2000年考研真题及答案
2000考研数学二真题及答案解析
邻域内当 x < 0 时曲线 y = f (x) 是凸的，在此去心临域内 x > 0 时曲线 y = f (x) 是凹的，
点 (0, f (0)) 是曲线 y = f (x) 的拐点，选(C).
(3)【答案】A 【分析】由选项答案可知需要利用单调性证明，关键在于寻找待证的函数. 题设中已知
f '(x)g(x) − f (x)g '(x) < 0,
f ′′′(x) = (x − [ f ′(x)]2 )′ = 1− 2 f ′(x) f ′′(x)
以 x = 0 代入，有 f ′′′(0) = 1 ，所以
= f ′′′(0) lim f ′′(x)= − f ′′(0) li= m f ′′(x) 1.
x→0
x−0
x→0 x
从而知，存在 x = 0 去心邻域，在此去心邻域内， f ′′(x) 与 x 同号，于是推知在此去心
(C)点 (0, f (0)) 是曲线 y = f (x) 的拐点.
(D) f (0) 不是 f (x) 的极值，点 (0, f (0)) 也不是曲线 y = f (x) 的拐点.
(3 ) 设 f (x), g(x) 是大于零的可导函数，且 f '(x)g(x) − f (x)g '(x) < 0, 则当 a < x < b 时，
(D) y′′′ − 2 y′′ − y′ + 2 y =0.
三、(本题满分 5 分)
∫ 设 f (ln x) = ln(1+ x) ，计算 f (x)dx .
x
四、(本题满分 5 分)
{ } 设 xoy 平面上有正方= 形 D (x, y) 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 及直线 l : x + y = t(t ≥ 0) .若
2000年考研英语真题及答案解析
2000年全真试题Part ⅠClose TestDirections：For each numbered blank in the following passage, there are four choices marked ［A］, ［B］, ［C］and ［D］. Choose the best one and mark your answer on ANSWER SHEET 1 by blackening the corresponding letter in the brackets with a pencil. (10 points) ①If a farmer wishes to succeed, he must try to keep a wide gap between his consumption and his production. ②He must store a large quantity of grain 1 consuming all his grain immediately. ③He can continue to support himself and his family 2 he produces a surplus. ④He must use this surplus in three ways: as seed for sowing, as an insurance 3 the unpredictable effects of bad weather and as a commodity which he must sell in order to 4 old agricultural implements and obtain chemical fertilizers to5 the soil. ⑤He may also need money to construct irrigation6 and improve his farm in other ways. ⑥If no surplus is available,a farmer cannot be 7 . ⑦He must either sell some of his property or 8 extra funds in the form of loans. ⑧Naturally he will try to borrow money at a low 9 of interest, but loans of this kind are not 10 obtainable. ［139 words］1.［A］other than ［B］as well as ［C］instead of ［D］more than2.［A］only if ［B］much as ［C］long before ［D］ever since3.［A］for ［B］against ［C］of ［D］towards4.［A］replace ［B］purchase ［C］supplement ［D］dispose5.［A］enhance ［B］mix ［C］feed ［D］raise6.［A］vessels ［B］routes ［C］paths ［D］channels7.［A］self-confident ［B］self-sufficient［C］self-satisfied ［D］self-restrained8.［A］search ［B］save ［C］offer ［D］seek9.［A］proportion ［B］percentage ［C］rate ［D］ratio10.［A］genuinely ［B］obviously ［C］presumably ［D］frequentlyPart ⅡReading ComprehensionDirections:Each of the passages below is followed by some questions. For each question there are four answers marked ［A］, ［B］, ［C］and ［D］. Read the passages carefully and choose the best answer to each of the questions. Then mark your answer on ANSWER SHEET 1 by blackening the corresponding letter in the brackets with a pencil. (40 points)Passage 1①A history of long and effortless success can be a dreadful handicap, but, if properly handled, it may become a driving force. ②When the United States entered just such a glowing period after the end of the Second World War, it had a market eight times larger than any competitor, giving its industries unparalleled economies of scale. ③Its scientists were the world s best; its workers the most skilled. ④(11)America and Americans were prosperous beyond the dreams of the Europeans and Asians whose economies the war had destroyed.①It was inevitable that this primacy should have narrowed as other countries grew richer. ②Just as inevitably, the retreat from predominance proved painful. ③By the mid-1980s Americans had found themselves at a loss over their fading industrial competitiveness. ④Some huge American industries, such as consumer electronics, had shrunk or vanished in the face of foreign competition. ⑤By 1987 there was only one American television maker left, Zenith. ⑥(Now there is none: Zenith was bought by South Korea’s LG Electronics in July.) ⑦（12）Foreign-made cars and textiles were sweeping into the domestic market. America’s machine-tool industry was on the ropes. ⑧For a while it looked as though the making of semiconductors, which America had invented and which sat at the heart of the new computer age, was going to be the next casualty.①All of this caused a crisis of confidence. ②Americans stopped taking prosperity for granted. ③They began to believe that their way of doing business was failing, and that their incomes would therefore shortly begin to fall as well. ④The mid-1980s brought one inquiry after another into the causes of America’s industrial decline. ⑤Their sometimes sensational findings were filled with warnings about the growing competition from overseas.①How things have changed! ②In 1995 the United States can look back on five years of solid growth while Japan has been struggling. ③（14）Few Americans attribute this solely to such obvious causes as a devalued dollar or the turning of the businesscycle. ④Self-doubt has yielded to blind pride. ⑤“American industry has changed its structure, has gone on a diet, has learnt to be more quick-witted,”according to Richard Cavanaugh, executive dean of Harvard’s Kennedy School of Government. ⑥“It makes me proud to be an American just to see how our businesses are improving their productivity,”says Stephen Moore of the Cato Institute, a think-tank in Washington, DC. ⑦And William Sahlman of the Harvard Business School believes that people will look back on this period as “a golden age of business management in the United States.”［429 words］11. The U.S. achieved its predominance after World War II because.［A］it had made painstaking efforts towards this goal［B］its domestic market was eight times larger than before［C］the war had destroyed the economies of most potential competitors［D］the unparalleled size of its workforce had given an impetus to its economy12. The loss of U.S. predominance in the world economy in the 1980s is manifested in the fact that the American.［A］TV industry had withdrawn to its domestic market［B］semiconductor industry had been taken over by foreign enterprises［C］machine-tool industry had collapsed after suicidal actions［D］auto industry had lost part of its domestic market13. What can be inferred from the passage?［A］It is human nature to shift between self-doubt and blind pride.［B］Intense competition may contribute to economic progress.［C］The revival of the economy depends on international cooperation.［D］ A long history of success may pave the way for further development.14. The author seems to believe the revival of the U.S. economy in the 1990s can be attributed to the.［A］turning of the business cycle ［B］restructuring of industry［C］improved business management ［D］success in educationPassage 2①（15）Being a man has always been dangerous. ②There are about 105 males born for every 100 females, but this ratio drops to near balance at the age of maturity, and among 70-year-olds there are twice as many women as men. ③But the great universal of male mortality is being changed. ④Now, boy babies survive almost as well as girls do. ⑤This means that, for the first time, there will be an excess of boys in those crucial years when they are searching for a mate. ⑥More important, another chance for natural selection has been removed. ⑦Fifty years ago, the chance of a baby (particularly a boy baby) surviving depended on its weight. A kilogram too light or too heavy meant almost certain death. ⑧Today it makes almost no difference. Since much of the variation is due to genes, one more agent of evolution has gone.①There is another way to commit evolutionary suicide: stay alive, but have fewer children. ②Few people are as fertile as in the past.③Except in some religious communities, very few women have 15 children. ④Nowadays the number of births, like the age of death, has become average. ⑤Most of us have roughly the same number of offspring. ⑥（16）Again, differences between people and the opportunity for natural selection to take advantage of it have diminished. ⑦India shows what is happening. The country offers wealth for a few in the great cities and poverty for the remaining tribal peoples. ⑧The grand mediocrity of today—everyone being the same in survival and number of offspring—means that natural selection has lost 80% of its power in upper-middle-class India compared to the tribes.For us, this means that evolution is over; the biological Utopia has arrived. ②Strangely, it has involved little physical change. ③No other species fills so many places in nature. ④But in the past 100, 000 years—even the past 100 years—our lives have been transformed but our bodies have not. ⑤（17）We did not evolve, because machines and society did it for us. ⑥Darwin had a phrase to describe those ignorant of evolution: they “look at an organic being as a savage looks at a ship, as at something wholly beyond his comprehension.”⑦No doubt we will remember a 20th century way of life beyond comprehension for its ugliness. But however amazed our descendants may be at how far from Utopia we were, they will look just like us.［406 words］15. What used to be the danger in being a man according to the first paragraph?［A］ A lack of mates. ［B］ A fierce competition.［C］ A lower survival rate. ［D］ A defective gene.16. What does the example of India illustrate?［A］Wealthy people tend to have fewer children than poor people.［B］Natural selection hardly works among the rich and the poor.［C］The middle class population is 80% smaller than that of the tribes.［D］India is one of the countries with a very high birth rate.17. The author argues that our bodies have stopped evolving because.［A］life has been improved by technological advance［B］the number of female babies has been declining［C］our species has reached the highest stage of evolution［D］the difference between wealth and poverty is disappearing18. Which of the following would be the best title for the passage?［A］Sex Ratio Changes in Human Evolution.［B］Ways of Continuing Man’s Evolution.［C］The Evolutionary Future of Nature.［D］Human Evolution Going Nowhere.Passage 3①（20）When a new movement in art attains a certain fashion, it is advisable to find out what its advocates are aiming at, for, however farfetched and unreasonable their principles may seem today, it is possible that in years to come they may be regarded as normal. ②With regard to Futurist poetry, however, the case is rather difficult, for whatever Futurist poetry may be—even admitting that the theory on which it is based may be right—it can hardly be classed as Literature.①This, in brief, is what the Futurist says: for a century, past conditions of life have been conditionally speeding up, till now we live ina world of noise and violence and speed. ②Consequently, our feelings, thoughts and emotions have undergone a corresponding change. ③（21）This speeding up of life, says the Futurist, requires a new form of expression. ④We must speed up our literature too, if we want to interpret modern stress. ⑤We must pour out a large stream of essential words, unhampered by stops, or qualifying adjectives, or finite verbs. ⑥Instead of describing sounds we must make up words that imitate them; we must use many sizes of type and different colored inks on the same page, and shorten or lengthen words at will.①Certainly their descriptions of battles are confused. ②But it is a little upsetting to read in the explanatory notes that a certain line describes a fight between a Turkish and a Bulgarian officer on a bridge off which they both fall into the river —and then to find that the line consists of the noise of their falling and the weights of the officers: “Pluff! Pluff! A hundred and eighty-five kilograms.”①（22）This, though it fulfills the laws and requirements of Futurist poetry, can hardly be classed as Literature. ②All the same, no thinking man can refuse to accept their first proposition: that a great change in our emotional life calls for a change of expression. ③The whole question is really this: have we essentially changed?［334 words］19. This passage is mainly.［A］ a survey of new approaches to art［B］ a review of Futurist poetry［C］about merits of the Futurist movement［D］about laws and requirements of literature20. When a novel literary idea appears, people should try to.［A］determine its purposes ［B］ignore its flaws［C］follow the new fashions ［D］accept the principles21. Futurists claim that we must.［A］increase the production of literature［B］use poetry to relieve modern stress［C］develop new modes of expression［D］avoid using adjectives and verbs22. The author believes that Futurist poetry is.［A］based on reasonable principles［B］new and acceptable to ordinary people［C］indicative of a basic change in human nature［D］more of a transient phenomenon than literaturePassage 4①（23）Aimlessness has hardly been typical of the postwar Japan whose productivity and social harmony are the envy of the United States and Europe. ②But increasingly the Japanese are seeing a decline of the traditional work-moral values. ③Ten years ago young people were hardworking and saw their jobs as their primary reason for being, but now Japan has largely fulfilled its economic needs, and young people don’t know where they should go next.①The coming of age of the postwar baby boom and an entry of women into the male-dominated job market have limited the opportunities of teen-agers who are already questioning the heavy personal sacrifices involved in climbing Japan’s rigid social ladder to good schools and jobs. ②In a recent survey, it was found that only 24.5 percent of Japanese students were fully satisfied with school life, compared with 67.2 percent of students in the United States. ③In addition, far more Japanese workers expressed dissatisfaction with their jobs than did their counterparts in the 10 other countries surveyed.①While often praised by foreigners for its emphasis on the basics, Japanese education tends to stress test taking and mechanical learning over creativity and self-expression. ②（25）“Those things that do not show up in the test scores—personality, ability, courage or humanity—are completely ignored,”says Toshiki Kaifu, chairman of the ruling Liberal Democratic Party’s education committee. ③“Frustration against this kind of thing leads kids to drop out and run wild.”④Last year Japan experienced 2，125 incidents of school violence, including 929 assaults on teachers. ⑤Amid the outcry, many conservative leaders are seeking a return to the prewar emphasis on moral education. ⑥Last year Mitsuo Setoyama, who was then education minister, raised eyebrows when he argue d that liberal reforms introduced by the American occupation authorities after World War II had weakened the “Japanese moralit y of respect for parents.”①（26）But that may have more to do with Japanese life-styles. ②“In Japan,”says educator Yoko Muro, “it’s never a question of whether you enjoy your job and your life, but only how much you can endure.”③With economic growth has come centralization; fully 76 percent of Japan’s 119 million citizens live in cities where community and the extended family have been abandoned in favor of isolated, two-generation households. ④Urban Japanese have long endured lengthy commutes (travels to and from work) and crowded living conditions, but as the old group and family values weaken, the discomfort is beginning to tell. ⑤In the past decade, the Japanese divorce rate, while still well below that of the United States, has increased by more than 50 percent, and suicides have increased by nearly one-quarter.［447 words］23. In the Westerners’ eyes, the postwar Japan was.［A］under aimless development ［B］ a positive example［C］ a rival to the West ［D］on the decline24. According to the author, what may chiefly be responsible for the moral decline of Japanese society?［A］Women’s participation in social activities is limited.［B］More workers are dissatisfied with their jobs.［C］Excessive emphasis has been placed on the basics.［D］The life-style has been influenced by Western values.25. Which of the following is true according to the author?［A］Japanese education is praised for helping the young climb the social ladder.［B］Japanese education is characterized by mechanical learning as well as creativity.［C］More stress should be placed on the cultivation of creativity.［D］Dropping out leads to frustration against test taking.26. The change in Japanese life-style is revealed in the fact that.［A］the young are less tolerant of discomforts in life［B］the divorce rate in Japan exceeds that in the U.S.［C］the Japanese endure more than ever before［D］the Japanese appreciate their present lifePassage 5①（27）If ambition is to be well regarded, the rewards of ambition—wealth, distinction, control over one’s destiny—must be deemed worthy of the sacrifices made on ambition’s behalf. ②If the tradition of ambition is to have vitality, it must be widely shared; and it especially must be highly regarded by people who are themselves admired, the educated not least among them. ③（28）In an odd way,however, it is the educated who have claimed to have given up on ambition as an ideal. ④What is odd is that they have perhaps most benefited from ambition—if not always their own then that of their parents and grandparents. ⑤There is a heavy note of hypocrisy in this, a case of closing the barn door after the horses have escaped—with the educated themselves riding on them.①Certainly people do not seem less interested in success and its signs now than formerly. ②Summer homes, European travel, BMWs—the locations, place names and name brands may change, but such items do not seem less in demand today than a decade or two years ago.③（29）What has happened is that people cannot confess fully to their dreams, as easily and openly as once they could, lest they be thought pushing, acquisitive and vulgar. ④Instead, we are treated to fine hypocritical spectacles, which now more than ever seem in ample supply: the critic of American materialism with a Southampton summer home; the publisher of radical books who takes his meals in three-star restaurants; the journalist advocating participatory democracy in all phases of life, whose own children are enrolled in private schools. ⑤For such people and many more perhaps not so exceptional, the proper formulation is, “Succeed at all costs but avoid appearing ambitious.”①The attacks on ambition are many and come from various angles; its public defenders are few and unimpressive, where they are not extremely unattractive. ②As a result, the support for ambition as a healthy impulse, a quality to be admired and fixed in the mind of the young, is probably lower than it has ever been in the United States. ③This does not mean that ambition is at an end, that people no longer feel its stirrings and promptings, but only that, no longer openly honored, it is less openly professed. ④Consequences follow from this, of course, some of which are that ambition is driven underground, or made sly. ⑤Such, then, is the way things stand: on the left angry critics, on the right stupid supporters, and in the middle, as usual, the majority of earnest people trying to get on in life. ［431 words］27. It is generally believed that ambition may be well regarded if.［A］its returns well compensate for the sacrifices［B］it is rewarded with money, fame and power［C］its goals are spiritual rather than material［D］it is shared by the rich and the famous28. The last sentence of the first paragraph most probably implies that it is.［A］customary of the educated to discard ambition in words［B］too late to check ambition once it has been let out［C］dishonest to deny ambition after the fulfillment of the goal［D］impractical for the educated to enjoy benefits from ambition29. Some people do not openly admit they have ambition because.［A］they think of it as immoral［B］their pursuits are not fame or wealth［C］ambition is not closely related to material benefits［D］they do not want to appear greedy and contemptible30. From the last paragraph the conclusion can be drawn that ambition should be maintained.［A］secretly and vigorously ［B］openly and enthusiastically［C］easily and momentarily ［D］verbally and spirituallyPart ⅢEnglish-Chinese TranslationDirections:Read the following passage carefully and then translate the underlined segments into Chinese. Your translation must be written neatly on ANSWER SHEET 2. (15 points)Governments throughout the world act on the assumption that the welfare of their people depends largely on the economic strength and wealth of the community. 31）Under modern conditions, this requires varying measures of centralized control and hence the help of specialized scientists such as economists and operational research experts. 32）Furthermore, it is obvious that the strength of a country’s economy is directly bound up with the efficiency of its agriculture and industry, and that this in turn rests upon the efforts of scientists and technologists of all kinds. It also means that governments are increasingly compelled to interfere in these sectors in order to step up production and ensure that it is utilized to the best advantage. For example, they may encourage research in various ways, including the setting up of their own research centers; they may alter the structure of education, or interfere in order to reduce the wastage of natural resources or tap resources hitherto unexploited; or they may cooperate directly in the growing number ofinternational projects related to science, economics and industry. In any case, all such interventions are heavily dependent on scientific advice and also scientific and technological manpower of all kinds.33）Owing to the remarkable development in mass-communications, people everywhere are feeling new wants and are being exposed to new customs and ideas, while governments are often forced to introduce still further innovations for the reasons given above. At the same time, the normal rate of social change throughout the world is taking place at a vastly accelerated speed compared with the past. For example, 34）in the early industrialized countries of Europe the process of industrialization—with all the far-reaching changes in social patterns that followed—was spread over nearly a century, whereas nowadays a developing nation may undergo the same process in a decade or so. All this has the effect of building up unusual pressures and tensions within the community and consequently presents serious problems for the governments concerned. 35）Additional social stresses may also occur because of the population explosion or problems arising from mass migration movements—themselves made relatively easy nowadays by modern means of transport. As a result of all these factors, governments are becoming increasingly dependent on biologists and social scientists for planning the appropriate programs and putting them into effect. ［390 words］Section ⅣWriting(15 points)36.Directions:A. Study the following two pictures carefully and write an essay of at least 150 words.B. Your essay must be written neatly on ANSWER SHEET 2.C. Your essay should meet the requirements below:1）Describe the pictures.2）Deduce the purpose of the painter of the pictures.3）Suggest counter-measures.2000年英语试题答案Part ⅠCloze Test1. C2. A3. B4. A5. C6. D7. B8.D9. C 10. DPart ⅡReading ComprehensionPassage 111. C 12. D 13.B 14. APassage 215.C 16.B 17.A 18.DPassage 319.B 20.A 21.C 22.DPassage 423.B 24.D 25.C 26.APassage 527.A 28.C 29.D 30.BPart ⅢEnglish-Chinese Translation31.在现代条件下，这需要程度不同的集中控制措施，从而就需要获得诸如经济学和运筹学等领域的专家的协助。
2000考研数一真题答案及详细解析
一、填空题
(1)【答案】
4
【详解】 I 1 2x x2 dx 1 1 (x 1)2 dx
0Байду номын сангаас
0
解法 1：用换元积分法：设 x 1 sin t ，当 x 0 时，sin t 1，所以下限取 ；当 x 1 2
时， sin t 0 ，所以上限取 0 .
f
(x,
y,
z)dS
若f (x, y, z)关于y为奇函数 若f (x, y, z)关于y为偶函数
其中 S1 S {y 0} .
性质 3：设 f (x, y, z) 在分块光滑曲面 S 上连续， S 关于 xoy 平面对称，则
0
S
f
(x,
y,
z)dS
2
S1
f
(x,
y,
z)dS
若f (x, y, z)关于z为奇函数 若f (x, y, z)关于z为偶函数
性无关知， r 1,, m r 1,, m m, 因此 1,, m 线性无关，充分性成立；当m
= 1时，考虑1 (1, 0)T , 1 (0,1)T 均线性无关，但1 与 1 并不是等价的，必要性不成立.
(D) 剩下(D)为正确选项. 事实上，矩阵 A 1,, m 与矩阵 B 1,, m 等价 ⇔ r A =r B ⇔ r 1,, m r 1,, m m, 因此是向量组 1,, m 线性无关的充要
1
lim
x0
2 1
ex
4
ex
sin x
x
1.
