，即t2﹣36t+288≤0，解得12≤t≤24．答：12小时后该城市开始受到台风侵袭．22．（14分）已知常数a＞0，在矩形ABCD中，AB＝4，BC＝4a，O为AB的中点，点E、F、G分别在BC、CD、DA上移动，且，P为GE与OF的交点（如图），问是否存在两个定点，使P到这两点的距离的和为定值？若存在，求出这两点的坐标及此定值；若不存在，请说明理由．【解答】解：根据题设条件，首先求出点P坐标满足的方程，据此再判断是否存在两定点，使得点P到定点距离的和为定值．按题意有A（﹣2，0），B（2，0），C（2，4a），D（﹣2，4a）设k（0≤k≤1），由此有E（2，4ak），F（2﹣4k，4a），G（﹣2，4a﹣4ak）．直线OF的方程为：2ax+（2k﹣1）y＝0，①直线GE的方程为：﹣a（2k﹣1）x+y﹣2a＝0．②从①，②消去参数k，得点P（x，y）坐标满足方程2a2x2+y2﹣2ay＝0，整理得．当时，点P的轨迹为圆弧，所以不存在符合题意的两点；当时，点P轨迹为椭圆的一部分，点P到该椭圆焦点的距离的和为定长；当时，点P到椭圆两个焦点的距离之和为定值；当时，点P到椭圆两个焦点的距离之和为定值2a．。
全国计算机等级考试三级网络技术真题2003年9月
全国计算机等级考试三级网络技术真题2003年9月(总分：100.00，做题时间：90分钟)一、{{B}}选择题{{/B}}(总题数：60，分数：60.00)1.我国第一条与国际互联网连接的专线是从中科院高能所到斯坦福大学直线加速器中心，它建成于（分数：1.00）A.1989年6月B.1991年6月√C.1993年6月D.1995年6月解析：【解析】本题考核应试者对Internet在中国的发展历史的认识。
1991年，中国科学院高能物理研究所通过租用国际卫星信道建立了与美国SLAC国家实验室的64Kbps的专线连接。
2.系统的可靠性通常用MTBF和MTTR来表示。
其中MTBF的意义是（分数：1.00）A.每年故障发生次数B.每年故障维修时间C.平均无故障时间√D.平均故障修复时间解析：【解析】系统性的可靠性通常用平均无故障时间(MTBF)和平均故障修复时间(MTTR)来表示。
如果系统的MTBF很长而MTTR很短，则该系统的可靠性就很高。
3.所谓“超流水线”技术的含义是（分数：1.00）A.两条流水线同时工作B.把指令与数据分开C.动态预测分支转移D.细化流水、提高主频√解析：【解析】本题考查超流水线技术的概念。
4.奔腾采用了增强的64位数据总线，它的含义是（分数：1.00）A.内部总线是32位的，而与存储器之间的外部总线是64位的√B.内部总线是64位的，而与存储器之间的外部总线是32位的C.内部总线是32位的，而与输出设备之间的外部总线是64位的D.内部总线是64位的，而与输出设备之间的外部总线是32位的解析：【解析】本题考查奔腾芯片的技术特点。
奔腾芯片的内部总线是32位的，但与存储器之间的外部总线增为64位。
5.主机板有多种分类方法，按系统总线带宽分类的是（分数：1.00）A.16MB主板、32MB主板、64MB主板等B.66MHz主板、100MHz主板等√C.SCSI主板、EDO主板、AGP主板等D.EISA主板、PCI主板、USB主板等解析：【解析】本题考查主板的分类方法：A)是按存储器容量分；B)是按系统总线的带宽分；C)是按数据端口分；D)是按扩展槽分。
2003年英语真题+答案解析
2003年英语真题+答案解析河南省2003年普通高等学校选拔优秀专科毕业生进入本科阶段学习考试公共英语Part Ⅰ Vocabulary and Structure (1×30 points)Directions:There are 30 incomplete sentences in this part. For each sentence there are 4 choices marked A,B,C and D. Choose the one that best completes the sentence. Then mark the corresponding letter on the Answer Sheet with a single line through the center.1._______that the trade between the two countries reached its highest point.A. It was the 1990sB. That it was in the 1990sC. It was in the 1990sD. During the 1990s2. Mary said that Hemingway was her_______writer and that she had read nearly all of his novels.A. FamousB. popularC. CommonD. favorite3. She_______English since she was a child.A. LearningB. had learnedC. is learningD. has been learning4. Not until ten o’clock _______it was too late to return.A. they realizedB. they did realizeC. did they realizeD. realized they5. Once a decision is made，it must be_______ firmly.A. carried onB. carried outC. carried awayD. carried off6. _______born in Chicago，the writer is most famous for his stories about New York City.A. AlthoughB. SinceC. WhenD. As7. The pen I am writing my letter with is different from______.A. that oneB. oneC. the oneD. the ones8. Without his wise mother，he______such a successful scientist.A. would not becomeB. should not have becomeC. may not have becomeD. could not have become9. ______，we shall go out for a picnic on Sunday.A. Weather permitsB. Weather permittingC. With weather permitsD. With weather permitting10. The color______ from yellow through green to black.A. RangesB. constitutesC. ComposesD. consists11. Experiments ______that accurate measurements be made.A. ShowB. proveC. DemandD. head12. I tried to catch the ball but it was ______my reach.A. BeyondB. besidesC. in addition toD. as well as13. Those scientists were conducting an experiment and expected a good______ .A. EffectB. resultC. ConsequenceD. affect14. Under heavy gunfire，those cameramen risked their lives to give______battle field reports.A. word-for -wordB. face- to- faceC.down -to- earthD. on- the- spot15. I appreciate ______to your birthday party.A. to be invitedB. to have invitedC. having invitedD. being invited16. The rain was heavy and ______the house was damaged.A. ConsequentlyB. constantlyC. ConsistentlyD. continuously17. When it ______school work，the professor will treat his students equally.A. comes to evaluateB. comes to evaluatingC. will come to evaluateD. will come to evaluating18. _____, you will never convince him.A. However long you argueB. However you argue longC. How you argue longD. How long you argue19. ______new products have been successfully trial produced.A. A great dealB. A large amount ofC. A plenty ofD. A large number of20. He has made a discovery，______of great importance to the progress of science and technology.A. I think which isB. which I think isC. that I think isD. of which I think it is21. Smoking is so harmful to personal health that it killspeople ______each year than automobile accidents.A. six more timesB. six times moreC. over six timesD. six times22. What they have done for us can’t be measured in ______of money.A. WayB. meansC. TermsD. place23. Go and get some water，______?A. don’t youB. shall youC. can’t youD. won’t you24. You must pack plenty of food for the journey，______，you’ll need warm clothes，so pack them too.A. on the wholeB. otherwiseC. HoweverD. likewise25. Meat ______easily in hot weather.A. turns badlyB. becomes badlyC. goes badD. changes worse26. I couldn’t