全国大学生08年首届中图杯大赛试卷
矿产资源开发利用方案编写内容要求及审查大纲
矿产资源开发利用方案编写内容要求及《矿产资源开发利用方案》审查大纲一、概述
㈠矿区位置、隶属关系和企业性质。
如为改扩建矿山, 应说明矿山现状、
特点及存在的主要问题。
㈡编制依据
(1简述项目前期工作进展情况及与有关方面对项目的意向性协议情况。
(2 列出开发利用方案编制所依据的主要基础性资料的名称。
如经储量管理部门认定的矿区地质勘探报告、选矿试验报告、加工利用试验报告、工程地质初评资料、矿区水文资料和供水资料等。
对改、扩建矿山应有生产实际资料, 如矿山总平面现状图、矿床开拓系统图、采场现状图和主要采选设备清单等。
二、矿产品需求现状和预测
㈠该矿产在国内需求情况和市场供应情况
1、矿产品现状及加工利用趋向。
2、国内近、远期的需求量及主要销向预测。
㈡产品价格分析
1、国内矿产品价格现状。
2、矿产品价格稳定性及变化趋势。
三、矿产资源概况
㈠矿区总体概况
1、矿区总体规划情况。
2、矿区矿产资源概况。
3、该设计与矿区总体开发的关系。
㈡该设计项目的资源概况
1、矿床地质及构造特征。
2、矿床开采技术条件及水文地质条件。
2008高教社杯全国大学生数学建模竞赛题目c
2008高教社杯全国大学生数学建模竞赛题目C题地面搜索5.12汶川大地震使震区地面交通和通讯系统严重瘫痪。
救灾指挥部紧急派出多支小分队，到各个指定区域执行搜索任务，以确定需要救助的人员的准确位置。
在其它场合也常有类似的搜索任务。
在这种紧急情况下需要解决的重要问题之一是：制定搜索队伍的行进路线，对预定区域进行快速的全面搜索。
通常，每个搜索人员都带有GPS定位仪、步话机以及食物和生活用品等装备。
队伍中还有一定数量的卫星电话。
GPS可以让搜索人员知道自己的方位。
步话机可以相互进行通讯。
卫星电话用来向指挥部报告搜索情况。
下面是一个简化的搜索问题。
有一个平地矩形目标区域，大小为11200米×7200米，需要进行全境搜索。
假设：出发点在区域中心；搜索完成后需要进行集结，集结点（结束点）在左侧短边中点；每个人搜索时的可探测半径为20米，搜索时平均行进速度为0.6米/秒；不需搜索而只是行进时,平均速度为1.2米/秒。
每个人带有GPS定位仪、步话机，步话机通讯半径为1000米。
搜索队伍若干人为一组,有一个组长，组长还拥有卫星电话。
每个人搜索到目标，需要用步话机及时向组长报告，组长用卫星电话向指挥部报告搜索的最新结果。
现在有如下问题需要解决：1．假定有一支20人一组的搜索队伍, 拥有1台卫星电话。
请设计一种你认为耗时最短的搜索方式。
按照你的方式，搜索完整个区域的时间是多少? 能否在48小时内完成搜索任务? 如果不能完成，需要增加到多少人才可以完成。
2．为了加快速度，搜索队伍有50人，拥有3台卫星电话，分成3组进行搜索。
每组可独立将搜索情况报告给指挥部门。
请设计一种你认为耗时最短的搜索方式。
按照你的搜索方式，搜索完整个区域的时间是多少?地面搜索问题分析与评述摘要：本文介绍2008年高教杯全国大学生数模竞赛C题“地面搜索”题的评卷情况，首先概括地介绍了这个问题的背景、评卷要点、问题的解决方法和答卷中存在的问题。
2008年全国大学生英语竞赛样题及答案
2008年全国大学生英语竞赛样题及答案（C级）Part II V ocabulary and Structure (10 minutes, 15 marks)Directions: There are 15 incomplete sentences in this part. For each blank there are four choices marked A, B, C and D. Choose the one that best completes the sentence. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.31. So nervous ________ that she didn't know how to start her speech.A. since she becameB. would she becomeC. that she becameD. did she become32. He ________ another career but, at the time, he just wanted to earn money to study abroad.A. might have chosenB. might chooseC. had to chooseD. must have chosen33. The second report was ________ by August 2005, but one year later it was still nowhere in sight.A. submittedB. to have submittedC. to submitD. to have been submitted34. In this experiment, the students studied are stopped several times during the listening test and asked to report what they ________ during the pause before answering the questions.A. had just been thinking aboutB. have just been thinking aboutC. are just thinking aboutD. had just thought about35. I was always taught that it was ________ to interrupt.A. rudeB. coarseC. roughD. crude36. Small boys are ________ questioners. They ask questions all the time.A. originalB. peculiarC. imaginativeD. persistent37. We regret to inform you that the materials you ordered are ________.A. out of workB. out of reachC. out of stockD. out of practice38. The bomb will ________ the moment it is touched.A. go onB. go offC. go outD. go over39. The car won't ________; I've tried it several times, but it won't work.A. beginB. launchC. startD. drive40. Children and old people do not like having their daily ________ upset.A. habitB. routineC. practiceD. custom41. In your first few days at school you'll be given a test to help the teachers to ________ you to a class at your level.A. locateB. assignC. deliverD. place42. China only started its nuclear power industry in recent years, and should ________ no time in catching up.A.loseB.delayC. spareD. relieve43. — You did an excellent job yesterday, Jim! I really enjoyed your presentation.— ________— Oh yeah, it was fabulous. It seems the English program is a great way to practice English.— Yeah. It is fun and motivating.A. Did you really?B. Oh, thank you. You are so kind.C. Really? What about yours?D. Not at all. My pleasure.44. — What kind of music do you like?— Well, I like different kinds.— ________— Er, I especially like punk rock.A. I beg your pardon?B. Are you serious?C. Any in particular?D. Why do you think so?45. — How did you like the fashion show last night?— ________— I didn't see anything wrong with the clothes; they looked pretty nice to me.— Do you really think people can wear that stuff and walk around in streets?A. Impressive. It's a good way to show off women's sense of style and wealth.B. It was cool. The clothes are more beautiful than the people wearing them.C. Nothing serious. It's only a show to attract the eyes of fashion fans.D. It was dumb. I think it's stupid for women to wear clothes like that.Part III Reading Comprehension (20 minutes, 40 marks)Section A (4 marks)Directions: There is one passage in this section with 4 questions. For each question, there are four choices marked A, B, C and D. You should decide on the best choice. Then mark the corresponding letter on the Answer Sheet with a single line through the centre.Questions 46-49 are based on the following passage.The ability to “see” oneself in the future is a remarkable human trait - some would say unique - that is not well understood. That's despite the fact that we probably spend as much time thinking about the future as we do thinking about the present.Now new research from Washington University in St. Louis suggests that it's precisely because we can remember the past that we can visualize the future. “Our findings provide convincing support for t he idea that memory and future thought are highly interrelated and help explain why future thought may be impossible without