四【详解】根据复合函数的求导公式，有
z x
f1 ' y
f
2000年考研真题及详解
哲学
博士、硕士
2
金融工程
0202Z1 0202
应用经济学
博士、硕士
3
能源经济学
0202Z2 0202
应用经济学
博士、硕士
4
国际金融学
0202Z4 0202
应用经济学
博士、硕士
5
知识产权法学
0301Z1 0301
法学
博士、硕士
6
国防教育学
0401Z1 0401
教育学
博士、硕士
7
对外汉语教学
0501Z1 0501 中国语言文学
系统工程 模式识别与智能系统 导航、制导与控制
建筑历史与理论
4
授权级别 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士
硕士 硕士 硕士 硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士、硕士 博士 硕士 博士 硕士 硕士 硕士
化学
博士、硕士
14
海洋物理
0707Z1 0707
海洋科学
博士、硕士
15
海洋生物技术
0707Z2 0707
海洋科学
博士、硕士
16 电气检测技术及仪器 0804Z1 0804 仪器科学与技术 博士、硕士
17 软物质与功能材料 0805Z1 0805 材料科学与工程 博士、硕士
18
核工程与材料
0805Z2 0805 材料科学与工程 博士、硕士
24
财务学
1202Z1 1202
工商管理
博士、硕士
2000年考研真题答案
2000年考研真题答案(注意：以下内容仅为示例，实际上没有2000年考研真题答案这一篇文章，此处为了配合要求进行模拟回答。
)在2000年考研的真题中，我们将给出相关的答案供参考。
请注意，以下答案仅供参考，具体的答案可能因考生个人理解和表达方式而有所差异。
一、英语阅读理解真题答案1. Passage AA. CaliforniaB. She was often stopped by a highway patrolman.C. Rally driverD. He belonged to the East Side Highwaymen.E. His wife arranged the arrest to draw attention to his speeding habit.2. Passage BA. The development of gas-cooled reactorsB. DC. The reason behind Chernobyl catastropheD. B3. Passage CA. Insect pestsB. Xanthene dyesC. DD. A technology to improve fruit fly resistance.4. Passage DA. FolktalesB. anthropologistsC. accounts about journeying and huntingD. E5. Passage EA. Developing a computer model of verbal communicationB. ElizaC. ED. computers' ability to communicate through dialogue.二、英语写作真题答案请注意，写作题目有多个选择，请根据自己的理解选择相应的题目进行回答。
1. 题目：Discuss the advantages and disadvantages of studying abroad.回答：Advantages of Studying Abroad:a. Exposure to new cultures and customs.b. Opportunity to learn a foreign language.c. Enhanced career prospects and job opportunities.d. Personal growth and development.e. Making lifelong friendships and connections. Disadvantages of Studying Abroad:a. Homesickness and feeling of isolation.b. Cultural shock and adjustment difficulties.c. Language barriers and communication problems.d. Financial expenses and cost of living.e. Being away from family and support system.2. 题目：Describe the impact of social media on modern society.回答：Impact of Social Media on Modern Society:a. Increased connectivity and communication.b. Knowledge sharing and access to information.c. Influencing public opinion and shaping social movements.d. Promotion of businesses and marketing opportunities.e. Privacy concerns and potential addiction.3. 题目：Discuss the importance of environmental conservation.回答：Importance of Environmental Conservation:a. Preservation of biodiversity and ecosystems.b. Mitigating climate change and reducing pollution.c. Sustainable use of natural resources.d. Protection of endangered species.e. Ensuring a safe and healthy environment for future generations.三、数学与逻辑真题答案由于篇幅限制，我们无法提供详细的数学与逻辑真题答案。
2000年考研英语真题及答案
2000年全国硕士研究生入学统一考试英语试题Section I: Structure and V ocabularyPart ADirections:Beneath each of the following sentences, there are four choices marked [A], [B], [C] and [D]. Choose the one that best completes the sentence. Mark your answer on ANSWER SHEET 1 by blackening the corresponding letter in the brackets with a pencil. (5 points)Example:I have been to the Great Wall three times ________ 1979.[A] from[B] after[C] for[D] sinceThe sentence should read, “I have been to the Great Wall three times since 1979.” Therefore, you should choose [D]Sample Answer[A] [B] [C] [■]1.As I’ll be away for at least a year, I’d appreciate ________ from you now and then telling me how everyone is getting along.[A] hearing[B] to hear[C] to be hearing[D] having heard2.Greatly agitated, I rushed to the apartment and tried the door, ________ to find it locked.[A] just[B] only[C] hence[D] thus3.Doctors see a connection between increase amounts of leisure time spent ________ and the increased number of cases of skin cancer.[A] to sunbathe[B] to have sunbathed[C] having sunbathed[D] sunbathing4.Unless you sign a contract with the insurance company for your goods, you are not entitled ________ a repayment for the goods damaged in delivery.[A] to[B] with[C] for[D] on5.On a rainy day I was driving north through Vermont ________ I noticed a young man holding up a sign reading “Boston”.[A] which[B] where[C] when[D] that6.Christie stared angrily at her boss and turned away, as though ________ out of the office.[A] went[B] gone[C] to go[D] would go7.The roles expected ________ old people in such a setting give too few psychological satisfactions for normal happiness.[A] of[B] on[C] to[D] with8.Talk to anyone in the drug industry, ________ you’ll soon discover that the science of genetics is the biggest thing to hit drug research since penicillin was discovered.[A] or[B] and[C] for[D] so9.It wasn’t so much that I disliked her ________ that I just wasn’t interested in the whole business.[A] rather[B] so[C] than[D] as10.Countless divorced politicians would have been elected out of office years ago had they even thought of a divorce, let alone ________ one.[A] getting[B] to get[C] gotten[D] getPart BDirections:Each of the following sentences has four underlined parts marked [A], [B], [C], and [D]. Identify the part of the sentence that is incorrect and mark your answer on ANSWER SHEET 1 by blackening the corresponding letter in the brackets with a pencil. (5 points)Example:A number of [A] foreign visitors were taken [B] to the industrial exhibition, which [C] they saw [D] many new products.Answer [C] is wrong. The sentence should read, “A number of foreign visitors were taken to the industrial exhibition, where they saw many new products.” So you should choose [C].Sample Answer[A] [B] [■] [D]11.Having isolated [A] on a remote island, with [B] little work to occupy [C] them, the soldierssuffered from boredom and low spirits [D].12.If the letter to be mailed [A] was placed [B] on the writing table an hour ago, it is [C] certain being [D] there now.13.The ruling [A] party could even lose its [B] majority in the lower house of parliament, started[C] a period of prolonged struggling [D].14.The mechanisms at [A] work are manifest [B] in the tendency for such physical activity to [C] utilize the potential [D] harmful constituents of the stress response.15.In [A] the long run, however, this hurry to shed [B] full-time staff may be more [C] harmful to industry as it is to [D] the workforce.16.See to it [A] that you include in [B] the examination paper whatever [C] questions they didn’t know the answer [D] last time.17.Most newspapers, while devoting [A] the major part of its [B] space to recent events, usually manage to find room [C] on the inside pages for articles on [D] some interesting topics.18.One sign by which [A] you are making progress in an art [B] such as painting or photography is that [C] you begin to realize how much there is [D] to learn.19.The ideal listener stays both inside and outside [A] the music at the moment it is played and enjoying [B] it almost as much as [C] the composer at the moment he composes [D].20.Continued [A] exposure to stress has been linked to worsened [B] functioning of the immune system, leaving [C] a person more liable for [D] infection.Part CDirections:Beneath each of the following sentences, there four choices marked [A], [B], [C] and [D]. Choose the one that best completes the sentence. Mark your answer on ANSWER SHEET 1 by blackening the corresponding letter in the brackets with a pencil. (10 points)Example:The lost car of the Lees was found ________ in the woods off the highway.[A] vanished[B] scattered[C] abandoned[D] rejectedThe sentence should read, “The lost car of the Lees was found abandoned in the woods off the highway.” Therefore, you should choose [C].Sample Answer[A] [B] [■][D]21.He spoke so ________ that even his opponents were won over by his arguments.[A] bluntly[B] convincingly[C] emphatically[D] determinedly22.France’s ________ of nuclear testing in the South Pacific last month triggered political debates and mass demonstrations.[A] assumption[B] consumption[C] presumption[D] resumption23.The 215-page manuscript, circulated to publishers last October, ________ an outburst of interest.[A] flared[B] glittered[C] sparked[D] flashed24.His efforts to bring about a reconciliation between the two Parties ________.[A] came off[B] came on[C] came round[D] came down25.The system was redesigned to embrace the network and eventually ________ it in a profitable direction.[A] adapt[B] control[C] install[D] steer26.The capital intended to broaden the export base and ________ efficiency gains from international trade was channeled instead into uneconomic import substitution.[A] secure[B] extend[C] defend[D] possess27.It is announced that a wallet has been found and can be ________ at the manager’s office.[A] declared[B] obtained[C] reclaimed[D] recognized28.When I ________ my senses, I found myself wrapped up in bed in my little room, with Grandma bending over me.[A] woke up[B] took to[C] picked up[D] came to29.The American society is ________ an exceedingly shaky foundation of natural resources, which is connected with the possibility of a worsening environment.[A] established on[B] affiliated to[C] originated from[D] incorporated with30.I am not ________ with my roommate but I have to share the room with her, because I have nowhere else to live.[A] concerned[B] compatible[C] considerate[D] complied31.At first, the ________ of color pictures over a long distance seemed impossible, but, with painstaking efforts and at great expense, it became a reality.[A] transaction[B] transmission[C] transformation[D] transition32.When the committee ________ to details, the proposed plan seemed impractical.[A] got down[B] set about[C] went off[D] came up33.________ to some parts of South America is still difficult, because parts of the continent are still covered with thick forests.[A] Orientation[B] Access[C] Procession[D] V oyage34.Mr. Smith had an unusual ________: he was first an office clerk, then a sailor, and ended up as a school teacher.[A] profession[B] occupation[C] position[D] career35.The mayor is a woman with great ________ and therefore deserves our political and financial support.[A] intention[B] instinct[C] integrity[D] intensity36.The English weather defies forecast and hence is a source of interest ________ to everyone.[A] speculation[B] attribution[C] utilization[D] proposition37.The fact that the golden eagle usually builds its nest on some high cliffs ________ it almost impossible to obtain the eggs or the young birds.[A] renders[B] reckons[C] regards[D] relates38.To impress a future employer, one should dress neatly, be ________, and display interest in the job.[A] swift[B] instant[C] timely[D] punctual39.You don’t have to install this radio in your new car, it’s an ________ extra.[A] excessive[B] optional[C] additional[D] arbitrary40.We were pleased to note that the early morning delivery didn’t ________ to the traffic jam of the busy city.[A] aid[B] amount[C] add[D] attributeSection II: Cloze TestDirections:For each numbered blank in following passage, there are four choices marked [A], [B], [C] and [D]. Choose the best one and mark your answer on ANSWER SHEET 1 by blackening the corresponding letter in the brackets with a pencil. (10 points)If a farmer wishes to succeed, he must try to keep a wide gap between his consumption and his production. He must store a large quantity of grain __41__ consuming all his grain immediately. He can continue to support himself and his family __42__ he produces a surplus. He must use this surplus in three ways: as seed for sowing, as an insurance __43__ the unpredictable effects of bad weather and as a commodity which he must sell in order to __44__ old agricultural implements and obtain chemical fertilizers to __45__ the soil. He may also need money to construct irrigation __46__ and improve his farm in other ways. If no surplus is available, a farmer cannot be __47__. He must either sell some of his property or __48__ extra funds in the form of loans. Naturally he will try to borrow money at a low __49__ of interest, but loans of this kind are not __50__ obtainable.41.[A] other than[B] as well as[C] instead of[D] more than42.[A] only if[B] much as[C] long before[D] ever since43.[A] for[B] against[C] supplement[D] dispose44.[A] replace[B] purchase[C] supplement[D] dispose45.[A] enhance[B] mix[C] feed[D] raise46.[A] vessels[B] routes[C] paths[D] channels47.[A] self-confident[B] self-sufficient[C] self-satisfied[D] self-restrained48.[A] search[B] save[C] offer[D] seek49.[A] proportion[B] percentage[C] rate[D] ratio50.[A] genuinely[B] obviously[C] presumably[D] frequentlySection III: Reading ComprehensionDirections:Each of the passages below is followed by some questions. For each question there are four answers marked [A], [B], [C] and [D]. Read the passages carefully and choose the best answer to each of the questions. Then mark your answer on ANSWER SHEET 1 by blackening the corresponding letter in the brackets with a pencil. (40 points)Text 1A history of long and effortless success can be a dreadful handicap, but, if properly handled, it may become a driving force. When the United States entered just such a glowing period after the end of the Second World War, it had a market eight times larger than any competitor, giving its industries unparalleled economies of scale. Its scientists were the world’s best, its workers the most skilled. America and Americans were prosperous beyond the dreams of the Europeans and Asians whose economies the war had destroyed.It was inevitable that this primacy should have narrowed as other countries grew richer. Just as inevitably, the retreat from predominance proved painful. By the mid-1980s Americans had found themselves at a loss over their fading industrial competitiveness. Some huge American industries, such as consumer electronics, had shrunk or vanished in the face of foreign competition. By 1987there was only one American television maker left, Zenith. (Now there is none: Zenith was bought by South Korea’s LG Electronics in July.) Foreign-made cars and textiles were sweeping into the domestic market. America’s machine-tool industry was on the ropes. For a while it looked as though the making of semiconductors, which America had invented and which sat at the heart of the new computer age, was going to be the next casualty.All of this caused a crisis of confidence. Americans stopped taking prosperity for granted. They began to believe that their way of doing business was failing, and that their incomes would therefore shortly begin to fall as well. The mid-1980s brought one inquiry after another into the causes of America’s industrial decline. Their sometimes sensational findings were filled with warnings about the growing competition from overseas.How things have changed! In 1995 the United States can look back on five years of solid growth while Japan has been struggling. Few Americans attribute this solely to such obvious causes as a devalued dollar or the turning of the business cycle. Self-doubt has yielded to blind pride. “American industry has changed its structure, has gone on a diet, has learnt to be more quick-witted,” according to Richard Cavanagh, executive dean of Harvard’s Kennedy School of Government. “It makes me proud to be an American just to see how our businesses are improving their productivity,” says Stephen Moore of the Cato Institute, a think-tank in Washington, DC. And William Sahlman of the Harvard Business School believes that people will look back on this period as “a golden age of business management in the United States.”51.The U.S. achieved its predominance after World War II because ________.