find ______，and so I took this one.A. a large coat enoughB. a large enough coatC. an enough large coatD. a coat enough large27. What_____ would happen if he knew you felt that way?A. will you thinkB. do you thinkC. you thinkD. you would think28. It is _____impossible to find a good educational computer program in this city.A. BarelyB. merelyC. HardlyD. nearly29. _____him do this job by himself?A. Why not letB. Why not to letC. Why don’t letD. Why you not to let30. Almost _____of the population of this country is literate.A. Ninth tenthsB. nine tenthC. Nine tenthsD. ninth tenPart Ⅱ Cloze (1×20 points)Directions：In this section，there are 20 blanks in the following. For each blank there are four choices marked A，B，C and D. You should choose the right answer that best fits into the passage.Scientists used to explore 31 the surface of the ocean. Now they 32 below the surface，too. They want to know about theocean water and the 33 and animal life 34 in the ocean.In 1934 the scientist William Beebe dived 3，000 feet below the surface in a hollow steel ball. In 1935 Auguste Picard dived 10，330 feet. In 1960 his son Jean dived to a 35 of 35，800 feet.All these early dives were deep. But the divers could not stay down for very long. They had to 36 the surface after a few seconds. Scientists needed to stay down longer to study life below the surface. 37 ，they succeeded. Cousteau，a Frenchman，was able to 38 down to a depth of 36 feet for one month and to a depth of 90 feet for a week.Now scientists are developing even 39 equipment. With this new equipment，men can stay 40 the surface for days or 41 weeks. In 1962 Consteau 42 a research station 35 feet below the surface. Then in 1964 he set up another station on the ocean floor of the Red Sea.This was the first undersea station to operate 43 help from the surface.Many countries are now studying undersea 44 . The Soviet Union has an undersea laboratory in the Crimean Sea. The United States has a laboratory 50 feet 45 on the ocean floor 46 the Virgin Islands. In 1970 five men lived there for two weeks. Then a 47 of five women scientists stayed in the laboratory. 48 came other teams of men. All were there to explore the ocean depths and 49 plans for the use of its resources. Scientists hope to find enough 50 ，vegetable，and animal wealth there to provide food for the whole world.31. A. in B. to C. On D. under32. A. explore B. are exploring C. explored D. will explore33. A. plant B. situation C.surroundings D. circumstance34. A. deeply B. normally C. commonly D. deep35. A. depth B. range C. length D. rate36. A. come to B. come over C. come across D. come back up to37. A. However B. Gradually C. moreover D. Though38. A. stay B. remain C. keep D. take39. A. good B. well C. better D. more40. A. below B. on C. in D. to41. A. even B. more C. several D. some42. A. put up B. came up C. made up D. set up43. A. with B. under C. for D. without44. A. living B. situation C. structure D. environment45. A. below B. down C. under D. deep46. A. away B. on C. off D. to47. A. group B. gang C. crowd D. team48. A. after B. next C. later D. then49. A. make B. to make C. making D. made50. A. mineral B. mine C. minefield D. mine layerPart Ⅲ Reading Comprehension(3×20 points)Directions: There are four passages in this part. Each passage is followed by some questions orunfinished statements. For each of them there are four choices marked A, B, C and D, you should decide the best choice.Passage 1With a tremendous roar from its rocket engine，the satellite is sent up into the sky. Minutes later，at an altitude of 300 miles，this tiny electronic moon begins to orbit about the earth. Its radio begins to transmit a staggering amount of information about the satellite’s orbital path，the amount of radiation it detects，and the presence of meteorites. Information of all kinds races back to the earth. No human being could possibly copy down all thesefacts，much less remember and organize them. But an electronic computer can.The marvel of the machine age，the electronic computer has been in use only since 1946. It can do simple computations—add，subtract，multiply，and divide—with lightning speed and perfect accuracy. It can multiply two 10digit numbers in 1／1，000 second，a problem that would take an average person five minutes to do with pencil and paper. Some computers can work 500，000 times faster than any person can.Once it is given a program，that is，a carefully worked out set of instructions devised by a technician trained in computer language，a computer can gather a wide range of information for many purposes. For the scientist it can get information from outer space or from the depths of the ocean. In business and industry the computer prepares factory inventories，keeps track of sales trends and production needs，mails dividend checks，and makes out company payrolls. It can keep bank accounts up to date and make out electric bills. If you are planning a trip by plane，the computer will