新泰一中北校高二上学期第一次阶段性考试-数学试题

2024年山东省新泰第一中学北校数学高三上期末复习检测试题注意事项：1．答卷前，考生务必将自己的姓名、准考证号填写在答题卡上。

2．回答选择题时，选出每小题答案后，用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，再选涂其它答案标号。

回答非选择题时，将答案写在答题卡上，写在本试卷上无效。

3．考试结束后，将本试卷和答题卡一并交回。

一、选择题：本题共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。

1．已知数列满足：.若正整数使得成立，则( ) A ．16B ．17C ．18D ．192．已知函数()()1xe a axf x e ⎛⎫=-+ ⎪⎝⎭，若()()0f x x R ≥∈恒成立，则满足条件的a 的个数为（ ）A ．0B ．1C ．2D ．33．向量1,tan 3a α⎛⎫= ⎪⎝⎭，()cos ,1b α=，且//a b ，则cos 2πα⎛⎫+=⎪⎝⎭（ ） A ．13B ．22C ．2D ．13-4．已知函数()f x 的定义域为[]0,2，则函数()()282x g x f x =- ） A ．0,1 B ．[]0,2 C ．[]1,2D ．[]1,35．已知函数()()sin 06f x A x a a A ωπ⎛⎫=+-<< ⎪⎝⎭在区间70,3ωπ⎡⎤⎢⎥⎣⎦有三个零点1x ，2x ，3x ，且123x x x <<，若123523x x x π++=，则()f x 的最小正周期为（ ） A ．2πB ．23πC ．πD ．43π 6．在ABC ∆中，角,,A B C 所对的边分别为,,a b c ，已知4cos sin 3b B C c =，则B =（ ）A ．6π或56πB ．4πC ．3π D ．6π或3π 7．已知函数()22018tan 1xx m f x x x m =+++()0,1m m >≠，若()13f =，则()1f -等于（ ）8．中国古代用算筹来进行记数，算筹的摆放形式有纵横两种形式（如图所示），表示一个多位数时，像阿拉伯记数一样，把各个数位的数码从左到右排列，但各位数码的筹式需要纵横相间，其中个位、百位、方位……用纵式表示，十位、千位、十万位……用横式表示，则56846可用算筹表示为（ ）A ．B ．C ．D ．9．已知等比数列{}n a 的各项均为正数，设其前n 项和n S ，若14+=nn n a a （n *∈N ），则5S =（ ）A ．30B ．312C ．152D ．6210．某几何体的三视图如图所示，若侧视图和俯视图均是边长为2的等边三角形，则该几何体的体积为A ．83B ．33C ．1D ．211．已知复数z 534i=+，则复数z 的虚部为（ ） A ．45B ．45-C ．45iD ．45-i 12．已知点(m ,8)在幂函数()(1)n f x m x =-的图象上，设,(ln ),()m a f b f c f n n π⎛⎫===⎪⎝⎭，则（ ）A ．b ＜a ＜cB ．a ＜b ＜cC ．b ＜c ＜aD ．a ＜c ＜b二、填空题：本题共4小题，每小题5分，共20分。