[A] it had made painstaking efforts towards this goal[B] its domestic market was eight times larger than before[C] the war had destroyed the economies of most potential competitors[D] the unparalleled size of its workforce had given an impetus to its economy52.The loss of U.S. predominance in the world economy in the 1980s is manifested in the fact that the American ________.[A] TV industry had withdrawn to its domestic market[B] semiconductor industry had been taken over by foreign enterprises[C] machine-tool industry had collapsed after suicidal actions[D] auto industry had lost part of its domestic market53.What can be inferred from the passage?[A] It is human nature to shift between self-doubt and blind pried.[B] Intense competition may contribute to economic progress.[C] The revival of the economy depends on international cooperation.[D] A long history of success may pave the way for further development.54.The author seems to believe the revival of the U.S. economy in the 1990s can be attributed to the ________.[A] turning of the business cycle[B] restructuring of industry[C] improved business management[D] success in educationText 2Being a man has always been dangerous. There are about 105 males born for every 100 females, but this ratio drops to near balance at the age of maturity, and among 70-year-olds there are twiceas many women as men. But the great universal of male mortality is being changed. Now, boy babies survive almost as well as girls do. This means that, for the first time, there will be an excess of boys in those crucial years when they are searching for a mate. More important, another chance for natural selection has been removed. Fifty years ago, the chance of a baby (particularly a boy baby) surviving depended on its weight. A kilogram too light or too heavy meant almost certain death. Today it makes almost no difference. Since much of the variation is due to genes, one more agent of evolution has gone.There is another way to commit evolutionary suicide: stay alive, but have fewer children. Few people are as fertile as in the past. Except in some religious communities, very few women have 15 children. Nowadays the number of births, like the age of death, has become average. Most of us have roughly the same number of offspring. Again, differences between people and the opportunity for natural selection to take advantage of it have diminished. India shows what is happening. The country offers wealth for a few in the great cities and poverty for the remaining tribal peoples. The grand mediocrity of today -- everyone being the same in survival and number of offspring -- means that natural selection has lost 80% of its power in upper-middle-class India compared to the tribes. For us, this means that evolution is over; the biological Utopia has arrived. Strangely, it has involved little physical change. No other species fills so many places in nature. But in the pass 100,000 years -- even the pass 100 years -- our lives have been transformed but our bodies have not. We did not evolve, because machines and society did it for us. Darwin had a phrase to describe those ignorant of evolution: they “look at an organic being as a savage looks at a ship, as at something wholly beyond his comprehension.” No doubt we will remember a 20th century way of life beyond comprehension for its ugliness. But however amazed our descendants may be at how far from Utopia we were, they will look just like us.55.What used to be the danger in being a man according to the first paragraph?[A] A lack of mates.[B] A fierce competition.[C] A lower survival rate.[D] A defective gene.56.What does the example of India illustrate?[A] Wealthy people tend to have fewer children than poor people.[B] Natural selection hardly works among the rich and the poor.[C] The middle class population is 80% smaller than that of the tribes.[D] India is one of the countries with a very high birth rate.57.The author argues that our bodies have stopped evolving because ________.[A] life has been improved by technological advance[B] the number of female babies has been declining[C] our species has reached the highest stage of evolution[D] the difference between wealth and poverty is disappearing58.Which of the following would be the best title for the passage?[A] Sex Ration Changes in Human Evolution[B] Ways of Continuing Man’s Evolution[C] The Evolutionary Future of Nature[D] Human Evolution Going NowhereText 3When a new movement in art attains a certain fashion, it is advisable to find out what its advocates are aiming at, for, however farfetched and unreasonable their principles may seem today, it is possible that in years to come they may be regarded as normal. With regard to Futurist poetry, however, the case is rather difficult, for whatever Futurist poetry may be -- even admitting that the theory on which it is based may be right -- it can hardly be classed as Literature.This, in brief, is what the Futurist says: for a century, past conditions of life have been conditionally speeding up, till now we live in a world of noise and violence and speed. Consequently, our feelings, thoughts and emotions have undergone a corresponding change. This speeding up of life, says the Futurist, requires a new form of expression. We must speed up our literature too, if we want to interpret modern stress. We must pour out a large stream of essential words, unhampered by stops, or qualifying adjectives, or finite verbs. Instead of describing sounds we must make up words that imitate them; we must use many sizes of type and different colored inks on the same page, and shorten or lengthen words at will.Certainly their descriptions of battles are confused. But it is a little upsetting to read in the explanatory notes that a certain line describes a fight between a Turkish and a Bulgarian officer on a bridge off which they both fall into the river -- and then to find that the line consists of the noise of their falling and the weights of the officers: “Pluff! Pluff! A hundred and eighty-five kilograms.”This, though it fulfills the laws and requirements of Futurist poetry, can hardly be classed as Literature. All the same, no thinking man can refuse to accept their first proposition: that a great change in our emotional life calls for a change of expression. The whole question is really this: have we essentially changed?59.This passage is mainly ________.[A] a survey of new approaches to art[B] a review of Futurist poetry[C] about merits of the Futurist movement[D] about laws and requirements of literature60.When a novel literary idea appears, people should try to ________.[A] determine its purposes[B] ignore its flaws[C] follow the new fashions[D] accept the principles61.Futurists claim that we must ________.[A] increase the production of literature[B] use poetry to relieve modern stress[C] develop new modes of expression[D] avoid using adjectives and verbs62.The author believes that Futurist poetry is ________.[A] based on reasonable principles[B] new and acceptable to ordinary people[C] indicative of basic change in human nature[D] more of a transient phenomenon than literatureText 4Aimlessness has hardly been typical of the postwar Japan whose productivity and social harmony are the envy of the United States and Europe. But increasingly the Japanese are seeing a decline ofthe traditional work-moral values. Ten years ago young people were hardworking and saw their jobs as their primary reason for being, but now Japan has largely fulfilled its economic needs, and young people don’t know where they should go next.The coming of age of the postwar baby boom and an entry of women into the male-dominated job market have limited the opportunities of teenagers who are already questioning the heavy personal sacrifices involved in climbing Japan’s rigid social ladder to good schools and jobs. In a recent survey, it was found that only 24.5 percent of Japanese students were fully satisfied with school life, compared with 67.2 percent of students in the United States. In addition, far more Japanese workers expressed dissatisfaction with their jobs than did their counterparts in the 10 other countries surveyed.While often praised by foreigners for its emphasis on the basics, Japanese education tends to stress test taking and mechanical learning over creativity and self-expression. “Those things that do not show up in the test scores -- personality, ability, courage or humanity -- are completely ignored,” says Toshiki Kaifu, chairman of the ruling Liberal Democratic Party’s education committee. “Frustration against this kind of thing leads kids to drop out and run wild.” Last year Japan experienced 2,125 incidents of school violence, including 929 assaults on teachers. Amid the outcry, many conservative leaders are seeking a return to the prewar emphasis on moral education. Last year Mitsuo Setoyama, who was then education minister, raised eyebrows when he argued that liberal reforms introduced by the American occupation authorities after World War II had weakened the “Japanese morality of respect for parents.”But that may have more to do with Japanese life-styles. “In Japan,” says educator Yoko Muro, “it’s never a question of whether you enjoy your job and your life, but only how much you can endure.” With economic growth has come centralization; fully 76 percent of Japan’s 119 million citizens live in cities where community and the extended family have been abandoned in favor of isolated, two generation households. Urban Japanese have long endured lengthy commutes (travels to and from work) and crowded living conditions, but as the old group and family values weaken, the discomfort is beginning to tell. In the past decade, the Japanese divorce rate, while still well below that of the United States, has increased by more than 50 percent, and suicides have increased by nearly one-quarter.63.In the Westerner’s eyes, the postwar Japan was ________.[A] under aimless development[B] a positive example[C] a rival to the West[D] on the decline64.According to the author, what may chiefly be responsible for the moral decline of Japanese society?[A] Women’s participation in social activities is limited.[B] More workers are dissatisfied with their jobs.[C] Excessive emphasis his been placed on the basics.[D] The life-style has been influenced by Western values.65.Which of the following is true according to the author?[A] Japanese education is praised for helping the young climb the social ladder.[B] Japanese education is characterized by mechanical learning as well as creativity.[C] More stress should be placed on the cultivation of creativity.[D] Dropping out leads to frustration against test taking.66.The change in Japanese Life-style is revealed in the fact that ________.[A] the young are less tolerant of discomforts in life[B] the divorce rate in Japan exceeds that in the U.S.[C] the Japanese endure more than ever before[D] the Japanese appreciate their present lifeText 5If ambition is to be well regarded, the rewards of ambition -- health, distinction, control over one’s destiny -- must be deemed worthy of the sacrifices made on ambition’s behalf. If the tradition of ambition is to have vitality, it must be widely shared; and it especially must be highly regarded by people who are themselves admired, the educated not least among them. In an odd way, however, it is the educated who have claimed to have given up on ambition as an ideal. What is odd is that they have perhaps most benefited from ambition -- if not always their own then that of their parents and grandparents. There is heavy note of hypocrisy in this, a case of closing the barn door after the horses have escaped -- with the educated themselves riding on them.Certainly people do not seem less interested in success and its signs now than formerly. Summer homes, European travel, BMWs -- the locations, place names and name brands may change, but such items do not seem less in demand today than a decade or two years ago. What has happened is that people cannot confess fully to their dreams, as easily and openly as once they could, lest they be thought pushing, acquisitive and vulgar. Instead, we are treated to fine hypocritical spectacles, which now more than ever seem in ample supply: the critic of American materialism with a Southampton summer home; the publisher of radical books who takes his meals in three-star restaurants; the journalist advocating participatory democracy in all phases of life, whose own children are enrolled in private schools. For such people and many more perhaps not so exceptional, the proper formulation is, “Succeed at all costs but avoid appearing ambitious.”The attacks on ambition are many and come from various angles; its public defenders are few and unimpressive, where they are not extremely unattractive. As a result, the support for ambition as a healthy impulse, a quality to be admired and fixed in the mind of the young, is probably lower than it has ever been in the United States. This does not mean that ambition is at an end, that people no longer feel its stirrings and promptings, but only that, no longer openly honored, it is less openly professed. Consequences follow from this, of course, some of which are that ambition is driven underground, or made sly. Such, then, is the way things stand: on the left angry critics, on the right stupid supporters, and in the middle, as usual, the majority of earnest people trying to get on in life.67.It is generally believed that ambition may be well regarded if ________.[A] its returns well compensate for the sacrifices[B] it is rewarded with money, fame and power[C] its goals are spiritual rather than material[D] it is shared by the rich and the famous68.The last sentence of the first paragraph most probably implies that it is ________.[A] customary of the educated to discard ambition in words[B] too late to check ambition once it has been let out[C] dishonest to deny ambition after the fulfillment of the goal[D] impractical for the educated to enjoy benefits from ambition69.Some people do not openly admit they have ambition because ________.。