memories,” says doctoral candidate Karl Szpunar. The findings are consistent with other research showing that persons with little memory of the past, such as young children or individuals suffering from loss of memory, are less able to see themselves in the future.The researchers base their conclusions on brain scans of 21 college students who were cued to think about something in their past, and anticipate the same event in the future, like a birthday or getting lost. The experiment was carried out as each student lay on their stomach in a magnetic resonance imaging machine, a dreadful but very useful piece of equipment that can show which areas of the brain are stimulated during specific thought processes. The students were also asked to picture former President Bill Clinton in a past and future setting. Clinton was chosen because he was easily recognized and familiar to all the students.The researchers found a “surprisingly complete overlap” among regions of the brain used for remembering the student's past and those used for picturing the future. And every region involved in remembering was also used in anticipating the future.In short, the researchers isolated the area of the brain that “lit up” when the students thought about an event in their own past. And more importantly, that same area lit up again when they thought about a similar event in their future. In fact, the rese archers report that the brain activity was so similar in both cases that it was “indistinguishable.” The findings were reinforced when students imagined Bill Clinton. Since none of them knew him personally, their memories were not autobiographical. And th e brain scans showed “significantly less” correlation between memories of having seen pictures of Clinton in the White House and projecting him into the future.So this “time machine,” as the researchers describe it, allows us to use the past to see ourse lves in the future, and both our memories and our anticipation are interdependent.46. A remarkable human trait that is not well understood is the ability ________.A. to think about the pastB. to see the futureC. to remember the pastD. to control the present47. The findings support that ________.A. future goals will greatly influence a person's present performanceB. a person's present performance is determined by his / her past knowledgeC. future thought depends to a great degree on the memory of the pastD. present thought is impossible without the ability to imagine the future48. The conclusion of the experiment on students was that ________.A. the students could picture themselves better than Bill Clinton in a past and future settingB. the students could imagine themselves as well as Bill Clinton in a past and future settingC. the students could anticipate Bill Clinton better than themselves in a past and future settingD. the students could only picture themselves in a past and future setting but not Bill Clinton49. This “time machine” in the last paragraph most probably refers to ________.A. clockB. brain scanningC. magnetic resonance imagingD. memorySection B (14 marks)Directions: There is one passage in this section with 10 questions. Go over the passage quickly and answer the questions on the Answer Sheet. For questions 50-55, mark Y (for YES)if the statement agrees with the information given in the passage; N (for NO)if the statement contradicts the information given in the passage; NG (for NOT GIVEN)if the information is not given in the passage. For questions 56-59, complete the sentences with the information given in the passage.Visiting the White HouseWhite House Tours Public tours of the White House are available for groups of 10 or more people. Requests must be submitted through one's Member of Congress and are accepted up to six months in advance. These self-guided tours are available from 7:30 a.m. to 12:30 p.m. Tuesday through Saturday, and are scheduled on a first come, first served basis approximately one month in advance of the requested date. We encourage you to submit your request as early as possible since a limited number of tours are available. All White House tours are free of charge. For the most current tour information, please call the 24-hour line at 202-456-7041. Please note that White House tours may be subject to last minute cancellation.White House Visitor CenterAll tours are significantly enhanced if visitors stop by the White House Visitor Center located at the southeast corner of 15th and E Streets, before or after their tour. The Center is open seven days a week from 7:30 a.m. until 4:00 p.m. and features many aspects of the White House, including its architecture, furnishings, first families, social events, and relations with the press and world leaders, as well as a thirty-minute video. Allow between 20 minutes to one hour to explore the exhibits. The White House Historical Association also sponsors a sales area. Please note that restrooms are available, but food service is not.Mobility-Impaired / Using a WheelchairGuests requiring the loan of a wheelchair should notify the officer at the Visitors Entrance Building upon arrival. Wheelchairs loans are offered on a first-come, first-served basis. Reservations are not possible.Visitors in wheelchairs, or with other mobility disabilities, on the Congressional guided or self-guided tours, between 8:00 a.m. and 12 noon, use the same Visitor entrance and, with