2020-2021学年山东省泰安市新泰一中北校高二（上）第一次段考英语试卷一、阅读理解（本大题共15小题，共37.5分）AIstanbul & FacesTitled "Timeless City：Istanbul & Faces"，the Turkey photography exhibition celebrates the 30th anniversary of Istanbul and Shanghai's sistership.The 42-year-old photographer focuses on the distinctive（独特的）historical quarters and the atmosphere that spreads in Istanbul.His work also pays respects to its people as well,with well-defined portraits.Date：Till Jan.18，10 a.m. - 4 p.m.Venue：Shanghai Art Collection Museum.Knitted WorksThe knitted（编织的）show is aimed at raising awareness of post natal depression.The works were created by more than 20 mothers from different cities.All the knitted dolls and other works will be sold with the profits（利益）going to charity when the exhibition ends.If you are interested in the knitted works,you can order a special piece.Date：Till Jan.1，7 a.m. -9 p.m.Venue：Bl Jinxiu Fun.Old Shanghai TeahouseThe newly-opened zone at Shanghai Dungeon features nostalgic sets and experiences for visitors to have "scary fun".Visitors can wear traditional Chinese costumes and take photos with the performers.Date：Daily,11 a.m. -8 p.m.Venue：Mosaic Shanghai Mail.Musical TitanicThe Tony Awards winning musical "Titanic" is in Shanghai with more than 100 characters cast.The impressive stage effects promise to take the audience to the bottom of the ocean. Date：Till Jan 22，2 p.m. -7：30 p.m.Venue：SAIC Shanghai Culture Square.1.Where should you go if you want to learn more about Turkey's history？______A. Shanghai Art Collection Museum.B. Mosaic Shanghai Mail.C. B1 Jinxiu Fun.D. SAIC Shanghai Culture Square.2.What can we know about knitted works？______A. People can have a talk with the creators.B. All the works can be sold when the show begins.C. The show offers people chances to learn how to make one.D. They were made by people from different places.3.What makes the Musical Titanic special？______A. Its opening time is the longest.B. It raises money for a special purpose.C. It offers a lifelike experience under water.D. It allows interacting with performers.BRobert Spring,a 19th century forger（伪造者），was so good at his profession that he was able to make his living for 15 years by selling false signatures of famous Americans.Spring was born in England in 1838 and arrived in Philadelphia in 1858 to open a bookstore.At first he prospered（繁荣）by selling his small but genuine collection of early US autographs（亲笔签名）.Discovering his ability at copying handwriting he began imitating the signatures of George Washington and Benjamin Franklin and writing them on the title pages of old books.To lessen the chance of detection（被发现）.He sent his forgeries to England and Canada for sale,Forgers have a hard time selling products.A forger cannot approach a respectable buyer but must deal with people who don't have much knowledge in the field.Forgers have many ways of making their works look real.For example,they buy old books and use the aged paper of the title page,and they can treat paper and ink with chemicals.In Spring's time,right after the Civil War,Britain was still fond of the southern states,so Spring invented a respectable young lady known as Miss Fanny Jackson,the only daughter of General "Stonewall" Jackson.For several years Miss Fanny's financial problems forced her to sell a great number of letters and manuscripts（手稿）belonging to her famous father.Spring had to work very hard to satisfy the demand.However,all this activity did not prevent Spring from dying in poverty,leaving sharp-eyed experts the difficult task of separating his forgeries from the originals.4.Robert Spring spent 15 years ______ .A. running a bookstore in PhiladelphiaB. as owners of old booksC. selling real signatures of famous AmericansD. as a forger5.Why did Spring sell his false autographs in England and Canada？______A. There was less chance that his forgeries would be discovered there.B. The prices were much higher in England and Canada.C. There was a greater demand there than in America.D. Britain was Spring's birthplace.6.After the Civil War there was a great demand in Britain for ______ .A. Civil War battle plansB. southern manuscripts and lettersC. the signatures of George Washington and Ben FranklinD. southern money7.Which of the following about Miss Fanny Jackson is TRUE？______A. She was Robert Spring's customer.B. She was an imaginary person created by Spring.C. She was a little-known girl who sold her father's papers to make money.D. She was the only daughter of General Stonewall Jackson.CHave you ever dreamed of visiting a planet in the Milk Way（银河系）？While the trip sounds exciting,it would take years and years to reach your destination.So in the future,bedtime for astronauts may be more than a few hours of regular shut-eye.They would have to sleep for years.European researchers are now conducting hibernation（冬眠）experiments.The study may help them understand whether humans could ever sleep through the years it would take for a spaceflight to distant planets. "If there was an effective technology,it could make deep-space travel a reality," said Mark Ayre of the European Space Agency last month.What seems like science fiction is not completely unlikely.Researchers have been able to use chemicals to put living cells into a sleep-like state.They have now moved on tosmall,non-hibernating mammals like rats.A major challenge is the fact that cells can be very simple systems,whereas body organs are far more complex. "It's like moving from a simple Apple computer to a supercomputer," said Marco Biggiogera,a hibernation researcher at Italy's University of Pavia.Just like bears andfrogs,the hibernation of human beings would cause a person's metabolism（新陈代谢）to lower so they would need less energy.Medical research,however,is just half of a spaceflight hibernation system.There is the challenge of designing a suitable protective shelter（栖身所）.Such a shelter would provide the proper environment for hibernation,such as the proper temperature.It would also have to monitor（监控）life functions and serve the physiological needs of the hibernator. According to Ayre,the six-person human Outer Planets Exploration Mission to Jupiter's moon （木星的卫星），Callisto,could be an opportunity to use human hibernation.The mission aims to send six humans on a five-year flight to Callisto,where they will spend 30 days,in 2045.8.According to the article,the hibernation research ______ .A. is just an ideaB. is always a science fictionC. has already finished successfullyD. has made some progress9.There are ______ major challenges facing scientists who study hibernation.A. twoB. threeC. fourD. five10.The first use of the hibernation technology ______ the six-person human Outer PlanetsExploration Mission to Jupiter's moon Callisto.A. must heB. has been planned forC. is certain to beD. may be11.What is the best title for the article？______A. Hibernation Study for Space TravelB. Welcome to Our Space TravelC. To Hibernate,to Live LongerD. Welcome to Milk WayDCall it a mobility system or a wheelchair—it can be seen at the Tokyo airport moving on its own.It is helping with social distancing while fighting the spread of the coronavirus.It also helps older passengers and those who otherwise need assistance walking the last few steps to the boarding gate.The mobility system seats one person and runs on its own without hitting anything.It has a pre-programmed path of about 600 meters at Tokyo's Haneda International Airport.It wasshown on Monday by WHILL，the company behind the technology.WHILL Chief Executive Satoshi Sugie said robotics and self-driving technology reduce the need for a human at the wheel. They are good for these times of "living with" coronavirus.The ride lasts several minutes.It travels from the security area to the boarding gate at a speed of 3.5 kilometers per hour.But many people hope the technology can help in other places,such as hospitals or parks.The technology uses sensors and cameras to see a clear path and avoid hitting those walking toward it.It could also help Japan free up laborers.Japan,like many other nations,suffers from a labor shortage.The technology could free a worker to do another job. Tests have been carried out at several airports since last year,including John F.Kennedy International Airport in New York.The company hopes to put it in airports around the world. The person on the machine can start or stop it through a simple controller.It runs on batteries and is programmed to return to where it started when its passenger gets off.Anyone needing help walking long distances can use it at Terminal One at Haneda airport.It is called the "last-mile mobility"，says WHILL.12.What do we know about the mobility system？______A. It can seat more than one person.B. It runs with the assistance of a controller.C. It helps older passengers to the boarding gate.D. It has a flexible path of about 600 meters.13.What does "They" in Paragraph 3 refer to？______A. Humans at the wheel.B. The needs for human beings.C. These times of "living with" coronavirus.D. Robotics and self-driving technology.14.What plays an essential part in the technology？______A. Sensors and cameras.B. The speed of the machine.C. The path of the ride.D. The distance of the ride.15.What can we infer from the last two paragraphs？______A. The system runs completely on its own.B. The system has been put into use in the US.C. WHILL will continue to test the system in airports worldwide.D. The system is controlled by a passenger to return to the start.二、阅读七选五（本大题共5小题，共12.5分）Ways to improve your mental health in 2020 Looking on the bright side of life really is beneficial.Optimists（乐观者）have a less chance of having a heart attack or stroke（中风），a bigger chance of a healthy and long life.However,being an optimist doesn't mean you ignore the stress of daily life. (1) .How to do that？• Practice optimismFaced with a challenge,optimists more often see it as temporary（临时的）or even positive,allowing themselves to learn and grow. (2) .Imagining yourself with all of your problems solved in a future may work.• (3)Studies have shown putting the well-being of others before our own without expecting anything in return is useful.Even if you have little time to offer,just the act of giving has been shown to improve our health,possibly by temporarily reducing our sense of pain.• Be gratefulBeing thankful protects us against anxiety and depression and boosts（增加）optimism. (4) .Before going to bed,write down any positive experience you had that day,however small. • Improve your social connectionsGood relationships keep us happier and healthier. (5) Instead,it's the quality of the relationship that matters.High-conflict（高冲突）marriages,for example,without much affection（感情），turn out to be bad for our health,perhaps worse than getting divorced（离婚）though it's not right to make light of divorce.A.Start volunteeringB.Find your purposeC.And you don't have to have scores of friends to get this benefitD.One of the best ways to make it a part of your life is to keep a daily journalE.It simply means when unpleasant things happen,don't blame yourself unnecessarilyF.People who are more socially connected to family,to friends,or to community,are happierG.They also believe they have control over their fate（命运）and can create opportunities for good things to happen16. A. A B. B C. C D. D E.E F.F G. G17. A. A B. B C. C D. D E.E F.F G. G18. A. A B. B C. C D. D E.E F.F G. G19. A. A B. B C. C D. D E.E F.F G. G20. A. A B. B C. C D. D E.E F.F G. G三、完形填空（本大题共15小题，共15.0分）Mary was always dreaming of graduating from college in cap and gown,accepting her diploma and ready to start a career.Since Mary came from a low-income family,her dream seemed out of(21) .There would never be enough money to send Mary to college.(22)，no member of her family had ever gone to college.But Mary(23) that thought.In secondary school she studied hard,and spoke(24) of her dream to teachers and friends.When her(25) day at school came,her principal（校长）called her into his office."I have a（n）(26) for you," he said.It contained a scholarship to a nearby college.The power of a thought had(27) its first bonus（奖金）.But,the scholarship could cover only part of her college(28).Mary took every part-time job she could find.Mary's dream came true when she graduated from college.Then she took a course in insurance.When she(29) herself to a local insurance company for a job,she was(30).She applied again.The answer was no.She kept going back until the manager(31) her.Twenty-five years later Mary Crowe's colleagues in that company(32) at a special dinner in her honor,recognizing her for her outstanding achievements as an insurance saleswoman. "Each of us is constantly in a state of becoming," she said,giving her formula for(33). "Through(34) and faith you can become what you think.Not that your life will be without problems,but along the way problems will be(35).Ask and believe；dream and believe；work and believe."21. A. control B. question C. sight D. thought22. A. However B. Instead C. Therefore D. Besides23. A. held on to B. took pride in C. made up for D. came up with24. A. selfishly B. freely C. unwillingly D. jokingly25. A. first B. formal C. final D. unique26. A. book B. envelope C. souvenir D. test27. A. joined B. wasted C. consumed D. produced28. A. expenses B. secrets C. incomes D. ambitions29. A. limited B. related C. accustomed D. introduced30. A. picked up B. put aside C. turned down D. taken in31. A. believed B. praised C. accepted D. noticed32. A. gathered B. declared C. resigned D. parted33. A. reputation B. independenceC. responsibilityD. success34. A. suspension B. enthusiasm C. politeness D. status35. A. corrected B. overcome C. forgotten D. collected四、语法填空（本大题共1小题，共15.0分）36.I rolle dover and turned off my alarm clock.It was still dark outside so I felt my way to thewall switch and turned (1) the light. I got (2) （dress）quickly because I knew both of my dogs would be eager (3) （get）outside for their morning walk. I reached over to take a coat.However,I failed to check the weather report for today,so I wasn't sure what kind of coat I should wear.I pulled on my light jacket, and (4) （lead）my bigger dog, Fluffy to walk first. I openedthe door and stepped outside. (5) （instant）I knew I had made the wrong choice.The grass was covered in frost and I tried to keep (6) （I）balance.I trembled with cold. Then I looked up at (7) sky and saw it was full of stars. It was sobeautiful and (8) （peace）．Suddenly,I felt warmth inside and stopped trembling. In all of its sweet seasons it had never failed to touch my soul. It had heated my heart on the (9) （cold）days and lighted my spirit on the darkest nights. I thanked God for the love (10) created the moon, the stars and me.Our lives here are full of sweet seasons as well. Make each day of them a creation of your love.五、书面表达（本大题共2小题，共40.0分）37.假如你是李华，请你写一封邮件给你的英国朋友Henry,邀请他参加你校国庆节举办的建国70周年庆祝活动．内容包括：1．时间和地点；2．有关活动："歌颂祖国"歌舞晚会、"改革开放成就"图片展…注意：1．词数100左右；2．可以适当增加细节，以使行文连贯．参考词汇：anniversary周年纪念日38.阅读下面短文，根据所给情节进行续写，使之构成一个完整的故事。

山东省新泰一中2020学年高二数学上学期期中试题本试卷分第Ⅰ卷(选择题)和第Ⅱ卷(非选择题)两卷，满分150分，测试时间120分钟，第Ⅰ卷将正确的选项填涂在答题卡的相应地点，第Ⅱ卷直接答在试卷上。

第Ⅰ卷(选择题，共60分)一、选择题：本大题共12小题，每题5分，共60分.在每题给出的四个选项中，只有一项为哪一项切合题目要求的.1．命题“存在x Z，使x22x m0”的否认是（）．A．存在x Z，使x22x m0B．不存在x Z，使x22x m0C．关于随意x Z，都有x22x m0D．关于随意xZ，都有x22x m02．等差数列｛a｝中，已知前15项的和S1590，则a8等于（）．nA．45B．12C．45D．6243、抛物线y216x的焦点坐标为（）A.(0,4)B.(4,0)C.(0,4)D.(4,0)4．在ABC中，“A”是“cosA 1”的（）23A．充足而不用要条件B．必需而不充足条件C．充足必需条件D．既不充足也不用要条件5.已知一个等比数列首项为1，项数是偶数，其奇数项之和为85，偶数项之和为170，则这个数列的公比和项数分别为（）A．8，2B．2，4C．4，10D．2，86．已知m0,n0，则112mn的最小值是（）m nA．5B．4C．22D．27．若b a0，则以下不等式①ab ab；②|a||b|;③11；④b a2中，a b a b正确的不等式有（）．A．1个B．2个C．3个D．4个8．已知F1(1,0),F2(1,0)是椭圆的两个焦点，过F1的直线l交椭圆于M,N两点，若MF2N 的周长为8，则椭圆方程为（）（A）x 2y21（B）y2x21（C）x2y21（D）y2x21434316151615 9.等差数列的前n项和为30，前2n项的和为100，则它前3n项的和为（）A.130 C.21010、探照灯反射镜的轴截面是抛物线y22pxx0)的一部分，光源位于抛物线的焦点处，(已知灯口圆的直径为60cm，灯深40cm，则抛物线的焦点坐标为（）A、45,0B、45,0C、45,0D、45,0 2481611．数列{a n}的前n项和为s n，若，则s5等于（）A．1B．C．D．12、双曲线C的左右焦点分别为F1,F2，且F2恰巧为抛物线y24x的焦点，设双曲线C与该抛物线的一个交点为A，若AF1F2是以AF1为底边的等腰三角形，则双曲线C的离心率为（）A、2B、12C、13D、23二、填空题：本大题共4小题，每题5分，共20分.13．若椭圆x2y21，则实数x的取值范围是. 5414．已知x＜0，则24的最大值等于. 3xx15．将全体正整数摆列成一个三角形数阵：依据以上摆列的规律，第16行从左向右的第3个数为．16、已知双曲线x2y21的一条渐近线和圆x2y24x30相切，则该双曲线的m离心率为三、解答题：本大题共6小题，共70分．解答应写出说明文字、演算式、证明步骤．（此题满分10分）等差数列a n的前n项和记为S n，已知a1030，a2050．（１）求通项a n；（2）若S n242，求n．18．（此题满分12分）若不等式a2x22a2x40对x R恒建立，务实数a的取值范围。