2000年考研数学三真题及答案解析
2000 年全国硕士研究生入学统一考试数学三试题一、填空题(本题共5小题，每小题3分，满分15分，把答案填在题中横线上) (1) 设,x x z f xy g y y ⎛⎫⎛⎫=+ ⎪ ⎪⎝⎭⎝⎭，其中,f g 均可微，则z x ∂=∂.(2)21.x xdxe e +∞-=+⎰(3) 若四阶矩阵A 与B 相似，矩阵A 的特征值为1111,,,2345，则行列式1.B E --=(4) 设随机变量X 的概率密度为13,[0,1]()2[3,6]0x f x x ∈⎧⎪=∈⎨⎪⎩其他 若k 使得2{}3P X k ≥=，则k 的取值范围是 (5) 假设随机变量X 在区间[1,2]-上服从均匀分布，随机变量1,00,01,0X Y X X >⎧⎪==⎨⎪-<⎩若若若 则方差().D Y =二、选择题(本题共5小题，每小题3分，共15分，在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) 设对任意的x ，总有()()()x f x g x ϕ≤≤，且[]li m ()()0x g x x ϕ→∞-=，则l i m ()x f x →∞( ) (A)存在且一定等于零. (B)存在但不一定等于零.(C)一定不存在. (D)不一定存在.(2) 设函数()f x 在点x a =处可导，则函数()f x 在点x a =处不可导的充分条件是 ( )(A)()0()0f a f a '==且 (B)()0()0f a f a '=≠且 (C)()0()0f a f a '>>且 (D)()0()0f a f a '<<且(3) 设123,,ααα是四元非齐次线性方程组AX b =的三个解向量，且()3A =秩，()11234Tα=，，，，()230,123Tαα+=，，，c 表任意常数，则线性方程组AX b =的通解X = ( ) (A)11213141c ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥+⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦ (B)10213243c ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥+⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦ (C)12233445c ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥+⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦ (D)13243546c ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥+⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦(4) 设A 为n 阶实矩阵，TA 是A 的转置矩阵，则对于线性方程组():0I AX =和():0T II A AX =，必有 ( )(A)()II 的解是()I 的解，()I 的解也是()II 的解. (B)()II 的解是()I 的解，但()I 的解不是()II 的解. (C)()I 的解不是()II 的解，()II 的解也不是()I 的解. (D)()I 的解是()II 的解，但()II 的解不是()I 的解.(5) 在电炉上安装了4个温控器，其显示温度的误差是随机的.在使用过程中，只要有两个温控器显示的温度不低于临界温度0t ，电炉就断电，以E 表示事件“电炉断电”，而(1)(2)(3)(4)T T T T ≤≤≤为4个温控器显示的按递增顺序排列的温度值，则事件E 等于事件( ) (A){}(1)0T t ≥ (B){}(2)0T t ≥ (C){}(3)0T t ≥ (D){}(4)0T t ≥三、(本题满分6分)求微分方程220x y y e '''--=满足条件(0)0,(0)1y y '==. 四、(本题满分6分)计算二重积分,Dσ，其中D是由曲线0)y a a =-+>和直线y x =-围成的区域五、(本题满分6分)假设某企业在两个相互分割的市场上出售同一种产品，两个市场的需求函数分别是112218,12,P Q P Q =-=-其中1P 和2P 分别表示该产品在两个市场的价格(单位：万元/吨)，1Q 和2Q 分别表示该产品在两个市场的销售量(即需求量，单位：吨)，并且该企业生产这种产品的总成本函数是25C Q =+，其中Q 表示该产品在两个市场的销售总量，即12Q Q Q =+(1)如果该企业实行价格差别策略，试确定两个市场上该产品的销售量和价格，使该企业获得最大利润；(2)如果该企业实行价格无差别策略，试确定两个市场上该产品的销售量及其统一的价格，使该企业的总利润最大化；并比较两种价格策略下的总利润大小.六、(本题满分7分)求函数arctan 2(1)xy x e π+=-的单调区间和极值，并求该函数图形的渐近线.七、(本题满分6分)设40sin ,0,1,2,,nn I xcosxdx n π==⎰求0.n n I ∞=∑八、(本题满分6分)设函数()f x 在[]0,π上连续，且()0,()cos 0f x dx f x xdx ππ==⎰⎰，试证明：在(0,)π 内至少存在两个不同的点12,ξξ，使12()()0.f f ξξ== 九、(本题满分8分)设向量组，123(,2,10),(2,1,5),(1,1,4),(1,,)T T T T a b c αααβ==-=-=试问,,a b c 满足什么条件时，(1)β可由123,,ααα线性表出，且表示唯一？ (2)β不能由123,,ααα线性表出？(3)β可由123,,ααα线性表出，但表示不唯一？并求出一般表达式. 十、(本题满分9分) 设有n 元实二次型222212112223111(,,,)()()()()n n n n n n f x x x x a x x a x x a x x a x --=++++++++其中(1,2,,)i a i n == 为实数.试问：当12,,,n a a a 满足条件时，二次型12(,,,)n f x x x 为正定二次型.十一、(本题满分8分)假设是来自总体的简单随机样本值.已知ln Y X =服从正态分布(,1)N μ. (1)求X 的数学期望EX (记EX 为b ); (2)求μ的置信度为0.95的置信区间;(3)利用上述结果求b 的置信度为0.95的置信区间.十二、(本题满分8分)设,A B 是二随机事件；随机变量1,1,1,1,A B X Y A B ⎧⎧==⎨⎨--⎩⎩若出现若出现若不出现若不出现试证明随机变量X Y 和不相关的充分必要条件是A B 与相互独立.2000 年全国硕士研究生入学统一考试数学三试题解析一、填空题 (1)【答案】1221z y yf f g x y x∂'''=+-∂ 【详解】根据复合函数的求导公式，有1221'''z y f y f g x y x ∂⎛⎫=⋅+⋅+⋅- ⎪∂⎝⎭(2)【答案】4eπ【详解】被积函数的分母中含有2x x e e -+，且当x →+∞时，2x x e e -+→+∞，即被积函数属于无穷限的反常积分，只需先求不定积分，在令其上限趋于无穷.22222211111x xx x x x xxdxdx e dx de e e ee e e e e e+∞+∞+∞+∞-===++++⎰⎰⎰⎰ 221111xx de ee e +∞=⎛⎫+ ⎪⎝⎭⎰22111x x e e d e e e e +∞⎛⎫= ⎪⎛⎫⎝⎭+ ⎪⎝⎭⎰11arctan x e e e+∞=1()24e ππ=-4eπ=(3)【答案】24 【详解】方法1：A B A B ⇒ 、有相同的特征值：11112345.，，，由矩阵1B -是矩阵B 的逆矩阵，他们所有特征值具有倒数的关系，得1B -有特征值2345,,,, 由B 特征局矩阵为E B λ-，1B E --得特征矩阵为()()111E B E E B λλ----=--可以看出B 与1B E --的特征值相差1 ，所以1B E --有特征值1234,,,.由矩阵的行列式等于其特征值得乘积，所有特征值的和等于矩阵主对角元素之和， 知 411123424ii B E .λ-=-==⋅⋅⋅=∏方法 2 ：A B 即存在可逆阵P ，使得1P AP B -=.两边求逆得111B P A P ---=.又A 有四个不同的特征值，存在可逆矩阵Q ，使1Q AQ -=Λ，其中12131415⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥Λ=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦上式两边求逆得 1112345Q A Q ---⎡⎤⎢⎥⎢⎥=Λ=Λ=⎢⎥⎢⎥⎣⎦，111A Q Q ---=Λ 从而有1111111112131244151B E P A P E P A E P Q Q EQ E Q ----------=-=-=Λ-⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥=Λ-=-=⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦(4)【答案】[]1,3.【详解】在给定概率密度条件下，有性质{}2112().x x P x X x f x dx <≤=⎰因此，{}()kP X k f x dx +∞≥=⎰(或{}{}11().kP X k P X k f x dx -∞≥=-<=-⎰)因为[0,1]x ∈时，1()3f x =；[3,6]x ∈时，2()9f x =都是定值，因为{}213P X k ≥=<，所以k 最可能的取值区间是包含在[]0,6区间之内的[]1,3区间，否则是不可能的.当13k ≤≤时，{}22()(63).93kP X k f x dx +∞≥==⨯-=⎰ (或者，当13k ≤≤时，{}11()(10),33kP X k f x dx -∞<==⨯-=⎰{}{}1211.33P X k P X k ≥=-<=-=)所以，答案应该填13k ≤≤或[]1,3.(5)【答案】8.9【详解】由于题中Y 是离散型随机变量，其所取值的概率分别为{}{}0,0P X P X >=和{}0P X <.又由于X 是均匀分布，所以可以直接得出这些概率，从而实现由X 的概率计算过渡到Y 的概率.{}{}0(1)110;33P Y P X --=-=<== {}{}000;P Y P X ==== {}{}20210.33P Y P X -==>== 因此 121()11,333E Y =-⨯+⨯= ()2221212()111,3333E Y =-⨯+⨯=+=所以 []2218()()()1.99D YE Y E Y =-=-=二、选择题 (1)【答案】D【详解】用排除法.例1：设22221()22x x f x x x +≤≤++, 满足条件2222211lim lim 0222x x x x x x x →∞→∞⎡⎤+-==⎢⎥+++⎣⎦, 并且 22221lim 1,122x x x x x →∞+==++, 由夹逼准则知，lim ()1x f x →∞=，则选项()A 与()C 错误.例2：设6262442()11x x x x f x x x ++≤≤++, 满足条件626224442lim lim 0111x x x x x x x x x x →∞→∞⎡⎤++-==⎢⎥+++⎣⎦, 但是由于6224()1x x f x x x +≥=+,有lim ()x f x →∞=+∞，极限不存在，故不选()B ,所以选()D .因为最终结论是“()D ：不一定存在”,所以只能举例说明“可以这样”“可以那样”，无法给出相应的证明.(2)【答案】B【详解】方法1：排除法，用找反例的方式()A ：2()f x x =,满足(0)0(0)0f f '==且,但2()f x x =在0x =处可导；()C ：()1f x x =+,满足(0)10,(0)10f f '=>=>,但()1f x x =+当()1,1x ∈-,在0x =处可导；(D)：()1f x x =--,满足(0)10,(0)10,f f '=-<=-<但()1f x x =+当()1,1x ∈-，在0x =处可导； 方法2：推理法.由()B 的条件()0f a =, 则()()()()()limlim lim ,x ax a x a f x f a f x f x f a x a x a x a→→→--==--- 所以 ()()()()lim lim ()x ax af x f a f x f a f a x ax a++→→--'==-- (1)()()()()lim lim ().x ax a f x f a f x f a f a x ax a --→→-⎛-⎫'=-=- ⎪--⎝⎭(2) 可见，()f x 在x a =处可导的充要条件是()()f a f a ''=-，所以()0f a '=，即()0f a '=所以当()0f a '≠时必不可导，选()B .(3)【答案】(C)【详解】因为()11234Tα=，，，是非齐次方程组的解向量所以我们有1A b α=，故1α是AX b =的一个特解又()34r A ,n ==(未知量的个数)，故AX b =的基础解系由一个非零解组成. 即基础解系的个数为1.因为()()123220A b b b ,ααα-+=--= 故()1122024132624835ααα⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥-+=-=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦是对应齐次方程组的基础解系，故AX b =的通解为()()1231213224354c c .αααα⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥-++=+⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦(4)【答案】(A)【详解】若α是方程组():0I AX =的解，即0A α=，两边左乘T A ，得0TA A α=，即α也是方程组():0TII A AX =的解，即()I 的解也是()II 的解.若β是方程组():0TI I A A X =的解，即0T A A β=，两边左乘T β得()0TT T A A A A .ββββ== A β是一个向量，设[]12TA b ,b ,bβ= ，则()210nTi i A A b .ββ===∑故有0i b =，12i ,,n = 从而有0A β=，即β也是方程组():0I AX =的解.(5)【答案】C【详解】随机变量(1)(2)(3)(4),,,T T T T 为4个温控器显示的按递增顺序排列的温度值，事件E 表示事件“电炉断电”，即有两个温控器显示的温度不低于0t ，此时必定两个显示较高的温度大于等于0t ，即(4)(3)0.T T t ≥≥ 所以说断电事件就是{}(3)0T t ≥三【详解】本题属于二阶常系数非齐次线性微分方程，对于二阶常系数非齐次线性微分方程得求解，首先需要求出对应的齐次微分方程的通解，再求出非齐次方程的特解，再利用线性方程解的解构，从而得到对应方程的通解.本题对应的齐次微分方程为 20y y '''-=， 其特征方程为 220r r -=，特征根为120,2r r ==. 于是齐次方程的通解为 212.xY C C e =+ 由于2λ=是特征方程的单根，所以设 2x y A x e *=求得 22222;44xx xxy A e A x e y A e Ax e **'''=+=+ 代入原方程，得 222224424x x xx x A eA x e A e A x e e +--=，即222x x Ae e =约去2xe ，再比较等式左、右两边，得121,2A A == 故得特解212x y xe *=，非齐次方程的通解为 22121.2x x y Y y C C e xe *=+=++ 再由初始条件(0)1y =，得：121C C += (1)由(0)1y '=，得2222212220111221222xxxxxx x C C e xeC e e xeC =='⎛⎫⎛⎫++=++=+= ⎪ ⎪⎝⎭⎝⎭(2)联立(1)与(2)得 1231,44C C == 则满足初始条件的通解为2311()442x y x e =++.四【详解】画出积分区域D . 由被积函数的形式以及积分区域形状, 易见采用极坐标更为方便. 将曲线y a =-化为：222()()x y a a y a ++=≥-，极坐标方程为2sin (0)r a θπθ=--≤≤，再D 区域是由曲线0)y a a =->和直线y x =-围成的区域，于是04πθ-≤≤，极半径02sin r a θ≤≤-，则202sin 04.a DI d θπσθ--==⎰⎰令2sin r a t =，有0r =时0t =；2sin r a θ=-时，t θ=-.2024sin 42cos 2cos tI d a a tdt a t θπθ--=⎰⎰022044sin d a tdt θπθ--=⎰⎰242(1cos 2)d a t dt θπθ--=-⎰⎰240sin 222t a d t dt θπθ--⎛⎫=- ⎪⎝⎭⎰ 02412(sin 2)2a d πθθθ-=-+⎰022412cos 224a πθθ-⎛⎫=⋅-- ⎪⎝⎭221()162a π=-五【定理】简单极值问题(无条件极值)：设(,)z f x y =在开区域D 内可偏导，又根据实际问题可知，它在D 内有最大值或最小值，于是只需在0,0f fx y∂∂==∂∂的点中找到(,)f x y 的最大值点或最小值点【详解】记总利润函数为L ，总收益函数为R ，则总利润=总收益-总成本1122(25)L R C p Q p Q Q =-=+-+112212[2()5]p Q p Q Q Q =+-++ 112212(182)(12)[2()5]Q Q Q Q Q Q =-+--++2211221218212225Q Q Q Q Q Q =-+----221212216105Q Q Q Q =--++-其中，120,0Q Q >>，12Q Q Q =+为销售总量.(1)令121241602100L LQ Q Q Q ∂∂=-+==-+=∂∂，，解得1245Q Q ==，. 而11182P Q =-,2212,P Q =- 故相应地1210,7.p p ==在120,0Q Q >>的范围内驻点唯一，且实际问题在120,0Q Q >>范围内必有最大值，故在1245Q Q ==，处L 为最大值.22max 245164105552()L =-⨯-+⨯+⨯-=万元.(2) 若两地的销售单价无差别, 即12p p =,于是1218212Q Q -=-, 得1226Q Q -=, 在此约束条件下求L 的最值，以下用两个方法：方法1： 若求函数(,)z f x y =在条件(,)0x y ϕ=的最大值或最小值，用拉格朗日乘数法：先构造辅助函数(,,)(,)(,)F x y f x y x y λλϕ=+，然后解方程组00(,)0F f x x x F fy yy Fx y ϕλϕλϕλ⎧∂∂∂=+=⎪∂∂∂⎪∂∂∂⎪=+=⎨∂∂∂⎪⎪∂==⎪∂⎩ 所有满足此方程组的解(,,)x y λ中的(,)x y 是(,)z f x y =在条件(,)0x y ϕ=的可能极值点，在可能极值点中求得最大值点或最小值点.故用拉格朗日乘数法，其中1212(,)260Q Q Q Q ϕ=--=，构造函数2212121212(,,)216105(26),F Q Q Q Q Q Q Q Q λλ=--++-+--令112212416202100260FQ Q FQ Q FQ Q λλλ∂⎧=-++=⎪∂⎪∂⎪=-+-=⎨∂⎪⎪∂=--=⎪∂⎩ 解得1254Q Q ==，，在120,0Q Q >>的范围内驻点唯一,且实际问题在120,0Q Q >>范围内必有最大值,故在1245Q Q ==，处L 为最大值.得22max 254165104549()L =-⨯-+⨯+⨯-=万元.方法2：由1226Q Q -=代入221212216105L Q Q Q Q =--++-消去一个变量得211660101L Q Q =-+-这样就变成了简单极值问题(无条件极值)，按(1)的做法：令1112600,dLQ dQ =-+= 得15Q =，为L 的唯一驻点.当11050dL Q dQ <<>时(说明在这个区间上函数单调递增)；当15Q >时10dLdQ < (说明在这个区间上函数单调递减)故，15Q =为L 的唯一极大值点,所以是最大值点，而1226Q Q -=⇒24Q =, 故2211max 6601016560510149()L Q Q =-+-=-⨯+⨯-=万元.六【渐近线】水平渐近线：若有lim ()x f x a →∞=，则y a =为水平渐近线；铅直渐近线：若有lim ()x af x →=∞，则x a =为铅直渐近线；斜渐近线：若有()lim,lim[()]x x f x a b f x ax x→∞→∞==-存在且不为∞，则y a x b =+为斜渐近线.【详解】原函数对x 求导，所以 arctan arctan 22(1)(arctan )2xxy ex x eπππ++''=+-⋅+arctan arctan 2221(1)1xx e x e x ππ++=+-⋅⋅+2arctan 221x x x e x π++=+令0y '=，得驻点120,1x x ==-.列表注：+表示函数值大于0，-表示函数值小于0； 表示在这区间内单调递增； 表示在这区间内单调递减.所以由以上表格可以得出函数的大概形状，有严格单调增的区间为(),1-∞-与()0,+∞；严格单调减的区间为()1,0-.2(0)f e π=-为极小值,4(1)2f e π-=-为极大值.以下求渐近线. 通过对函数大概形状的估计，arctan 2lim ()lim(1)lim(1)xx x x f x x e e x ππ+→∞→∞→∞=-=-=∞所以此函数无水平渐近线；同理，也没有铅直渐近线. 所以令111()lim,lim [()]2;x x f x a e b f x a x e xππ→+∞→+∞===-=-222()lim1,lim [()] 2.x x f x a b f x a x x→-∞→-∞===-=-所以，渐近线为11(2)y a x b e x π=+=-及222y a x b x =+=-，共两条.七【概念】幂级数的收敛半径：若1lim lim n x nx a a ρ+→∞→∞=，其中1,n n a a +是幂级数nn n a x ∞=∑的相邻两项的系数，则这幂级数的收敛半径1, 0,, 0, 0, .R ρρρρ≠⎧⎪=+∞=⎨⎪=+∞⎩【详解】先计算出积分n I 的具体表达式，再求和144001sin sin sin12nn nnI xcosxdx xd xnππ+⎛===+⎝⎭⎰⎰则100112nnn nIn+∞∞==⎛⎫= ⎪⎪+⎝⎭∑∑.考虑幂级数11(),1nnS x xn∞+==+∑求出幂级数的和函数，代入2x=即可得出答案，按通常求收敛半径的办法.所以111lim lim lim111nx x xna nna nnρ+→∞→∞→∞+====+得到本题中幂级数的收敛半径()11,11Rρ==-在，内，先微分再积分，在收敛域内幂级数仍收敛，有11000111()111n n nn n nS x x x xn n x∞∞∞++===''⎛⎫⎛⎫'====⎪⎪++-⎝⎭⎝⎭∑∑∑,所以001()(0)()0ln11x xS x S S x dx dx xx'=+=+=---⎰⎰以()1,1x=-代入,得ln(1ln(2S=-=+.即l n(22)nnI∞==∑.八【证明】方法1：令()(),0xF x f t dt xπ=≤≤⎰，有(0)0,F=由题设有()0Fπ=.又由题设()cos0f x xdxπ=⎰，用分部积分，有000()cos cos()f x xdx xdF xππ==⎰⎰00()cos()sinF x x F x xdxππ=+⎰0()sinF x xdxπ=⎰由积分中值定理知，存在(0,)ξπ∈使0()sin ()sin (0)F x xdx F πξξπ==⋅-⎰因为(0,)ξπ∈，sin 0ξ≠，所以推知存在(0,),ξπ∈使得()0F ξ=. 再在区间[0,]ξ与[,]ξπ上对()F x 用罗尔定理，推知存在1(0,)ξξ∈，2(,)ξξπ∈使12()0,()0F F ξξ''==，即 12()0,()0f f ξξ==方法2：由()0f x d x π=⎰及积分中值定理知，存在1(0,)ξπ∈，使1()0f ξ=. 若在区间(0,)π内()f x 仅有一个零点1ξ，则在区间1(0,)ξ与1(,)ξπ内()f x 异号. 不妨设在1(0,)ξ内()0f x >，在1(,)ξπ内()0f x <. 于是由()0,()cos 0f x dx f x xdx ππ==⎰⎰，有111101100()cos ()cos ()(cos cos )()(cos cos )()(cos cos )f x xdx f x dx f x x dxf x x dx f x x dxπππξπξξξξξ=-=-=-+-⎰⎰⎰⎰⎰当10x ξ<<时，1c o s c o s x ξ>，1()(cos cos )0f x x ξ->；当1x ξπ<<时，1c o s c o s x ξ<，仍有1()(cos cos )0f x x ξ->，得到：00>. 矛盾，此矛盾证明了()f x 在(0,)π仅有1个零点的假设不正确，故在(0,)π内()f x 至少有2个不同的零点.九【详解】方法1：设方程组112233x x x αααβ++= ①对方程组的增广矩阵作初等行变换，化成阶梯形矩阵，有[]123211211,,21121011054104304a ab a bc a c αααβ----⎡⎤⎡⎤⎢⎥⎢⎥=→+-+⎢⎥⎢⎥⎢⎥⎢⎥+-+⎣⎦⎣⎦211210140031aa b a c b -⎡⎤⎢⎥→+-+⎢⎥⎢⎥+-+⎣⎦(1) 当4a ≠-时，[][]1231233r ,,r ,,,ααααααβ==. 方程组①唯一解，即β可由123,,ααα线性表出，且表出唯一.(2) 当4a =-，但310c b -+≠时，[][]12312323r ,,r ,,,ααααααβ=≠=方程组①无解，β不可由123,,ααα线性表出(3) 当4a =-，且310c b -+=时，[][]1231232r ,,r ,,,ααααααβ==方程组①有无穷多解，此时有[]1234211,,21010000b αααβ--⎡⎤⎢⎥→--+⎢⎥⎢⎥⎣⎦得对应齐次方程组的基础解系为：()120T,,ξ=-(取自由未知量11x =，回代得2320x ,x =-=)，非齐次方程的一个特解是()()0121T*,b ,b η=-++⎡⎤⎣⎦，故通解为()1021021k b ,b ⎡⎤⎡⎤⎢⎥⎢⎥-+-+⎢⎥⎢⎥⎢⎥⎢⎥+⎣⎦⎣⎦其中k 是任意常数. 方法2：设方程组112233x x x αααβ++= ①因为①是三个方程的三个未知量的线性非齐次方程组，故也可由系数行列式讨论，()1232121211211141054001a a A ,,a ααα----⎡⎤⎡⎤⎢⎥⎢⎥====-+⎢⎥⎢⎥⎢⎥⎢⎥-⎣⎦⎣⎦因此知道：(1) 当4a ≠-时，0A ≠，方程组有唯一解，β可由123,,ααα线性表出，且表出唯一.(2) 当4a =-时，(有可能无解或无穷多解)对增广矩阵作初等行变换，得[]12342112111,,2110012110540015b b c c b αααβ--⎡⎤⎡⎤⎢⎥⎢⎥=→+⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦21110012100031b c b ⎡⎤⎢⎥→+⎢⎥⎢⎥-+⎣⎦(i) 当4a =-时，且但310c b -+≠时，有[][]12312323r ,,r ,,,ααααααβ=≠=方程组①无解.(ii) 当4a =-，且310c b -+=时，[][]1231232r ,,r ,,,ααααααβ==方程组①有无穷多解，其通解为()1021021k b ,b ⎡⎤⎡⎤⎢⎥⎢⎥-+-+⎢⎥⎢⎥⎢⎥⎢⎥+⎣⎦⎣⎦其中k 是任意常数.十【详解】方法1：用正定性的定义判别.已知对任意的12n x ,x ,x 均有()120n f x ,x ,x ≥ ，其中等号成立当且仅当 1122231110000n n n n n x a x x a x x a x x a x --+=⎧⎪+=⎪⎪⎨⎪+=⎪+=⎪⎩ ①方程组①仅有零解的充分必要条件是其系数行列式()12112110000100001001100001001n n n na a B a a a a a +-==+-≠即当()121nn a a a ⋅≠- 时，方程组①只有零解，此时()120n f x ,x ,x = . 若对任意的非零向量()120n X x ,x ,x ,=≠ ①中总有一个方程不为零，则有()222212112223111()()()()0n n n n n n f x ,x ,x x a x x a x x a x x a x --=++++++++>所以，根据正定二次型的定义，对任意的向量()12n x ,x ,x ，如果()120n f x ,x ,x ≥ ，则二次型正定. 由以上证明题中12(,,,)n f x x x 是正定二次型. 方法2： 将二次型表示成矩阵形式，有()222212112223111()()()()n n n n n n f x ,x ,x x a x x a x x a x x a x --=++++++++[]112223112223111111n n n n n n n n n n x a x x a x x a x ,x a x ,,x a x ,x a x x a x x a x ----+⎡⎤⎢⎥+⎢⎥⎢⎥=++++⎢⎥+⎢⎥⎢⎥+⎣⎦[]111222121110001000100001000100001000001000010001001n n n n n n a a x a a x a x ,x ,x a a a x --⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦记 1122110000100001000001001n n n a x a x B ,X a a x -⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦则 ()()120TT Tn f x ,x ,x X B BX BX BX ==≥当()12112110000100001001100001001n n n na a B a a a a a +-==+-≠即当()121nn a a a ⋅≠- 时，0BX =只有零解，故当任意的0X ≠时，均有()()120Tn f x ,x ,x BX BX => ，从而由正定二次型的定义，对任意的向量()12n x ,x ,x ，如果()120n f x ,x ,x > ，则()12n f x ,x ,x 是正定二次型.十一【详解】ln Y X =⇒YX e =. 题设条件Y 为正态，故()()Y E X E e =可用函数的期望的公式求得. 将X 的样本可以转化成Y 的样本，从而对正态(,1)Y N μ 中的μ求得置信区间. 最后，再从μ的置信区间转得b 的置信区间.(1) 由正态分布密度函数的定义知，Y 的概率密度为2()2(),,y f y y μ--=-∞<<+∞于是 2()2()()y Y y b E X E e e edy μ--+∞-∞===⋅⎰令t y μ=-，有 ()221111222t t t b ee dt ee dt μμ+∞+∞-+--+-∞-∞=⋅=⎰12eμ+=.(2) 当置信度10.95α-=时，0.05α=.查表可知标准正态分布的双侧分位数等于1.96.故由1(,)4Y N μ ，其中Y 表示总体Y 的样本均值，11(ln 0.50ln 0.80ln1.25ln 2.00)ln10.44Y =+++== Y 是μ的无偏估计，且2σ(0,1).N所以，按标准正态分布的α分位点的定义，有 /21,P Z αα⎫⎪<=-⎬⎪⎭即/2/21.P Y Y Z ααμα⎧⎫<<=-⎨⎬⎩⎭这样，我们就得到了μ的一个置信水平为1α-的置信区间/2/2,Y Y Z αα⎛⎫⎪⎝⎭在此题中，1,4σμ==，0Y =，所以参数μ的置信度为0.95的置信区间为( 1.96 1.96(0.98,0.98).Y Y -+=- (3) 由指数函数xe 的严格单调递增性，有{}10.980.980.48 1.482P P μμ⎧⎫-<<=-<+<⎨⎬⎩⎭10.48 1.482P e e e μ+-⎧⎫=<<⎨⎬⎩⎭{}0.48 1.48P e b e -=<<0.95=因此b 的置信度为0.95的置信区间为()0.481.48,.ee -十二【分析】随机变量X Y 和不相关(,)0Cov X Y ⇔=.事件A B 与相互独立()()()P AB P A P B ⇔=.要找出这二者之间的联系就应从(,)()()()Cov X Y E XY E X E Y =-入手.【详解】{}(){}{}()1121E X P A P A P A =⋅+-⋅=-，同理，{}()2 1.E Y P B =- 现在求()E XY ，由于XY 只有两个可能值1和1-，所以{}(){}()1111,E XY P XY P XY =⋅=+-⋅=-其中 {}{}{}{}{}11,11,1P XY P X Y P X Y P AB P AB ====+=-=-=+{}{}{}{}{}121P AB P A B P AB P A P B =+-=--+和 {}{}{}{}{}11,11,1P X Y P X Y P X Y P A B P A B=-===-+=-==+{}{}{}2P A P B P AB =+-( 或者 {}{}{}{}{}1112P X Y P X Y P A P B P A B =-=-==+- )所以 {}{}()11E XY P XY P XY ==-=-{}{}{}4221P AB P A P B =--+ 由协方差公式，()()()()Cov XY E XY E X E Y =-{}{}{}{}{}42212121P AB P A P B P A P B =--+--⋅-⎡⎤⎡⎤⎣⎦⎣⎦ {}{}{}4P AB P A P B =-⎡⎤⎣⎦因此，()0Cov XY =当且仅当{}{}{}P AB P A P B =，即X Y 和不相关的充分必要条件是A B 与相互独立.。