find out what route to take and what space is available.51. Why does the author regard the electronic computer as the marvel of the machine age?A. Because electronic computers are rare.B. Because people know little about electronic computers.C. Because electronic computers can do much more kinds of work that human beings can’t.D. Because electronic computers have been widely suspected.52. Electronic computers were given the name in the beginning just for the simple reason that_______ .A. they could do computationsB. they could work much faster than human beingsC. they looked like an abacusD. they could remember data quickly53. According to the passage，which of the following is NOT true?A. Computers began to be used in 1946.B. Computers can add，subtract，multiply and divide.C. Computers were widely used during World War Ⅱ.D. No human being can work as fast as a computer.54. A computer can calculate_______.A. with lightning speedB. accuratelyC. with a speed 500，000 times faster than a personD. all above55. In the first sentence of last paragraph，“program” would probably mean_______.A. a performanceB. a showC. a worked out set of instructionsD. a TV playPassage 2On the old time farm in America there were chickens and turkeys. There were also cows，pigs，and other livestock. But there were very few machines. Most of the work was done by the entire farm family with the h elp of a “hired” man. Sometimes extra labourers were needed in busy seasons. Horses provided 79 percent of the power used，human labor 15 percent，and machines only 6 percent.Today all that has changed. On many modern farms machines now supply 96 percent of the power，human labor 3percent，and horses 1 percent. Modern farms are enterprising businesses which keep only the livestock that can pay its way. The children go to school by bus every morning，the parents work on the farm or in the house，and hired help is seldom needed. Their work has been replaced by a whole army of farm machines.Farmers in the machine age also use the new fertilizers，new sprays，new feeds，new hybrid seeds，and other helps developed by farm sciences. As a result the farmers are able to produce more food with less labour. This means fewer but larger farms and fewer but moreprosperous farmers.56. In old days，most of the work on a farm was done by_______.A. all the farm familyB. livestockC. farm machinesD. both A and B57. From the first paragraph，we learn that in old days the most important farming power was____.A. LivestockB. farmersC. MachinesD. extra laborers58. On today’s farms，the chief reason why livestock is still raised would probably be____.A. that farmers conventions are difficult to be given upB. that farmers want to gain more profitC. that farmers have to do soD. that farmers work has been replaced by machines59. On modern farms，farmers need mostly all thefollowings EXCEPT_______.A. farm scienceB. seedsC. FertilizersD. hard work60. In the last sentence of Paragraph 3，“fewer but larger farms and fewer but more prosperous farmers” means_______.A. hired help is seldom neededB. productivity on the farms has been raised greatlyC. productivity on the farms has fallen sharplyD. more food is never neededPassage 3City traffic jam—one of the least wanted effects of the motor vehicle—is something with which we’re all familiar and for which most of us have an answer. But which solution is best?Some people suggest for better roads，others for cars to be banned(禁止) from city centers and yet others say better public transport would attract drivers from their lonely and boring journeys.But the important question is what natural power creates a big city center. We are，after all，in an age of electronic communication; our big shopping areas have moved out of city centers，and our living areas moved out of them long ago.Yet some force causes offices and service industries related to them to gather in London or New York or Tokyo. This suggests that far from the problems of a crowded environment forcing companies and people to move out，there is a critical (重大的) size beyond which more companies are attracted to move in. Nobody seems to know why，yet the answer is important to theway traffic jam is dealt with.61. Which of the following is NOT mentioned in the passage as the cause of the traffic problem?A. The poor public transport.B. The over concentration of shopping centers.C. The great number of cars in the city centers.D. The bad road conditions.62. According to the passage，offices and their associated service industries move to big cities because of_______.A. the convenience of communicationB. the gathering of companiesC. some unknown forceD. the convenience of shopping63. The word “This”(Line 2，Para.4) refers to_______.A. some forceB. traffic jamC. the force that causes offices to move in or outD. the fact that companies gather in big cities64. This passage is most likely taken from_______.A. a novelB. a TV playC. an instructionD. a news report65. The topic of