新泰一中北校2024级高一上学期期中考试物理试卷考试范围：必修一第四章第3节之前；考试时间：90分钟；2024.11注意事项：1.答题前填写好自己的姓名、班级、考号等信息2.请将答案正确填写在答题卡上第I 卷（选择题40分）一、单选题（本大题共有8个小题，每小题3分，共24分。

每个小题只有一个选项符合题目要求，选对得3分，错选、多选、不选均得0分。

）1.根据你对物理概念和规律的理解，下列说法中正确的是（ ）A.枪膛中子弹加速度约为，射出速度可达，所以加速度大，速度变化就大B.“秋风萧萧，落叶勿勿”，“秋风”可以扫落叶”，但是“秋风”不能扫“汽车”，就是因为落叶质量小，惯性小；汽车质量大，惯性大的缘故。

C.牛顿第一定律是牛顿通过大量实验总结出来的实验结论D.大小相等，方向相反，作用在同一条直线上的两个力一定是一对平衡力2.如图所示为甲、乙两物体运动的位移-时间图像，在内，下列说法正确的是（ ）A.甲沿曲线运动，乙沿直线运动B.两物体运动路程均为C.两物体的平均速度大小均为D.两物体的运动方向均没有变化3.下面对于物体的受力情况描述正确的是（ ）A.图（1）光滑球静止在两夹板之间，小球受到两夹板的弹力作用B.图（2）竖直细线悬挂光滑小球在斜面上静止，斜面对小球可能有弹力C.图（3）物体保持相对静止向右匀速，间摩擦力与大小相等D.图（4）对木块施加一沿斜面向上的拉力，木块静止，斜面对木块可能没有摩擦力4.如图所示，两位同学利用自由落体运动测量反应时间，同学用手捏住直尺上端，同学在直尺下方做好1000m /s 0~30s 1m /sA B 、A B 、F A B准备，但手不碰到尺，此时两手间的距离为，重力加速度为。

当同学看见同学放开直尺后，立即捏住直尺，下列说法正确的是（）A.本实验一个人用两个手也能完成B.本实验C.本实验同学准备捏尺时必须对准尺的0刻度线D.只需记录同学准备捏尺时和捏住尺时大拇指下边缘的刻度即可5.如图所示，做匀加速直线运动的物体依次经过三点。

新泰一中北校高一期中考试语文试题一、现代文阅读（35分）（一）现代文阅读Ⅰ（本题共5小题，19分）阅读下面的文字，完成1—5小题。

材料一：取材于中国古典小说《西游记》的《黑神话：悟空》是一款单机动作冒险游戏，由中国游戏开发商游戏科学于2018年立项，足足历时6年开发完成。

游戏上市首日，热度迅速破圈，相关话题在抖音、微博等多个平台榜单上均位居前列。

玩家的游戏体验如何？网友“杜梁”的游戏体验可以代表不少人的心声：中式美学音画体验，独此一家别无分号。

优美的自然风景、随着光线呈现不同色泽的怪物毛发、炫酷的法术特效，让玩家感叹：“妥妥的电影大片级享受。

”战斗时激昂的鼓点、旅行时如同流水淙淙的配乐，高品质的游戏配乐让玩家直观感受到中式音乐美学的魅力。

作为一款动作类游戏，既要让玩家对重要的敌方角色产生“敬畏之心”，又要避免因为战斗难度太大而“劝退”玩家，如何把控游戏难度是一个精妙而复杂的工程。

《黑神话：悟空》花了很大精力对游戏难度进行调节，保持了足够的弹性和宽容度，以期让玩家保持良好的游戏体验。

游戏科学CEO冯骥很有信心地告诉大家：“如果你以前从没玩过单机动作冒险游戏，那么《黑神话：悟空》非常适合作为你入门的第一款。

”玩家群体因何热爱？不少游戏圈外的人，被《黑神话：悟空》上市首日的热度“震住了”：为什么这游戏会火到刷屏？其实，用触乐网创始人祝佳音的一句话就可以回答：对游戏玩家来说，单机3A游戏就像是奥运会上的金牌。

《黑神话：悟空》表现出的高品质，对于缺乏国产3A大作而耿耿于怀多年的游戏玩家们，终于得以为情怀“激情买单”。

祝佳音形容《黑神话：悟空》的出现“可能就是送给我们看着《大闹天宫》《阿凡提》《天书奇谈》，乃至《幽灵公主》长大的人的一份情书。

”许多国外3A级单机游戏大作，深度植入了各国的历史文化元素。

例如《刺客信条》系列对中世纪地中海、法国、意大利等国家和地区历史风貌和文化的呈现，《荒野大镖客》对西部牛仔生活的描绘等等，让玩家在不知不觉中接受了异域文化的影响。