up to four members of their party, are admitted without waiting in line and without tickets.Visitors in wheelchairs are escorted by ramp from the entrance level to the ground floor, and by elevator from the ground to the state floor. Guests generally wait in line with their family or group.Hearing-ImpairedTours for hearing-impaired groups may be arranged in advance by writing to the Visitors Office, White House, Washington, DC 20502. Tours are usually scheduled at 9:30 a.m., between the Congressional and public tour times. Participants enter at the East Appointment gate. A U.S. Secret Service / Uniformed Division Tour Officer conducts the tour in sign language. Signed tours are available to groups of 8 to 20. Groups are also encouraged to bring their own interpreters.Signing interpretation is also available for individual visitors with advance notice. A Congressional office first issues guided tour tickets to a guest who is hearing-impaired and then contacts the Visitors Office at least 2 weeks in advance to request interpreter service.The Visitors Office TDD (telephone device for the deaf) is 202-456-2121. Messages may be left outside normal business hours.Visually-ImpairedTours for visually-impaired groups may be arranged in advance by writing to the Visitors Office, White House, Washington, DC 20502. The tours are usually scheduled at 9:30 a.m., between the Congressional and public tour times. Participants enter at the East Appointment gate. A U.S. Secret Service / Uniformed Division Tour Officer permits visitors to touch specific objects in the House. Touch tours are currently available only to groups of 8 to 20, not to individual visitors. Guide animals are permitted in the White House.General Tour InformationAll White House tours are free. Changes in tour schedules are occasionally made because of official events. Notice may not be given until that morning. The Visitors Office 24-hour Information Line recording at 202-456-7041 provides the most up-to-date information. The TDD is 202-456-2121. Visitors should confirm tour schedules by calling the information line the night before and the morning that they plan to visit. It is occasionally necessary to close individual rooms on the tour; however, notice about closed rooms is not possible.Prohibited ItemsProhibited items include, but are not limited to, the following: handbags, book bags, backpacks, purses, food and beverages of any kind, strollers, cameras, video recorders or any type of recording device, tobacco products, personal grooming items (make-up, hair brush or comb, lip or hand lotions, etc.), any pointed objects (pens, knitting needles, etc.), aerosol containers, guns, ammunition, fireworks, electric stun guns, mace, martial arts weapons / devices, or knives of any size. The U.S. Secret Service reserves the right to prohibit any other personal items. Umbrellas, wallets, cell phones and car keys are permitted.Please note that no storage facilities are available on or around the complex. Individuals who arrive with prohibited items will not be permitted to enter the White House.ParkingThe closest Metrorail stations to the White House are Federal Triangle (blue and orange lines), Metro Center (blue, orange, and red lines) and McPherson Square (blue and orange lines). On-street parking is not available near the White House, and use of public transportation is strongly encouraged.Restrooms / Public TelephonesThe nearest restrooms and public telephones to the White House are in the Ellipse Visitor Pavilion (the park area south of the White House) and in the White House Visitor Center. Restrooms or public telephones are not available at the White House.50. Both Congressional guided and self-guided tours need to be scheduled in advance.51. All White House tours are free of charge except on federal holidays.52. The White House Visitor Center provides free drinks but not food service.53. Wheelchair reservation service is provided by the officer at the Visitors Entrance Building.54. Hearing-impaired visitors can request signing interpretation service from the Visitors Office.55. Touch tours are currently only offered to visually-impaired groups of 8 to 20.56. Sometimes official events make it necessary to close ________________ without notice.57. The personal items permitted to be carried into the White House are ________________.58. The transportation visitors are encouraged to use is ________________.59. Inside the White House, visitors cannot find or use restrooms or ________________.Section C (10 marks)Directions: In this section, there is one passage followed by 5 questions. Read the passage carefully, then answer the questions in as few words as possible (not more than 10 words). Remember to write the answers on the Answer Sheet.Questions 60-64 are based on the following passage.If you were on a distant planet, and if you had instruments that could tell you the composition of Earth's atmosphere, how would you know there was life on this planet?Water in the atmosphere would suggest there could be water on the surface, and as we all know water is considered crucial to life. But water would only suggest that life is possible. It wouldn't prove it's there.Carbon? That basic component of “life as we know it?” Not necessarily. A diamond is pure carbon, and it may be pretty, but it isn't alive.What really sets Earth apart is nitrogen, which makes up 80 