this passage would probably be_______.A. Best Answer to Traffic ProblemB. The Banning of Cars in the City CenterC. The Formation of City CenterD. Searching for Ways to Solve Traffic ProblemPassage 4Mrs. Cox teaches English in a large high school located in theinner area of a big city on the West Coast. Ever since she was a young girl，Mrs. Cox had wanted to become a teacher. She has taught eight years now and hasn’t changed her mind.After she graduated from high school，Mrs. Cox went on to college. Four years later，she received her bachelor’s degree (B.A.) in English and her teaching certificate. Then she was qualified to teach in the secondary schools of her state. In the summers，Mrs. Cox takes more classes. Someday she hopes to get a master’s degree (M.A.).With an M.A.，she will receive a higher salary.The school day at Mrs. Cox’s high school，like that in many high schools in the United States，is divided into six periods of one hour each. Mrs. Cox must teach five of these six periods. During her free period，which for her is from 2 to 3 p.m.，Mrs. Cox must meet with parents，order supplies，make out examinations，check assignments，and take care of many other things. In short，her free period isn’t really free at all. Mrs. Cox works steadily from the time she arrives at school in the morning until the time she leaves for home late in the afternoon.66. Mrs. Cox wants to be a teacher because_______.A. she likes teachingB. she is a young girlC. she has many problems to deal withD. she doesn’t mind what she is doing67. From paragraph two，we can infer_______.A. Mrs. Cox has received a teaching certificateB. Mrs. Cox likes teaching very muchC. Mrs. Cox takes more classesD. Mrs. Cox is a qualified teacher68. She will receive a higher salary if_______.A. she gets a master’s degreeB. she takes more classesC. she has studies normal teachingD. she gets a bachelor’s degree69. According to the passage, all the followings are true EXCEPT_______.A. she graduated from high schoolB. she graduated from collegeC. she has got her bachelor’s degreeD. she teaches five periods for a school day70. The best title for the passage would probably be_______.A. A Typical School DayB. Teacher Mrs. CoxC. Mrs. Cox Likes T eachingD. Mrs. Cox Has Got Her Teaching CertificatePart Ⅳ Translation (2.5×8 points)Section ADirections：There are 4 sentences in this section. You are required to translate them from Chinese into English.71.这口大钟大约有三个人那么高。
2003考研数一真题解析
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(4)【答案】
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【详解】 n 维向量空间中，从基1, 2 ,, n 到基 1, 2 ,, n 的过渡矩阵 P 满足
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(5)【答案】(B) 【分析】本题可找反例用排除法进行分析，但①、②两个命题的反例比较复杂一些，关键是 抓住③、④，迅速排除不正确的选项．
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(3)【答案】 ( A)
【详解】由 lim x0, y0
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2003年国家公务员考试行测真题答案解析
参考答案及解析第一部分 数字运算一、数字推理1.B【解析】该数列相邻两数的差成3、4、5一组循环的规律，所以空缺项应为20＋5＝25，故选B。
2.C【解析】该数列相邻两数的差为2 的n次方（n=1,2,3……），分别为21，22，23，24……因此，空缺项应为31＋25＝63。
故选C。
3.C【解析】该数列是n的n次方（n=1,2,3……），11，22，33……55，所以要选的数应该是4的4次方即256，故选C。
4.B【解析】该数列的规律比较难找，需要相邻两数做差后再次做差，我们从给出的五个数相邻两数做差得到17、9、5、3，再将这四个数做差得到8、4、2，可以发现它们都是2的n次方（n=1,2,3……），所以空缺项应为36＋17＋24＝69，故答案选B。
5.A【解析】该数列的奇数项的分子都为2，分母是首项为3，公差为2的等差数列3、5、7……；偶数项的分子都为1，分母是首项为2，公差为1的等差数列2、3、4……，故选A。
二、数学运算6.D【解析】设原价为1，进价为x。
则售价为0.8，毛利为0.2x。
0.8-0.2x=x，x= 。
如果以原价出售，则售毛利为1- = 。
所以，毛利相当于进价的 ÷ =12。
7.C【解析】由题中可知大号衬衫、小号衬衫各50件，白色衬衫共25件，蓝色衬衫共75件。
题中已告诉大号白色衬衫有10件，可知大号蓝色衬衫有50-10=40件，则剩余的蓝色衬衫全是小号的，共75-40=35（件）。
8.C【解析】票价为14元时，不算空位的总售价为14×100=1 400（元），若算上空位可知总售价应低于1 360元，所以可排除A、B；票价为16元时的总售价为：16×100-16×15=1 360（元）与题意相符，故C正确。
9.A【解析】设2000年销售台数为x，则2001年销售台数为：x（1+20%），即1.2x。
设2000年每台的售价为y，则2001年为：y(1-20%)，即，0.8y。
2003年全国硕士研究生入学统一考试数学一真题及答案
2003年全国硕士研究生入学统一测试数学一试题解析一、填空题(此题共6小题,每题4分,总分值24分.)12~71 2Tln(1 x )(1)㈣(cos x) =.1【答案】【考点】两个重要极限【难易度】★【详解】此题涉及到的主要知识点：- lim对于1型不定式,可以采取lim(1 ) lim(1 ) e x 0( 0, ),进而转化为0 ,0,—,通过等价无穷小或洛必达法那么来计算.1x21 1 1 cosx 1 q 1--- 丁 ---- - (cosx1) 亍lim2 lim_2_ _ln(1 x2) cosx 1 ln(1 x2) x 01n(1 x2) x 0 x2 2 解析：1im o(cosx) (1im(1 cosx 1) () e () e x e 22 2,(2)曲面z x y与平面2x 4y z 0平行的切平面的万程是 .【答案】2x 4y z 5【考点】曲面的切平面【难易度】★★【详解】解析：令F(x, y,z) z x2 y2,贝U F x2x , F y 2y , F z 1 .设切点坐标为小.力...),那么切平面的法矢量为{ 2X O, 2y0,1},其与平面2x 4y z 0平行,因此有30 30 —,可解得X O1, y0 2,相应地有Z O x2 y2 5.2 4 1故所求的切平面方程为2(x 1)4(y 2) (z 5) 0,即2x 4y z 5.(3)设x2a n cosnx(冗x 力,那么a2 =【考点】函数在［0,1］上的余弦级数 【难易度】★★【详解】此题涉及到的主要知识点:将f(x)( x )展开为余弦级数f(x) a n cosnx( x ),其系数计算公式 n 042为 a n — ° f (x) cos nxdx .解析：根据余弦级数的定义,有= -[x 2sin2x0 Sin2x 2xdx【考点】向量空间及其相关概念 【难易度】★★【详解】此题涉及到的主要知识点:a 2 - x 2 cos2xdx 212 ,x dsin2xo(4)从R 2到基-xdcos2x -[xcos2x到基 ° cos2xdx]的过渡矩阵为n 维向量空间中,从基2,n 到基n 的过渡矩阵P 满足n ］P,因此过渡矩阵P 为:P=[ 1, 2 , , n ] [ 1 , 2 ,,一 ___ 一.1 11解析：根据7E 义,从 R 的基12到基10 ,1 11 的过渡矩阵为22]11 11 11.=112 0(5)设二维随机变量(X ,Y)的概率密度为f (x, y)6x, 0 x y 1, 0,其他,那么P{ X Y 1} =.