2020-2021学年山东省泰安市新泰一中北校高二（上）第一次段考英语试卷一、阅读理解（本大题共15小题，共37.5分）AIstanbul & FacesTitled "Timeless City：Istanbul & Faces"，the Turkey photography exhibition celebrates the 30th anniversary of Istanbul and Shanghai's sistership.The 42-year-old photographer focuses on the distinctive（独特的）historical quarters and the atmosphere that spreads in Istanbul.His work also pays respects to its people as well,with well-defined portraits.Date：Till Jan.18，10 a.m. - 4 p.m.Venue：Shanghai Art Collection Museum.Knitted WorksThe knitted（编织的）show is aimed at raising awareness of post natal depression.The works were created by more than 20 mothers from different cities.All the knitted dolls and other works will be sold with the profits（利益）going to charity when the exhibition ends.If you are interested in the knitted works,you can order a special piece.Date：Till Jan.1，7 a.m. -9 p.m.Venue：Bl Jinxiu Fun.Old Shanghai TeahouseThe newly-opened zone at Shanghai Dungeon features nostalgic sets and experiences for visitors to have "scary fun".Visitors can wear traditional Chinese costumes and take photos with the performers.Date：Daily,11 a.m. -8 p.m.Venue：Mosaic Shanghai Mail.Musical TitanicThe Tony Awards winning musical "Titanic" is in Shanghai with more than 100 characters cast.The impressive stage effects promise to take the audience to the bottom of the ocean. Date：Till Jan 22，2 p.m. -7：30 p.m.Venue：SAIC Shanghai Culture Square.1.Where should you go if you want to learn more about Turkey's history？______A. Shanghai Art Collection Museum.B. Mosaic Shanghai Mail.C. B1 Jinxiu Fun.D. SAIC Shanghai Culture Square.2.What can we know about knitted works？______A. People can have a talk with the creators.B. All the works can be sold when the show begins.C. The show offers people chances to learn how to make one.D. They were made by people from different places.3.What makes the Musical Titanic special？______A. Its opening time is the longest.B. It raises money for a special purpose.C. It offers a lifelike experience under water.D. It allows interacting with performers.BRobert Spring,a 19th century forger（伪造者），was so good at his profession that he was able to make his living for 15 years by selling false signatures of famous Americans.Spring was born in England in 1838 and arrived in Philadelphia in 1858 to open a bookstore.At first he prospered（繁荣）by selling his small but genuine collection of early US autographs（亲笔签名）.Discovering his ability at copying handwriting he began imitating the signatures of George Washington and Benjamin Franklin and writing them on the title pages of old books.To lessen the chance of detection（被发现）.He sent his forgeries to England and Canada for sale,Forgers have a hard time selling products.A forger cannot approach a respectable buyer but must deal with people who don't have much knowledge in the field.Forgers have many ways of making their works look real.For example,they buy old books and use the aged paper of the title page,and they can treat paper and ink with chemicals.In Spring's time,right after the Civil War,Britain was still fond of the southern states,so Spring invented a respectable young lady known as Miss Fanny Jackson,the only daughter of General "Stonewall" Jackson.For several years Miss Fanny's financial problems forced her to sell a great number of letters and manuscripts（手稿）belonging to her famous father.Spring had to work very hard to satisfy the demand.However,all this activity did not prevent Spring from dying in poverty,leaving sharp-eyed experts the difficult task of separating his forgeries from the originals.4.Robert Spring spent 15 years ______ .A. running a bookstore in PhiladelphiaB. as owners of old booksC. selling real signatures of famous AmericansD. as a forger5.Why did Spring sell his false autographs in England and Canada？______A. There was less chance that his forgeries would be discovered there.B. The prices were much higher in England and Canada.C. There was a greater demand there than in America.D. Britain was Spring's birthplace.6.After the Civil War there was a great demand in Britain for ______ .A. Civil War battle plansB. southern manuscripts and lettersC. the signatures of George Washington and Ben FranklinD. southern money7.Which of the following about Miss Fanny Jackson is TRUE？______A. She was Robert Spring's customer.B. She was an imaginary person created by Spring.C. She was a little-known girl who sold her father's papers to make money.D. She was the only daughter of General Stonewall Jackson.CHave you ever dreamed of visiting a planet in the Milk Way（银河系）？While the trip sounds exciting,it would take years and years to reach your destination.So in the future,bedtime for astronauts may be more than a few hours of regular shut-eye.They would have to sleep for years.European researchers are now conducting hibernation（冬眠）experiments.The study may help them understand whether humans could ever sleep through the years it would take for a spaceflight to distant planets. "If there was an effective technology,it could make deep-space travel a reality," said Mark Ayre of the European Space Agency last month.What seems like science fiction is not completely unlikely.Researchers have been able to use chemicals to put living cells into a sleep-like state.They have now moved on tosmall,non-hibernating mammals like rats.A major challenge is the fact that cells can be very simple systems,whereas body organs are far more complex. "It's like moving from a simple Apple computer to a supercomputer," said Marco Biggiogera,a hibernation researcher at Italy's University of Pavia.Just like bears andfrogs,the hibernation of human beings would cause a person's metabolism（新陈代谢）to lower so they would need less energy.Medical research,however,is just half of a spaceflight hibernation system.There is the challenge of designing a suitable protective shelter（栖身所）.Such a shelter would provide the proper environment for hibernation,such as the proper temperature.It would also have to monitor（监控）life functions and serve the physiological needs of the hibernator. According to Ayre,the six-person human Outer Planets Exploration Mission to Jupiter's moon （木星的卫星），Callisto,could be an opportunity to use human hibernation.The mission aims to send six humans on a five-year flight to Callisto,where they will spend 30 days,in 2045.8.According to the article,the hibernation research ______ .A. is just an ideaB. is always a science fictionC. has already finished successfullyD. has made some progress9.There are ______ major challenges facing scientists who study hibernation.A. twoB. threeC. fourD. five10.The first use of the hibernation technology ______ the six-person human Outer PlanetsExploration Mission to Jupiter's moon Callisto.A. must heB. has been planned forC. is certain to beD. may be11.What is the best title for the article？______A. Hibernation Study for Space TravelB. Welcome to Our Space TravelC. To Hibernate,to Live LongerD. Welcome to Milk WayDCall it a mobility system or a wheelchair—it can be seen at the Tokyo airport moving on its own.It is helping with social distancing while fighting the spread of the coronavirus.It also helps older passengers and those who otherwise need assistance walking the last few steps to the boarding gate.The mobility system seats one person and runs on its own without hitting anything.It has a pre-programmed path of about 600 meters at Tokyo's Haneda International Airport.It wasshown on Monday by WHILL，the company behind the technology.WHILL Chief Executive Satoshi Sugie said robotics and self-driving technology reduce the need for a human at the wheel. They are good for these times of "living with" coronavirus.The ride lasts several minutes.It travels from the security area to the boarding gate at a speed of 3.5 kilometers per hour.But many people hope the technology can help in other places,such as hospitals or parks.The technology uses sensors and cameras to see a clear path and avoid hitting those walking toward it.It could also help Japan free up laborers.Japan,like many other nations,suffers from a labor shortage.The technology could free a worker to do another job. Tests have been carried out at several airports since last year,including John F.Kennedy International Airport in New York.The company hopes to put it in airports around the world. The person on the machine can start or stop it through a simple controller.It runs on batteries and is programmed to return to where it started when its passenger gets off.Anyone needing help walking long distances can use it at Terminal One at Haneda airport.It is called the "last-mile mobility"，says WHILL.12.What do we know about the mobility system？______A. It can seat more than one person.B. It runs with the assistance of a controller.C. It helps older passengers to the boarding gate.D. It has a flexible path of about 600 meters.13.What does "They" in Paragraph 3 refer to？______A. Humans at the wheel.B. The needs for human beings.C. These times of "living with" coronavirus.D. Robotics and self-driving technology.14.What plays an essential part in the technology？______A. Sensors and cameras.B. The speed of the machine.C. The path of the ride.D. The distance of the ride.15.What can we infer from the last two paragraphs？______A. The system runs completely on its own.B. The system has been put into use in the US.C. WHILL will continue to test the system in airports worldwide.D. The system is controlled by a passenger to return to the start.二、阅读七选五（本大题共5小题，共12.5分）Ways to improve your mental health in 2020 Looking on the bright side of life really is beneficial.Optimists（乐观者）have a less chance of having a heart attack or stroke（中风），a bigger chance of a healthy and long life.However,being an optimist doesn't mean you ignore the stress of daily life. (1) .How to do that？• Practice optimismFaced with a challenge,optimists more often see it as temporary（临时的）or even positive,allowing themselves to learn and grow. (2) .Imagining yourself with all of your problems solved in a future may work.• (3)Studies have shown putting the well-being of others before our own without expecting anything in return is useful.Even if you have little time to offer,just the act of giving has been shown to improve our health,possibly by temporarily reducing our sense of pain.• Be gratefulBeing thankful protects us against anxiety and depression and boosts（增加）optimism. (4) .Before going to bed,write down any positive experience you had that day,however small. • Improve your social connectionsGood relationships keep us happier and healthier. (5) Instead,it's the quality of the relationship that matters.High-conflict（高冲突）marriages,for example,without much affection（感情），turn out to be bad for our health,perhaps worse than getting divorced（离婚）though it's not right to make light of divorce.A.Start volunteeringB.Find your purposeC.And you don't have to have scores of friends to get this benefitD.One of the best ways to make it a part of your life is to keep a daily journalE.It simply means when unpleasant things happen,don't blame yourself unnecessarilyF.People who are more socially connected to family,to friends,or to community,are happierG.They also believe they have control over their fate（命运）and can create opportunities for good things to happen16. A. A B. B C. C D. D E.E F.F G. G17. A. A B. B C. C D. D E.E F.F G. G18. A. A B. B C. C D. D E.E F.F G. G19. A. A B. B C. C D. D E.E F.F G. G20. A. A B. B C. C D. D E.E F.F G. G三、完形填空（本大题共15小题，共15.0分）Mary was always dreaming of graduating from college in cap and gown,accepting her diploma and ready to start a career.Since Mary came from a low-income family,her dream seemed out of(21) .There would never be enough money to send Mary to college.(22)，no member of her family had ever gone to college.But Mary(23) that thought.In secondary school she studied hard,and spoke(24) of her dream to teachers and friends.When her(25) day at school came,her principal（校长）called her into his office."I have a（n）(26) for you," he said.It contained a scholarship to a nearby college.The power of a thought had(27) its first bonus（奖金）.But,the scholarship could cover only part of her college(28).Mary took every part-time job she could find.Mary's dream came true when she graduated from college.Then she took a course in insurance.When she(29) herself to a local insurance company for a job,she was(30).She applied again.The answer was no.She kept going back until the manager(31) her.Twenty-five years later Mary Crowe's colleagues in that company(32) at a special dinner in her honor,recognizing her for her outstanding achievements as an insurance saleswoman. "Each of us is constantly in a state of becoming," she said,giving her formula for(33). "Through(34) and faith you can become what you think.Not that your life will be without problems,but along the way problems will be(35).Ask and believe；dream and believe；work and believe."21. A. control B. question C. sight D. thought22. A. However B. Instead C. Therefore D. Besides23. A. held on to B. took pride in C. made up for D. came up with24. A. selfishly B. freely C. unwillingly D. jokingly25. A. first B. formal C. final D. unique26. A. book B. envelope C. souvenir D. test27. A. joined B. wasted C. consumed D. produced28. A. expenses B. secrets C. incomes D. ambitions29. A. limited B. related C. accustomed D. introduced30. A. picked up B. put aside C. turned down D. taken in31. A. believed B. praised C. accepted D. noticed32. A. gathered B. declared C. resigned D. parted33. A. reputation B. independenceC. responsibilityD. success34. A. suspension B. enthusiasm C. politeness D. status35. A. corrected B. overcome C. forgotten D. collected四、语法填空（本大题共1小题，共15.0分）36.I rolle dover and turned off my alarm clock.It was still dark outside so I felt my way to thewall switch and turned (1) the light. I got (2) （dress）quickly because I knew both of my dogs would be eager (3) （get）outside for their morning walk. I reached over to take a coat.However,I failed to check the weather report for today,so I wasn't sure what kind of coat I should wear.I pulled on my light jacket, and (4) （lead）my bigger dog, Fluffy to walk first. I openedthe door and stepped outside. (5) （instant）I knew I had made the wrong choice.The grass was covered in frost and I tried to keep (6) （I）balance.I trembled with cold. Then I looked up at (7) sky and saw it was full of stars. It was sobeautiful and (8) （peace）．Suddenly,I felt warmth inside and stopped trembling. In all of its sweet seasons it had never failed to touch my soul. It had heated my heart on the (9) （cold）days and lighted my spirit on the darkest nights. I thanked God for the love (10) created the moon, the stars and me.Our lives here are full of sweet seasons as well. Make each day of them a creation of your love.五、书面表达（本大题共2小题，共40.0分）37.假如你是李华，请你写一封邮件给你的英国朋友Henry,邀请他参加你校国庆节举办的建国70周年庆祝活动．内容包括：1．时间和地点；2．有关活动："歌颂祖国"歌舞晚会、"改革开放成就"图片展…注意：1．词数100左右；2．可以适当增加细节，以使行文连贯．参考词汇：anniversary周年纪念日38.阅读下面短文，根据所给情节进行续写，使之构成一个完整的故事。