percent of the planet's atmosphere. And it's there only because there is abundant life on Earth, say scientists at the University of Southern California.The report grew out of a class discussion two years ago in a course taught by Capone and Kenneth Nealson, professor of earth sciences. Students were asked to come up with different ideas about searching for life on other planets. What is a distinct “signature,” as Capone puts it, that would show there is life on another planet?That's a question that has been kicked around in many quarters in recent decades, especially since all efforts to find some form of life, no matter whether on Mars or in the distant reaches of space, have failed. At least so far.The current effort to search for some evidence of life on Mars focuses primarily on the search for water, because it has long been believed that water, or at least some fluid, is necessary for the chemical processes that lead life to take place. But that's probably the wrong approach, the USC group argues.“It's hard to imagine life without water, but it's easy to imagine water without life,” sa ys Nealson, who was on the Mars team before moving to USC.But nitrogen would be a much clearer signature of life. Only about 2 percent to 3 percent of the Martian atmosphere is nitrogen. That's just a trace, and it probably means there is no life on Mars today, and if there was in the past, it probably ended many, many years ago.But, the USC team adds quickly, that doesn't mean there's no life anywhere else in the universe. They don't know where, of course, but they may have found a way to narrow down the search. Look first for nitrogen, then look for biological activity that should be there.So if life exists elsewhere, and is similar to life as we know it, there should be nitrogen, and that's what we shouldbe looking for first, the researchers say.If they don't find nitrogen on Mars, Capone says, “that will probably bring us to the conclusion that there likely never was life on Mars.”But how about elsewhere? Could this technique be used to search for life in other solar systems?Maybe. It might be possible to detect a nitrogen-rich atmosphere around a planet orbiting another star, but not yet. Current instruments aren't that sensitive.If they ever are, the search for life might be narrowed down to the most promising prospects, chiefly because of the presence of nitrogen. And won't that be fun!Questions:Section D (12 marks)Directions: In this section, there is one passage followed by a summary. Read the passage carefully and complete the summary below by choosing a maximum of three words from the passage to fill in the spaces 65-70. Remember to write the answers on the Answer Sheet.Questions 65-70 are based on the following passage.In August 2008, athletes from the United States and around the world will compete in the Beijing Olympics. But did you know that in September of next year, disabled athletes will compete in the Paralympic Games in Beijing? The Olympics and the Paralympics are separate movements. But they have always been held in the same year, and since 1988, they have also been held in the same city. The International Olympic Committee and the International Paralympic Committee signed an agreement in 2001 to secure this connection. The next winter games will take place in Vancouver, Canada, in 2010.The Paralympic Games grew out of a sports competition held in 1948 in England and a doctor named Ludwig Guttmann organized it for men who suffered spinal cord injuries in World War II. Four years later, it became an international event as competitors from the Netherlands took part. Then, in 1960, the first Paralympics were held in Rome. 400 athletes from 23 countries competed. By 2004, the Paralympic Games in Athens had almost 4000 athletes from 136 countries, who may have physical or mental limitations and may be blind or in wheelchairs. Yet sometimes they perform better than athletes without disabilities.In 1968, Eunice Kennedy Shriver, the sister of former President John F. Kennedy, started the Special Olympics, which are just for children and adults with mental limitations and whose programs currently serve more than two million people in 160 countries. In November 2006, in Mumbai, India, teams competed in the First Special Olympics International Cricket Cup. In addition to India, there were men's teams from Afghanistan, Australia, Bangladesh, Nepal, Pakistan, Sri Lanka and the West Indies. There were also women's cricket teams from India and Pakistan.There are many organizations in the United States that help people with disabilities play sports. Wheelchair tennis is a popular sport. So is basketball. In fact, there are more than one hundred professional teams playing wheelchair basketball thanks to the special wheelchairs for athletes that are lightweight and designed for quick moves. For people who want to go really fast in their chairs, there is a Power Wheelchair Racing Association.In the state of Utah there is a place called the National Ability Center, which teaches all kinds of sports to people with all kinds of physical and mental disabilities and even gives friends and family members a chance to try a sport as if they were disabled.A reporter from the Washington Post wanted to know what it would be like for a blind person to use a climbing wall. So, protected by a safety line, the newspaper reporter closed