- 1【答案】14【考点】二维连续型随机变量【难易度】★【详解】此题涉及到的主要知识点:二维随机变量(X, Y)的概率密度f(x, y),求满足一定条件的概率P{g(X,Y) z0},一般可转化为二重积分P{g(X,Y) z o}= f(x, y)dxdy进行at算.g(x,y) Z01 1 (1 x 一2 1解析：P{X Y 1} f(x, y)dxdy ： dx * 6xdy= 2 (6x 12x )dx .4x y 1(6)一批零件的长度X (单位：cm)服从正态分布N( ,1),从中随机地抽取16个零件,得到长度的平均值为40(cm),那么的置信度为0.95的置信区间是.(注：标准正态分布函数值(1.96) 0.975, (1.645) 0.95.)【答案】(39.51,40.49)【考点】区间估计的概念【难易度】★★【详解】此题涉及到的主要知识点：X①方差___________________,1 ,对正态总体的数学期望进行估计,可根据—-------- N(0,1),由nu } 1 确定临界值u ,进而确定相应的置信区间.万万②在单个正态总体方差条件下,求期望值的置信区间为 & u /2-j=其中P{U u } 1 ,U : N(0,1). 万解析：方法1：由题设,1 0.95,可见0.05.查标准正态分布表知u 1.96.此题2二、选择题(此题共 6小题,每题4分,总分值24分.每题给出的四个选项中,只有 项符合题目要求,把所选项前的字母填在题后的括号内.)内连续,其导函数的图形如下图,那么 f(x)有()(A) 一个极小值点和两个极大值点.(B)两个极小值点和一个极大值点.(C)两个极小值点和两个极大值点. (D)三个极小值点和一个极大值点. 【答案】C【考点】函数的极值 【难易度】★★【详解】此题涉及到的主要知识点： ①根据导函数的符号,确定原函数的单调性； ②可能的极值点应是导数为零或导数不存在的点；解析：根据导函数的图形可知,一阶导数为零的点有 3个,而x 0那么是导数不存在的点.三 个一阶导数为零的点左右两侧导数符号不一致, 必为极值点,且两个极小值点,一个极大值 点；在x 0左侧一阶导数为正,右侧一阶导数为负,可见 x 0为极大值点,故f(x)共有 两个极小值点和两个极大值点.⑵设a n , b n , g 均为非负数列,且lim a n 0,limb n 1,limg ,那么必有() n nnn 16, X 40,因此,根据 P{ 1.96} 0.95,有0.95,即 P{39.51,40.49} 0.95 ,故的置信度为0.95的置信区间是 (39.51,40.49).方法2: 由题设,0.95,P{U u } P{ u22(u ) 1 0.95,2(u ) 0.975查得 u 1.96.221, n 16, x 40代入(xu2「n ,x u得置信区间(39.51,40.49).(1)设函数f (x)在(【考点】数列极限的性质 【难易度】★★【详解】此题涉及到的主要知识点:①极限只研究n 时的情况,前面的情况不能确定;根据所给条件无法判断点 (0,0)是否为f(x, y)的极值点.【考点】多元函数的极值 【难易度】★★(4)设向量组I: 1, 2,, 「可由向量组II : 1, 2,, s 线性表示,那么()(A)当r s 时,向量组II 必线性相关. (B)当r s 时,向量组II 必线性相关. (C)当r s 时,向量组I 必线性相关.(D)当r s 时,向量组I 必线性相关.D(A) a n b n 对任意n 成立. (B) b n C n 对任意n 成立.(C)极限lim a n c n 不存在.n(D)极限lim b n C n 不存在.n②类似函数极限,不定式的结果是不确定的, 常见不定式包括：0,—,0 0解析：由知识点①知, A, B 不正确,由知识点②知, C 不正确,应选,1 ,D .(3)函数f (x, y)在点(0,0)的某个邻域内连续,且limx 0 y 0f(x,y) xy22、2(x y )1,那么((A) 点(0,0)不是f(x, y)的极值点. (B) 点(0,0)是f (x,y)的极大值点.(C) 点(0,0)是f (x,y)的极小值点.,00;(D) 【详解】解析：由 xlim y f(x, y) xy0 / 22\2(x y )1知, 分子的极限必为零,从而有f(0,0) 0,且f(x, y)xy : (x 2y 2)2 (x,y 充分小时),于是 f(x, y) f(0,0) : xy可见当y x 且x 充分小时,f (x, y) f (0,0)2 一 4 一 ... ...... x 4x 0 ；而当yx 且x 充分小时,f(x, y) 一一一2f(0,0) : x 4x 4 0.故点(0,0)不是f (x, y)的极值点【考点】向量组的秩、向量组的线性相关【难易度】★★【详解】此题涉及到的主要知识点：① r(A m n) m,r(A m n) n ,类似地,向量组的秩不大于向量的个数或维数；②向量组的秩<向量的个数向量组线性相关解析：由题干易知：r(I) r,r(II) s,由于向量组I可由向量组II线性表示,所以r(I) r(II) s(A)当r s时, r(II) s；(B)当r s时, r(II) s(C)当r s时, r(I) r ；(D)当r s时,r(I) r(II) s r r(I) r( 5)设有齐次线性方程组Ax 0和Bx 0,其中A,B 均为m n 矩阵,现有4个命题：①假设Ax 0的解均是Bx 0的解,那么秩(A);^<(B)；②假设秩(A) n秩(B),那么Ax 0的解均是Bx 0的解；③假设Ax 0与Bx 0同解,那么秩瓜)=秩(8)；④假设秩6)=秩(8),那么Ax 0与Bx 0同解.以上命题中正确的选项是( )(A)①②.(B)①③.(C)②④.(D)③④【答案】B【考点】线性方程组的公共解、同解【难易度】★★【详解】此题涉及到的主要知识点：①齐次线性方程组Ax 0解向量的秩为n r(A) ；②向量组A可由向量组B表示,那么有r(A) r(B)；③向量组等价,即向量组的极大无关组等价,有r(A) r(B).解析：①假设Ax 0的解均是Bx 0的解,即方程组Ax 0的解向量可由方程组Bx 0的解向量表示,所以n r(A) n r(B),即秩(A) ＞秩(B),反之不一定成立;③假设Ax 0与Bx 0同解,即方程组 Ax 0的解向量与方程组 Bx 0的解向量等价,所 〜一——一 一一、,1 以n r(A) n r(B),即秩(A)=秩(B),反之不一定成立；如A那么秩(A 尸秩(B )=1 ,但AX 0与BX 0不同解.(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V .【考点】平面曲线的切线、定积分的几何应用 【难易度】★★ 【详解】解析:(6) 设随机变量X ~ t(n)(n(A)2(n) .(B)2(n 1).(C) Y~ F(n,1). (D) Y~ F(1,n) .【考点】2分布、F 分布【难易度】★★【详解】此题涉及到的主要知识点:① x 2 x 22 2,x n ~ (n),其中 x i (i 1,2, ,n)~ N(0,1)且相互独立;② F (m, n)2(m)m 2(n) n解析：由题设知,X —U =,其中U ~ N(0,1),V2(n),于是Y 」Y I 2 X 2% U 2这里U 2 ~212(1),根据F 分布的定义知Y —X 2F(n,1).故应选(C).、(此题总分值10分)过坐标原点作曲线 y lnx 的切线,该切线与曲线y lnx 及x 轴围成平面图形(1)设切点的横坐标为 X o ,那么曲线y ln x 在点(x o , ln X o )处的1 . ——(XX o ).由该切线过原点知X o1 2x ................................. arctan 展开成x 的帚级数,并求级数 1 2x【考点】初等函数的骞级数展开式、简单骞级数的和函数的求法 【难易度】★★【详解】此题涉及到的主要知识点: ①哥级数展开一般通过适当的恒等变形、 求导或积分等,转化为可利用哥级数展开的情形.11②函数的哥级数展开1 X x 2x n1 X1 X2 1 1解析：由于 f(x)22 ( 1)n 4n x 2n ,x (1,1).又 f(0)-,1 4x2 no2 24X X _所以 f (x) f (0) f (t)dt 2 [( 1)n 4n t ]dt 04n 0n n(1) 4 2n 1---------- X , X n o 2n 1切线方程是y ln X oIn X o 10 ,从而 X o e.所以该切线的方程为(2)切线y1 ____ _ ______ 1 yy -x.平面图形D 的面积 Ao(e y ey)dye“1-X 与X 轴及直线X e 所围成的三角形绕直线 Xe e 旋转所得的圆锥体 旋转体体积为 e 2.曲线y Inx 与x 轴及直线x e 所围成的图形绕直线x e 旋转所得的1V 2 o (e e y ) dy,因此所求旋转体的体积为V V 1 V 2四、(此题总分值 1 2e312分)(e e y )2dy -(5e 2 12e 3).将函数f (x)o 羽的和•,1 ................................... 在X 一处右边级数成为由于级数21)n n o2n 1 1%收敛,左边函数f(x)连续,所以成立范围sin ysinxsin ysin x ।白xe y dy ye dx 臼xe dy ye dx;L L sin ysinx2(2)oxe dy ye dx 2 冗L【考点】第二类曲线积分的计算、格林公式 【难易度】★★★ 【详解】 解析:方法1 :一」sin ysinxsinx sinx 、(1)左边=0 e dy e dx =0 (e e )dx ,0 sin ysin xsin x sin x 、右边=0 e dy e dx =(e e )dx,sin ysin xsin ysin x ।所以；xe dy ye dx ：xedy ye dx .(2)由于 e s1nx e sinx 2,故由(1)得sin ysin x sin x sin x 2\xedy ye dx 0 (e e )dx 2 .方法2:(1)根据格林公式,得sin y sin x sin y sin x 、 °Lxe dy ye dx (e e )dxdy, D sin ysin xsin ysin x 、xe dy ye dx (e e )dxdy .D由于D 具有轮换对称性,所以sin y sin x sin ysin x 、(e e )dxdy = (e e)dxdy,m11可扩大到x —处.而在x—处,右边级数虽然收敛,但左边函数2 2,,-1 1立范围只能是x (--].2’2f (x)不连续,所以成,1 1 令 x 一,得 f (一)22_ 2 [( 1)4n 1 ] _ ( 1)n4 n 0 2n 1 22n 14 n o 2n 1一 , 1 ( 1)n 1 再由 f (-) 0,得■(一)- — f (-)—.2no2n 1424五、(此题总分值10分)平面区域D (x,y)0 x ,0 y山为D 的正向边界.试证:故xe sin y dy ye sin x dx xe siny dy ye sinx dx.(2)由(1)知sin y sin x sin y sin x、L xe dy ye dx (e e )dxdyDsin y sinx=e dxdy e dxdyD D=e sinx dxdy e sinx dxdy (利用轮换对称性)D Dsin x sin x 2= (e e ) dxdy 2dxdy 2 .D D六、(此题总分值10分)某建筑工程打地基时, 需用汽锤将桩打进土层. 汽锤每次击打,都将克服土层对桩的阻力而作功.设土层对桩的阻力的大小与桩被打进地下的深度成正比(比例系数为k,k 0),汽锤第一次击打将桩打进地下a(m) .