新泰一中高二上学期第一次大单元考试数学试题(文) 2015-10注意事项：1、 本试卷分选择题和非选择题两部分，满分150分，时间120分钟.2、 答题前，考生务必将密封线内的项目填写清楚.答题前先将自己的姓名、考号、考试科目用2B 铅笔涂写在答题卡上.3、 选择题每小题选出答案后，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.4、 非选择题要写在答题纸对应的区域内，超出部分无效，严禁在试题或草稿纸上答题.第Ⅰ卷（选择题，共50分）一、选择题：（本大题共10小题，每小题5分，共50分.在每题给出的四个选项中，只有一个是符合题目要求的.）1．在等差数列}{n a 中，352676a a a a ==+=，，则 （ ）A ．10B ．13C ．15D ．252．数列⋯--,924,715,58,1的一个通项公式是 （ ）A ．12)1(3++-=n nn a nnB ．12)3()1(++-=n n n a n nC ．121)1()1(2--+-=n n a nnD ．12)2()1(++-=n n n a nn3．在ABC ∆中，60A ∠=，a =3b =，则ABC ∆解的情况 （ ）A ．无解B ．有一解C ．有两解D ．不能确定4．已知锐角三角形的边长分别为2、3、x ，则x 的取值范围是 （ ） A ．135<<xB ．513<<xC ．52<<xD ．55<<x5．在ABC ∆中，若B b A a cos cos =，则ABC ∆的形状是 （ ）A ．等腰三角形B ．直角三角形C ．等腰直角三角形D ．等腰或直角三角形 6．等比数列}{n a 中，233,9a a ==，若243=k a ，则k 等于 （ ）A ．4B ．5C ．6D ．427．ABC ∆中，若60A a ∠==，则sin sin sin a b c A B C-+-+等于（ ）A ．2B ．21 C ．3 D ．23 8．△ABC 的三个内角A 、B 、C 的对边分别a 、b 、c ，且a cos C ,b cos B ,c cos A 成等差数列，则角B 等于( ) A ．30B ．60C ．90D ．1209．在等比数列{a n }中，4813S S ==，，则20191817a a a a +++的值是 （ ）A ．14B ．16C ．18D ．2010．已知nn a )31(=，把数列{}n a 的各项排列成如下的三角形状，记),n m A （表示第m 行的第n 个数，则）（12,10A = （ ）A ．9231）（ B ．9331）（ C ．9431）（ D ．11231）（第Ⅱ卷（共100分）二、填空题: （本大题共5小题，每小题5分，共25分）11．在ABC ∆，如果4:3:2sin :sin :sin =C B A ，那么C cos 等于 .12．在ABC ∆中，已知15030b c B ===，，则边长=a .13．在数列{}n a 中，其前n 项和227n S n n =-+，则n a = .14．ABC ∆中，a 、b 、c 成等差数列，∠B=30°，ABC S ∆=23，那么b = .15．两个等差数列{}{}n n a b 、，1212 (72)...3n n a a a n b b b n ++++=++++，则55b a =__________.三、解答题：（本大题共6题，满分75分．解答须写出文字说明、证明过程和演算步骤）16．（本小题12分）已知：ABC ∆中角A 、B 、C 所对的边分别为a 、b 、c 且sin cos sin cos sin 2A B B A C ⋅+⋅=.(1)求角C 的大小；(2)若,,a c b 成等差数列，且18CA CB ⋅=，求c 边的长.17．（本小题12分）数列{}n a 满足112323(2)n n a a a n n -==-+≥，且. （1）求23a a ，，并证明数列{}n a n -是等比数列； （2）求n a .18．(本小题12分)已知A 、B 、C 为ABC ∆的三内角，且其对边分别为a 、b 、c ，若21sin sin cos cos =-C B C B ． （1）求A ； （2）若4,32=+=c b a ，求ABC ∆的面积．19．（本小题12分）已知{a n }为等差数列，且36a =-，60a =.(1) 求{a n }的通项公式；(2) 若等比数列{}n b 满足18b =-，2123b a a a =++，求{}n b 的前n 项和n S ； (3) 求数列|}{|n a 的前n 项和n T .20．（本小题13分）一缉私艇发现在北偏东45°方向，距离12nmile 的海面上有一走私船正以10nmile/h 的速度沿东偏南15°方向逃窜.缉私艇的速度为14nmile/h ，若要在最短的时间内追上该走私船，缉私艇应沿北偏东45°+α的方向去追，求追及所需的时间和α角的正弦值.21．（本小题13分）在等比数列｛a n ｝中，a n ＞0 (n ∈N ＊），公比q ∈(0,1)，且153528225,a a a a a a ++=35a a 与的等比中项为2.（1）求数列｛a n ｝的通项公式；（2）设b n ＝2log a n ，数列｛b n ｝的前n 项和为S n ，当1212n S S S n++∙∙∙+最大时，求n 的值. 新泰一中高二上学期第一次大单元考试 数学（文）试题 （参考答案）一、选择题：1—5：BDAAD 6—10：CABBB二、填空题：11. 41- 12. 3503100或 13. 8,143,2n n n =⎧⎨-≥⎩1 15. 6512三、解答题：16．解：(1) ∵sin cos sin cos sin 2A B B A C ⋅+⋅=∴sin()sin 2A B C += ………………………………………………（2分） ∵,sin()sin A B C A B C π+=-∴+=∴sin sin 22sin cos C C C C == ………………………………………（4分） ∵0C π<< ∴sin 0C > ∴1cos 2C =∴.3C π= ………………………………………（6分） (2)由,,a c b 成等差数列，得.2b a c += …………………………（7分）∵18CA CB ⋅=，即.36,18cos ==ab C ab ……………………………………………（9分） 由余弦弦定理ab b a C ab b a c 3)(cos 22222-+=-+=，36,3634222=⨯-=∴c c c ，.6=∴c ……………………………………………………………（12分）17.解：12,532==a a …………………………………………（2分）323.1+-=-n a a n n∴[])1(3333.11--=+-=---n a n a n a n n n ∴3)1(1=----n a na n n∴数列{}n a n -是等比数列 ………………………………………（8分） （2）13-+=n n n a ………………………………………………（12分） 18．解:（1）21sin sin cos cos =-C B C B 21)cos(=+∴C B 又π<+<C B 0 ，3π=+∴C Bπ=++C B A ，32π=∴A ． …………………………………（8分） （2）由余弦定理A bc c b a cos 2222⋅-+= 得 32cos22)()32(22π⋅--+=bc bc c b 即：)21(221612-⋅--=bc bc ，4=∴bc323421sin 21=⋅⋅=⋅=∴∆A bc S ABC ……………………………（12分） 19. 解：（1）设等差数列{}n a 的公差d . 因为366,0a a =-=所以112650a d a d +=-⎧⎨+=⎩解得110,2a d =-=所以10(1)2212n a n n =-+-⋅=- ………………（4分） （2）设等比数列{}n b 的公比为q . 因为212324,8b a a a b =++=-=- 所以824q -=-即q =3.所以{}n b 的前n 项和公式为1(1)4(13)1n n n b q S q-==-- ………（8分）(3)2211,61160,6n n n n T n n n ⎧-≤⎪=⎨-+>⎪⎩ ……………………………（12分）20．解：设A 、C 分别表示缉私艇、走私船的位置，设经过x 小时后在B 处追上 则有AB =14x ，BC =10x ，∠ACB =120°…………（2分） 222(14)12(10)240cos120x x x ∴=+-︒……………（8分） 22820x AB BC ∴===，，………………………（10分）sin12020sin120sin 28BC AB α︒︒∴===2sin α∴=所需时间小时，……………………（13分）21. 解：（1）153528225,a a a a a a ++=∴23a + 2a 3a 5 +25a ＝25350,5n a a a >∴+= …………………………………………（2分） 35a a 与的等比中项为2，35a a ∴＝4∵q ∈（0,1）， 353511,4,1,,162a a a a q a ∴>∴====…………（4分） ∴1511622n n n a --⎛⎫=⨯= ⎪⎝⎭…………………………………………（6分）（2）21log 5,1n n n n b a n b b -==-∴-=-…………………………（8分） ∴{b n }是以4为首项，－1为公差的等差数列。