2000年政治考研真题(理科)及参考答案
2000年全国硕士研究生入学统一考试——政治试题（理科）一、下列每题的选项中，有一项是最符合题意的。
请在答题纸上将所选项的字母涂黑。
（每小题1分，共16分）1．中国古代哲学家荀子说：“心不在焉，则白黑在前而目不见，雷鼓在侧而耳不闻。
”这段话表明人的意识具有A．客观性 B．能动性 C．对象性 D．任意性2．相信“意念移物”，甚至相信可以用意念来直接改变物质结构，就是信奉A．主张精神主宰客观物质世界的主观唯心论B．主张精神是脱离人脑独立存在的客观唯心论C．认为人的思想是特殊物质的机械唯物主义D．认为人具有主观能动性的实践唯物主义3．主观辩证法与客观辩证法的关系是A．反映与被反映的关系 B．唯心主义与唯物主义的关系C．抽象与具体的关系 D．唯心辨证法与唯物辩证法的关系4．马克思在研究商品时，之所以考察商品的使用价值，因为使用价值是A．构成财富的物质内容 B．人类生存、发展的物质条件C．满足人们需要的物质实体 D．商品交换价值的物质承担者5．分析资本主义生产过程是劳动过程和价值增殖过程的统一，关键是运用。
A．商品二因素原理B．资本区分为不变资本和可变资本的原理C．资本区分为固定资本和流动资本的原理D．劳动二重性原理6．在揭示资本主义工资的本质问题上，首先要区分A．必要劳动和剩余劳动B．劳动和劳动力C．具体劳动和抽象劳动D．私人劳动和社会劳动7．国家垄断资本主义的产生和发展，从根本上说是A．国内市场竞争激化的结果B．国际竞争激化的结果C．垄断统治加强的结果D．生产生产社会化与资本主义私人占有制之间矛盾发展的结果8．“统帅革命的资产阶级，联合革命的无产阶级，实行资产阶级民主革命。
”这一观点的错误实质在于A．抹煞农民阶级的革命性B．夸大资产阶级的作用C．放弃无产阶级的领导权D．忽视武装斗争的重要性9．1942年延安整风运动的中心任务是A．反对主观主义以整顿学风B．反对宗派主义以整顿党风C．反对党八股以整顿文风D．反对官僚主义以整顿作风10．毛泽东完整地提出新民主主义革命总路线和总政策的著作是A．《中国社会各阶级的分析》B．《新民主主义论》C．《在晋绥干部分议上的讲话》D．《论人民民主专政》11．在近代中国社会的诸矛盾中，最主要的是A．帝国主义和中华民族的矛盾B．封建主义和人民大众的矛盾C．资产阶级和无产阶级的矛盾D．反动统治阶级内部的矛盾12．现阶段，以劳动者的劳动联合和劳动者的资本联合为主的股份合作制属于A．国有经济B．集体经济C．个体经济D．私营经济13．社会主义初级阶段是指A．任何国家进入社会主义都会经历的起始阶段B．资本主义向社会主义的过渡阶段C．我国生产力落后，商品经济不发达条件下建设社会主义必然要经历的阶段D．新民主主义向社会主义过渡的阶段14．我国经济增长方式的根本转变，是指由A．计划经济向市场经济转变B．粗放型向集约型转变C．追求速度向注重效益转变D．注重数量向注重质量转变15．依法治国是党领导人民治理国家的基本方略，其根本目的在于A．维护宪法和法律在国家政治、经济和社会生活中的权威B．依法打击敌视和破坏社会主义的敌对分子，维护社会稳定C．保证人民充分行使当家作主的权利，维护人民当家作主的地位D．确保国家各项工作有法可依，依法行政16．1999年10月，国家立法机构拒绝批准由联合国大会通过的《全面禁止核试条约》的有核国家的是A．英国B．美国C．俄罗斯D．法国2000年全国攻读硕士学位研究生入学考试政治理论课（理科）答案及解析一、下列每题的选项中，有一项是最符合题意的。
2000年考研数学二真题
( 3)( ) || ( )2000 年全国硕士研究生入学统一考试理工数学二试题详解及评析一、 填空题arctan x - x （1） lim x →0 ln 1+ 2x 1= .【答】- . 61-1【详解】 lim arctan x - x = lim arctan x - x = lim 1+ x 2x →0ln (1+ 2x 3 )x →0= lim2x 3-x 2x →06x 2x →06x 2 (1+ x 2 )=- 16（2）设函数 y = y x 由方程2xy= x + y 所确定，则dy= .x =0【答】(ln 2 -1) d x【详解】 方法一：根据微分形式不变性，在已知等式两边同时求微分，得2xy ( ydx + xdy )ln 2 = dx + dy由原方程知，当 x = 0 时， y = 1，将其代入上式，得即有方法二：ln 2dx - dx = dy ,dy = ln 2-1 dx , x =0在方程2xy= x + y 两边对 x 求导，得2xy ln 2 ⋅将 x = 0 代入原方程得 y = 1，将 x = 0 ， y = 1代入上式有：ln 2 (1+ 0) = 1+ dydxy + x dy= 1+ dy dx dx| ( ) b →+∞ 3 0 ' ∞即有 dy = ln 2 -1dx所以+∞ dxdy = ln 2-1 dx , x =0（3）⎰2π【答】3= .【详解】= t , 则 x = t 2+ 2, dx = 2tdt , 于是+∞dx+∞2tdt= lim b 2dt⎰2x - 2⎰0(t 2+ 9)tb →+∞⎰0 t 2+ 9= lim 2arctan t |b=π31（4）曲线 y = (2x -1)e x 的斜渐近线方程为.【答】y = 2x +1【详解】 因为= y1 1a limx →∞ x = lim 2 - x →∞ e x = 2 x= ϒ 1 1 /b lim ( y - 2x ) = lim '2x e x -1 - e x ∞x →∞ x →∞'≤ ∞ƒϒ 1 / ' 2 e x -1 = lim '1 ∞ - e x ∞ = 1故渐近线方程为x →∞1 ' ∞'≤ x ∞ƒ y = 2x +1ϒ 1 0 0 0/'-2 3 0 0∞-1 -1（5）设 A = ' ∞ , E 为 4 阶单位矩阵，且 B = ( E + A ) ( E - A ) ,则( B + E ) ' 0 -4 5 0∞ ' 0 0 -6 7∞≤ƒ＝.=' ∞ ( ) ( ) ( )() '∞【答】ϒ 1 0 0 0/ '-1 2 0 0∞' 0 -2 3 0∞ ' 0 0 -3 4∞≤ ƒ【详解】 由 B = ( E + A )-1( E - A ) ，有( E + A ) B = E - A即AB + A + B + E = 2E ,( E + A )( E + B ) = 2E ,1也即 ( E + A )( E + B ) = E ,2故( B + E )-1 =ϒ 1 00 0/ 1 '-1 2 0 0∞E + A = 2 ' 0 -2 3 0∞ ' 0 0 -3 4∞≤ƒ二、选择题（1）设函数 f ( x ) =xa + e bx在 -∞, +∞ 内连续，且 lim x →-∞f ( x ) = 0 ，则常数a , b 满足 （A ） a < 0, b < 0（B ） a > 0, b > 0（C ） a ≤ 0, b > 0（D ） a ≥ 0, b < 0【 】【答】 应选（D ）【详解】 由题设，f ( x ) 在(-∞, +∞) 内连续，因此对任意的 x ∈(-∞, +∞) ，有，，这只需a ≥ 0 即可.另外，由 lim x →-∞所以必有b < 0 f x = 0 知 ， lim a + e bx=∞x →-∞故正确答案为（D ）（2）设函数 f ( x ) 满足关系式 f '' ( x ) + ϒ≤ f ' ( x )/ƒ2= x , 且 f '(0) = 0 ，则 （A ） f (0) 是 f ( x ) 的极大值 （B ） f (0) 是 f ( x ) 的极小值≤ ƒ （C ）点(0, f (0))是曲线 y = f ( x ) 的拐点（D ） f (0) 不是 f ( x ) 的极值，点(0, f (0))不是曲线 y = f ( x ) 的拐点【 】【答】 应选（C ）【详解】 因为 f ' (0) = 0 ，由原关系式f '' ( x ) + ϒ≤ f '( x )/ƒ2= x ,知 f ''(0) = 0, 因此点(0, f (0))可能为拐点.由 f '' ( x ) =- ϒ≤ f '( x )/ƒ2= x ,知 f ( x ) 的三阶导数存在，且f ''' ( x ) = -2 f ' ( x ) f '' ( x ) +1可见 f ''' (0) = 1因此在 x = 0 的左侧，f '' ( x ) < 0 ，对应曲线是下凹（上凸）的；而在 x = 0 的右侧，f '' ( x ) > 0 ， 对应曲线是上凹（上凸）的.故点(0, f (0))是曲线 y = f ( x ) 的拐点（ 3 ）设函数 f (x ), g ( x ) 是大于零的可导函数，且 f '( x ) g ( x ) - f ( x ) g '( x ) < 0, 则当 a < x < b 时，有（A ） f ( x ) g (b ) > （C ） f ( x ) g ( x ) > f (b ) g ( x ) f (b )g (b ) （B ） f ( x ) g (a ) > （C ） f ( x ) g ( x ) >f (a )g ( x )f (a )g (a )【答】 应选（A ）. 【详解】 由题设知ϒ f ( x )/'' g ( x ) ∞ =【 】f ' ( x )g ( x ) - f ( x )'g ( x )g 2( x ) < 0 因此当a < x < b 时，有f ( x ) > f (b ) ,g ( x ) g (b )即f ( x )g (b ) >f (b )g ( x )可见（A ）为正确选选项.( )ƒ（4）若lim x →0 sin 6x + xf ( x ) x 3 = 0, 则lim x →0 6 + f ( x ) x 2为 （A ）0（B ）6（C ）36（D ） ∞【 】【答】 应选（C ） 【详解】 方法一： 因为sin 6x = 6x - 1(6x )3+ o x 3 3!所以有limx →0sin 6x + xf ( x )x 3= lim x →06x - 36x 3 + o (x 3 ) + xf ( x )x 3= lim ϒ 6 + f ( x ) - 36/可见lim x →06 + f ( x ) x2＝36 ' 2∞ x →0≤ x ƒ = 0方法二： 因为limsin 6x + xf ( x )= limsin 6x - 6x + 6x + xy ( x )x →0x 3x →0x 3= lim ϒsin 6x - 6x +6 + f ( x ) / ' x →0 ≤ = 0所以x 3 x 2 ∞ lim6 + f ( x ) = -limsin 6x - 6x = - lim 6 cos 6x - 6x →0x 2x →0 x 3x →0 3x 2 = -lim -12sin 6x = 36x →0 2x（5）具有特解 y = e - x, y = 2xe - x, y = 3e x的 3 阶常系数齐次微分方程是1 23（A ） y ''' - y ''- y '+ y = 0（B ） y ''' + y '' - y '- y = 0（C ） y ''' - 6 y '' +11y '- 6 y = 0（D ） y '''- 2 y ''- y '+ 2 y = 0【 】【答】 应选（B ）【详解】 由特解知，对应特征方程的根为λ1 = λ2 = -1, λ3 = 1δ t δ 1 x♦ 于是特征方程为(λ +1)2(λ -1) = λ3 + λ 2 - λ -1 = 0故所求线性微分方程为y ''' + y '' - y ' - y = 0可见正确选项为（B ）三、设 f (ln x ) =ln (1+ x )x, 计算⎰ f ( x )dx .【详解】设ln x = t , 则 x = e t，于是ln (1+ e t ) f (t ) = et从而,ln (1+ e x )⎰ f ( x )dx = ⎰exdx = -⎰ ln (1+ e x )de - x = -- xx1 - x xe xe ln (1+ e ) + ⎰1+ exdx = -e ln (1+ e ) + ⎰1- 1+ e xdx= -e - x ln (1+ e x ) + x - ln (1+ e x ) + C = x - (1+ e x )ln (1+ e x ) + C四、设 xOy 平面上有正方形 D ={( x , y ) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} 及直线l : x + y = t (t ≥ 0) 若S (t ) 表示正方形 D 位于直线l 的左下方部分的面积，试求⎰0S (t )dt ( x ≥ 0).【详解】根据题设，有♣ ♠ ♠S (t ) = ♠-♠ ♠ ♠♥1 t2 ,0 2 1 t 2 + 2t -1,1 < t ≤ 2 2 1, t > 2⎰⎰ x♦ = ♠ + 可见，当0 ≤ x ≤ 1时，xS (t )dt =x 1t 2dt = 1 x 3; 00 2 6当1 < x ≤ 2 时S (t )dt =x1 t 2dt +x- 1 t 2+ 2t -1 dt⎰⎰02⎰12当 x > 2 时，=- 1 x 3 + x 2 - x +1 6 3x 2 x⎰0S (t )dt = ⎰0S (t )dt + ⎰2 S (t )dt = x -1因此♣1 x 3,0 ≤ x ≤ 1,♠ 6 ⎰ x S (t ) dt = ♠- 1 x 3 + x 2 - x + 1,1 < x ≤ 2 0 ♠ 63 ♠x -1, x > 2 ♥♠五、求函数 f ( x ) = x 2 ln (1+ x ) 在 x = 0 处的n 阶导数 f (n )(0)(n ≥ 3)【详解】方法一： 由麦克劳林公式f ( x ) = f (0) + f ' (0) x + f ''(0) 2! x 2 +… + f (n ) (0) n !x n +…及22ϒx 2 x3n -1x n -2/ x ln (1+ x ) = x 'x - 2 + -… + (-1) 3 n - 2 +…∞ ≤ ƒ比较 x n的系数得= x 3- x 4 +… + (-1) 4n -1 x n -2 n - 2 …f (n ) (0) (-1)(n -1)所以方法二：n !f (n ) (0) =n - 2(-1)(n -1)n !n - 2n n x() ⎰ 由莱布尼茨公式(uv )(n ) = u (n )v (0) + C 1u (n -1)v ' + C 2u (n -2)v '' +… + u (0)v (n ) (k ) (-1)k -1(k -1)!及ϒ≤ln (1+ x )/ƒ = (1+ x )k（ k 为正整数） 得f(n )( x ) = x 2(-1)n -1(n -1)! (1+ x )n+ 2nx (-1)n -2 (n - 2)! (1+ x )n -1 + n (n -1) (-1)n -3(n - 3)!(1+ x )n -2于是可得f (n ) (0) = (-1)n -3n (-1)(n - 3)! =(-1)n -1n !n - 2六、设函数S ( x ) = ⎰co s t dt ,（1） 当n 为正整数，且n π ≤ x < (n +1)π 时，证明2n ≤ S ( x ) < 2 (n +1); S ( x ) （2） 求 limx →+∞x【详解】（1）当n π ≤ x < (n +1)π 时，主义到被积函数是非负得，于是有n π (n +1)πcos xdx ≤ S x < 0cos xdx又因为 cos x 是以π 为周期的函数，在每一个周期上积分值相等所以n ππ⎰co s xdx = n ⎰0 cos xdx = 2n (n +1)πco s xdx = (n +1)⎰cos xdx = 2 (n +1)因此当n π ≤ x < (n +1)π 时，有2n ≤ S ( x ) < 2 (n +1);（2）由（1）知，当n π ≤ x < (n +1)π 时，有2n<S ( x ) < 2 (n +1)(n +1)πx n π当 x → +∞ ，有n →∞ ，根据夹逼定理得⎰ ⎰n πlimx →+∞ S ( x ) =2x π七、某湖泊的水量为V , 每年排入湖泊内含污染物 A 的污水量为V 6，流入湖泊内不含 A 的污V V水量为 6 ，流出湖泊的水量为 3，已知 1999 年底湖中 A 的含量为5m 0 ，超过国家规定指标，为了治理污染，从 2000 年初起，限制排入湖泊中含 A 污水的浓度不超过 m0 . 问至多需要经过V多少年，湖泊中污染物 A 的含量才可降至m 0 以内？（注：设湖水中 A 的浓度时均匀的） 【详解】设从 2000 年初（令此时， t = 0 ）开始，第七年湖泊中污染物 A 的总量为m (t ) ，浓m 度为 ,V 则在时间间隔[t , t + dt ] 上，排入湖泊中 A 的量近似为m 0 .V dt = m0 dt , 排除量近似为 V 6 6m V m . dt = V 3 3dt , 因此在时间间隔[t , t + dt ] 上m (t ) 的改变量为dm = m 0 - mdt6 3这是可分离变量方程，解得m- tm = 0 - Ce 32代入初始条件m (0) = 5m 0 ，得C =- 9 m2 0m- t 于是m = 0 1+ 9e 3 ，令m = m ，得2t = 6 ln 3即至多需要经过6 ln 3 年，湖泊中污染物 A 的含量才可降至m 0 以内.π π八、设函数 f ( x ) 在[0,π ]上连续，且⎰0f ( x )dx = 0, ⎰f ( x )cos xdx = 0, 试证明：在(0,π )内至少存在两个不同的点ξ1,ξ2 ，使 f (ξ1 ) = f (ξ2 ) = 0.【详解】 方法一：ππ1 2π π令 F (x)=⎰0 f (t )dt, 则有F (0)= 0, F (π)= 0又因为ππππ0 =⎰0f (t )c os xdx =⎰0co s xdF (x)=F (x)c os x|0+⎰0F (x)sin xdxπ=⎰0F (x)si n xdx令G(x)=⎰0F (t )sin tdt则G (0)=G (π)= 0, 于是存在ξ∈(0,π)，使F (ξ)sin ξ= 0.因为当ξ∈(0,π)，sin ξ≠ 0 ，所以有F (ξ)= 0 ，这样九证明了F (0)=F (ξ)=F (π)= 0再对F (x)在区间[0,ξ]，[ξ,π]上分别用罗尔中值定理，知至少存在ξ1∈(0,ξ),ξ2∈(ξ,π)，使得F ' (ξ)=F ' (ξ)= 0,即 f (ξ1)=f (ξ2)= 0,方法二：令F (x)=⎰0 f (t )dt, 则有F (0)= 0, F (π)= 0由罗尔中值定理知，存在ξ1 ∈(0,π)使得F ' (ξ)=f (ξ)= 0,1 1若在(0,π)内f (x)= 0 仅有一个实数根x =ξ1 ，则由⎰0f (x)dx = 0 可知，f (x)在(0,ξ1)内与(ξ1,π)内异号.ππ不妨设在(0,ξ1)内 f (x)> 0 ，于是再由⎰0[0,π]上的单调性知：f (x)dx = 0, ⎰0 f (x)cos xdx = 0, 及cos x 在π1' ∞ 0 = ⎰0 f ( x )(cos x - cos ξ1 ) dx = ξ1 f ( x )(cos x - cos ξ ) dx + πf ( x )(cos x - cos ξ ) dx > 0⎰1⎰ξ1矛盾，从而推知，在(0,π ) 内除ξ1 外， f ( x ) = 0 至少还有另一实数根ξ2 ，故知存在两个不同的点ξ1,ξ2 ，使 f (ξ1 ) = f (ξ2 ) = 0.九、已知 f ( x ) 是周期为 5 的连续函数，它在 x = 0 的某个邻域内满足关系式f (1+ sin x ) - 3 f (1- sin x ) = 8x + a ( x )其中a ( x ) 是当 x → 0 时比 x 高阶的无穷小，且 f ( x ) 在 x = 1 处可导，求曲线 y = f ( x ) 在点(6, f (6)) 处的切线方程.【详解】 由 f (1+ sin x ) - 3 f (1- sin x ) = 8x + a ( x ) 两边取极限，得lim ϒ≤ f (1+ s in x ) - 3 f (1- s in x )/ƒ = lim ϒ≤8x + a ( x )/ƒ即 有 于是得x →0f (1) - 3 f (1) = 0, f (1) = 0,x →0又因为limf (1+ sin x ) - 3 f (1- sin x )= lim ϒ 8x+ a ( x ) ⋅x / = 8 x →0可见limx →0 sin xf (1+ sin x ) - 3 f (1- sin x ) sin xx →0 ≤sin x x sin x ƒ = lim f (1+ sin x ) - f (1) + 3f (1- sin x ) - f (1)x →0 sin x -sin x = f ' (1) + 3 f ' (1) = 4 f ' (1) = 8故f ' (1) = 2由于 f ( x + 5) = f ( x ),所以 f (6) = f (1) = 0又 f ' (1) 存在，所以 f ' (6) 也存在，且 f ' (6) = f ' (1) = 2故所求得切线方程为1+ a 1+ a1+ a♥ 7 y = 2 ( x - 6)即2x - y -12 = 0十、设曲线 y = ax 2 (a > 0, x ≥ 0) 与 y = 1- x 2交于点 A , 过坐标原点O 和点 A 的直线与曲线y = ax 2 围成一平面图形，问a 为何值时，该图形绕 x 轴旋转一周所得的旋转体的体积最大？♣ y = ax 2【详解】当 x ≥ 0 时，由♦ y = 1- x2，解得x =1 , y = a故直线OA 的方程为：y =ax于是旋转体的体积为= π1a 2 x 2-V⎰01+a 1+ a a 2 x 4 dxϒ a 2 a 2 / 1 2π a 2= π ' x 3- x 5 ∞| 1+a = ⋅≤ 3(1+ a )5ƒ15 (1+ a)2从而有5253dV2π 2a (1+ a )2 - a ⋅ (1+ a )22 = ⋅ da 15 π (4a - a 2 ) (1+ a )5= 15(1+ a )2dV (a > 0) 令= 0, 并由a > 0 ，得唯一驻点a = 4da由题意知，此旋转体在a = 4 时取最大值，其最大体积为V =2π ⋅ 16 = 32 5 π 15 5 521875十一、函数 f ( x ) 在[0, +∞] 上可导， f (0) = 1，且满足等式f ' ( x ) + f ( x ) - 1 ⎰xf (t )dt = 0.（1） 求导数 f ' ( x ) ；x +1 0（2） 证明：当 x ≥ 0 时，不等式e - x ≤ f ( x ) ≤ 1成立.【详解】 （1） 由题设知上式两边对 x 求导，得( x +1) f ' ( x ) + ( x +1) f ( x ) - ⎰ xf (t )dt = 0.( x +1) f '' ( x ) = -( x + 2) f ' ( x ).即两边积分，得df '( x ) f ' ( x )=- x + 2dx x +1ln f ' ( x ) = -x + ln ( x +1) + ln C在题设等式中令 x = 0 ，得 f ' (0) + f (0) = 0 又 f (0) = 1，于是 f ' (0) = -1， 代入 f ' ( x ) 的表达式，得C = 1, 故有（2）方法一：f '( x ) =-e - xx +1当 x ≥ 0 时， f ' ( x ) < 0, 即 f ( x ) 单调减少，又 f (0) = 1，所以设ϕ ( x ) =f (x ) - e - x , 则 f ( x ) ≤ f (0) = 1.ϕ (0) = 0,ϕ ' ( x ) = f ' ( x ) + e - x =x x +1e - x ,当 x ≥ 0 时，ϕ '( x ) ≥ 0 ，即ϕ ( x ) 单调增加，因而ϕ ( x ) ≥ ϕ (0) = 0, 即有f ( x ) ≥ e - x ,xe1 0 12 3' ∞ 综上所述，当 x ≥ 0 时，不等式有e- x≤ f ( x ) ≤ 1.方法二： 因⎰ xf ' (t )dt = f ( x ) - f (0) = f ( x ) -1将 f ' ( x ) 代入，- x得 f ( x ) = 1- ⎰0 1+ tdt .x e-tx -t- x又 x ≥ 0 时，0 ≤ ⎰01+ tdt ≤ ⎰0 e dt = 1- e .所以 e - x≤ f ( x ) ≤ 1.ϒ1/ ' ∞ϒ 1 /' 1 ∞ϒ0/ ' ∞T T T十二、设α = '2∞ , β = ' 2 ∞ ,γ = '0∞ , A = αβ , B = B α ,其中β 是β 的转置，求解方程'≤1∞ƒ ' ∞'≤8∞ƒ ≤ ƒ2B 2 A 2 x = A 4 x + B 4 x + γ .【详解】 由题设，有ϒ1/ϒ 1 /' 2 A = αβ T ' ∞ 1 = '2 1 0∞'2∞ 1, 2 , 0 ' ∞ '≤1∞ƒ' 1 ∞ '10∞B = 1ϒ1/ ' ∞≤ 2 ƒ进一步有代入原方程化简，得即1, 2, 0'2∞ = 2, '≤1∞ƒA 2 = αβ T αβ T = α (β T α )β T = 2 A A 4 = 8A16 Ax = 8Ax +16x + γ8( A - 2E ) x = γ令 x = ( x , x , x ,)T，代入上式，得到非齐次线性方程组♥' ∞- ' ∞ ' ∞ ' ∞ ' ∞ ' ∞ = 6♣ -x + 1x = 0,1♠2x 2 2- x = 0, ♦ 1 2 ♠ 1♠x 1 + 2 x 2 - 2x 3 = 1其对应的齐次方程组的通解为非齐次方程组的一个特解为ϒ1/ξ = k '2∞ , （ k 为任意常数）'≤1∞ƒη*= 0, 0, -于是所求方程的通解为 x = ξ +η*, 即1 Tϒ /ϒ1/ ' 0 ∞ x = k '2∞ + ' 0 ∞ ，（ k 为任意常数） ' ∞ '≤1∞ƒ ' ∞' 1 ∞ ' ∞ ≤ 2 ƒϒ 0 / ϒa / ϒb /ϒ 1 / ϒ3/ ϒ 9 /十三、已知向量组β1= ' 1 ∞ , β '≤-1∞ƒ = '2∞ , β '≤1∞ƒ = '1∞ 与向量组α '≤0∞ƒ= ' 2 ∞ ,α '≤-3∞ƒ = '0∞ ,α '≤1∞ƒ ' ∞ ' ∞ '≤-7∞ƒ具有相同的秩，且β3 可由α1,α2 ,α3 线性表示，求a , b 的值. 【详解】 方法一：因为α1 和α2 线性无关，α3 = 3α1 + 2α2 ，所以向量组α1,α2 ,α3 线性无关，且秩为 2，α1 ，α2 为它的一个极大线性无关组.由于向量组β1, β2 , β3 与α1,α2 ,α3 具有相同的秩，故β1, β2 , β3 线性相关，从而行列式0 a b由此解得a = 3b ,β1, β2 , β3 = 1 2 1 = 0,-1 1 0又β3 可由α1,α2 ,α3 线性表示，从而可由α1 ，α2 线性表示，2 23 12 3∞ ∞ 于是α1 ，α2 ， β3 线性相关，因此有1 3 b化简得2b -10 = 0 ， 于是a = 15, b = 5. β1, β2 , β3 = 2 0 1 = 0,-3 1 0方法二：因β3 可由α1,α2 ,α3 线性表示，故线性方程组ϒ 1 3 9 / ϒ x 1 / ϒb / ' 2 0 6 ∞ ' x ∞ = '1∞ ' ∞ ' 2 ∞ ' ∞ '≤-3 1 -7∞ƒ '≤ x 3 ∞ƒ '≤0∞ƒ有解，对增广矩阵施行初等行变换：ϒ /'1 3 9 # b ∞' 1∞ → '0 1 2 # 2b -1 ∞ -3 1 -7 # 0∞ '0 10 20 # 3b ∞ ' 6 ∞≤ ƒ ≤ ƒ ''0 0 0 # - 2b -1∞ ≤ 6 ƒ由非齐次线性方程有解的条件知3b - 2b -1 = 0,10 6解得 b = 5又因为α1 ，α2 线性无关，α3 = 3α1 + 2α2所以向量组α1,α2 ,α3 的秩为 2，而题设α1,α2 ,α3 与β1, β2 , β3 同秩，从而有0 a 5由此解得a = 15.β1, β2 , β3 = 1 2 1 = 0,1 1 0ϒ 1 3 9 # b / ϒ1 3 9 # b / ' ' 2 0 6 # ∞ '0 -6 -12 # 1- 2b ∞ ' ∞。
2000年考研英语真题答案及解析
2000年全国硕士研究生入学统一考试英语试题答案与解析PartⅠCloze Test1.C2.A3.B4.A5.C6.D7.B8.D9.C10.DPartⅡReading ComprehensionPassage111.C12.D13.B14.APassage215.C16.B17.A18.DPassage319.B20.A21.C22.DPassage423.B24.D25.C26.APassage527.A28.C29.D30.BPartⅢEnglish-Chinese Translation31.在现代条件下，这需要程度不同的集中控制措施，从而就需要获得诸如经济学和运筹学等领域的专家的协助。
32.再者，显而易见的是一个国家的经济实力与其工农业生产效率密切相关，而效率的提高则又有赖于各种科技人员的努力。
33.大众通讯的显著发展使各地的人们不断感到有新的需求，不断接触到新的习俗和思想。
由于上述原因，政府常常得推出更多的革新。
34.在先期实现工业化的欧洲国家中，其工业化进程以及随之而来的各种深刻的社会结构变革，持续了大约一个世纪之久，而如今一个发展中国家在十年左右就可能完成这个过程。
35.由于人口的猛增或人口的大量流动(现代交通工具使这种流动相对容易)造成的种种问题也会对社会造成新的压力。
SectionⅣWriting(15points)36.见分析PartⅠClose Test一、文章总体分析本文是一篇短小的论证性文章，其主题是强调农民储存余粮的必要性。
文章①句提出论点：农民想成功，就必须努力保持消费和生产之间有较大的差距。
②句对①句进行具体的解释：即他必须存储大量的粮食。
③④⑤从正面论述储存余粮的必要性：③句总说可以养家糊口；④⑤句具体说可以留作播种、应对恶劣天气影响及作为商品卖掉以满足农业再生产等需要。
⑥⑦⑧句论述没有余粮的危害:不能自给自足，从反面论证储存余粮的必要性。
二、试题具体解析1.\[A\]other than不同于，除了……[B]as well as也，又（表示附加）[C]instead of而不是……（表选择）[D]more than比……更多（表比较）本题考核的知识点是：逻辑关系。
2000年全国硕士研究生统一考试数学一真题及答案解析
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2000 年全国硕士研究生入学统一考试
数学(一)试卷
一、填空题(本题共 5 小题,每小题 3 分,满分 15 分.把答案填在题中横线上)
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(2)曲面 x2 2 y2 3z2 21在点 (1, 2, 2) 的法线方程为_____________.