根据设计方案,要求汽锤每次击打桩时所作的功与前一次击打时所作的功之比为常数r(0 r 1).问(1)汽锤击打桩3次后,可将桩打进地下多深？(2)假设击打次数不限,汽锤至多能将桩打进地下多深？( 注：m表示长度单位米.)【考点】定积分的物理应用、数列极限的定义及其性质【难易度】★★★【详解】解析：(1)设第n次击打后,桩被打进地下x n,第n次击打时,汽锤所作的功为W n(n 1,2,3,).由题设,当桩被打进地下的深度为x时,土层对桩的阻力的大小为kx ,xx2) ;2(x f a2).1 k 2k 2&k 2所以皿° kxdx —x1 —a , W2 x kxdx —(x2由川2 rW1 可彳x x2 a2 ra2即x2 (1 r)a2.W3 x'kxdx ^(x12x2) ；[x2 (1 r)a2]. 由川3 rW2 r2W1 可得x； (1 r)a2 r2a2,从而x3 V1 r r2a,即汽锤击打3次后,可将桩打进地下W r r2am.(2)由归纳法,设x n V1 r r2r n 1a,那么x n 1k 2 2k 2n 1 2W nix kxdx -(X 21 xn2) =-[x 2i (1 r r )a 2]. x n2 2由于 W n 1rW n r 2W n 1r n W 1,故得 x 2 1 (1 r r n1)a 2 r n a 2,---------------------------- n1 r n 1从而 x n 1.. 1 r r a ----------------------- a.:1 r于是lim x 11L a ,即假设击打次数不限,汽锤至多能将桩打进地下J —a m .n 11 r . 1 r七、(此题总分值12分) )内具有二阶导数,且 y 0,x x(y)是y y(x)的反函数.j d 「, 、 、▼ d 2x (1)试将x x( y)所满足的微分方程d-4 ( y dy 2分方程;3(2)求变换后的微分万程满足初始条件 y( 0) 0, y (0)-的解.【考点】反函数的求导法那么、简单的二阶常系数非齐次线性微分方程 【难易度】★★★【详解】此题涉及到的主要知识点：2dx 11d xd ,dx 、 d , 1、dxy 1y ①一 二=一,—r——(——)=一(—)—=2-dy dyy dy dy dy dx y dy y y(y)dx②二阶常系数非齐次线性微分方程y py q f(x)的通解为：二阶常系数齐次线性微分方程y py q 0的通解加二阶常系数非齐次线性微分方程 解.一 , ........ ........ ,, dx 1 一, 解析：(1)由反函数的求导公式知 dx 工,于是有dy y代入原微分方程得 y y sin x.设函数y y(x)在(dx osin x)(——)0&i 换为dyy y(x)满足的微y py q f(x)的特d 2x dy 2dy(dy)嗫中dx dyy (y)3.(2)方程(* )所对应的齐次方程y y0的通解为Y xC2 eC〔e x*1设方程〔* 〕的特解为y Acosx Bsinx,代入方程〔* 〕,求得A 0, B故y *1 . … -sinx ,从而 y y2 …、_. . . . _ _ * _ V _sinx 的通解是 y Y y C 1e C 2e 2,一一 sinx. 2X⑼e y 由 x ey12分) 2c, 11c得31.故所求初值问题的解为 八、〔此题总分值 设函数f 〔x 〕连续且恒大于零, F(t)f(x 2 Q(t) 其中 (t) (x,y,z)x 2 (1) 讨论F 〔t 〕在区间〔0, (2) 证实当t 0时,F 〔t 〕【考点】二重积分的计算、 【难易度】 ★★★★ 解析:〔1〕 F (t) 所以在〔0, (2) 要证实ty 2 z 2)dV 2—2 --------------------- ,G(t) f(x y )d D(t)f(x 2 y 2)dD(t) t 2 t f(x 2)dx t 2 ,D(t) (x,y)x 2 〕内的单调性. 2 -G(t).冗 t 2重积分的计算、积分上限的函数及其导数 2d 由于F(t) — d 0_ 2 d. 2t . 2tf(t 2) 0 f(r 2)r(t r)dr t 2 2 [0 f (r )rdr ] -2 2f (r )r sin dr_ t _2 22 ° f(r )r dr-2f(r )rdr-2f (r )rdr)± F (t) 0,故 F(t)在(0, 〕内单调增加._ 2 f (r )rdr - 2f(r )dr2 _ 一0时F(t) —G(t),只需证实t一 一 2 一 一.0时,F(t) —G(t) 0,即-2 2 t - 2 t -2f (r )r dr f (r )dr [ f (r )rdr]20.入t 2 2t _ 2 令 g(t) 0 f(r )r dr 0 f (r )drt L 22[0f(r )rdr],那么 g (t) f(t 2) f(r 2)(t r)2dr 0,故 g(t)在(0,由于g(t)在t 0处连续,所以当 t 0时,有g(t))内单调增加g(0) .又g(t) 0,故当 t 0 时,g(t) 0,因此,当t 0时,F(t) 2 -G(t). 九、(此题总分值 10分) 设矩阵A 向量,其中A 为A 的伴随矩阵, P ,求B 2E 的特征值与特征E 为3阶单位矩阵 【考点】矩阵的特征值的计算、矩阵的特征向量的计算【难易度】★★★【详解】此题涉及到的主要知识点: 设B P 1AP, 假设是A 的特征值,对应特征向量为 B 与A 有相同的特征值,但对应特征向量不同, B 对应特征值的特征向量为解析:方法1: 经计算可得 A* 2EE (B 2E) 9)2(3)故B 2E 的特征值为29, 3.121当12 9时,解〔9E A 〕x 0 ,得线性无关的特征向量为k 1 *2是不全为零的任意常数.当3 3时,解〔3E A 〕x 0 ,得线性无关的特征向量为31 1所以属于特征值 33的所有特征向量为k 3 3卜3 1 ,其中k 30为任意常数1方法2：设A 的特征值为,对应特征向量为,即 A0.又因A* A AE,故有A*一,,1、 1.,1、 A,1、1A 于是有 B(P ) P A*P(P ) —(P ), (B 2E)P(— 2)P因此,四 2为B 2E 的特征值,对应的特征向量为 P3 2 223 2 ( 1)2( 7), 223故A 的特征值为121,3 7.1 21时,对应的线性无关特征向量可取为当37时,对应的一个特征向量为31 .所以属于特征值29的所有特征向量为k 1 1 k 21k 1 1 02 k 2 0 ,其中 1由于A 7 0 ,所以由于 E A1全为零的任意常数;十、(此题总分值8分)平面上三条不同直线的方程分别为11 : ax 2by 3c 0, l 2：bx 2cy 3a 0, l 3:cx 2ay 3b 0.试证这三条直线交于一点的充分必要条件为 a b c 0.【考点】线性方程组有解和无解的判定 【难易度】★★【详解】此题涉及到的主要知识点: 三条直线相交于一点,相当于对应线性方程组有唯一解ax 2by3 c, 解析:方法1：必要性：设三条直线11,12,13交于一点,那么线性方程组bx 2cy 3a, (*) cx 2ay 3b,a 有唯一解,故系数矩阵 Abc 2ba 2b 2c 与增广矩阵Ab 2c 2ac 2a3c2223a 6( a b c)[a b c ab ac bc] 3b3(a b c)[( a b)2 (b c)2 (c a)2] 0,0,那么a b c,三条直线相同,故 a b c 0.0,得 P 1 11 因此,B 2E 的三个特征值分别为 9, 9, 3.对应于牛I 征值9的全部特征向量为 k 1P k 2P k 1k 2 1,其中k 1,k 2是不1对应于牛I 征值3的全部特征向量为k 3 1 1其中 k 3是不为零的任意常数3c3a 的秩均为2,于是 3ba 2b A| b 2c c 2a假设(a b)2 (b c)2 (c a)2…a2b 22123 2 由于2(ac b )2[a(a b) b ] = 2[(a 一 b) —b [ 0,b 2c24故秩(A) 2.于是,秩(A 尸秩(A) =2.因此方程组(*)有唯一解,即三直线11,12,13交1 点.X oax 2by 3c,充分性：考虑线性方程组bx 2cy 3a, cx2ay3b,将方程组(*)的三个方程相加,并由 a b c0.可知,方程组(*)等价于方程组22222[a(a b) b ] =- [a b (a b) ] 0,故方程组(* *)有唯一解,所以方程组(*)有唯一解,即三直线l i 」2/3交于一点.卜一、(此题总分值10分)有3件合格品.从甲箱中任取 3件产品放入乙箱后,求: (1)乙箱中次品件数 X 的数学期望；充分性：由a b c 0,那么从必要性的证实可知,0,故秩(A) 3. 方法2:必要性：设三直线交于一点(x o ,y .),那么y .为AX 0的非零解,其中a 2b 3c A b 2c 3a. c2a 3b a 2b 3c于是A b 2c 3a c2a 3b6( a bc)[ a 2 b 2 c 2 ab ac bc]3( a bc )[( a b )2 (b c )2 (c a)2] 0,假设(a b)2(b c)22(c a) 0 ,那么 ab c,三条直线相同,故 a bc 0.ax 2by3G bx 2cy3a.(* *)(*)由于 a 2b 2(ac b 2)甲、乙两箱中装有同种产品,其中甲箱中装有3件合格品和3件次品,乙箱中仅装(2)从乙箱中任取一件产品是次品的概率.【考点】随机变量的数学期望、离散型随机变量的概率分布【难易度】★★★解析：(1)X的可能取值为0, 1, 2, 3,X的概率分布为因此(2 ){Xk 3 kP{X k} C7VC6EXc 1/ 9 c 9 c 10——1——2——3——20202020k=0,1,2,3.设A表示事件“从乙箱中任取一件产品是次品〞3｝构成完备事件组,因此根据全概率公式,有3P(A) P{Xk 0k}P{AX3k}= P{Xk 0,由于｛X0},{X 1} ,{X 2},3kP{X k}k 01EX6十二、(此题总分值8分)设总体X的概率密度为2e 2(xf(x)0,其中0是未知参数.从总体X中抽取简单随机样本X i,X2,(1)(2)(3)min( X i,X2, ,X n)求总体X的分布函数F (x)；求统计量的分布函数F?(x)；如果用作为的估计量,讨论它是否具有无偏性.【考点】连续型随机变量分布函数的计算、随机变量的分布函数的概念、估计量的无偏性【难易度】★★★【详解】此题涉及到的主要知识点:作为无偏性估计量就是E()x 1 e 2(x) x解析：(1) F(x) f(t)dt , '0, x .(2) F?(x) P{? x} P{min(X i,X2, ,XJ x}=1 P{min(X「X2, ,X n) x)=1 P{X i x,X2 x, ,X n x)=1 [1 F(x)]n2n(x )e ,x0, x一、dF?(x) 2ne2n(x),x ,f ?(x)dx 0, x .由于E? xf ?(x)dx 2nxe 2n(x)dx= —2n 所以？作为的估计量不具有无偏性。
2003年山东专升本(计算机)真题试卷(题后含答案及解析)
2003年山东专升本（计算机）真题试卷(题后含答案及解析)题型有：1.0101e+007C．1．011le+007D．1．01lle+007正确答案：B解析：将各十六进制位分别对应转换成二进制数即可。
A→1010、B→1011，所以AB→l0l01011，答案为B。