his eyes and started to feel for places to put his hands and feet. Trainers on the ground urged him on: “Take your time. You can do it.” Finally he reached the top. At the National Ability Center people can learn to ride horses and mountain bikes. They can try winter mountain sports, and learn scuba diving and other water activities. The center also prepares athletes for the Paralympics. These days, the first place many people go when they want to travel is the Internet, where they can get information about hotels, transportation and services like tour companies. The Internet can also help travelers find special services for the disabled. For example, there are groups that help young people with disabilities travel to different countries.Susan Sygall, who uses a wheelchair herself, leads an organization called Mobility International USA, and has traveled to more than twenty-five countries to talk about the rights of people with disabilities. She says people with disabilities are all members of a global family and working together across borders is the most powerful way of making changes.Summary:Part IV Cloze (15 minutes, 15 marks)Section A: There are 5 blanks in the passage. Use the word given on the right side to form a word that fits in each blank. Remember to write the answers on the Answer Sheet.Maria Callas was one of the best-known opera singers in the world, who became famous internationally for her beautiful voice and intense (71)________ during the 1950s, and the recordings of her singing the well-known operas remain very popular today.Maria Callas was born in New York City in 1923 and her real name was Maria Kalogeropoulous. Her parents were Greek and when she was fourteen, she and her mother returned to Greece, where Maria studied singing at the national conservatory in Athens and the well-known opera (72)________ Elvira de Hidalgo chose Maria as her student.In 1941, when she was 17, Maria Callas was paid to sing in a major opera for the first time. She sang the (73)________ role in several operas in Athens during the next three years. In 1943, Callas was invited to perform in Italy, which was the real beginning of her profession as an opera singer. She performed major parts in several of the most (74)________ operas. In 1949, she married an Italian (75)________, Giovanni Battista Meneghini, who was twenty years older and became her adviser and manager. person singlead fame industrySection B: There are 10 blanks in the passage. For each blank, some letters of the word has been given (not exceeding 3 letters). Read the passage below and think of the word which best fits each blank. Use only one word in each blank. Remember to write the answers on the Answer Sheet.Part V Translation (15 minutes, 15 marks)Section A (8 marks)Directions: Translate the underlined sentences of the following passage into Chinese. Remember to write the answers on the Answer Sheet.When you talk about China and India's seemingly unstoppable growth, nobody's surprised. After all, the increasing economic strength of the two countries has dominated the news for the past few years. But both China and India face some substantial demographic hurdles to continued expansion. Despite the billion-plus citizens of each country, both may simply lack sufficient qualified workers.(86) China is a rapidly aging society whose current challenges of unemployment and overpopulation will, within a decade, shift to different problems: labor shortages and an elderly population with too few children. In fact, China may be the first country to go gray before it reaches developed status. The World Bank estimates that by 2020 the mainland will face a lack of even unskilled labor due to aging.China is already facing a shortage of skilled labor. Construction sites lack welders, skilled machine operators, and plumbers. And a recent report said the country is short some 750,000 managers. Despite a 95% literacy rate among all but the oldest citizens, there are not enough well-educated Chinese. (87) This is in part due to a lack of schools that combine basic theory with practical skills and a focus on passing the elite university exams. So despite rising salaries, many of those entering China's workforce cannot learn the skills they need.India seems to have an age advantage, with half its population under 25. In the long run, this gives them the upper hand. But at present, India shares China's problem of an insufficiently educated workforce. Citigroup reports that India's talent pool isn't deep enough to meet demand in industries including textiles, aviation, telecom, retail, and engineering.By some other measures, both countries look pretty good. China graduates about 1.7 million students from 1,500 colleges and universities annually, of whom 350,000 are engineers. India produces about 3 million college and university graduates, including 440,000 engineers. Those numbers, though, don't tell the whole story.(88) Only 10% to 25% of these graduates are employable by multinationals, and this is only partially because of language challenges, which every human resources executive I've met who works in China or India agrees with.The source of the problem is cultural. Young people in these countries are highly motivated to study and learn, but only in theory. There is no tradition of practical application. So engineers don't go into operations, factories, or mines, and don't really know the machines and conditions for which they are designing processes. (89) Chinese and Indian students who have just earned BAs want to pursue MBAs right away, and neither they nor their parents understand the value of practical experience. The degree is seen as a trophy, a traditional ticket to higher status. Section B (7 marks)Directions: Translate the following sentences into English, using the words given in the brackets. Remember to write your answers on the Answer Sheet.90. 如果你告诉他们真相，你就可以获得这次机会而不被他们误解。