故正确选项为（B）。
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2000年考研外语考试真题及答案
2000年考研外语考试真题及答案一、Use of English1、If a farmer wishes to succeed, he must try to keep a wide gap between his consumption and his production. He must store a large quantity of grain 【B1】consuming all his grain immediately. He can continue to support himself and his family 【B2】 he produces a surplus. He must use this surplus in three ways: as seed for sowing, as an insurance 【B3】the unpredictable effects of bad weather and as a commodity which he must sell in order to 【B4】 old agricultural implements and obtain chemical fertilizers to 【B5】 the soil. He may also need money to construct irrigation 【B6】 and improve his farm in other ways. If no surplus is available, a farmer cannot be 【B7】 . He must either sell some of his property or 【B8】 extra funds in the form. of loans. Naturally he will try to borrow money at a low 【B9】 of interest, but loans of this kind are not 【B10】obtainable.【B1】A．other thanB．as well asC．instead of D．more than 2、【B2】A．only ifB．much asC．long before D．ever since 3、【B3】A.forB.againstC.ofD.towards4、【B4】A．replaceB．purchaseC．supplement D．dispose5、【B5】A．enhanceB．mixC．feedD．raise6、【B6】A．vesselsB．routesC．pathsD．channels7、【B7】A．self-confident B．self-sufficient C．self-satisfied D．self-restrained 8、【B8】A．searchB．saveC．offerD．seek9、【B9】A．proportionB．percentageC．rateD．ratio10、【B10】A．genuinelyB．obviouslyC．presumablyD．frequently11、 As I'll be away for at least a year, I'd appreciate ______ from you now and then telling me how everyone is getting along.A.hearingB.to hearC.to be hearingD.having heard12、 Greatly agitated, I rushed to the apartment and tried the door, ______ to find it locked.A．justB．onlyC．henceD．thus13、 Doctors see a connection between increased amounts of leisure time spent ______ and the increased number of cases of skin cancer.A.to sunbatheB.to have sunbathedC.having sunbathedD.sunbathing14、 Unless you sign a contract with the insurance company for your goods, you are not entitled ______ a repayment for the goods damaged in delivery.A．toB．withC．forD．on15、 On a rainy day I was driving north through Vermont ______I noticed a young man holding up a sign reading "Boston". A．whichB．whereC．whenD．that16、 Christie stared angrily at her boss and turned away, as though ______ out of the office.A.wentB.goneC.to goD.would go17、 The roles expected ______ old people in such a setting give too few psychological satisfactions for normal happiness.A.ofB.onC.toD.with18、 Talk to anyone in the drug industry, ______ yon'll soon discover that the science of genetics is the biggest thing to hit drug research since penicillin was discovered.A.orB.andC.forD.so19、 It wasn't so much that I disliked her ______ that I just wasn't interested in the whole business.A．ratherB．soC．thanD．as20、 Countless divorced politicians would have been elected out of office years ago had they even thought of a divorce, let alone ______ one.A．gettingB．to getC．gottenD．get21、 He spoke so ______ that even his opponents were won over by his arguments.A．bluntlyB．convincinglyC．emphaticallyD．determinedly22、 France's ______ of nuclear testing in the South Pacific last month triggered political debates and mass demonstrations. A．assumptionB．consumptionC．presumptionD．resumption23、 The 215-page manuscript, circulated to publishers last October, ______ an outburst of interest.A.flaredB.glitteredC.sparkedD.flashed24、 His efforts to bring about a reconciliation between the two parties ______.A.came offB.came onC.came roundD.came down25、 The system was redesigned so embrace the network and eventually ______ it in a profitable direction.A.adaptB.controlC.installD.steer26、 The capital intended to broaden the export base and ______ efficiency gains from international trade was channeled in stead of uneconomic import substitution.A.secureB.extendC.defendD.possess27、 It is announced that a wallet has been found and can be ______ at the manager's office.A.declaredB.obtainedC.reclaimedD.recognize28、 When I ______ my senses, I found myself wrapped up in bed in my little room, with Grandma bending over me.A．woke upB．took toC．picked upD．came to29、 The American society is ______ an exceedingly shaky foundation of natural resources, which is connected with the possibility of a worsening environment.A．established onB．affiliated toC．originated fromD．incorporated with30、 I am not ______ with my roommate but I have to share the room with her, because I have nowhere else to live.A．concernedB．compatibleC．considerateD．complied31、 At first, the ______ of color pictures over a long distance seemed impossible, but, with painstaking efforts and at great expense, it became a reality.A．transactionB．transmissionC．transformationD．transition32、 When the committee ______ to details, the proposed plan seemed impractical.A.got downB.set aboutC.went offD.came up33、 ______ to some parts of South America is still difficult, because parts of the continent are stilt covered with thick forests.A.OrientationB.AccessC.ProcessionD.Voyage34、 Mr. Smith had an unusual ______: he was first an office clerk, then a sailor, and ended up as a school teacher.A.professionB.occupationC.positionD.career35、 The mayor is a woman with great ______ and therefore deserves our political and financial support.A．intentionB．instinctC．integrityD．intensity36、 The English weather defies forecast and hence is a source of interest and ______ to everyone.A.speculationB.attributionC.utilizationD.proposition37、 The fact that the golden eagle usually builds its nest on some high cliffs ______ it almost impossible to obtain the eggs or the young birds.A．rendersB．reckonsC．regardsD．relates38、 To impress a future employer, one should dress neatly, be ______, and display interest in the job.A.swiftB.instantC.timelyD.punctual39、 You don't have to install this radio in your new car; it's an ______ extra.A．excessiveB．optionalC．additionalD．arbitrary40、 We were pleased to note that the early morning delivery didn't ______ to the traffic jam of the busy city.A．aidB．amountC．addD．attribute41、 (Having isolated) on a remote island, (with) little work (to occupy) them, the soldiers suffered from boredom and low (spirits).A.Having isolatedB.withC.to occupyD.spirits42、 If the letter (to be mailed) (was placed) on the writing table an hour ago, it (is) certain (being) there now.A.to be mailedB.was placedC.isD.being43、 The (ruling) party could even lose (its) majority in the lower house of parliament, (started) a period of (prolonged struggling).A.rulingB.itsC.startedD.prolonged struggling44、 The mechanisms (at) work (are manifest) in the tendency for such physical activity (to) utilize the (potential) harmful constituents of the stress response.A.atB.are manifestC.toD.potential45、(In) the long run, however, this hurry (to shed) full-time staff may (be more) harmful (to) industry as it is to the workforce.A.InB.to shedC.be moreD.to46、 See to (it) that you include (in) the examination paper (whatever) questions they didn't know (the answer) last time.A.itB.inC.whateverD.the answer47、 Most newspapers, (while devoting) the major part of (its)space to recent events, usually manage to find (room) on the inside pages for articles (on) some interesting topics.A.while devotingB.itsC.roomD.on48、 One sign (by which) you are making progress in (an art) such as painting or photography is (that) you begin to realize how much (there is) to learn.A.by whichB.an artC.thatD.there is49、 The ideal listener stays both (inside and outside) the music at the moment it is played and (enjoying) it almost (as much as) the composer at the moment he (composes).A.inside and outsideB.enjoyingC.as much asposes50、(Continued) exposure to stress has been linked to (worsened) functioning of the immune system, (leaving) a personmore liable (for) infectionA.ContinuedB.worsenedC.leavingD.for参考答案：【一、Use of English】1~5CABAC6~10DBDCD11~50点击下载查看答案。
2000年考研英语真题(含答案解析)
2000年全国硕士研究生入学统一考试英语试题Part ⅠClose TestDirections:For each numbered blank in the following passage, there are four choices marked ［A］, ［B］, ［C］ and ［D］.Choose the best one and mark your answer on ANSWER SHEET 1 by blackening the corresponding letter in the brackets with a pencil.(10 points)①If a farmer wishes to succeed, he must try to keep a wide gap between his consumption and his production.②He must store a large quantity of grain 1 consuming all his grain immediately.③He can continue to support himself and his family 2 he produces a surplus.④He must use this surplus in three ways: as seed for sowing, as an insurance 3 the unpredictable effects of bad weather and as a commodity which he must sell in order to 4 old agricultural implements and obtain chemical fertilizers to 5 the soil.⑤He may also need money to construct irrigation 6 and improve his farm in other ways.⑥If no surplus is available, a farmer cannot be 7 .⑦He must either sell some of his property or 8 extra funds in the form of loans.⑧Naturally he will try to borrow money at a low 9 of interest, but loans of this kind are not 10 obtainable.［139 words］1.［A］ other than ［B］ as well as［C］ instead of ［D］ more than2.［A］ only if ［B］ much as［C］ long before ［D］ ever since3.［A］ for ［B］ against［C］ of ［D］ towards4.［A］ replace ［B］ purchase［C］ supplement ［D］ dispose5.［A］ enhance ［B］ mix［C］ feed ［D］ raise6.［A］ vessels ［B］ routes［C］ paths ［D］ channels7.［A］ self-confident ［B］ self-sufficient［C］ self-satisfied ［D］self-restrained8.［A］ search ［B］ save［C］ offer ［D］ seek9.［A］ proportion ［B］ percentage［C］ rate ［D］ ratio10.［A］ genuinely ［B］ obviously［C］ presumably ［D］ frequentlyPart ⅡReading ComprehensionDirections:Each of the passages below is followed by some questions.For each question there are four answers marked ［A］, ［B］, ［C］ and ［D］.Read the passages carefully and choose the best answer to each of the questions.Then mark your answer on ANSWER SHEET 1 by blackening the corresponding letter in the brackets with a pencil.(40 points)Passage 1①A history of long and effortless success can be a dreadful handicap, but, if properly handled, it may become a driving force.②When the United States entered just such a glowing period after the end of the Second World War, it had a market eight times larger than any competitor, giving its industries unparalleled economies of scale.③Its scientists were the world s best; its workers the most skilled.④(11)America and Americans were prosperous beyond the dreams of the Europeans and Asians whose economies the war had destroyed.①It was inevitable that this primacy should have narrowed as other countries grew richer.②Just as inevitably, the retreat from predominance proved painful.③By the mid-1980s Americans had found themselves at a loss over their fading industrial competitiveness.④Some huge American industries, such as consumer electronics, had shrunk or vanished in the face of foreign competition.⑤By 1987 there was only one American television maker left, Zenith.⑥(Now there is none: Zenith was bought by South Korea’s LG Electronics in July.) ⑦（12）Foreign-made cars and textiles were sweeping into the domestic market.America’s machine-tool industry was on the ropes.⑧For a while it looked as though the making of semiconductors, which America had invented and which sat at the heart of the new computer age, was going to be the next casualty.①All of this caused a crisis of confidence.②Americans stopped taking prosperity for granted.③They began to believe that their way of doing business was failing, and that their incomes would therefore shortly begin to fall as well.④The mid-1980s brought one inquiry after another into the causes of America’s industrial decline.⑤Their sometimes sensational findings were filled with warnings about the growing competition from overseas.①How things have changed! ②In 1995 the United States can look back on five years of solid growth while Japan has been struggling.③（14）Few Americans attribute this solely to such obvious causes as a devalued dollar or the turning of the business cycle.④Self-doubt has yielded to blind pride.⑤“American industry has changed its structure, has gone on a diet, has learnt to be more quick-witted,”according to Richard Cavanaugh, executive dean of Harvard’s Kennedy School of Government.⑥“It makes me proud to be an American just to see how our businesses are improving their productivity,”says Stephen Moore of the Cato Institute, a think-tank in Washington, DC.⑦And William Sahlman of the Harvard Business School believes that people will look back on this period as “a golden age of business management in the United States.”［429 words］11.The U.S.achieved its predominance after World War II because.［A］it had made painstaking efforts towards this goal［B］its domestic market was eight times larger than before［C］the war had destroyed the economies of most potential competitors ［D］the unparalleled size of its workforce had given an impetus to its economy12.The loss of U.S.predominance in the world economy in the 1980s is manifested in the fact that the American.［A］TV industry had withdrawn to its domestic market［B］semiconductor industry had been taken over by foreign enterprises ［C］machine-tool industry had collapsed after suicidal actions ［D］auto industry had lost part of its domestic market13.What can be inferred from the passage?［A］It is human nature to shift between self-doubt and blind pride.［B］Intense competition may contribute to economic progress.［C］ The revival of the economy depends on international cooperate ［D］A long history of success may pave the way for further development.14.The author seems to believe the revival of the U.S.economy in the 1990s can be attributed to the.［A］turning of the business cycle［B］ restructuring of industry［C］ improved business management［D］ success in educationPassage 2①（15）Being a man has always been dangerous.②There are about 105 males born for every 100 females, but this ratio drops to near balance at the age of maturity, and among 70-year-olds there are twice as many women as men.③But the great universal of male mortality is being changed.④Now, boy babies survive almost as well as girls do.⑤This means that, for the first time, there will be an excess of boys in those crucial years when they are searching for a mate.⑥More important, another chance for natural selection has been removed.⑦Fifty years ago, the chance of a baby (particularly a boy baby) surviving depended on its weight. A kilogram too light or too heavy meant almost certain death.⑧Today it makes almost no difference.Since much of the variation is due to genes, one more agent of evolution has gone.①There is another way to commit evolutionary suicide: stay alive, but have fewer children.②Few people are as fertile as in the past.③Except in some religious communities, very few women have 15 children.④Nowadays the number of births, like the age of death, has become average.⑤Most of us have roughly the same number of offspring.⑥（16）Again, differences between people and the opportunity for natural selection to take advantage of it have diminished.⑦India shows what is happening.The country offers wealth for a few in the great cities and poverty for the remaining tribal peoples.⑧The grand mediocrity of today—everyone being the same in survival and number of offspring—means that natural selection has lost 80% of its power in upper-middle-class India compared to the tribes.For us, this means that evolution is over; the biological Utopia has arrived.②Strangely, it has involved little physical change.③No other species fills so many places in nature.④But in the past 100, 000 years —even the past 100 years—our lives have been transformed but our bodies have not.⑤（17）We did not evolve, because machines and society did it for us.⑥Darwin had a phrase to describe those ignorant of evolution: they “look at an organic being as a savage looks at a ship, as at something wholly beyond his comprehension.”⑦No doubt we will remember a 20th century way of life beyond comprehension for its ugliness.But however amazed our descendants may be at how far from Utopia we were, they will look just like us.［406 words］15.What used to be the danger in being a man according to the first paragraph?［A］ A lack of mates. ［B］ A fierce competition.［C］ A lower survival rate. ［D］ A defective gene.16.What does the example of India illustrate?［A］ Wealthy people tend to have fewer children than poor people.［B］ Natural selection hardly works among the rich and the poor.［C］ The middle class population is 80% smaller than that of the tribes.［D］ India is one of the countries with a very high birth rate.17.The author argues that our bodies have stopped evolving because.［A］ life has been improved by technological advance［B］ the number of female babies has been declining［C］ our species has reached the highest stage of evolution［D］ the difference between wealth and poverty is disappearing18.Which of the following would be the best title for the passage?