20．计算机最主要的工作特点是( )。
A．高速度B．高精度C．存记忆能力D．存储程序和程序控制正确答案：D21．和广域网相比,局域网( )。
A．有效性好但可靠性差B．有效性差但可靠性高C．有效性好可靠性也高D．有效性差可靠性也差正确答案：C解析：比较来讲，局域网的有效性和可靠性都要好于广域网。
22．一个完整的计算机系统应包括( )。
A．系统硬件和系统软件B．硬件系统和软件系统C．主机和外部设备D．主机、键盘、显示器和辅助存储器正确答案：B23．按照网络分布和覆盖的地理范围，可将计算机网络分为( )。
A．局域网、互联网和internet网B．广域网、局域网和城域网C．广域网、互联网和城域网D．Internet网、城域网和Novell网正确答案：B24．目前微型计算机中采用的逻辑元件是( )。
A．小规模集成电路B．中规模集成电路C．大规模和超大规模集成电路D．分立元件正确答案：C25．软件与程序的区别是( )。
A．程序价格便宜、软件价格昂贵B．程序是用户自己编写的，而软件是由厂家提供的C．程序是用高级语言编写的，而软件是由机器语言编写的D．软件是程序以及开发、使用和维护所需要的所有文档的总称，而程序是软件的一部分正确答案：D26．下列四个不同进制的无符号整数中，数值最小的是( )。
A．10010010(B)B．22l(O)C．147(D)D．94(H)正确答案：B解析：把各种进制的数转换成同一进制后再进行比较，比如都转换为我们熟悉的十进制。
选项A为二进制，对应146t选项B为八进制，对应145；选项D 为十六进制，对应148。
都与选项C比较可知，答案为B。
2003年考研数学三真题及全面解析
2003年全国硕士入学统考数学(三)试题及答案一、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）（1）设,0,0,0,1cos )(=≠⎪⎩⎪⎨⎧=x x xx x f 若若λ其导函数在x=0处连续，则λ的取值范围是2>λ. 【分析】 当≠x 0可直接按公式求导，当x=0时要求用定义求导.【详解】 当1>λ时，有,0,0,0,1sin 1cos )(21=≠⎪⎩⎪⎨⎧+='--x x xx x x x f 若若λλλ 显然当2>λ时，有)0(0)(lim 0f x f x '=='→，即其导函数在x=0处连续.（2）已知曲线b x a x y +-=233与x 轴相切，则2b 可以通过a 表示为=2b 64a .【分析】 曲线在切点的斜率为0，即0='y ，由此可确定切点的坐标应满足的条件，再根据在切点处纵坐标为零，即可找到2b 与a 的关系.【详解】 由题设，在切点处有03322=-='a x y ，有 .220a x =又在此点y 坐标为0，于是有0300230=+-=b x a x ,故 .44)3(6422202202a a a x a x b =⋅=-=（3）设a>0，，x a x g x f 其他若,10,0,)()(≤≤⎩⎨⎧==而D 表示全平面，则⎰⎰-=Ddxdy x y g x f I )()(= 2a .【分析】 本题积分区域为全平面，但只有当10,10≤-≤≤≤x y x 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】 ⎰⎰-=D dxdy x y g x f I )()(=dxdy a x y x ⎰⎰≤-≤≤≤10,102=.])1[(2102112a dx x x a dy dx ax x=-+=⎰⎰⎰+（4）设n 维向量0,),0,,0,(<=a a a Tα；E 为n 阶单位矩阵，矩阵TE A αα-=， T aE B αα1+=， 其中A 的逆矩阵为B ，则a= -1 .【分析】 这里Tαα为n 阶矩阵，而22a T=αα为数，直接通过E AB =进行计算并注意利用乘法的结合律即可.【详解】 由题设，有)1)((T Ta E E AB αααα+-= =TT T T a a E αααααααα⋅-+-11=TT T T a a E αααααααα)(11-+-=TT T a a E αααααα21-+-=E aa E T=+--+αα)121(,于是有 0121=+--a a ，即 0122=-+a a ，解得 .1,21-==a a 由于A<0 ,故a=-1.（5）设随机变量X 和Y 的相关系数为0.9, 若4.0-=X Z ，则Y 与Z 的相关系数为0.9 .【分析】 利用相关系数的计算公式即可. 【详解】 因为)4.0()()]4.0([()4.0,cov(),cov(---=-=X E Y E X Y E X Y Z Y =)(4.0)()()(4.0)(Y E X E Y E Y E XY E +-- =E(XY) – E(X)E(Y)=cov(X,Y), 且.DX DZ =于是有 cov(Y ,Z)=DZDY Z Y ),cov(=.9.0),cov(==XY DYDXY X ρ（6）设总体X 服从参数为2的指数分布，n X X X ,,,21 为来自总体X 的简单随机样本，则当∞→n 时，∑==n i i n X n Y 121依概率收敛于 21.【分析】 本题考查大数定律：一组相互独立且具有有限期望与方差的随机变量n X X X ,,,21 ，当方差一致有界时，其算术平均值依概率收敛于其数学期望的算术平均值：).(1111∞→→∑∑==n EX n X n ni i pn i i【详解】 这里22221,,,n X X X 满足大数定律的条件，且22)(i i i EX DX EX +==21)21(412=+，因此根据大数定律有 ∑==n i i n X n Y 121依概率收敛于.21112=∑=n i i EX n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，只有一项符合题目要求，把所选项前的字母填在题后的括号内）（1）设f(x)为不恒等于零的奇函数，且)0(f '存在，则函数xx f x g )()(=(A) 在x=0处左极限不存在. (B) 有跳跃间断点x=0.(C) 在x=0处右极限不存在. (D) 有可去间断点x=0. [ D ] 【分析】 由题设，可推出f(0)=0 , 再利用在点x=0处的导数定义进行讨论即可. 【详解】 显然x=0为g(x)的间断点，且由f(x)为不恒等于零的奇函数知，f(0)=0. 于是有 )0(0)0()(lim )(lim)(lim 00f x f x f x x f xg x x x '=--==→→→存在，故x=0为可去间断点.（2）设可微函数f(x,y)在点),(00y x 取得极小值，则下列结论正确的是(A) ),(0y x f 在0y y =处的导数等于零. （B ）),(0y x f 在0y y =处的导数大于零. (C) ),(0y x f 在0y y =处的导数小于零. (D) ),(0y x f 在0y y =处的导数不存在. [ A ]【分析】 可微必有偏导数存在，再根据取极值的必要条件即可得结论.【详解】 可微函数f(x,y)在点),(00y x 取得极小值，根据取极值的必要条件知0),(00='y x f y ，即),(0y x f 在0y y =处的导数等于零, 故应选(A).（3）设2nn n a a p +=，2nn n a a q -=， ,2,1=n ，则下列命题正确的是(A) 若∑∞=1n na条件收敛，则∑∞=1n np与∑∞=1n nq都收敛.(B) 若∑∞=1n na绝对收敛，则∑∞=1n np与∑∞=1n nq都收敛.(C) 若∑∞=1n na条件收敛，则∑∞=1n np与∑∞=1n nq敛散性都不定.(D) 若∑∞=1n na绝对收敛，则∑∞=1n np与∑∞=1n nq敛散性都不定. [ B ]【分析】 根据绝对收敛与条件收敛的关系以及收敛级数的运算性质即可找出答案. 【详解】 若∑∞=1n na绝对收敛，即∑∞=1n na收敛，当然也有级数∑∞=1n na收敛，再根据2nn n a a p +=，2nn n a a q -=及收敛级数的运算性质知，∑∞=1n np与∑∞=1n nq都收敛，故应选(B).（4）设三阶矩阵⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=a b b b a b b b a A ，若A 的伴随矩阵的秩为1，则必有 (A) a=b 或a+2b=0. (B) a=b 或a+2b ≠0.(C) a ≠b 且a+2b=0. (D) a ≠b 且a+2b ≠0. [ C ] 【分析】 A 的伴随矩阵的秩为1, 说明A 的秩为2，由此可确定a,b 应满足的条件. 【详解】 根据A 与其伴随矩阵A*秩之间的关系知，秩(A)=2，故有0))(2(2=-+=b a b a ab b b a bbb a ，即有02=+b a 或a=b.但当a=b 时，显然秩(A)2≠, 故必有 a ≠b 且a+2b=0. 应选(C).（5）设s ααα,,,21 均为n 维向量，下列结论不正确的是(A) 若对于任意一组不全为零的数s k k k ,,,21 ，都有02211≠+++s s k k k ααα ，则s ααα,,,21 线性无关.(B) 若s ααα,,,21 线性相关，则对于任意一组不全为零的数s k k k ,,,21 ，都有.02211=+++s s k k k ααα(C) s ααα,,,21 线性无关的充分必要条件是此向量组的秩为s.(D) s ααα,,,21 线性无关的必要条件是其中任意两个向量线性无关. [ B ] 【分析】 本题涉及到线性相关、线性无关概念的理解，以及线性相关、线性无关的等价表现形式. 应注意是寻找不正确的命题.【详解】(A): 若对于任意一组不全为零的数s k k k ,,,21 ,都有02211≠+++s s k k k ααα ，则s ααα,,,21 必线性无关，因为若s ααα,,,21 线性相关，则存在一组不全为零的数s k k k ,,,21 ,使得 02211=+++s s k k k ααα ，矛盾. 可见（A ）成立.(B): 若s ααα,,,21 线性相关，则存在一组，而不是对任意一组不全为零的数s k k k ,,,21 ，都有.02211=+++s s k k k ααα (B)不成立.(C) s ααα,,,21 线性无关,则此向量组的秩为s ；反过来，若向量组s ααα,,,21 的秩为s ，则s ααα,,,21 线性无关，因此(C)成立.(D) s ααα,,,21 线性无关,则其任一部分组线性无关，当然其中任意两个向量线性无关，可见(D)也成立.综上所述，应选(B).（6）将一枚硬币独立地掷两次，引进事件：1A ={掷第一次出现正面}，2A ={掷第二次出现正面}，3A ={正、反面各出现一次}，4A ={正面出现两次}，则事件(A) 321,,A A A 相互独立. (B) 432,,A A A 相互独立.(C) 321,,A A A 两两独立. (D) 432,,A A A 两两独立. [ C ] 【分析】按照相互独立与两两独立的定义进行验算即可，注意应先检查两两独立，若成立，再检验是否相互独立.【详解】 因为21)(1=A P ，21)(2=A P ，21)(3=A P ，41)(4=A P ， 且 41)(21=A A P ，41)(31=A A P ，41)(32=A A P ，41)(42=A A P 0)(321=A A A P ，可见有)()()(2121A P A P A A P =，)()()(3131A P A P A A P =，)()()(3232A P A P A A P =， )()()()(321321A P A P A P A A A P ≠，)()()(4242A P A P A A P ≠.