山东省泰安新泰市第一中学（实验部）2023-2024学年高二上学期第一次阶段性测试数学试题
学校:___________姓名：___________班级：___________考号：___________一、单选题
A．
23
,
33
⎡⎤
⎢⎥
⎣⎦
B．
11,
32
⎡⎤
⎢⎥
⎣⎦
二、多选题
9．关于直线:310
l x y--=，下列说法正确的有（A．过点(3,2)
-
C．倾斜角为60°
三、填空题
四、双空题
16．在平面直角坐标系中，()1,2A 的最小值是
；（2）AC 五、解答题
17．已知ABC 的顶点()3,1B ，边BC 上的中线AD 所在直线方程为470x y +-=，边AB 上的高CM 所在直线方程为2320x y -+=．求：(1)直线AB 的方程；(2)顶点C 的坐标．
18．如图所示，在四棱锥M ABCD -中，底面ABCD 是边长为2的正方形，侧棱AM 的
长为3，且60MAB MAD ∠=∠=︒，N 是CM 的中点，设a AB =
，b AD =uuu r r ，c AM =uuu r r ，用
19．如图，四棱锥P
M为BC的中点.
⊥；
(1)求证：AD PC
(2)求平面PAM与平面20．已知圆心为C的圆经过
(1)求圆C的标准方程；
(2)设P为圆C上的一个动点，21．已知直线方程为(
（1）证明：直线恒过定点；（2）m为何值时，点
（3）若直线分别与x轴，
线的方程.
22．如图，在三棱柱ABC
的正方形，D为AB中点，且
ABB A；
(1)求证：CD⊥平面11。

2014级高二年级上学期期中模块结业考试 数学试题 （理倾） 2015-11 注意事项：1、 本试卷分选择题和非选择题两部分，满分150分，时间120分钟.2、 答题前，考生务必将密封线内的项目填写清楚.答选择题前先将自己的姓名、考号、考试科目用2B 铅笔涂写在答题卡上.3、 选择题的每小题选出答案后，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.4、 非选择题要写在答题纸对应的区域内，超出部分无效，严禁在试题或草稿纸上答题.第Ⅰ卷（选择题，共50分）一、选择题：（本大题共10小题，每小题5分，共50分.在每题给出的四个选项中，只有一个是符合题目要求的.）1．已知011<<ba ，给出下列四个结论： ①a ＜b ②a+b ＜ab ③|a|＞|b| ④ab ＜b 2其中正确结论的序号是（ ）A ．①② B. ②④ C ．②③ D ．③④2. 在△ABC 中，BC=5，B=120°，AB=3，则△ABC 的周长等于 ( )A. 7B. 58C. 49D. 153．已知一个等差数列的前四项之和为21，末四项之和为67，前n 项和为286，则项数n 为（ ）A ． 24B ． 26C ． 27D ． 284．的”是，则“已知411,≤=+∈xy y x R y x ( ) A.充分不必要条件 B 必要不充分条件C 充要条件D 既不充分也不必要条件5、已知命题“若a,b,c 构成等比数列，则2b ac =”，在它的逆命题、否命题，逆否命题中，真命题的个数是（ ）A ．0B ．1C ．2D ．36.在△ABC 中，030,23,2B AB AC ∠===，则△ABC 的面积是 ( )A ． 32B ． 3C ． 32或34D ．3或327．已知F 1，F 2是椭圆191622=+y x 的两焦点，过点F 2的直线交椭圆于A ，B 两点．在△AF 1B 中，若有两边之和是10，则第三边的长度为( )A ．6B ．5C ．4D ．38．设△ABC 的内角A ，B ，C 所对的边分别为a ，b ，c ，若A a B c C b sin cos cos =+，则△ABC 的形状为（ ）A ． 锐角三角形B ． 直角三角形C ． 钝角三角形D ． 不确定9. 已知x ，y 满足，则使目标函数z=y ﹣x 取得最小值﹣4的最优解为（ ）A ． （2，﹣2）B ． （﹣4，0）C ． （4，0）D ． （7，3）10.已知0,0>>b a ，4112=+b a ，若不等式m b a 42≥+恒成立，则m 的最大值为A ．10B ．9C ．8D ．7第Ⅱ卷（共100分）二、填空题: （本大题5小题，每小题5分，共25分）11．已知椭圆C ：+=1（a ＞b ＞0）的两个焦点分别为F 1（﹣1，0），F 2（1，0），且椭圆C 经过点P （，）, 椭圆C 的方程为12．不等式<--b ax x 20的解集是(2,3)，则不等式012>--ax bx 的解集是__________. {}{}._________,,241n 2,,.131561118310=+++∈-+=*b b a b b a N n n T S n b a T S n n n n n n 则项和，已知的前分别是等差数列设 14.已知函数12)(22+-++-=b b x x x f ，若当[]恒成立，时，0)(1,1>-∈x f x 则b 的取值范围是 .15. 下列命题中真命题为 .（1）命题“20,0x x x ∀>-≤”的否定是“20,0x x x ∃≤->” （2）在三角形ABC 中，A>B,则sinA>sinB.（3）已知数列{n a }，则“12,,n n n a a a ++成等比数列”是“221++=n n n a a a ”的充要条件（4）已知函数()1lg lg f x x x =+，则函数()f x 的最小值为2三、解答题：（本大题共6题，满分75分．解答须写出文字说明、证明过程和演算步骤）16．（12分）在ABC ∆中，,,a b c 分别为角,,A B C 所对的边，角C 32sin a c A =.（Ⅰ）求角C 的值；（Ⅱ）若1a =，ABC ∆的面积为32,求c 的值.17.（12分）已知p:01322≤+-x x ,q ：0)1()12(2≤+++-a a x a x （1）若a=21，且q p ∧为真，求实数x 的取值范围;（2）若p 是q 的充分不必要条件，求实数a 的取值范围.18.（12分）等差数列{}n a 中，31=a ，其前n 项和为n S . 等比数列{}n b 的各项均为正数，11=b ，且1222=+S b ，33a b =.（Ⅰ）求数列{}n a 与{}n b 的通项公式； （Ⅱ）求数列⎭⎬⎫⎩⎨⎧n S 1的前n 项和n T .19. （12分）经过长期观察得到：在交通繁忙的时段内，某公路汽车的车流量y （千辆/小时）与汽车的平均速度v （千米/小时）之间的函数关系为2710(0)3900v y v v v =>++ （1）在该时段内，当汽车的平均速度v 为多少时，车流量最大，最大车流量为多少？（精确到0.1千辆/小时）（2）若要求在该时段内车流量超过10千辆/小时，则汽车的平均速度应在什么范围内？20、（本小题满分13分）设ABC ∆中的内角,,A B C 的对应边分别为,,a b c ，已知11,2,cos 4a b C ===（1）求ABC ∆的边长c ； （2）求cos()A C -的值21.（本小题14分）设数列{}n a 前n 项和为n S ，且2n n S a +=.（Ⅰ）求数列{}n a 的通项公式；（Ⅱ）若数列{}n b 满足11113,, 2.3n n n b b a b n b --==≥+ 求证1n b ⎧⎫⎨⎬⎩⎭为等比数列，并求数列{}n b 的通项公式；（Ⅲ）设nn n b a c =，求数列{}n c 的前n 和n T .2014级高二年级上学期期中模块结业考试 数学试题 参考答案（理倾） 2015-11一、选择题 1-5.BDBAB 6-10.DABCB二、填空题11.+y 2=1；12.⎪⎭⎫ ⎝⎛--31,21；137841；14.()()∞+∞，，21-- ；15. （2） 三、解答题16.解：（Ⅰ）因为32sin a c A =，由正弦定理得，3sin 2sin sin A C A =，所以sin (2sin 3)0A C -=，…………………………3分 因为sin 0A ≠，所以3sin 2C =， 因为C 是锐角，所以3C π=. ………………………… 6分( Ⅱ)因为133sin 2S ab C ===，2b =，…………………………9分由余弦定理，2222cos 1423c a b ab C =+-=+-=，3c =. 即c 的值为3. …………………………………………………12分17、解：（1）∵q p ∧为真∴p 真q 真 ……………………1分P 真：则设A={x|01322≤+-x x }=}121|{≤≤x x ，…………………2分q 真：B={x|0)1()12(2≤+++-a a x a x }=}1|{+≤≤a x a x …………………3分 ∵21=a ∴B=}2321|{≤≤x x …………………4分∴=⋂B A }121|{≤≤x x∴实数x 的取值范围为:}121|{≤≤x x …………………6分 (2)由（1）知设A={x|}121|{≤≤x x ，B=}1|{+≤≤a x a x∵p 是q 的充分不必要条件，∴A 是B 的真子集 …………………8分∴⎪⎩⎪⎨⎧>+≤1121a a 或⎪⎩⎪⎨⎧≥+<1121a a 解得210≤≤a ，…………………11分 ∴实数a 的取值范围为：}210|{≤≤a a . …………………12分 18. 解：（Ⅰ）设{}n a 公差为d ，数列{}n b 的公比为q ，由已知可得2331232q d q d+++=⎧⎨=+⎩， ………………………………2分 又0>q ⎩⎨⎧==∴33q d . ………………………… 4分 所以33(1)3n a n n =+-=，13n n b -=. …………………………6分（Ⅱ）由（Ⅰ）知数列{}n a 中，31=a ，n a n 3=，(33),2n n n S +∴= …………8分 )111(32)33(21+-=+=∴n n n n S n ， ……………………10分 12111+n nT S S S ∴=++211111[(1)()(]32231n n =-+-++-+） 212(1)3131n n n =-=++（）. …………………………12分 19.解：由题意有2710710710900390063329003v y v v v v==≤=+++++ （3分） 当且仅当900v v=，即30v =时上式等号成立， 此时max 71011.363y =≈（千辆/小时） （6分） （2）由条件得2710103900v v v >++，整理得2689000v v -+<， （8分） 即(50)(18)0v v --< ，∴1850v << （11分）故当30v =千米/小时时车流量最大，且最大车流量为11.3千辆/小时若要求在该时段内车流量超过10千辆/小时，则汽车的平均速度应在1850v <<所表示的范围内. （12分）21. 解：（Ⅰ）由2n n S a +=，得112n n S a +++=，两式相减，得12n n a a +=，∴112n n a a +=（常数），所以，{}n a 是等比数列，-----------------2分又n=1时，1112,1S a a +=∴=，∴112n n a -=. -------------------4分 （Ⅱ）由111,b a ==，且2n ≥时，1133n n n b b b --=+，得111111333n n n n n n b b b b b b ---+=⇒-=，--------------------------------------------------------------------6分 ∴1n b ⎧⎫⎨⎬⎩⎭是以1为首项，13为公差的等差数列， ∴112133n n n b -+=+=，故32n b n =+.-----------------------8分 （Ⅲ） 121()32n n n n a n c b -+==，-----------------9分 012111111[3()4()5()...(2)()]32222n n T n -=+++++ 12311111111[3()4()5()...(1)()(2)()]2322222n n n T n n -=+++++++---------11分 以上两式相减得，1231111111111[3()()()...()(2)()] (122322222)11[1()]1122[3(2)()]13212111[4()(2)()]322n n n n n n n T n n n ---=+++++-+-=+-+-=--+分 184332n n n T -+∴=-------------------14分。