2008年普通高等学校招生全国统一考试全国卷Ⅰ数学文科试题及答案（河北、河南、山西、广西）.doc
2008年普通高等学校招生全国统一考试（全国卷Ⅰ：河北、河南、山西、广西）文科数学（必修+选修Ⅰ）第Ⅰ卷考生注意： 1．答题前，考生在答题卡上务必用直径0.5毫米黑色墨水签字笔将自己的姓名、准考证号、填写清楚 ，并贴好条形码．请认真核准条形码上的准考证号、姓名和科目． 2．每小题选出答案后，用2Ｂ铅笔把答题卡上对应题目的答案标号涂黑，如需改动，用橡皮擦干净后，再选涂其他答案标号．在试题卷上作答无效．．．．．．．．．．3．本卷共12小题，每小题5分，共60分．在每小题给出的四个选项中，只有一项是符合题目要求的． 参考公式： 如果事件A B ，互斥，那么 球的表面积公式()()()P A B P A P B +=+24πS R =如果事件A B ，相互独立，那么 其中R 表示球的半径()()()P A B P A P B =球的体积公式如果事件A 在一次试验中发生的概率是P ，那么34π3V R =n 次独立重复试验中事件A 恰好发生k 次的概率其中R 表示球的半径()(1)(01,2)k k n kn n P k C P P k n -=-=，，，一、选择题1．函数y = ） A ．{|1}x x ≤B ．{|0}x x ≥C ．{|10}x x x ≥或≤D ．{|01}x x ≤≤2．汽车经过启动、加速行驶、匀速行驶、减速行驶之后停车，若把这一过程中汽车的行驶路程s 看作时间t 的函数，其图像可能是（ ）A ．B ．C ．D ．3．512x ⎛⎫+ ⎪⎝⎭的展开式中2x 的系数为（ ）A ．10B ．5C ．52D ．14．曲线324y x x =-+在点(13)，处的切线的倾斜角为（ ） A ．30°B ．45°C ．60°D ．120°5．在ABC △中，AB c =，AC b =．若点D 满足2BD DC =，则AD =（ ） A ．2133b c + B ．5233c b -C ．2133b c - D ．1233b c +6．2(sin cos )1y x x =--是（ ） A ．最小正周期为2π的偶函数 B ．最小正周期为2π的奇函数 C ．最小正周期为π的偶函数D ．最小正周期为π的奇函数7．已知等比数列{}n a 满足122336a a a a +=+=，，则7a =（ ） A ．64B ．81C ．128D ．2438．若函数()y f x =的图象与函数1y =的图象关于直线y x =对称，则()f x =（ ） A ．22ex -B ．2e xC ．21ex +D ．2+2ex9．为得到函数πcos 3y x ⎛⎫=+ ⎪⎝⎭的图象，只需将函数sin y x =的图像（ ） A ．向左平移π6个长度单位 B ．向右平移π6个长度单位 C ．向左平移5π6个长度单位 D ．向右平移5π6个长度单位10．若直线1x y a b+=与圆221x y +=有公共点，则（ ）A ．221a b +≤B ．221a b +≥ C ．22111a b+≤D ．2211a b +≥1 11．已知三棱柱111ABC A B C -的侧棱与底面边长都相等，1A 在底面ABC 内的射影为ABC △的中心，则1AB 与底面ABC 所成角的正弦值等于（ ）A ．13B C D ．2312．将1，2，3填入33⨯的方格中，要求每行、每列都没有重复数字，下面是一种填法，则不同的填写方法共有（ ）A ．6种B ．12种C ．24种D ．48种第Ⅱ卷注意事项：1．答题前，考生先在答题卡上用直径0.5毫米黑色墨水签字笔将自己的姓名、准考证号填写清楚，然后贴好条形码．请认真核准条形码上的准考证号、姓名和科目．2．第Ⅱ卷共7页，请用直径0.5毫米黑色墨水签字笔在答题卡上各题的答题区域内作答，在试题卷上作答无效．．．．．．．．．． 3．本卷共10小题，共90分．二、填空题：本大题共4小题，每小题5分，共20分．把答案填在题中横线上．（注意：在试题卷上作答无效．．．．．．．．．） 13．若x y ，满足约束条件03003x y x y x ⎧+⎪-+⎨⎪⎩，，，≥≥≤≤则2z x y =-的最大值为 ．14．已知抛物线21y ax =-的焦点是坐标原点，则以抛物线与两坐标轴的三个交点为顶点的三角形面积为 ． 15．在ABC △中，90A ∠=，3tan 4B =．若以A B ，为焦点的椭圆经过点C ，则该椭圆的离心率e = ．16．已知菱形ABCD 中，2AB =，120A ∠=，沿对角线BD 将ABD △折起，使二面角A BD C --为120，则点A 到BCD △所在平面的距离等于 ．三、解答题：本大题共6小题，共70分．解答应写出文字说明，证明过程或演算步骤． 17．（本小题满分12分）（注意：在试题卷上作答无效．．．．．．．．．） 设ABC △的内角A B C ，，所对的边长分别为a b c ，，，且cos 3a B =，sin 4b A =． （Ⅰ）求边长a ；（Ⅱ）若ABC △的面积10S =，求ABC △的周长l ．（注意：在试题卷上作答无效．．．．．．．．．） 四棱锥A BCDE -中，底面BCDE 为矩形，侧面ABC ⊥底面BCDE ，2BC =，CD =AB AC =．（Ⅰ）证明：AD CE ⊥；（Ⅱ）设侧面ABC 为等边三角形，求二面角C AD E --的大小．19．（本小题满分12分）（注意：在试题卷上作答无效．．．．．．．．．） 在数列{}n a 中，11a =，122n n n a a +=+． （Ⅰ）设12nn n a b -=．证明：数列{}n b 是等差数列； （Ⅱ）求数列{}n a 的前n 项和n S ． 20．（本小题满分12分）（注意：在试题卷上作答无效．．．．．．．．．） 已知5只动物中有1只患有某种疾病，需要通过化验血液来确定患病的动物．血液化验结果呈阳性的即为患病动物，呈阴性即没患病．下面是两种化验方案： 方案甲：逐个化验，直到能确定患病动物为止．方案乙：先任取3只，将它们的血液混在一起化验．若结果呈阳性则表明患病动物为这3只中的1只，然后再逐个化验，直到能确定患病动物为止；若结果呈阴性则在另外2只中任取1只化验．求依方案甲所需化验次数不少于依方案乙所需化验次数的概率．CDE AB（注意：在试题卷上作答无效．．．．．．．．．） 已知函数32()1f x x ax x =+++，a ∈R ． （Ⅰ）讨论函数()f x 的单调区间；（Ⅱ）设函数()f x 在区间2133⎛⎫-- ⎪⎝⎭，内是减函数，求a 的取值范围． 22．（本小题满分12分）（注意：在试题卷上作答无效．．．．．．．．．） 双曲线的中心为原点O ，焦点在x 轴上，两条渐近线分别为12l l ，，经过右焦点F 垂直于1l 的直线分别交12l l ，于A B ，两点．已知OA AB OB 、、成等差数列，且BF 与FA 同向． （Ⅰ）求双曲线的离心率；（Ⅱ）设AB 被双曲线所截得的线段的长为4，求双曲线的方程．2008年普通高等学校招生全国统一考试（全国卷Ⅰ：河北、河南、山西、广西）文科数学（必修+选修Ⅰ）参考答案一、1．D 2．A 3．C 4．B 5．A 6．D 7．A 8．A 9．C 10．D 11．B 12．B二、13．9 14．12 15．12 16 三、17．解：（1）由cos 3a B =与sin 4b A =两式相除，有：3cos cos cos cot 4sin sin sin a B a B b BB b A A b B b ==== 又通过cos 3a B =知：cos 0B >，则3cos 5B =，4sin 5B =，则5a =．（2）由1sin 2S ac B =，得到5c =．由222cos 2a c b B ac+-=，解得：b =最后10l =+18．解：（1）取BC 中点F ，连接DF 交CE 于点O ， AB AC =，∴AF BC ⊥，又面ABC ⊥面BCDE ， ∴AF ⊥面BCDE ， ∴AF CE ⊥．tan tan 2CED FDC ∠=∠=， ∴90OED ODE ∠+∠=，90DOE ∴∠=，即CE DF ⊥，CE ∴⊥面ADF ， CE AD ∴⊥．（2）在面ACD 内过C 点做AD 的垂线，垂足为G ． CG AD ⊥，CE AD ⊥，AD ∴⊥面CEG ， EG AD ∴⊥，则CGE ∠即为所求二面角．233AC CD CG AD ==，DG =，EG==，CE =则222cos 2CG GE CE CGE CG GE +-∠== πarccos CGE ∴∠=-⎝⎭．19．解：（1）122n n n a a +=+，11122n nn n a a +-=+， 11n n b b +=+，则n b 为等差数列，11b =，n b n =，12n n a n -=．（2）01211222(1)22n n n S n n --=+++-+12121222(1)22n n n S n n -=+++-+两式相减，得01121222221n n n n n S n n -=---=-+．20．解：依方案甲所需化验次数ξ为：1、2、3、4； P(ξ＝1)＝15； P(ξ＝2)＝142515A A =；P(ξ＝3)＝243515A A =；P(ξ＝4)＝32155-=(第4次是，结束，不是时，就是最后的哪一个，也结束)； 依方案乙所需化验次数η为：2、3.P(η＝2)＝32114411331553315C C C A C C A ⨯⨯+⨯=；(3只血液混在一起化验结果呈阴性,另两只再化验一次就可以了，或者3只血液混在一起化验结果呈阳性, 再逐个化验时的第一次查出)P(η＝3)＝21141131532(1)5C C A C A ⨯⨯-= （3只血液混在一起化验结果呈阳性, 再逐个化验时的第一次没有查出，第二次是，结束，不是时，就是最后的哪一个，也结束)依方案甲所需化验次数不少于依方案乙所需化验次数的概率：P(ξ≥η)＝P(ξ＝2) ×P(η＝2) +[P(ξ＝3)+ P(ξ＝4)][ P(η＝2)+ P(η＝3)]=33255+. 1825=＝0.72（由于标准答案不好看懂，我特用用此法与提供的标准答案的方法不一致，但结论是一致的！――――王新敞）21．解：（1）32()1f x x ax x =+++ 求导：2()321f x x ax '=++ 当23a≤时，0∆≤，()0f x '≥()f x 在R 上递增当23a >，()0f x '=求得两根为3a x -±=即()f x在⎛-∞ ⎝⎭递增，⎝⎭递减，⎫+∞⎪⎪⎝⎭递增 （2）23133a -⎨-+⎪-⎪⎩，且23a >解得：74a ≥22．解：（1）设OA m d =-，AB m =，OB m d =+ 由勾股定理可得：222()()m d m m d -+=+ 得：14d m =，tan b AOF a ∠=，4tan tan 23AB AOB AOF OA ∠=∠== 由倍角公式∴22431ba b a =⎛⎫- ⎪⎝⎭，解得12b a =则离心率2e =（2）过F 直线方程为()ay x c b=-- 与双曲线方程22221x y a b-=联立将2a b =，c =代入，化简有22152104x x b+=124x =-=将数值代入，有4=解得3b =最后求得双曲线方程为：221369x y -=． 点评：本次高考题目难度适中，第12道选择题是2007年北京市海淀区第二次模拟考试题，新东方在2008年寒假强化班教材的220页33题选用此题进行过详细讲解，在2008年春季冲刺班教材30页33题也选用此题，新东方的老师曾在多种场合下对此题做过多次讲解．第19道计算题也是一个非常典型的题型，在2007年12月31日，新东方在石家庄的讲座上曾经讲过这类问题的解法，在2008年的讲课中也多次提过此题型是重点．其他的题型也都很固定，没有出现偏题怪题，应该说，本次高考题的难度，区分度都非常恰当．。