故321,,A A A 两两独立但不相互独立；432,,A A A 不两两独立更不相互独立，应选(C).三 、（本题满分8分）设).1,21[,)1(1sin 11)(∈--+=x x x x x f πππ 试补充定义f(1)使得f(x)在]1,21[上连续.【分析】 只需求出极限)(lim 1x f x -→，然后定义f(1)为此极限值即可.【详解】 因为)(lim 1x f x -→=])1(1sin 11[lim 1x x x x --+-→πππ =xx xx x πππππsin )1(sin )1(lim 111---+-→=xx x xx ππππππππcos )1(sin cos lim 111-+---+-→=xx x x xx ππππππππππsin )1(cos cos sin lim 11221----+-→=.1π由于f(x)在)1,21[上连续，因此定义π1)1(=f ，使f(x)在]1,21[上连续.四 、（本题满分8分）设f(u,v)具有二阶连续偏导数，且满足12222=∂∂+∂∂v f u f ，又)](21,[),(22y x xy f y x g -=,求.2222ygx g ∂∂+∂∂ 【分析】 本题是典型的复合函数求偏导问题：),(v u f g =，)(21,22y x v xy u -==，直接利用复合函数求偏导公式即可，注意利用.22uv fv u f ∂∂∂=∂∂∂ 【详解】vf x u f y xg ∂∂+∂∂=∂∂， .vf y u f x yg ∂∂-∂∂=∂∂ 故 vf v f x v u f xy u f y xg ∂∂+∂∂+∂∂∂+∂∂=∂∂2222222222，.2222222222v f vf y u v f xy u f x yg ∂∂-∂∂+∂∂∂-∂∂=∂∂ 所以 222222222222)()(vf y x u f y x yg x g ∂∂++∂∂+=∂∂+∂∂ =.22y x +五 、（本题满分8分） 计算二重积分 .)sin(22)(22dxdy y x e I Dy x +=⎰⎰-+-π其中积分区域D=}.),{(22π≤+y x y x【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：θθsin ,cos r y r x ==，有 dxdy y x e e I Dy x )sin(22)(22+=⎰⎰+-π=.sin 2022dr r re d e r ⎰⎰-πππθ令2r t =，则 tdt e e I t sin 0⎰-=πππ.记 tdt e A t sin 0⎰-=π，则t t de e A --⎰-=int 0π=]cos sin [0⎰----ππtdt e t e t t=⎰--πcos t tde=]sin cos [0tdt e t e t t⎰--+-ππ=.1A e -+-π因此 )1(21π-+=e A ， ).1(2)1(2πππππe e e I +=+=-六、（本题满分9分）求幂级数∑∞=<-+12)1(2)1(1n nnx n x 的和函数f(x)及其极值. 【分析】 先通过逐项求导后求和，再积分即可得和函数，注意当x=0时和为1. 求出和函数后，再按通常方法求极值.【详解】.1)1()(1212∑∞=-+-=-='n n n xxx x f 上式两边从0到x 积分，得).1ln(211)0()(202x dt t t f x f x+-=+-=-⎰ 由f(0)=1, 得).1(),1ln(211)(2<+-=x x x f 令0)(='x f ，求得唯一驻点x=0. 由于,)1(1)(222x x x f +--='' 01)0(<-=''f ，可见f(x)在x=0处取得极大值，且极大值为 f(0)=1.七、（本题满分9分）设F(x)=f(x)g(x), 其中函数f(x),g(x)在),(+∞-∞内满足以下条件： )()(x g x f ='，)()(x f x g ='，且f(0)=0, .2)()(xe x g xf =+(1) 求F(x)所满足的一阶微分方程； (2) 求出F(x)的表达式.【分析】 F(x)所满足的微分方程自然应含有其导函数，提示应先对F(x)求导，并将其余部分转化为用F(x)表示，导出相应的微分方程，然后再求解相应的微分方程.【详解】 (1) 由)()()()()(x g x f x g x f x F '+'=' =)()(22x f x g +=)()(2)]()([2x g x f x g x f -+ =(22)x e -2F(x), 可见F(x)所满足的一阶微分方程为.4)(2)(2x e x F x F =+'(2) ]4[)(222C dx e e e x F dx xdx +⎰⋅⎰=⎰-=]4[42C dx e e x x +⎰-=.22x xCe e-+将F(0)=f(0)g(0)=0代入上式，得 C=-1. 于是.)(22x xe e x F --=八、（本题满分8分）设函数f(x)在[0，3]上连续，在（0，3）内可导，且f(0)+f(1)+f(2)=3, f(3)=1.试证必存在)3,0(∈ξ，使.0)(='ξf【分析】 根据罗尔定理，只需再证明存在一点c )3,0[∈，使得)3(1)(f c f ==，然后在[c,3]上应用罗尔定理即可. 条件f(0)+f(1)+f(2)=3等价于13)2()1()0(=++f f f ，问题转化为1介于f(x)的最值之间，最终用介值定理可以达到目的.【详解】 因为f(x)在[0，3]上连续，所以f(x)在[0，2]上连续，且在[0，2]上必有最大值M 和最小值m ，于是M f m ≤≤)0(， M f m ≤≤)1(， M f m ≤≤)2(. 故.3)2()1()0(M f f f m ≤++≤由介值定理知，至少存在一点]2,0[∈c ，使.13)2()1()0()(=++=f f f c f因为f(c)=1=f(3), 且f(x)在[c,3]上连续，在(c,3)内可导，所以由罗尔定理知，必存在)3,0()3,(⊂∈c ξ，使.0)(='ξf九、（本题满分13分） 已知齐次线性方程组⎪⎪⎪⎩⎪⎪⎪⎨⎧=+++++=+++++=+++++=+++++,0)(,0)(,0)(,0)(332211332211332211332211nn nn n n n n x b a x a x a x a x a x b a x a x a x a x a x b a x a x a x a x a x b a 其中.01≠∑=ni ia试讨论n a a a ,,,21 和b 满足何种关系时，(1) 方程组仅有零解；(2) 方程组有非零解. 在有非零解时，求此方程组的一个基础解系.【分析】方程的个数与未知量的个数相同，问题转化为系数矩阵行列式是否为零，而系数行列式的计算具有明显的特征：所有列对应元素相加后相等. 可先将所有列对应元素相加，然后提出公因式，再将第一行的（-1）倍加到其余各行，即可计算出行列式的值.【详解】 方程组的系数行列式ba a a a a ba a a a ab a a a a a b a A n n n n++++= 321321321321 =).(11∑=-+ni i n a b b(1) 当0≠b 时且01≠+∑=ni iab 时，秩(A)=n ，方程组仅有零解.(2) 当b=0 时，原方程组的同解方程组为 .02211=+++n n x a x a x a 由01≠∑=ni ia可知，),,2,1(n i a i =不全为零. 不妨设01≠a ，得原方程组的一个基础解系为T a a )0,,0,1,(121 -=α，T a a )0,,1,0,(132 -=α，.)1,,0,0,(,1T n n a a -=α 当∑=-=ni iab 1时，有0≠b ，原方程组的系数矩阵可化为⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡----∑∑∑∑====n i i n nni inni inni ia a a a a a a a a a a a a a a a a a a a 1321132131213211（将第1行的-1倍加到其余各行，再从第2行到第n 行同乘以∑=-ni ia11倍）→ ⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡----∑=1001010100113211 n ni ia a a a a（ 将第n 行n a -倍到第2行的2a -倍加到第1行，再将第1行移到最后一行）→.0000100101010011⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡---由此得原方程组的同解方程组为12x x =，13x x =，1,x x n = . 原方程组的一个基础解系为 .)1,,1,1(T=α十、（本题满分13分）设二次型)0(222),,(31232221321>+-+==b x bx x x ax AX X x x x f T ，中二次型的矩阵A 的特征值之和为1，特征值之积为-12. (1) 求a,b 的值；(2) 利用正交变换将二次型f 化为标准形，并写出所用的正交变换和对应的正交矩阵. 【分析】 特征值之和为A 的主对角线上元素之和，特征值之积为A 的行列式，由此可求出a,b 的值；进一步求出A 的特征值和特征向量，并将相同特征值的特征向量正交化（若有必要），然后将特征向量单位化并以此为列所构造的矩阵即为所求的正交矩阵.【详解】 （1）二次型f 的矩阵为.200200⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-=b b a A 设A 的特征值为).3,2,1(=i i λ 由题设，有1)2(2321=-++=++a λλλ，.12242002002321-=--=-=b a b ba λλλ解得 a=1,b= -2.(2) 由矩阵A 的特征多项式)3()2(22202012+-=+----=-λλλλλλA E ，得A 的特征值.3,2321-===λλλ对于,221==λλ解齐次线性方程组0)2(=-x A E ，得其基础解系 T )1,0,2(1=ξ，.)0,1,0(2T=ξ对于33-=λ，解齐次线性方程组0)3(=--x A E ，得基础解系.)2,0,1(3T-=ξ由于321,,ξξξ已是正交向量组，为了得到规范正交向量组，只需将321,,ξξξ单位化，由此得T )51,0,52(1=η，T )0,1,0(2=η，.)52,0,51(3T -=η令矩阵[]⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡-==5205101051052321ηηηQ ，则Q 为正交矩阵. 在正交变换X=QY 下，有⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-=300020002AQ Q T ，且二次型的标准形为.322232221y y y f -+=十一、（本题满分13分） 设随机变量X 的概率密度为;],8,1[,0,31)(32其他若∈⎪⎩⎪⎨⎧=x x x fF(x)是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。