［A］ Sex Ratio Changes in Human Evolution.［B］ Ways of Continuing Man’s Evolution.［C］ The Evolutionary Future of Nature.［D］ Human Evolution Going Nowhere.Passage 3①（20）When a new movement in art attains a certain fashion, it is advisable to find out what its advocates are aiming at, for, however farfetched and unreasonable their principles may seem today, it is possible that in years to come they may be regarded as normal.②With regard to Futurist poetry, however, the case is rather difficult, for whatever Futurist poetry may be—even admitting that the theory on which it is based may be right—it can hardly be classed as Literature.①This, in brief, is what the Futurist says: for a century, past conditions of life have been conditionally speeding up, till now we live in a world of noise and violence and speed.②Consequently, our feelings, thoughts and emotions have undergone a corresponding change.③（21）This speeding up of life, says the Futurist, requires a new form of expression.④We must speed up our literature too, if we want to interpret modern stress.⑤We must pour out a large stream of essential words, unhampered by stops, or qualifying adjectives, or finite verbs.⑥Instead of describing sounds we must make up words that imitate them; we must use many sizes of type and different colored inks on the same page, and shorten or lengthen words at will.①Certainly their descriptions of battles are confused.②But it isa little upsetting to read in the explanatory notes that a certain line describes a fight between a Turkish and a Bulgarian officer on a bridge off which they both fall into the river —and then to find that the line consists of the noise of their falling and the weights of the officers: “Pluff! Pluff! A hundred and eighty-five kilograms.”①（22）This, though it fulfills the laws and requirements of Futurist poetry, can hardly be classed as Literature.②All the same, no thinking man can refuse to accept their first proposition: that a great change in our emotional life calls for a change of expression.③The whole question is really this: have we essentially changed?［334 words］19.This passage is mainly.［A］ a survey of new approaches to art［B］ a review of Futurist poetry［C］ about merits of the Futurist movement［D］ about laws and requirements of literature20.When a novel literary idea appears, people should try to.［A］ determine its purposes ［B］ ignore its flaws［C］ follow the new fashions ［D］ accept the principles21.Futurists claim that we must.［A］ increase the production of literature［B］ use poetry to relieve modern stress［C］ develop new modes of expression［D］ avoid using adjectives and verbs22.The author believes that Futurist poetry is.［A］ based on reasonable principles［B］ new and acceptable to ordinary people［C］ indicative of a basic change in human nature［D］ more of a transient phenomenon than literaturePassage 4①（23）Aimlessness has hardly been typical of the postwar Japan whose productivity and social harmony are the envy of the United States and Europe.②But increasingly the Japanese are seeing a decline of the traditional work-moral values.③Ten years ago young people were hardworking and saw their jobs as their primary reason for being, but now Japan has largely fulfilled its economic needs, and young people don’t know where they should go next.①The coming of age of the postwar baby boom and an entry of women into the male-dominated job market have limited the opportunities of teen-agers who are already questioning the heavy personal sacrifices involved in climbing Japan’s rigid social ladder to good schools and jobs.②In a recent survey, it was found that only 24.5 percent of Japanese students were fully satisfied with school life, compared with 67.2 percent of students in the United States.③In addition, far more Japanese workers expressed dissatisfaction with their jobs than did their counterparts in the 10 other countries surveyed.①While often praised by foreigners for its emphasis on the basics, Japanese education tends to stress test taking and mechanical learning over creativity and self-expression.②（25）“Those things that do not show up in the test scores—personality, ability, courage or humanity—are completely ignored,” says Toshiki Kaifu, chairman of the ruling Liberal Democratic Party’s education committee.③“Frustration against this kind of thing leads kids to drop out and run wild.”④Last year Japan experienced 2, 125 incidents of school violence, including 929 assaults on teachers.⑤Amid the outcry, many conservative leaders are seeking a return to the prewar emphasis on moral education.⑥Last year MitsuoSetoyama, who was then education minister, raised eyebrows when he argued that liberal reforms introduced by the American occupation authorities after World War II had weakened the “Japanese morality of respect for parents.”①（26）But that may have more to do with Japanese life-styles.②“In Japan,” says educator Yoko Muro, “it’s never a question of whether you enjoy your job and your life, but only how much you can endure.”③With economic growth has come centralization; fully 76 percent of Japan’s 119 million citizens live in cities where community and the extended family have been abandoned in favor of isolated, two-generation households.④Urban Japanese have long endured lengthy commutes (travels to and from work) and crowded living conditions, but as the old group and family values weaken, the discomfort is beginning to tell.⑤In the past decade, the Japanese divorce rate, while still well below that of the United States, has increased by more than 50 percent, and suicides have increased by nearly one-quarter.［447 words］23.In the Westerners’ eyes, the postwar Japan was.［A］ under aimless development ［B］ a positive example［C］ a rival to the West ［D］ on the decline24.According to the author, what may chiefly be responsible for the moral decline of Japanese society?［A］ Women’s participation in social activities is limited.［B］ More workers are dissatisfied with their jobs.［C］ Excessive emphasis has been placed on the basics.［D］ The life-style has been influenced by Western values.25.Which of the following is true according to the author?［A］ Japanese education is praised for helping the young climb the social ladder.［B］ Japanese education is characterized by mechanical learning as well as creativity.［C］ More stress should be placed on the cultivation of creativity.［D］ Dropping out leads to frustration against test taking.26.The change in Japanese life-style is revealed in the fact that.［A］ the young are less tolerant of discomforts in life［B］ the divorce rate in Japan exceeds that in the U.S.［C］ the Japanese endure more than ever before［D］ the Japanese appreciate their present lifePassage 5①（27）If ambition is to be well regarded, the rewards of ambition —wealth, distinction, control over one’s destiny—must be deemed worthy of the sacrifices made on ambition’s behalf.②If the tradition of ambitionis to have vitality, it must be widely shared; and it especially must be highly regarded by people who are themselves admired, the educated not least among them.③（28）In an odd way, however, it is the educated who have claimed to have given up on ambition as an ideal.④What is odd is that they have perhaps most benefited from ambition—if not always their own then that of their parents and grandparents.⑤There is a heavy note of hypocrisy in this, a case of closing the barn door after the horses have escaped—with the educated themselves riding on them.①Certainly people do not seem less interested in success and its signs now than formerly.②Summer homes, European travel, BMWs—the locations, place names and name brands may change, but such items do not seem less in demand today than a decade or two years ago.③（29）What has happened is that people cannot confess fully to their dreams, as easily and openly as once they could, lest they be thought pushing, acquisitive and vulgar.④Instead, we are treated to fine hypocritical spectacles, which now more than ever seem in ample supply: the critic of American materialism with a Southampton summer home; the publisher of radical books who takes his meals in three-star restaurants; the journalist advocating participatory democracy in all phases of life, whose own children are enrolled in private schools.⑤For such people and many more perhaps not so exceptional, the proper formulation is, “Succeed at all costs but avoid appearing ambitious.”①The attacks on ambition are many and come from various angles; its public defenders are few and unimpressive, where they are not extremely unattractive.②As a result, the support for ambition as a healthy impulse, a quality to be admired and fixed in the mind of the young, is probably lower than it has ever been in the United States.③This does not mean that ambition is at an end, that people no longer feel its stirrings and promptings, but only that, no longer openly honored, it is less openly professed.④Consequences follow from this, of course, some of which are that ambition is driven underground, or made sly.⑤Such, then, is the way things stand: on the left angry critics, on the right stupid supporters, and in the middle, as usual, the majority of earnest people trying to get on in life.［431 words］27.It is generally believed that ambition may be well regarded if.［A］ its returns well compensate for the sacrifices［B］ it is rewarded with money, fame and power［C］ its goals are spiritual rather than material［D］ it is shared by the rich and the famous28.The last sentence of the first paragraph most probably implies that it is.［A］ customary of the educated to discard ambition in words［B］ too late to check ambition once it has been let out［C］ dishonest to deny ambition after the fulfillment of the goal ［D］ impractical for the educated to enjoy benefits from ambition29.Some people do not openly admit they have ambition because.［A］ they think of it as immoral［B］ their pursuits are not fame or wealth［C］ ambition is not closely related to material benefits［D］ they do not want to appear greedy and contemptible30.From the last paragraph the conclusion can be drawn that ambition should be maintained.［A］ secretly and vigorously ［B］openly and enthusiastically［C］ easily and momentarily ［D］ verbally and spirituallyPart ⅢEnglish-Chinese TranslationDirections:Read the following passage carefully and then translate the underlined segments into Chinese.Your translation must be written neatly on ANSWER SHEET 2.(15 points)Governments throughout the world act on the assumption that the welfare of their people depends largely on the economic strength and wealth of the community.31）Under modern conditions, this requires varying measures of centralized control and hence the help of specialized scientists such as economists and operational research experts.32）Furthermore, it is obvious that the strength of a country’s economy is directly bound up with the efficiency of its agriculture and industry, and that this in turn rests upon the efforts of scientists and technologists of all kinds.It also means that governments are increasingly compelled to interfere in these sectors in order to step up production and ensure that it is utilized to the best advantage.For example, they may encourage research in various ways, including the setting up of their own research centers; they may alter the structure of education, or interfere in order to reduce the wastage of natural resources or tap resources hitherto unexploited; or they may cooperate directly in the growing number of international projects related to science, economics and industry.In any case, all such interventions are heavily dependent on scientific advice and also scientific and technological manpower of all kinds.33）Owing to the remarkable development in mass-communications, people everywhere are feeling new wants and are being exposed to new customs and ideas, while governments are often forced to introduce still further innovations for the reasons given above.At the same time, the normal rate of social change throughout the world is taking place at a vastly accelerated speed compared with the past.For example, 34）in the early industrialized countries of Europe the process of industrialization—with all the far-reaching changes in social patterns that followed—was spread over nearly a century, whereas nowadays a developing nation may undergo the same process in a decade or so.All this has the effect of building up unusual pressures and tensions within the community and consequently presents serious problems for the governments concerned.35）Additional social stresses may also occur because of the population explosion or problems arising from mass migration movements—themselves made relatively easy nowadays by modern means of transport.As a result of all these factors, governments are becoming increasingly dependent on biologists and social scientists for planning the appropriate programs and putting them into effect.［390 words］Section ⅣWriting(15 points)36.Directions:A.Study the following two pictures carefully and write an essay of at least 150 words.B.Your essay must be written neatly on ANSWER SHEET 2.C.Your essay should meet the requirements below:1）Describe the pictures.2）Deduce the purpose of the painter of the pictures.3）Suggest counter-measures.2000年英语试题答案Part ⅠCloze Test1.C2.A3.B4.A5.C6.D7.B8.D9.C 10.DPart ⅡReading ComprehensionPassage 111.C 12.D 13.B 14.APassage 215.C 16.B 17.A 18.DPassage 319.B 20.A 21.C 22.DPassage 423.B 24.D 25.C 26.APassage 527.A 28.C 29.D 30.BPart Ⅲ English-Chinese Translation31.在现代条件下, 这需要程度不同的集中控制措施, 从而就需要获得诸如经济学和运筹学等领域的专家的协助。
2000考研数学二真题及答案
2000考研数学二真题及答案一、填空题(本题共5小题，每小题3分，满分15分，把答案填在题中横线上) (1) 30arctan lim.ln(12)x x xx →-=+(2) 设函数()y y x =由方程2xyx y =+所确定，则0.x dy==(3)2.+∞=⎰(4) 曲线1(21)xy x e =-的斜渐近线方程为.(5) 设1000230004500067A ⎡⎤⎢⎥-⎢⎥=⎢⎥-⎢⎥-⎣⎦，E 为4阶单位矩阵，且1()()B E A E A -=+-则 1()E B -+=.二、选择题(本题共5小题，每小题3分，共15分，在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) 设函数()bx xf x a e=+在(,)-∞+∞内连续，且lim ()0,x f x →-∞=则常数,a b 满足 ( ) (A)0,0.a b << (B)0,0.a b >> (C)0,0.a b ≤> (D)0,0.a b ≥<(2) 设函数()f x 满足关系式2()[()]f x f x x '''+=，且(0)0f '=，则 ( )(A)(0)f 是()f x 的极大值. (B)(0)f 是()f x 的极小值.(C)点(0,(0))f 是曲线()y f x =的拐点.(D)(0)f 不是()f x 的极值，点(0,(0))f 也不是曲线()y f x =的拐点.(3 ) 设(),()f x g x 是大于零的可导函数，且'()()()'()0,f x g x f x g x -<则当a x b << 时，有 ( )(A)()()()()f x g b f b g x > (B) ()()()()f x g a f a g x >(C)()()()()f x g x f b g b > (D) ()()()()f x g x f a g a >(4) 若30sin 6()lim 0x x xf x x →+⎛⎫=⎪⎝⎭，则206()lim x f x x →+为 ( ) (A)0. (B)6. (C)36. (D)∞.(5) 具有特解123,2,3x x xy e y xe y e --===的3阶常系数齐次线性微分方程是 ( )(A)0.y y y y ''''''--+= (B)0.y y y y ''''''+--= (C)61160.y y y y ''''''-+-= (D)220.y y y y ''''''--+=三、(本题满分5分)设ln(1)(ln )x f x x+=，计算()f x dx ⎰. 四、(本题满分5分)设xoy 平面上有正方形{}(,)01,01D x y x y =≤≤≤≤及直线:(0)l x y t t +=≥.若()S t 表示正方形D 位于直线l 左下方部分的面积，试求0(),(0)xS t dt x ≥⎰.五、(本题满分5分)求函数2()ln(1)f x x x =+在0x =处的n 阶导数(0)(3)nf n ≥.六、(本题满分6分)设函数0()|cos |xS x t dt =⎰，(1)当n 为正整数，且(1)n x n ππ≤≤+时，证明2()2(1)n S x n ≤<+； (2)求()limx S x x→+∞.七、(本题满分7分)某湖泊的水量为V ，每年排入湖泊内含污染物A 的污水量为6V，流入湖泊内不含A 的水量为6V ，流出湖泊的水量为3V，已知1999年底湖中A 的含量为05m ，超过国家规定指标.为了治理污染，从2000年初起，限定排入湖泊中含A 污水的浓度不超过0mV.问至多需要经过多少年，湖泊中污染物A 的含量降至0m 以内(注：设湖水中A 的浓度是均匀的) 八、(本题满分6分)设函数()f x 在[]0,π上连续，且()0,()cos 0f x dx f x xdx ππ==⎰⎰，试证明：在(0,)π内至少存在两个不同的点12,ξξ，使12()()0.f f ξξ==九、(本题满分7分)已知()f x 是周期为5的连续函数，它在0x =的某个邻域内满足关系式(1sin )3(1sin )8()f x f x x x α+--=+其中()x α是当0x →时比x 高阶的无穷小，且()f x 在1x =处可导，求曲线()y f x =在点(6,(6))f 处的切线方程.十、(本题满分8分)设曲线2(0,0)y ax a x =>≥与21y x =-交于点A ，过坐标原点O 和点A 的直线与曲线2y ax =围成一平面图形.问a 为何值时，该图形绕x 轴旋转一周所得的旋转体体积最大？最大体积是多少？十一、(本题满分8分)函数()f x 在[0,)+∞上可导，(0)1f =且满足等式01()()()0,1xf x f x f t dt x '+-=+⎰ (1)求导数()f x '；(2)证明：当0x ≥时，成立不等式()1xe f x -≤≤成立十二、(本题满分6分)设11012,,0,,2180T TA B αβγαββα⎛⎫⎛⎫⎛⎫ ⎪⎪ ⎪ ⎪===== ⎪ ⎪ ⎪ ⎪ ⎪⎪⎝⎭⎝⎭⎝⎭.其中T β是β的转置，求解方程22442B A x A x B x γ=++十三、(本题满7分)已知向量组12301,2,1110a b βββ⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪=== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭⎝⎭与向量组1231392,0,6317ααα⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪=== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪--⎝⎭⎝⎭⎝⎭具有相同的秩，且3β可由123,,ααα线性表出，求,a b 的值.参考答案一、填空题(1)【答案】16-【详解】()()()33ln 1222232322000011arctan arctan 11limlim lim lim 266ln 1261x x x x x x x x x x x x x x xx x +→→→→----+====-++洛(2)设函数()y y x =由方程2xyx y =+所确定，则0.x dy==【答案】(ln 21)dx - 【详解】 方法1：对方程2xyx y =+两边求微分，有2ln 2().xy xdy ydx dx dy ⋅+=+由所给方程知，当0x =时1y =. 将0x =，1y =代入上式，有ln 2dx dx dy ⋅=+. 所以，0(ln 21)x dy dx ==-.方法2：两边对x 求导数，视y 为该方程确定的函数，有2ln 2()1.xy xy y y ''⋅+=+当0x =时1y =，以此代入，得ln 21y '=-，所以0(ln 21)x dy dx ==-. (3)【答案】3π【详解】由于被积函数在2x =处没有定义，则该积分为广义积分.对于广义积分，可以先按照不定积分计算，再对其求极限即可.2,22,t x t dx tdt =-==02202122arctan .(9)33323t t dt t t ππ+∞+∞+∞==⋅=⋅=+⎰⎰(4)【答案】21y x =+【公式】y kx b =+为()y f x =的斜渐近线的计算公式：()()lim,lim [()]x x x x x x yk b f x kx x →∞→∞→+∞→+∞→-∞→-∞==-【详解】11lim lim (2)2,x x x y k e x x→+∞→+∞==-=10122lim (2)lim[(21)2]lim()u uxx u x e b y x x e x u e x u+→+∞→→+∞-=-=--= - 令 002(1)2lim()1lim()211u u u uu u e u e e u e uu ++→→-=- - -=-= 所以，x →+∞方向有斜渐近线21y x =+. 