2008年度全国大学生数学建模竞赛
2008年度全国大学生数学建模竞赛郑州轻工业学院选拔赛备选题目A. 电梯控制问题我校教三楼有四部电梯。
等电梯的人给出要上下的信号，电梯只有在空闲或同方向行进时才接受这个指令。
然而，电梯经常出现十分拥挤的状况，特别在上下课的时候，要等很长的时间，所以埋怨声很多。
请为电梯设计一个调度方案，减少大家的等待时间，减少师生的不满。
并分析说明你所设计方案的合理性和可操作性。
请你撰写一份800—2000字之间的建议书，说明你的方案使得管理者能够接受你的方案。
B. 汽车车库库存的数学模型某汽车制造厂有一大型仓库存放成品小型汽车，厂方希望将尽可能多的汽车贮存在车库内。
在满足一定要求的条件下，尽可能提高仓库的利用率。
设车库形状为200米╳300米的矩形，仓库只有一个门，位于矩形长边的正中央，门宽5米。
假设汽车形状只有两种形式，如下图所示：从网上查出以上两种型号汽车的形状尺寸。
要求：1、在任何时刻只有一辆汽车开出仓库大门，开出过程中不得有任何碰撞；2、摆放时任意两辆汽车之间至少保持40cm的间距，不重叠；3、出门时必须车头先出，不得使用任何其他辅助设备。
试建立合理的数学模型，解决以下问题。
1、在每辆车都可顺利开出车库的条件下，如何摆放，可提高车库利用率。
2、假设在车辆无法调出时，可以先将阻碍的车辆开出车库外，在这种情况下，给出车辆摆放的优化数学模型。
3、对问题2的车俩摆放模型，假定每辆汽车开出仓库时的速度均相同，且汽车前轮可以左右转动90度，给出将车库4个角落的汽车全部开出所需最少时间的调运方案。
C. 自习教室开放的优化管理近年来，大学用电浪费比较严重，集中体现在学生上晚自习上，一种情况是去某个教室上自习的人比较少，但是教室内的灯却全部打开，第二种情况是晚上上自习的总人数比较少，但是开放的教室比较多，这要求我们提供一种最节约、最合理的管理方法。
下面是某学校收集的部分数据，请完成以下问题.管理人员只需要每天晚上开一部分教室供学生上自习，每天晚上从7：00---10：00开放（如果哪个教室被开放，则假设此教室的所有灯管全部打开）。
2008年全国高中数学联赛试题及答案详解
受中国数学会委托，2008年全国高中数学联赛由重庆市数学会承办。
中国数学会普及工作委员会和重庆市数学会负责命题工作。
2008年全国高中数学联赛一试命题范围不超出教育部2000年《全日制普通高级中学数学教学大纲》中所规定的教学要求和内容，但在方法的要求上有所提高。
主要考查学生对基础知识和基本技能的掌握情况，以及综合和灵活运用的能力。
全卷包括6道选择题、6道填空题和3道大题，满分150分。
答卷时间为100分钟。
全国高中数学联赛加试命题范围与国际数学奥林匹克接轨，在知识方面有所扩展，适当增加一些竞赛教学大纲的内容。
全卷包括3道大题，其中一道平面几何题，试卷满分150分。
答卷时问为120分钟。
一 试4．若三个棱长均为整数（单位：cm）的正方体的表面积之和为564 cm 2，则这三个正方体的体积之和为 （ ）。
（A）764 cm 3或586 cm 3（B） 764 cm 3（C）586 cm 3或564 cm 3（D） 586 cm 35．方程组0,0,0x y z xyz z xy yz xz y ++=⎧⎪+=⎨⎪+++=⎩的有理数解(,,)x y z 的个数为 （ ）。
（A） 1 （B） 2 （C） 3 （D） 46．设ABC ∆的内角A B C 、、所对的边a b c 、、成等比数列，则sin cot cos sin cot cos A C AB C B++的取值范围是（ ）。
（A）(0,)+∞ （B） 12（C） （D）)+∞二、填空题（每小题9分，共54分）11．设()f x 是定义在R 上的函数，若(0)2008f = ，且对任意x ∈R ，满足(2)()32x f x f x +-≤⋅，(6)()632x f x f x +-≥⋅，则)2008(f = .12．一个半径为1的小球在一个内壁棱长为的正四面体容器内可向各个方向自由运动，则该小球永远不可能接触到的容器内壁的面积是 . 三、解答题（每小题20分，共60分）13．已知函数|sin |)(x x f =的图像与直线y kx = )0(>k 有且仅有三个交点，交点的横坐标的最大值为α，求证：2cos 1sin sin 34ααααα+=+． 14．解不等式121086422log (3531)1log (1)x x x x x ++++<++．加 试一、（本题满分50分）如图，给定凸四边形ABCD ，180B D ∠+∠< ，P 是平面上的动点，令()f P PA BC PD CA PC AB =⋅+⋅+⋅．（1）求证：当()f P 达到最小值时，P A B C 、、、四点共圆；（2）设E 是ABC ∆外接圆O 的 AB上一点，满足：2AE AB =，1BC EC =，12ECB ECA ∠=∠，又,DA DC 是O的切线，AC ，求()f P 的最小值．二、（本题满分50分）设()f x 是周期函数，T 和1是()f x 的周期且01T <<．证明： （1）若T 为有理数，则存在素数p ，使1p是()f x 的周期； （2）若T 为无理数，则存在各项均为无理数的数列{}n a 满足110n n a a +>>>(1,2,)n =⋅⋅⋅，且每个(1,2,)na n =⋅⋅⋅都是()f x 的周期．三、（本题满分50分）设0k a >，1,2,,2008k = ．证明：当且仅当200811k k a =>∑时，存在数列{}n x 满足以下条件：（1）010n n x x x +=<<，1,2,3,n = ；答一图1（2）lim n n x →∞存在；（3）20082007111n n k n k k n k k k x x a x a x -+++==-=-∑∑，1,2,3,n = ．1. 【答案】C 【解析】当2x <时，20x ->，因此21(44)1()(2)22x x f x x x x +-+==+---2≥2=，当且仅当122x x=--时取等号．而此方程有解1(,2)x =∈-∞，因此()f x 在(,2)-∞上的最小值为2．故选C.3.【答案】B12125(2)()()9P P A A P A A ξ==+=， 1234123412341234(4)()()()()P P A A A A P A A A A P A A A A P A A A A ξ==+++332112202[()()()()]333381=+=， 1234123412341234(6)()()()()P P A A A A P A A A A P A A A A P A A A A ξ==+++2221164(()3381==，因此520162662469818181E ξ=⨯+⨯+⨯=．故选B。
2008年普通高等学校招生全国统一考试(全国I)理科综合能力测试试题及答案(全WORD版)2008年
绝密★启用前试卷类型：B2008年普通高等学校招生全国统一考试（全国Ⅰ）理科综合能力测试本试卷共12页，满分360分，考试时间150分钟。
★祝考试顺利★注意事项：1.答卷前，考生务必将自己的姓名，准考证号填写在试题卷和答题卡上，并将准考证号条形码粘巾在答题卡上指定位置。
2.选择题每小题选出答案后，用2B铅笔将答题卡上，对应题目的答案标号涂写，如写改动，用橡皮擦干净后，再选涂其它答案标号，答在试题卷上无效。
3.非选择题用0.5毫米的黑色墨水签字笔答在答题卡上每题对应的答题区域内，答在试题卷上无效。
4.考试结束，请将本试题卷和答题卡一并上交。
选择题共21小题，每小题6分，共126分。
以下数据可供解题时参考：相对原子质量（原子量）：H l C 12 O 16 Na 23 K 39Mn 55 Cu 64 Zn 65 Ag 108 Pb 207一、选择题（本题共13小题，在每小题给出的四个选项中，只有一项是符合题目要求的。
）1.为了验证胰岛素具有降低血糖含量的作用，在设计实验方案时，如果以正常小鼠每次注射药物前后小鼠症状的变化为观察指标，则下列对实验组小鼠注射药物的顺序。
正确的是A.先注射胰岛素溶液，后注射葡萄糖溶液B.先注射胰岛素溶液，再注射胰岛素溶液C.先注射胰岛素溶液，后注射生理盐水D.先注射生理盐水，后注射胰岛素溶液2.某水池有浮游动物和藻类两个种群，其种群密度随时间变化的趋势如图，若向水池中投放大量专食浮游动物的某种鱼（丙），一段时期后，该水池甲、乙、丙三个种群中公剩一个种群。
下列关于该水池中上述三个种群关系及变化的叙述，正确的是A.甲和丙既有竞争关系又有捕食关系，最终仅剩下甲种群B.甲和乙既有竞争关系又有捕食关系，最终仅剩下丙种群C.丙和乙既有竞争关系又有捕食关系，最终仅剩下甲种群D.丙和乙既有竞争关系又有捕食关系，最终仅剩下丙种群3.下列关于细菌的叙述，错误．．的是A.硝化细菌能以NH,作为氮源和能源物质B.某些细菌可以利用光能因定CO2合成有机物C.生长因子是某些细菌生长过程中需要额外补弃的营养物质D.含伊红和美蓝试剂的培养基不能用来签别牛奶中的大肠杆菌4.已知某种限制性内切酶在一线性DNA分子上有3个酶切位点，如图中箭头所指，如果该线性DNA分子在3个酶切位点上都被该酶切断，则会产生a、b、c、d四种不同长度的DNA片段。
2008-2010年全国卷试题+录音稿+答案
2008年普通高等学校招生全国统一考试（全国卷I）第一节（共5小题，每小题1.5分，满分7.5分）听下面5段对话，每段对话后有一个小题，从题中所给的A、B、C三个选项中选出最佳选项，并标在试卷的相应位置，听完每段对话后，你都有10秒钟时间来回答有关小题和阅读下一小题。
每段对话仅读一遍。
例：How much is the shirt?A. ￡19.15B. ￡9.15C. ￡9.18答案是B。
1. What is the weather like?A. It’s raining.B. It’s cloudy.C. It’s sunny.2. Who will go to China next month?A. Lucy.B. Alice.C. Richard.3. What are the speakers talking about?A. The man’s sister.B. A film.C. An actor.4. Where will the speakers meet?A. In Room 340.B. In Room 314.C. In Room 223.5. Where does the conversation most probably take place？A. In a restaurant.B. In an office.C. At home.第二节（共15小题；每小题1.5分，满分22.5分）听下面5段对话或独白，每段对话或独白后面有几个小题，从题中所给的A、B、C三个选项中选出最佳选项，并标在试卷的相应位置。
听每段对话或独白前，你将有时间阅读各小题，每小题5秒钟；听完后，各小题将给出5秒钟的作答时间。
每段对话或独白读两遍。
听第6段材料，回答第6至8题。
6. Why did the woman go to New York?A. To spend some time with the baby.B. To look after her sister.C. To find a new job.7. How old was the baby when the woman left New York?A. Two months.B. Five months.C. Seven months.8. What did the woman like doing most with the baby?A. Holding him.B. Playing with him.C. Feeding him.听第7段材料，回答第9至11题。
2008GCT真题A卷及答案
2008年GCT考试真题A卷第一部分语言表达能力测试(50题，每题2分，满分100分)一、选择题1．下列词语中加点字的读音完全相同的一组是A. 宦官豢养盥洗患得患失风云变幻B．莅临乖戾官吏呕心沥血不寒而栗C. 翌日对弈肄业苦心孤诣雄关险隘D．羡慕汗腺霰弹谄媚阿谀借花献佛2．下列成语中，加点字的意义都不相同的一组是A．沽名钓誉徒有虚名不可名状B．栉风沐雨风声鹤唳移风易俗C. 横生枝节起死回生无中生有D．动人心弦兴师动众动辄得咎3．下列句子中，没有语病的一句是A. 长达两个小时的淡心会，积存于代表们心中的疑虑得到了一定程度的消除。