新泰中学高二年级2020年第一次阶段性数学检测试题考试时间：120分钟 满分150分注意事项：1．答题前填写好自己的姓名、班级、考号等信息 2．请将答案正确填写在答题卡上第I 卷（选择题)一、单选题.(共8小题，每小题5分，共40分.在每小题给出的四个选项中，只有一项符合题目要求)1. 已知向量()()1,1,01,0,2a b ==-，且2ka b a b +-与互相垂直，则k 的值是 ( ) A.75B. 2C.53D. 1【答案】A 【解析】 【分析】由向量垂直，可得对应向量数量积为0，从而可求出结果. 【详解】因为()()1,1,01,0,2a b ==-，，所以1a b =-，25a b ==，，又2ka b a b +-与互相垂直，所以()()20ka b a b +-=，即22220k a ka b a b b -+-=，即4250k k +--=，所以75k =;故选A【点睛】本题主要考查向量的数量积的坐标运算，属于基础题型.2. {},,a b c 为空间向量的一组基底，则下列各项中，能构成空间向量的基底的一组向量是（ ）A. {},,a a b a b +-B. {},,b a b a b +-C. {},,c a b a b +-D. {},,2a b a b a b +-+【答案】C 【解析】直接利用基底的定义和共线向量的应用求出结果． 【详解】解：对于{a 、b 、}c 为空间的一组基底， 所以对于()()2a b a b a ++-=与a 共线，故选项A 错误． 对于()()2a b a b b +--=与b 共线，故选项B 错误．对于c 和a b a b +-与不共线向量，所以可以作为基底，故选项C 正确． 对于312()()22a b a b a b +=++-，所以不可以作为向量的基底，故选项D 错误． 故选：C ．【点睛】本题考查的知识要点：基底的定义，共线向量，主要考查学生的运算能力和转换能力及思维能力，属于基础题．3. 在空间直角坐标系O xyz -中，记点()1,2,3A 在xOz 平面内的正投影为点B ，则OB =（ ） 5101314【答案】B 【解析】 【分析】求出B 点坐标，然后计算OB ．【详解】点()1,2,3A 在xOz 平面内的正投影为点(1,0,3)B ，则2210310OB =++= 故选：B.【点睛】本题考查空间点在坐标平面上的投影，考查空间两点间距离．属于基础题． 4. 已知m 是实常数，若方程22240x y x y m ++++=表示的曲线是圆，则m 的取值范围为（ ） A. (),20-∞B. (),5-∞C. ()5,+∞D.()20,+∞【答案】B【分析】由方程表示的曲线为圆，可得出关于实数m 的不等式，解出即可.【详解】由于方程22240x y x y m ++++=表示的曲线为圆，则222440m +->，解得5m <.因此，实数m 的取值范围是(),5-∞. 故选：B.【点睛】本题考查利用圆的一般方程求参数，考查计算能力，属于基础题. 5. 已知点P (－1,1)与点Q (3,5)关于直线l 对称，则直线l 的方程为( ) A. x －y ＋1＝0 B. x －y ＝0 C. x ＋y －4＝0 D. x ＋y ＝0【答案】C 【解析】PQ 中点()1,3，直线斜率11PQk k =-=-，所以直线为()31y x -=--， 即40x y +-=，故选C ．6. 已知直线()1:21230l x a y a +-+-=，22:340l ax y a +++=，则“32a =”是“12l l //”的（ ） A. 充分不必要条件 B. 必要不充分条件C. 充要条件D. 既不充分也不必要条件 【答案】C 【解析】 【分析】先根据直线12l l //求出a 的值,再判断充要关系即可. 【详解】若12l l //,则()213a a -=,解得32a =或1a =-.当1a =-时,直线1l 的方程为350x y --=,直线2l 的方程为350x y -++=,两直线重合,所以32a =,所以“32a =”是“12l l //”的充要条件.易错警示:很多考生根据12l l //求出32a =或1a =-后,直接得出结论,而忽略排除两直线重合的情况,从而错选A. 故选：C.【点睛】本题主要考查充要关系的判断、两直线平行,考查的数学核心素养是数学运算、逻辑推理.7. 直线2cos 30,63x y ππαα⎛⎫⎡⎤--=∈ ⎪⎢⎥⎣⎦⎝⎭的倾斜角的取值范围是（ ） A. ,63ππ⎡⎤⎢⎥⎣⎦B. ,43ππ⎡⎤⎢⎥⎣⎦ C ,42ππ⎡⎤⎢⎥⎣⎦ D. 2,43ππ⎡⎤⎢⎥⎣⎦【答案】B 【解析】 【分析】根据直线方程求出直线的斜率2cos k α=，再由α的范围即可求解. 【详解】直线2x cos α－y －3＝0的斜率k ＝2cos α， 因为α∈,63ππ⎡⎤⎢⎥⎣⎦，所以12≤cos α≤32，因此k ＝2cos α∈3⎡⎣．设直线的倾斜角为θ，则有tan θ∈3⎡⎣．又θ∈[0，π)，且正切函数在0,2π⎡⎫⎪⎢⎣⎭上单调递增，在,2ππ⎛⎫⎪⎝⎭上为单调递增函数， 结合正切函数的图像可知 所以θ∈,43ππ⎡⎤⎢⎥⎣⎦，即倾斜角的取值范围是,43ππ⎡⎤⎢⎥⎣⎦. 故选：B【点睛】本题考查了直线的斜率与倾斜角，需熟记直线的斜率与倾斜角的关系，属于基础题. 8. 在如图的正方体ABCD ﹣A 'B 'C 'D '中，AB ＝3，点M 是侧面BCC 'B '内的动点，满足AM ⊥BD '，设AM 与平面BCC 'B '所成角为θ，则tan θ的最大值为（ ）A.22B.2C.43D.34【答案】B 【解析】 【分析】构建以B 为原点，,,CB AB BB '分别为,,x y z 轴的正方向构建空间直角坐标系，根据正方体棱长标识,,,A B B D ''，令(,0,)M x z 结合AM ⊥BD '有3z x =+且30x -≤≤，而AM 与平面BCC 'B '所成角的平面角为AMB ∠，即有2||tan ||269AB MB x x θ==++，即可求tan θ的最大值.【详解】如下图，以B 为原点，,,CB AB BB '分别为,,x y z 轴的正方向构建空间直角坐标系，则有(0,3,0),(0,0,0),(0,0,3),(3,3,3)A B B D ''---，令(,0,)M x z ，∴(,3,)AM x z =，(3,3,3)BD '=--，又AM ⊥BD '，有3z x =+且30x -≤≤， AM 与平面BCC 'B '所成角为θ，即AMB θ∠=，而(,0,3)BM x x =+，∴22tan 392692()22x x x θ==++++，30x -≤≤， ∴当32x =-时，max (tan )2θ= 故选： B.【点睛】本题考查了利用空间向量求线面角的最值，综合应用了向量垂直的坐标公式，线面角，以及利用二次函数求最值.二、多选题(共4小题，每小题5分，共20分，在每小题给出的四个选项中，至少有一个选项是符合题目要求，全部选对的得5分，选对但不全的得3分，有选错的不得分)9. 下面四个结论正确的是（ ）A. 向量(),0,0a b a b ≠≠，若a b ⊥，则0a b ⋅=．B. 若空间四个点P ，A ，B ，C ，1344PC PA PB =+，则A ，B ，C 三点共线． C. 已知向量()1,1,a x =，()3,,9b x =-，若310x <，则,a b 为钝角．D.任意向量a ，b ，c 满足()()a b c a b c ⋅⋅=⋅⋅． 【答案】AB 【解析】 【分析】由向量垂直的充要条件可判断A ；由题意11334444PC PA PB PC -=-，即可判断B ；举出反例可判断C ；由向量的数量积运算不满足结合律可判断D.即可得解. 【详解】由向量垂直的充要条件可得A 正确；1344PC PA PB =+，∴11334444PC PA PB PC -=-即3AC CB =， ∴A ，B ，C 三点共线，故B 正确；当3x =-时，两个向量共线，夹角为π，故C 错误； 由于向量的数量积运算不满足结合律，故D 错误．故选：A 、B【点睛】本题考查了向量垂直的判定、利用向量证明点共线和向量数量积的应用，属于基础题.10. 已知直线l ：2(1)10a a x y ++-+=，其中a R ∈，下列说法正确的是（ ） A. 当a ＝－1时，直线l 与直线x ＋y ＝0垂直 B. 若直线l 与直线x －y ＝0平行，则a ＝0 C. 直线l 过定点(0，1)D. 当a ＝0时，直线l 在两坐标轴上的截距相等 【答案】AC 【解析】 【分析】利用两直线平行、垂直以及过定点和在两轴上的截距分析直线方程的特征，逐项分析，得到结果.