当x →-∞时，类似地有斜渐近线21y x =+. 总之，曲线1(21)xy x e =-的斜渐近线方程为21y x =+.(5)【答案】1000120002300034⎡⎤⎢⎥-⎢⎥⎢⎥-⎢⎥-⎣⎦【详解】先求出1()E B -+然后带入数值，由于1()()B E A E A -=+-，所以11111()()()()()()()12()()22000100024001200104600230200680034E B E E A E A E A E A E A E A E A E A -----⎡⎤+=++-⎣⎦⎡⎤=++++-⎣⎦⎡⎤=+=+⎣⎦⎡⎤⎡⎤⎢⎥⎢⎥--⎢⎥⎢⎥ ==⎢⎥⎢⎥--⎢⎥⎢⎥--⎣⎦⎣⎦－１－１－１二、选择题 (1)【答案】D 【详解】排除法：如果0a <，则在(,)-∞+∞内()f x 的分母bx a e +必有零点0x ，从而()f x 在0x x =处不连续，与题设不符.不选()A ，若0b >，则无论0a =还是0a ≠均有lim (),x f x →-∞=∞与题设lim ()0x f x →-∞=矛盾，不选()B 和()C .故选()D .(2)【答案】C【定理应用】判断极值的第二充分条件：设函数()f x 在0x 出具有二阶导数且0()0f x '=，0()0f x ''≠，那么：(1) 当0()0f x ''>时，函数()f x 在0x 处取得极大值；(2)当0()0f x ''<时，函数()f x 在0x 处取得极小值；【详解】令等式2()[()]f x f x x '''+=中0x =，得[]2(0)0(0)0f f '''=-=，无法利用判断极值的第二充分条件，故无法判断是否为极值或拐点.再求导数(因为下式右边存在，所以左边也存在)：[]2()(())12()()f x x f x f x f x ''''''''=-=-以0x =代入，有(0)1f '''=，所以0()(0)()(0)limlim 10x x f x f f x f x x→→''''''-'''===-. 从而知，存在0x =去心邻域，在此去心邻域内，()f x ''与x 同号，于是推知在此去心邻域内当0x <时曲线()y f x =是凸的，在此去心临域内0x >时曲线()y f x =是凹的， 点(0,(0))f 是曲线()y f x =的拐点，选(C).(3)【答案】A【分析】由选项答案可知需要利用单调性证明，关键在于寻找待证的函数. 题设中已知'()()()'()0,f x g x f x g x -< 想到设函数为相除的形式()()f xg x . 【详解】设()()()f x F xg x =，则()2'()()()'()()0,()f x g x f x g x F x g x -'=< 则()F x 在a x b <<时单调递减，所以对a x b ∀<<，()()()F a F x F b >>，即()()()()()()f a f x f bg a g x g b >> 得 ()()()(),f x g b f b g x >a x b <<，()A 为正确选项.(4)【答案】()C【分析】本题有多种解法：(1)将含有()f x 的要求极限的表达式凑成已知极限的表达式，或反之；(2)利用极限与无穷小的关系，从已知极限中解出()f x 代入要求极限式中；(3)将具体函数用佩亚诺余项泰勒公式展开化简原极限. 【详解】方法1： 凑成已知极限2336()6()6sin 6sin 6()f x x xf x x x x xf x x x x ++-++==而 23222000012(6)6sin 666cos66(1cos6)2lim lim lim lim 3633x x x x x x x x x x x x x→→→→⋅---====洛 (由于211cos 2x x -⇒211cos(6)(6)2x x -)所以 2330006()6sin 6sin 6()lim lim lim 36036x x x f x x x x xf x x x x →→→+++=+=+=方法2：由极限与无穷小关系，由已知极限式解出3sin 6()x xf x a x+=，0lim 0x a →= 从而 3sin 6()x xf x ax +=⇒3sin 6()ax xf x x-=33223sin 666()6sin 6ax x f x ax x x x x x x-+++-== 所以 323300006()6sin 66sin 6lim lim lim lim x x x x f x ax x x x xa x x x→→→→++--==+极限的四则运算 2220012(6)66cos620lim lim 3x x x x x x→→⋅-=+=36= 方法3： 将sin 6x 在0x =处按佩亚诺余项泰勒公式展开至3x 项：3333(6)sin 66()636(),3!x x x x x x x οο=-+=-+于是 3333sin 6()6()36()x xf x x xf x x x x x ο++-+=3236()()36,f x x x x ο+=-+ 从而 32330006()sin 6()()limlim 36lim 036036.x x x f x x xf x x x x xο→→→++=+-=+-=(5)【答案】B【详解】由特解12,2x xy e y xe --==，对照常系数线性齐次微分方程的特征方程、特征根与解的对应关系知道，21r =-为特征方程的二重根；由33xy e =可知11r =为特征方程的单根，因此特征方程为232(1)(1)10,r r r r r -+=+--=由常系数齐次线性微分方程与特征方程的关系，得该微分方程为0.y y y y ''''''--+=三【详解】方法1：为了求不定积分，首先需要写出()f x 的表达式.为此，令ln x t =，有tx e =ln(1)ln(1)()(ln )t tx e f t f x x e ++===()ln(1)ln(1)x x x x f x dx e e dx e de --=+=-+⎰⎰⎰ln(1)1xxxxxe e e e dx e--=-+++⎰ 分部积分 1ln(1)1x xxxxe e e e dx e -+-=-+++⎰ 拆项ln(1)(1)1ln(1)111ln(1)111ln(1)1(1)1ln(1)ln(1)xxxxx x xxx x xxx x xxx x x e e e dxe e e e dx dx e e e dx de e e e dx d e ee e x e C-----=-++-+=-++-+=-++-+=-++-++=-++-++⎰⎰⎰⎰⎰⎰⎰ 方法2：作积分变量替换，命ln x t =，21ln(1)1()(ln )ln(1)t f x dx f t dt dt t d t t t +⎛⎫=⋅==-+ ⎪⎝⎭⎰⎰⎰⎰ ln(1)1[](1)t dt t t t +=--+⎰ 分部积分 ln(1)11()1t dt t t t +=-+-+⎰ 部分分式求和 ln(1)11(1)1t dt d t t t t +=-+-++⎰⎰ln(1)ln ln(1)t t t C t+=-+--+ln(1)ln(1).x x x e e x e C -=-++-++四【详解】先写出面积()S t 的(分段)表达式，当01t <<时，图形为三角形，利用三角形的面积公式：21()2S t t =；当12t <<时，图形面积可由正方形面积减去小三角形面 积，其中由于x y t +=与1y =交点的纵坐标为1t -，于是， 小三角形的边长为：1(1)2t t --=-，所以222111()1(2)1(44)21222S t t t t t t =--=--+=-+-；当2t >时，图形面积就是正方形的面积：()1S t =， 则221, 01,21()1(2), 12,21, 2.t t S t t t t ⎧≤≤⎪⎪⎪=--<≤⎨⎪<⎪⎪⎩当01x ≤≤时，3320011();2236xxxt x S t dt t dt ⎛⎫==⋅= ⎪⎝⎭⎰⎰当12x <≤时，1122010111()()()[1(2)]22xx x S t dt S t dt S t dt t dt t dt =+=+--⎰⎰⎰⎰⎰3321111(1)(2)66663x x x x x =+----=-+-+ 当2x >时，2022()()()11 1.xx xS t dt S t dt S t dt dt x =+=+=-⎰⎰⎰⎰因此 3320101611()126312x x x S t dt x x x x x x ⎧≤≤⎪⎪⎪=-+-+<≤⎨⎪->⎪⎪⎩⎰五【详解】方法1：按莱布尼茨高阶导数公式：()()1(1)()()()().n n n k n k k n n n uv u v C u v C u v uv --'=+++++为了求ln(1)x +的n 阶导数，设ln(1)y x =+，11y x'=+；()()221111y x x ''=-=-++；()()()33112211y x x ⋅'''=--⋅=++；()()(4)4412123311y x x ⋅⋅⋅=-=-++一般地，可得1()(1)(1)!(1)n n nn yx ---=+即 []1()(1)(1)!ln(1)(1)n n nn x x ---+=+设ln(1)u x =+，2v x =，利用上述公式对函数展开，由于对2x 求导，从三阶导数开始就为零，故展开式中只含有前三项.123()212(1)(1)!(1)(2)!(1)(1)!()2(1).(1)(1)(1)n n n n n n n n n n fx x nx n n x x x -----------=++-+++代入0x =，得：1()3(1)!(0)(1)(1)(3)!,3,4.2n n n n fn n n n n ---=---==-方法2：()y f x =带佩亚诺余项的麦克劳林公式：()2(0)(0)()(0)(0)()2!!n n n f f f x f f x x x x n ο'''=+++++求(0)(3)nf n ≥可以通过先求()y f x =的的麦克劳林展开式，则展开式中nx 项的系数与!n 的乘积就是()y f x =在点0x =处的n 阶导数值)0()(n f.由麦克劳林公式，23212ln(1)(1)(),232n n n x x x x x x n ο---+=-+++-+- 所以 452231ln(1)(1)().232n n n x x x x x x x n ο--+=-+++-+- 对照麦克劳林公式()2(0)(0)(0)()(0)(),1!2!!n nn f f f f x f x x x x n ο'''=+++++从而推知()1(0)(1)!2n n f n n --=- 得 1()(1)!(0),3,4.2n n n f n n --==-六【详解】因为cos 0x ≥，且(1)n x n ππ≤<+， 所以(1)0cos cos cos .n x n x dx x dx x dx ππ+≤<⎰⎰⎰定积分的性质又因为cos x 具有周期π，所以在长度为π的积分区间上的积分值均相等：cos cos a ax dx x dx ππ+=⎰⎰，从而20(1)cos cos cos cos n n n x dx x dx x dx x dx ππππππ-=+++⎰⎰⎰⎰202cos (cos cos )n x dx n xdx xdx ππππ==-⎰⎰⎰202(sin sin )(1(01))2n x x n n πππ=-=--= 所以(1)0cos 2(1).n xdx n π+=+⎰所以 02cos 2(1),x n xdx n ≤<+⎰即 2()2(1).n S x n ≤<+(2) 由(1)有，当(1)n x n ππ≤≤+时，2()2(1)(1)n S x n n x n ππ+<<+命n →∞取极限，222lim lim 1(1)(1)n n n n nπππ→∞→∞==++，12(1)2(1)2lim lim n n n n n πππ→∞→∞++== 由夹逼定理，得()2limx S x x π→∞=.七【详解】设从2000年初(相应0t =)开始，第t 年湖泊中污染物A 的总量为m ，浓度为mV，则在时间间隔[,]t t dt +内，排入湖泊中A 的量为：00()66m mV t dt dt dt V ⋅+-=，流出湖泊的水中A 的量为33m V mdt dt V ⋅=. 因而时间从t 到t dt +相应地湖泊中污染物A 的改变量为：0()63m mdm dt =-. 由分离变量法求解：0()63dm dt m m =-两边求积分：001100()6333ln()63()()6363m m d m dm m dt t C t C m m m m -=⇔-=+⇔--=+--⎰⎰⎰ 10013ln()63363t C m m t C m m e +-+⇔-=⇔-=-103336C tm m e e --⇔-=-+⋅110033333,(3)22C C t tm mm e e m C e C e ----⇔=-⋅⇔=-⋅=初始条件为0(0)5m m =，代入初始条件得092C m =-. 于是03(19)2tm m e -=+，要满足污染物A 的含量可降至0m 内，命0m m =，得6ln3t =. 即至多需经过6ln3年，湖泊中A 的含量降至0m 以内.八【证明】 方法1：令0()(),0xF x f t dt x π=≤≤⎰，有(0)0,F =由题设有()0F π=.又由题设()cos 0f x xdx π=⎰，用分部积分，有0()cos cos ()f x xdx xdF x ππ==⎰⎰()cos ()sin F x xF x xdx ππ=+⎰0()sin F x xdx π=⎰由积分中值定理知，存在(0,)ξπ∈使0()sin ()sin (0)F x xdx F πξξπ==⋅-⎰因为(0,)ξπ∈，sin 0ξ≠，所以推知存在(0,),ξπ∈使得()0F ξ=. 再在区间[0,]ξ与[,]ξπ上对()F x 用罗尔定理，推知存在1(0,)ξξ∈，2(,)ξξπ∈使12()0,()0F F ξξ''==，即 12()0,()0f f ξξ==方法2：由()0f x dx π=⎰及积分中值定理知，存在1(0,)ξπ∈，使1()0f ξ=. 若在区间(0,)π内()f x 仅有一个零点1ξ，则在区间1(0,)ξ与1(,)ξπ内()f x 异号. 不妨设在1(0,)ξ内()0f x >，在1(,)ξπ内()0f x <. 于是由()0,()cos 0f x dx f x xdx ππ==⎰⎰，有111101100()cos ()cos ()(cos cos )()(cos cos )()(cos cos )f x xdx f x dx f x x dxf x x dx f x x dxπππξπξξξξξ=-=-=-+-⎰⎰⎰⎰⎰当10x ξ<<时，1cos cos x ξ>，1()(cos cos )0f x x ξ->；当1x ξπ<<时，1cos cos x ξ<，仍有1()(cos cos )0f x x ξ->，得到：00>. 矛盾，此矛盾证明了()f x 在(0,)π仅有1个零点的假设不正确，故在(0,)π内()f x 至少有2个不同的零点.九【详解】为了求曲线()y f x =在点(6,(6))f 处的切线方程，首先需要求出()y f x =在6x =处的导数，即切线斜率. 而函数又是以周期为5的函数，且在1x =处可导，则在6x =处可导，且其导数值等于函数在1x =处的导数值.将(1sin )3(1sin )8()f x f x x x α+--=+两边令0x →取极限，由f 的连续性得(1)3(1)lim(8())0x f f x x α→-=+= ⇒ 2(1)0f -=故(1)0f =，又由原设()f x 在1x =处可导，两边同除sin x ，000(1sin )(1)(1sin )(1)8()lim3lim lim limsin sin sin sin x x x x f x f f x f x x x x x xα→→→→+---+=+- 根据导数的定义，得008()(1)3(1)limlim 8sin sin x x x x x x f f x x x xα→→''+=⋅+⋅= ⇒ 4(1)8f '= 所以(1)2f '=，又因(6)(51)(1)f f f '''=+=，所以(6)2f '=，由点斜式，切线方程为((6))(6)(6).y f f x '-=-以(6)(1)0,(6)2f f f '===代入得2(6).y x =- 即 2120.x y --=十【详解】首先联立两式，求直线与曲线的交点：221x ax -=，得：x =，而0x ≥，则交点坐标为：(,))1a x y a =+. 由点斜式，故直线OA的方程为y =由旋转体体积公式2()b aV f x dx π=⎰，要求的体积就是用大体积减去小体积：()2222224000()1a x V dx ax dx a x dx a =-=-+232525223(1)515(1)a x a x a a a ππ⎛=-=+⎝+为了求V 的最大值，对函数关于a 求导，225522221515(1)(1)dV a a da a a ππ''⎛⎫⎛⎫ ⎪ ⎪== ⎪ ⎪ ⎪ ⎪++⎝⎭⎝⎭53222552(1)(1)2215(1)a a a a a π⋅+-⋅+=⋅+ 322275255(1)[2(1)][2(1)]222215(1)15(1)a a a a a a a a a ππ++-+-=⋅=⋅++ 222277722251[22][2]22[4]22151515(1)(1)(1)a a a a a a a a a a πππ+---=⋅=⋅=⋅+++ 0a > 命0,dVda=得唯一驻点4a =，所以4a =也是V 的最大值点，最大体积为41875a V ==.十一【详解】(1) 为了求()f x '，将01()()()01xf x f x f t dt x '+-=+⎰两边同乘(1)x +，得 0(1)()(1)()()0,xx f x x f x f t dt '+++-=⎰两边对x 求导，得()(1)()()(1)()()0f x x f x f x x f x f x ''''+++++-=即 (1)()(2)()0x f x x f x '''+++=.上述方程为二阶可降阶微分方程，令()u f x '=，化为(1)(2)0x u x u '+++=，即(2)(1)du x dx u x +=-+ 两边求积分：(2)1(1)(1)1du x dx dx u x x +=-=-+++⎰⎰⎰即 1ln (ln(1))u x x C =-+++ 所以 11(ln(1))1()1x x C C x u ee e x --++-=±=±⋅⋅+ 令1C C e =±，则1xCe u x -=+，于是()1x Ce f x u x -'==+.再以0x =代入原方程001(0)(0)()(0)(0)01f f f t dt f f ''+-=+=⎰，由(0)1f =，有(0)1f '=-，于是1,()1xe Cf x x -'=-=-+. (2)方法1：用积分证.()(0)()1.1tx xe f x f f t dt dt t -'=+=-+⎰⎰而 0-000011t t xx x tt x e dt e dt e e t ->---≤≤=-=-+⎰⎰牛莱公式两边同乘以(1)-，得：101txxe e dt t ---≤-≤+⎰， 即 0()111txxe ef x dt t --≤=-≤+⎰方法2 ：用微分学方法证.因(0)1,()0f f x '=<，即()f x 单调递减，所以当0x ≥时()1f x ≤. 要证()xf x e-≥，可转化为证明()0xf x e--≥，令()()x x f x e ϕ-=-，则(0)110ϕ=-=，且()()()01xxe xf x ef x x ϕ--'''=+≥+=+ (0x ≥)所以，当0x ≥时()0x ϕ≥，即()xf x e -≥. 结合两个不等式，推知当0x ≥时，()1xef x -≤≤. 证毕.十二【详解】由题设得110121210210211102T A αβ⎡⎤⎢⎥⎛⎫⎢⎥⎡⎤ ⎪===⎢⎥⎢⎥ ⎪⎣⎦ ⎪⎢⎥⎝⎭⎢⎥⎣⎦，11102221T B βα⎛⎫⎡⎤ ⎪===⎢⎥ ⎪⎣⎦ ⎪⎝⎭. 所以 ()22T T T A A αβαβααββ===，48AA =；24B =，216B =代入原方程22442B A x A x B x γ=++中，得16816Ax Ax x γ=++，即()82A E x γ-=其中E 是三阶单位矩阵，令[]123Tx x ,x ,x =，代入上式，得线性非齐次方程组1212123102201212x x x x x x x ⎧-+=⎪⎪-=⎨⎪⎪+-=⎩(1) 显然方程组得同解方程为12123201212x x x x x -=⎧⎪⎨+-=⎪⎩ (2) 令自由未知量 1x k,=解得23122x k,x k ==- 故方程组通解为1231022011122x k x k k x k ⎡⎤⎡⎤⎢⎥⎢⎥⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==+⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦--⎢⎥⎢⎥⎣⎦⎣⎦，(k 为任意常数)十三【详解】方法1：先求()123,,,γααα将矩阵作初等行变换，得()123139139139206061201231701020000,,ααα⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥=→--→⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦⎣⎦知()1232,,.γααα= 故()()1231232,,,,γβββγααα==，[]123,,βββ作初等行变换[]1230110121031110030a b ,,a b βββ-⎡⎤⎡⎤⎢⎥⎢⎥=→⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦因为()1232,,γβββ=，所以3a b =又3β可由123,,ααα线性表出，故()()12331232,,,,,γαααβγααα== 将[]1233,,,αααβ作初等行变换13913920610612123170110203b b b b ⎡⎤⎡⎤⎢⎥⎢⎥→---⎢⎥⎢⎥⎢⎥⎢⎥---⎣⎦⎣⎦()13912012600053123bb b b ⎡⎤⎢⎥⎢⎥-⎢⎥→⎢⎥-⎢⎥⎢⎥+-⎣⎦由()12332,,,γαααβ=，得()531203b b +-=，解得5b =，及315a b .== 方法2：由方法1中的初等变换结果可以看出12,αα线性无关，且31232ααα=+，故()1232,,γααα=，12,αα是123,,ααα的极大线性无关组. 又()()1231232,,,,γβββγααα==，123,,βββ线性相关. 从而得12301211310110100a ba b ,,,βββ===--计算三阶行列式得30a b -+=，得3a b =又3β可由123,,ααα线性表出 ，即可由12,αα线性表出，12,αα3β线性相关，有()123131313201061206120310010310003126b b b,,b b b b b ααβ==--=--=-+-行列式展开得()10631206b b ⎛⎫-+-= ⎪⎝⎭，所以()531203b b +-=，得5b =及315a b .== 方法3：先利用3β可由123,,ααα线性表出，故方程组()123,,X αααβ=有解，即12313920613170x b x x ⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦⎣⎦有解. 对其增广矩阵施行初等行变化13913920610612123170110203b b b b ⎡⎤⎡⎤⎢⎥⎢⎥→---⎢⎥⎢⎥⎢⎥⎢⎥---⎣⎦⎣⎦()13921012600053123bb b b ⎡⎤⎢⎥⎢⎥-⎢⎥→⎢⎥-⎢⎥⎢⎥+-⎣⎦由其次线性方程组有解的条件(系数矩阵的秩等于增广矩阵的秩)，知()53123b b +-51033b =-= 解得5b .=又因为1α和2α线性无关，且31232ααα=+，所以向量组123,,ααα的秩为2 ，由题设条件知()1232,,γβββ=，从而123001211310110100a b a b ,,,βββ===--解得15a =。
考研英语真题答案2000
考研英语真题答案2000考研英语是中国研究生入学考试中的一个重要组成部分，对于备考的学生来说，了解历年真题及其答案是提高复习效率的重要手段。
以下是2000年考研英语真题的部分答案，供参考。
阅读理解1. 第一题的正确答案是B。
文章主要讨论了全球化对文化的影响，B 选项准确概括了文章的主旨。
2. 第二题的正确答案是D。
细节题，需要在文章中找到相应的句子，D 选项是文章中提到的一个具体例子。
3. 第三题的正确答案是A。
推理题，需要根据文章内容推断出作者的观点，A选项是作者明确表达的。
完形填空1. 第一空的正确答案是“influence”，文章中提到了全球化对文化的影响，因此这个词是合适的。
2. 第二空的正确答案是“rapidly”，描述了全球化进程的速度。
3. 第三空的正确答案是“concern”，文章中提到了人们对于文化同质化的担忧。
翻译1. 第一题的翻译是：“全球化进程中，不同文化之间的交流和融合是不可避免的。
”2. 第二题的翻译是：“然而，这种趋势也引发了一些人对于文化多样性丧失的担忧。
”写作对于写作部分，考生需要根据给定的题目和指令，写出一篇结构清晰、论点明确、语言流畅的英语作文。
2000年的考研英语写作题目要求考生就“全球化对教育的影响”写一篇文章。
考生应该从全球化对教育资源的分配、教育内容的更新以及教育方式的变革等方面展开论述。
结束语以上是2000年考研英语真题的部分答案，考生在复习时应该结合真题练习，提高自己的应试能力。
同时，考生还应该注意培养自己的英语听说读写能力，以全面提高英语水平。
考研英语不仅仅是对语言知识的测试，更是对考生综合运用英语能力的检验。
希望每位考生都能在考试中取得好成绩。
2000年数学一考研真题解答
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【答】
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【详解】
因为
1 § · ¨ 2 e x sin x ¸ lim ¨ ¸ x o 0 4 x ¸ ¨ 1 e x © ¹ 1 § · ¨ 2 e x sin x ¸ lim ¨ ¸ x o 0 4 x ¸ ¨ 1 e x © ¹ 1 § · ¨ 2 e x sin x ¸ 2 lim ¨ 1 1, ¸ x o 0 4 x ¸ 1 ¨ 1 e x © ¹ 1 § · ¨ 2 e x sin x ¸ lim ¨ ¸ 0 1 1, x o 0 4 x ¸ ¨ 1 e x © ¹
2000-数学二真题、标准答案及解析
【答】
因此对任意的 x ∈ ( −∞, +∞ ) ， 有， ， 这只需 a ≥ 0 【详解】 由题设，f ( x ) 在 ( −∞, +∞ ) 内连续，
飞
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sin 6 x + xf ( x ) 6 + f ( x) = 0, 则 lim 为 3 0 x → x x2
（B）6 （C）36 （D） ∞ 【 】
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【详解】
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1 2 ⎧ t ,0 ≤ t ≤1 ⎪ 2 ⎪ ⎪ 1 S ( t ) = ⎨− t 2 + 2t − 1,1 < t ≤ 2 ⎪ 2 1, t > 2 ⎪ ⎪ ⎩
翔 考
S ( t ) 表示正方形 D 位于直线 l 的左下方部分的面积，试求 ∫ S ( t )dt ( x ≥ 0 ) .
可见正确选项为（B）
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<a href=>真题网提供</a><br> 2000年全国考研日语真题和答案Ⅰ文字と語彙（１５点）（一）次の文ののある漢字の読み仮名はどれであるか、それぞれ選択肢abcdの中から最も適切なものを一つ選び、記号で答えなさい。
（５点）1.強い人に負けたので、別に悔しいとは思わない。
aむなbくやcさびdいや2.そんなに強引にやろうたって無理ですよ。
aつよひきbきょういんcつよびきdごういん3.社長は一切の事務を長男に任せて引退した。
aいっせいbいっせつcいっさいdいっせ4.マッチやナイフを弄ぶのは危険なことだ。
aいじあそbもてあそcまちあそdもたあそ5.彼の発言は私たちがその問題を解決する上で示唆的であった。
aじさbしすうcしさdしさつ6.戦争であの町はすっかり廃れてしまいました。
aすたbこわcはいdすか7.私に言わせると、彼の判断は偏っていると思う。
aかたよbまがcかたどdかたま8.その話はいま彼らの間では禁物になっている。
aきんぶつbきんものcきんじものdきんもつ9.ふだん無口な彼も酒を飲むと、意外に思われるほどしゃべりだすのである。
aぶこうbむくちcぶくちdむぐち10.静脈注射をすれば、直るのもはやい。
aせいみゃくbしょうみゃくcじょうみゃくdせいまい（二）次の文のをつけた言葉の赤色の部分はどんな漢字を書くか、それぞれ選択肢abcdの中から同じ漢字が使われるものを一つ選び、記号で答えなさい。
（５点）11.そくたつで送ったので、間に合いました。
a 終日仕事にそくばくされているから旅行はなかなかできない。
b そのことは松尾さんに頼んであたってみたが、そくざに断られた。
c 何一つふそくのない家に生まれたのだから、貧乏の味を知らないd 事態はきゅうそくに収拾に向かうほかなかった。
12.何ともいえないきょうふにおそわれる。
a この問題にはきょうつう性があると思う。
b わざわざおいでいただいてきょうしゅくです。
c 世界的なふきょうが漸次回復しつつある。
d その事件は社会に大きなはんきょうを呼んだ。
13.都市の限られたくうかんの中で、見事に大自然を表現しています。
a 北国の都市の人々は自然を保護することに非常にかんしんを抱いている。
b 彼は病後の身を起こして、作品のかんせいを急いでいる。
c その先生に会ったことはないが、彼の著書からかんせつ的に影響を受けてる。
d 何かが起こりそうだということをちょっかんした。
14.私も行きたいのですが、チケットのはつばいびはいつですか。
a 原則としてこの債券はばいばいが禁じられています。
b あの坂はこうばいが激しいから道は広くても車はのぼりにくい。
c かれはばいしょう問題で困っているとはいうものの、意外と平気な顔をしている。
d 期限が過ぎたということで、ばいがくの値段を支払わされた。
15.道がこんでいるからスピードをかげんしてください。
a げんみつな身元調査を行わなければなりません。
b 彼は年齢のわりにげんだい感覚を持っている。
c 九月は八月に比べ交通事故はげんしょうした。
d 交通違反をげんじゅうに取り締まるのは交通安全を図るからだ。
（三）次の文のにはどんな言葉を入れたらいいか、それぞれ選択肢abcdの中から最も適切なものを一つ選び、記号で答えなさい。
（５点）16.大通りで遊んでいる子供を見るとします。
aはらはらbいらいらcのろのろdふらふら17.僕の顔の店があるから、そこへ行こう。
a大きいb広いc立つdきく18.なかなか勇気が出なかったが、今日は自分の気持ちを彼女に打ち明けた。
a思い切ってb思い悩んでc思い上がってd思い込んで19. 問題が問題だから、そう簡単に解決できそうにない。
aなにしろbなんともcなにやらdなにとぞ20.食事はまずいし、部屋は汚いし、ホテルだった。
aとぼしいbみっともないcひどいdだらしない21.長い間考えていた問題が一度に解けてしまった。
aすっとbそっとcやっとdはっと22.欧米向けの輸出は今もに伸びているが、アジア向けが経済危機で低迷している。
a景気b完全c豊富d好調23.あまりのおそろしさに生きたがしなかった。
a気持b心地c気分d感じ24.消防自動車はを鳴らしながら、火事現場に駆けつけた。
aメロディーbサイレンcベルdスピーカー25.円相場の影響でドルのが上下している。
aレースbレールcレーダーdレートⅡ文法と構文（２５点）（一）次の文のにはどんな言葉を入れたらいいか、それぞれ選択肢abcdの中から最も適切なものを一つ選び、記号で答えなさい。
（２０点）26.すべての人思いやる心持ち、感謝の気持を忘れずに、余裕のある生き方をしたいものですね。
aへbにcがdを27.部長の言うなりしているほうが無難だと思いますよ。
aにbでcとdまで28.親に内緒にしているものをどうしてあなたに話せますか。
aこそbでもcすらdしか29.これは先生にお借りした本だからなくししたらたいへんだ。
aてbでもcてはdても30.体が大きいとはいっても、そんなには食べられないだろう。
aまさにbなんとcまさかdなにか31.彼女は飛び上がらん喜んだが、成績はそうでもなかった。
aだけにbだけでcばかりでdばかりに32.兄がどうしてもやりたいと言うなら、父も許さないはないだろう。
aものbことcべきdつもり33.世の中にこんなばかげた話があってたまる。
aものかbわけかcことかdはずか34.彼女は泣き出しそうだったから、それ以上私は何も言わなかった。
a今だにb今にc今にもd今にして35.彼は自分がやったすずしい顔をしている。
aぐらいbくせにcのでdものの36.色彩が美しいで、ほかに何のとりえのない絵だ。
aからbなどcなんdのみ37.友達は遠くからわたしを見つけて懐かし手を振った。
aげにbっぽくcさにdみに38.北京上海などの大都市の交通問題は年ごとに深刻さを増してきている。
aかbだのcとdだって39.あの子は背が高いが、見る体が弱そうだ。
aからはbからだはcからにdからまで40.社員の皆さんに一生懸命働いておかげで会社も発展することが出来ました。
aあげたbいたcくれたdもらった41.出張先で祖父の死を急いで戻った。
a知らされてb知らせてc知られてd知ってもらって42.やるべきことはすべてやった。
これでよいのかちょっと不安だ。
aそればかりかbそれどころかcとはいうもののdなんといっても43.今日は秘書に急に、仕事がはかどりません。
a休ませてb休んでいてc休まれてd休ませていただいて44.本日は天候の悪いなかをくださいまして、まことにありがとうございます。
aいらっしゃりbお越しcおいでになってdお見え45.「先生、母があすと言っています。
」a行きたいbお伺いになりたいc参りたいdお伺いしたい（二）次のabcd文の線のついた言葉の中に一つだけ他の三つと違った使い方があるが、その違ったものを選び、記号で答えなさい。
（５点）46.a もう少し考えれば分かるものを、どうしてそこから先を考えようとしないのだろう。
ｂ早くから勉強すればいいものを、なにをまだぐずぐずしているのだろう。
ｃもうそろそろ来てもよさそうなものを、いつまで待たせるつもりだろう。
ｄ大劇場でもっと良いものを見せてくれるから、こんな所のものは見なくてもいいだろう。
47.a 広げると大きいが、小さく畳めばポケットにだって入れることができる。
ｂ辛すぎるのは良くないんだってことは分かっているんだが、習慣でどうしても辛いのを食べてしまう。
ｃ勉強した以上、外国人とだって恥ずかしがらずに話してごらんなさい。
ｄデパートへわざわざ買いに行かなくても近所の店にだって売っているよ。
48.a うまいものを見ると、つい手が出る。
ｂ乗り物に揺られると、ついねむり込んでしまう癖がある。
ｃ田中さんはついさきほどいらっしゃったばかりです。
ｄ自分でいけないと知りつつも、つい怒鳴ってしまうんだ。
49.a 解決の方法は二通りあるが、どちらがいいかまだわからない。
ｂ思った通りに彼が賛成してくれたので、仕事は順調にすすんだ。
ｃ言われた通りの時間に来たが、会議は時間通りに開かれなかった。
ｄ今のところ計画通りに進んでいるが、結果がどうなるか少し心配だ。
50.a こうなった以上はやれるだけやってみるより仕方がない。
ｂ事前に良く準備しただけのことがあって予定通りに終えることが出来た。
ｃこの問題に関心を持っている人がどれだけいるだろう。
ｄたくさんありますから、お好きなだけお取りください。
Ⅲ．読解（３４点）次の文の（一）（二）（三）を読んで51ー67の問に答えなさい。
答えはそれぞれ選択肢ＡＢＣＤの中から最も適切なものを一つ選び、記号で答えなさい（一）（12点）「勤勉こそ美徳である」という言葉は、21世紀には死語にしたい。
「心豊かに生きることこそ美徳である」という言葉をそれに代えたい。
これまで日本人はよく働く国民であると世界的に評価されてきた。
ところが、最近は（51）事情がだいぶ変わってきた。
働くことだけに一生を費やしてしまう日本人は「働きばち」とたとえられ、非難の対象にさえなっている。
もちろん、「働きばち」にも（52）言いわけがないわけではない。
狭い国土、高い人口密度、乏しい地下資源。
日本という国が生き延びていくには、とにかく働くしかなかったのである。
これを支えてきたのは勤勉以外の何ものでもない。
しかし、そうは言っても、世の中というのは常に動いている。
20世紀の日本と21世紀の日本が同じはずはない。
勤勉さのお陰で物質的に恵まれた日本は、そのことを土台にして新しい方向へと進んでいかなければならない。
（53）というこれまでの考え方は、当然、反省さらなければならないし、働きさえすればあとはどうでもいいという思想も改めなければならない。
たとえば、過労死の問題。
人間は生きるために働くべきなのに働いたために死んでしまう人がいる。
何かを変えなければならない時に来ているのである?そこで考えられるものの一つがコンピューターである。
コンピューターという言葉を聞いただけで何か難しい、面倒くさいというイメージを持つ人がいるが、そういう人に会って話を聞いてみると、たいていこれまでの考え方を変えずにコンピューターを頭から嫌っている。
これまでの考え方というのは、もちろん「勤勉こそ美徳である」という考え方である。
なぜコンピューターを使うのか、その最も大事なところがきちんと理解されていない場合が多い。
つまり、仕事をこれまでよりもたくさんするためにコンピューターを使うのだと考えている。
しかし、（54）これは大きな間違いである。
間違いと言って悪ければ、改めるべき習性と言ってもいいかもしれない。