B. 公司引进国外先进的技术和设备，使原材料的消耗比过去节省了将近一倍。
C. 当地教育部门采取各种办法，努力培养和提高中学中青年教师的业务水平。
D．体育场馆的建设和对公众开放，为广泛开展群众文体活动提供了有利条件。
4．下列各句中，语义明确、没有歧义的一句是A．老张也喜欢到公园里去打太极拳。
B．他连你都不认识，又怎么会认识我呢?C．他抬头一看，发现那幅画上头有一只苍蝇。
D. 辩论赛上，正方对反方的反驳是有充分准备的。
5．下列句子中没有使比喻修辞手法的一句是A．生命如风，创造着春的精致；生命如月，辉映着夏的清凉；生命如水，荡涤着秋的萧瑟；生命如云，浮掠着冬的悲伤。
B．那些细碎的往事，当时只是寻常，如今却全变成钻石一样晶莹纯粹的回忆。
C．清晨，迎着朝阳，红领巾们带着小树苗去参加植树活动了。
D．儿子是箭，父亲是弓，要想把箭射得更远，父亲的背便愈弓。
6．下列关于文史知识的表述，错误的一项是A．元代王实甫的传奇剧《西厢记》，描写书生张珙和相国小姐崔莺莺从相爱到结合的曲折过程，表达了“有情人终成眷属”的美好愿望。
B．秦末战乱，刘邦于公元前206年攻占秦都咸阳，废除秦朝的严刑峻法，召集关中父老宣布法令：“杀人者死，伤人及盗抵罪。
”史称“约法三章”。
C．俄国诗人普希金的诗体小说《叶甫盖尼．奥涅金》中的贵族青年奥涅金，是俄国文学史上第一个所谓“多余人”的形象。
2008年全国数学竞赛试题
中国教育学会中学数学教学专业委员会“《数学周报》杯”2008年全国初中数学竞赛试题参考答案答题时注意：1．用圆珠笔或钢笔作答.2．解答书写时不要超过装订线. 3．草稿纸不上交.一、选择题（共5小题，每小题6分，满分30分. 以下每道小题均给出了代号为A ，B ，C ，D 的四个选项，其中有且只有一个选项是正确的. 请将正确选项的代号填入题后的括号里. 不填、多填或错填都得0分）1．已知实数x y ，满足42424233y y xx-=+=，，则444yx+的值为（ ）．（A ）7 （B ） 12+（C ）72+（D ）5【答】（A ）解：因为20x >，2y ≥0，由已知条件得212184x++==， 21122y -+-+==，所以444y x+=22233yx++- 2226y x=-+=7．另解：由已知得：2222222()()30()30x xy y ⎧-+--=⎪⎨⎪+-=⎩，显然222y x -≠，以222,y x -为根的一元二次方程为230t t +-=，所以 222222()1,()3yy xx-+=--⨯=- 故444yx+＝22222222[()]2()(1)2(3)7y y xx-+-⨯-⨯=--⨯-= 2．把一枚六个面编号分别为1，2，3，4，5，6的质地均匀的正方体骰子先后投掷2次，若两个正面朝上的编号分别为m ，n ，则二次函数2y x m x n =++的图象与x 轴有两个不同交点的概率是（ ）．（第3题）（A ）512（B ）49（C ）1736（D ）12【答】（C ）解：基本事件总数有6×6＝36，即可以得到36个二次函数. 由题意知∆＝24m n -＞0，即2m ＞4n .通过枚举知，满足条件的m n ，有17对. 故1736P =.3．有两个同心圆，大圆周上有4个不同的点，小圆周上有2个不同的点，则这6个点可以确定的不同直线最少有( )．（A ）6条 （B ） 8条 （C ）10条 （D ）12条 【答】（B ）解：如图，大圆周上有4个不同的点A ，B ，C ，D ，两两连线可以确定6条不同的直线；小圆周上的两个点E ，F 中，至少有一个不是四边形ABCD 的对角线AC 与BD 的交点，则它与A ，B ，C ，D 的连线中，至少有两条不同于A ，B ，C ，D 的两两连线．从而这6个点可以确定的直线不少于8条．当这6个点如图所示放置时，恰好可以确定8条直线． 所以，满足条件的6个点可以确定的直线最少有8条．4．已知A B 是半径为1的圆O 的一条弦，且1A B a =<．以A B 为一边在圆O 内作正△ABC ，点D 为圆O 上不同于点A 的一点，且D B AB a ==，D C 的延长线交圆O 于点E，则A E的长为（ ）． （A ）2 （B ）1 （C 2（D ）a【答】（B ）解：如图，连接OE ，OA ，OB ． 设D α∠=，则120EC A EAC α∠=︒-=∠． 又因为()1160180222A B O A B D α∠=∠=︒+︒-120α=︒-，所以A C E △≌A B O △，于是1A E O A ==．另解：如图，作直径EF ，连结AF ，以点B 为圆心，AB 作⊙B ，因为AB ＝BC ＝BD ，则点A ，C ，D 都在⊙B 上， 由11603022F ED A C BA ∠=∠=∠=⨯︒=︒所以2301AE EF sim F sim =⨯∠=⨯︒=（第4题）5．将1，2，3，4，5这五个数字排成一排，最后一个数是奇数，且使得其中任意连续三个数之和都能被这三个数中的第一个数整除，那么满足要求的排法有（ ）．（A ）2种 （B ）3种 （C ）4种 （D ）5种 【答】（D ）解：设12345a a a a a ，，，，是1，2，3，4，5的一个满足要求的排列．首先，对于1234a a a a ，，，，不能有连续的两个都是偶数，否则，这两个之后都是偶数，与已知条件矛盾．又如果i a （1≤i ≤3）是偶数，1i a +是奇数，则2i a +是奇数，这说明一个偶数后面一定要接两个或两个以上的奇数，除非接的这个奇数是最后一个数．所以12345a a a a a ，，，，只能是：偶，奇，奇，偶，奇，有如下5种情形满足条件：2，1，3，4，5； 2，3，5，4，1； 2，5，1，4，3； 4，3，1，2，5； 4，5，3，2，1． 二、填空题（共5小题，每小题6分，满分30分）6．对于实数u ，v ，定义一种运算“*”为：u v uv v *=+．若关于x 的方程1()4x a x **=-有两个不同的实数根，则满足条件的实数a 的取值范围是 ．【答】0a >，或1a <-．解：由1()4x a x **=-，得21(1)(1)04a x a x ++++=，依题意有 210(1)(1)0a a a +≠⎧⎨∆=+-+>⎩，， 解得，0a >，或1a <-．7．小王沿街匀速行走，发现每隔6分钟从背后驶过一辆18路公交车，每隔3分钟从迎面驶来一辆18路公交车．假设每辆18路公交车行驶速度相同，而且18路公交车总站每隔固定时间发一辆车，那么发车间隔的时间是 分钟．【答】4．解：设18路公交车的速度是x 米/分，小王行走的速度是y 米/分，同向行驶的相邻两车的间距为s 米．每隔6分钟从背后开过一辆18路公交车，则 s y x =-66． ① 每隔3分钟从迎面驶来一辆18路公交车，则s y x =+33． ②（第8题）（第9题答案）由①，②可得 x s 4=，所以4=xs ．即18路公交车总站发车间隔的时间是4分钟．8．如图，在△ABC 中，AB =7，AC =11，点M 是BC 的中点， AD 是∠BAC 的平分线，MF ∥AD ，则FC 的长为 ． 【答】9．解：如图，设点N 是AC 的中点，连接MN ，则MN ∥AB ． 又//M F AD ，所以 F M N B A D D A C M F N ∠=∠=∠=∠， 所以 12F N M N A B==．因此 1122F C F N N C A B A C =+=+=9．另解：如图，过点C 作AD 的平行线交BA 的延长线为E ，延长MF AE 于点N.则E B A D D A C A C E ∠=∠=∠=∠所以11A E A C ==. 又//F N C E ，所以四边形C E N F 是等腰梯形， 即11(711)922C F E N B E ===⨯+=9．△ABC 中，AB ＝7，BC ＝8，CA ＝9，过△ABC 的内切圆圆心I 作DE ∥BC ，分别与AB ，AC 相交于点D ，E ，则DE 的长为 ．【答】163．解：如图，设△ABC 的三边长为a ，b ，c ，内切圆I 的半径为r ， BC 边上的高为a h ，则11()22a A B C ah S abc r==++△，所以ar a h a b c=++．因为△ADE ∽△ABC ，所以它们对应线段成比例，因此a ah r D E h BC-=，所以 (1)(1)a aah r r aD E a a a h h a b c-=⋅=-=-++()a b c a b c +=++，故 879168793D E ⨯+==++（）．（第8题答案）另解：ABC S rp ∆===（这里2a b cp ++=）所以12r ==28ABCa S h a===△由△ADE ∽△ABC ，得23a ah r D E BCh -===，即21633D E B C ===10．关于x ，y 的方程22208()x y x y +=-的所有正整数解为 ． 【答】481603232.x x y y ==⎧⎧⎨⎨==⎩⎩，，， 解：因为208是4的倍数，偶数的平方数除以4所得的余数为0，奇数的平方数除以4所得的余数为1，所以x ，y 都是偶数．设2,2x a y b ==，则22104()a b a b +=-，同上可知，a ，b 都是偶数．设2,2a c b d ==，则2252()c dc d +=-，所以，c ，d 都是偶数．设2,2c s d t ==，则2226()s t s t +=-，于是 22(13)(13)s t -++＝2213⨯， 其中s ，t 都是偶数．所以222(13)213(13)s t -=⨯-+≤2222131511⨯-<．所以13s -可能为1，3，5，7，9，进而2(13)t +为337，329，313，289，257，故只能是2(13)t +＝289，从而13s -＝7．于是62044s s t t ==⎧⎧⎨⎨==⎩⎩，，；，因此 481603232.x x y y ==⎧⎧⎨⎨==⎩⎩，，，另解：因为222(104)(104)210421632x y -++=⨯= 则有2(104)21632,y +≤ 又y 正整数，所以 143y ≤≤令22|104|,|104|,21632a x b y a b =-=++= 则 因为任何完全平方数的个位数为：1，4，5，6，9由2221632a b +=知22,a b 的个位数只能是1和1或6和6； 当22,a b 的个位数是1和1时，则,a b 的个位数字可以为1或9但个位数为1和9的数的平方数的十位数字为偶数，与22a b +的十位数字为3矛盾。
2008年全国大学生数学建模竞赛选拔试题
2008年全国大学生数学建模竞赛选拔试题时量：180分钟 满分：200分系别： 专业： 学号： 姓名：一、数学模型部分(每题10分，共90分)1、 简述数学建模论文的基本结构。
答：应该主要包含论文标题，摘要，问题重述，问题分析，模型建立，模型求解，模型验证，模型分析与改进，模型评价，参考文献等内容。
2、 简述数学建模论文摘要的要求及其应包含的主要内容。
答：应该主要包含论文建立的模型，模型的求解，模型验证，模型的分析与改进，模型的评价等的简要说明，以及论文的主要创新点和模型的优势。
3、 简述插值和拟合的区别，并简要介绍常用的插值方法和拟合方法及其基本理论和Matlab 命令。
答：插值是指根据已有的数据（自变量及对应的因变量）计算一些新的自变量对应的因变量的值；而拟合则是指根据已有的数据（自变量及对应的因变量），确定自变量与因变量之间最为恰当的一个函数关系式。
4、 请把1~9共9个数字填入3乘以3的正方形格子，使3个行中每个行的数字总和为15，3个列中每个列的数字总和也15，两条对角线数字总和也15。
(1) 中间格的数字应该为多少？并证明之。
(2) 用推理或建立模型方法求出其它数字(建模只说明求解,不求具体解)，最终结果请填入右图。
解：（略）5、 设一个鞋店平均每天卖出鞋100双，批发一次差旅费为每次200元 ，每双鞋每存储一天的费用为0.01元。
请建立数学模型寻求最佳进货方式。
即该鞋店每隔多少天批发一次，每次进货量为多少时，使费用最少。
解：设鞋店第隔x 天批发一次货，每次进货量为y ，则在一个进货周期内的费用共有：∑=-+=xi i y y x C 0)100(01.0200),(只考虑不允许缺货的情况，即x y 100=，则平均每天的费用有：2100*01.0200),(xx x y x C +=， 考虑上式，当且仅当20100*01.0200*2==x (天)时，平均每天的费用最小，此时每次进货2000(双)。