【详解】对于A 项，当a ＝－1时，直线l 的方程为10x y -+=，显然与x ＋y ＝0垂直，所以正确；对于B 项，若直线l 与直线x －y ＝0平行，可知2(1)(1)1(1)a a ++⋅-=⋅-， 解得0a =或1a =-，所以不正确；对于C 项，当0x =时，有1y =，所以直线过定点(0,1)，所以正确； 对于D 项，当a ＝0时，直线l 的方程为10x y -+=， 在两轴上的截距分别是1,1-，所以不正确； 故选：AC.【点睛】该题考查的是有关直线的问题，涉及到的知识点有两直线平行，两直线垂直，直线过定点问题，直线在两轴上的截距的求解，属于简单题目. 11. 下列说法的正确的是 （ ） A. 经过定点的直线都可以用方程()00y y k x x -=-表示． B. 经过定点的直线都可以用方程y kx b =+表示．C. 不经过原点的直线都可以用方程表示．D. 经过任意两个不同的点的直线都可以用方程()()()()121121y y x x x x y y --=--表示．【答案】D 【解析】 【详解】 【分析】解：因为选项A 中缺少了斜率不存在的直线，因此错误 选项B 中，也是同上选项C 中，表示的缺少与x 轴平行和与y 轴平行的直线，因此错误，选D 12. 如图，正方体1111ABCD A B C D -的棱长为1，E 是1DD 的中点，则（ ）A. 直线1//B C 平面1A BDB. 11B C BD ⊥C. 三棱锥11C B CE -的体积为13D. 异面直线1B C 与BD 所成的角为60︒【答案】ABD 【解析】 【分析】建立空间直角坐标系，利用空间向量法一一验证即可；【详解】解：如图建立空间直角坐标系，()0,0,0A ，()1,0,0B ，()1,1,0C ，()0,1,0D ，()10,0,1A ，()11,0,1B ，()11,1,1C ，()10,1,1D ，10,1,2⎛⎫ ⎪⎝⎭E ，()1B C 0,1,1=-，()11,1,1BD =-，()1,1,0BD =-，()11,0,1BA =-所以()111011110B C BD=-⨯+⨯+-⨯=，即11BC BD⊥，所以11B C BD⊥，故B正确；()11011101B C BD=-⨯+⨯+-⨯=，12B C=，2BD=，设异面直线1B C与BD所成的角为θ，则111cos2B C BDB C BDθ==，又0,2πθ⎛⎤∈ ⎥⎝⎦，所以3πθ=，故D正确；设平面1A BD的法向量为(),,n x y z=，则1·0·0n BAn BD⎧=⎨=⎩，即x yx z-+=⎧⎨-+=⎩，取()1,1,1n=，则()10111110n B C=⨯+⨯+⨯-=，即1Cn B⊥，又直线1B C⊄平面1A BD，所以直线1//B C平面1A BD，故A正确；111111111111113326C B CE B C CE C CEV B C SV-∆-===⨯⨯⨯⨯=⋅，故C错误；故选：ABD【点睛】本题考查空间向量法在立体几何中的应用，属于中档题.第II卷（非选择题)三、填空题(本题共4小题，每小题5分，共20分)13 已知A（1，－2，11）、B（4，2，3）、C（x，y，15）三点共线，则xy=___________.【答案】2．【解析】试题分析：由三点共线得向量AB与AC共线，即AB k AC=，(3,4,8)(1,2,4)k x y-=-+，124348x y-+==-，解得12x=-，4y=-，∴2xy=．考点：空间三点共线．14. 已知圆C 的圆心在直线230x y --=上，且过点3(2,)A -，(2,5)B --，则圆C 的标准方程为_________【答案】22(1)(2)10x y +++= 【解析】 【分析】由圆心在直线230x y --=上有(23,)C m m +，设半径为r 结合所过点,A B 即可求圆C 的标准方程.【详解】圆C 的圆心在直线230x y --=上，令(23,)C m m +，半径为r ， ∴圆C 的方程为：222(23)()x m y m r --+-=，又3(2,)A -，(2,5)B --，有()()()()222222213{255m m r m m r+++=+++=，解得2210m r =-⎧⎨=⎩，有(1,2)C --， 故答案为：22(1)(2)10x y +++=；【点睛】本题考查了求圆的标准方程，根据圆心位置、所过的点求圆的方程，属于简单题. 15. 已知一个等腰三角形ABC 的一个顶点是A (4,2)，底边的一个端点B (3,5)，底边另一个端点C 的轨迹方程是___________.【答案】22(4)(2)10x y -+-=（去掉（3,5），（5,-1）两点） 【解析】 【分析】根据等腰三角形和已知顶点A (4,2)，一个端点B (3,5)，利用腰相等且能构成三角形即可求端点C 的轨迹方程；【详解】由题意知：设另一个端点(,)C x y ，腰长为22(34)(52)10r =-+-=∴C 的轨迹方程：22(4)(2)10x y -+-=，又由A 、B 、C 构成三角形，即三点不可共线，∴需要去掉重合点（3,5），反向共线点（5,-1），故答案为：22(4)(2)10x y -+-=（去掉（3,5），（5,-1）两点）【点睛】本题考查了轨迹方程，利用等要三角形的性质及三角形三点不共线求轨迹方程，属于基础题.16. 已知正方体1111ABCD A B C D -的棱长为2，点M ，N 分别是棱BC ，1CC 的中点，则二面角C AM N --的余弦值为__．若动点P 在正方形11BCC B （包括边界）内运动，且1//PA 平面AMN ，则线段1PA 的长度范围是__．【答案】 (1).23 (2). 3252⎡⎢⎣， 【解析】 【分析】 易知NQC ∠为二面角C AM N --的平面角，利用相似的性质可求得CQ ，进而求得NQ ，由此得解二面角C AM N --的余弦值；建立空间直角坐标系，可求得点P 的轨迹为经过1BB ，11B C 中点的线段，再根据对称性即可求得线段1PA 长度的最值，进而得到取值范围．【详解】解：延长AM 至Q ，使得CQ AQ ⊥，连接NQ ，如图，由于1111ABCD A B C D -为正方体，由三垂线定理易知NQC ∠为二面角C AM N --的平面角， 而2sin sin 521CQ AB CMQ AMB CM AM ∠=∠====+，故55CQ == ∴22()155NQ =+=， ∴2cos 3CQ NQC NQ ∠==； 以点D 为坐标原点，建立如图所示的空间直角坐标系，设(P m ，2，)(0n m ，2)n ，(2A ，0，0)，(1M ，2，0)，(0N ，2，1)，1(2A ，0，2)，则(1,2,0),(2,2,1)AM AN =-=-，1(2,2,2)A P m n =--，设平面AMN 的一个法向量为(,,)v x y z =，则·20·220v AM x y v AN x y z ⎧=-+=⎨=-++=⎩， 故可取(2,1,2)v =，又1//PA 平面AMN ，∴12(2)22(2)30A P v m n m n =-++-=+-=， ∴点P 的轨迹为经过1BB ，11BC 中点的线段，根据对称性可知，当点P 在两个中点时，21||215max PA =+=，当点P 在两个中点的中点时，221232||(5)()22min PA =-=， 故选段1PA 的长度范围是32[,5]2． 故答案为：23，32[,5]2．四、解答题(共6小题，70分)17. 已知空间中三点(2,0,2)A -，(1,1,2)B -，(3,0,4)C -，设a AB =，b AC =.（1）求向量a 与向量b 的夹角的余弦值；（2）若ka b +与2ka b -互相垂直，求实数k 的值.【答案】（1）1010-；（2）52k =-或2k =. 【解析】【分析】（1）先写出a ，b ，再根据空间向量的夹角公式直接求解即可；（2）根据空间向量垂直的坐标表示直接求解即可得答案.【详解】（1）∵()1,1,0a AB ==，()1,0,2b AC ==-，设a 与b 的夹角为θ，∴10cos 10|a ba b θ⋅===-∣； （2）∵()1,,2ka b k k +=-，()22,,4ka b k k -=+-且()()2ka b ka b +⊥-，∴2(1)(2)80k k k -++-=，即：52k =-或2k =. 【点睛】本题考查空间向量的夹角的计算，空间向量的垂直求参数，考查运算能力，是基础题.18. 如图，已知M 、N 分别为四面体ABCD 的面BCD 与面ACD 的重心，且G 为AM 上一点，且:1:3GM GA =，设AB a =，AC b =，AD c =，试用a ，b ，c 表示BG ，BN ．【答案】BG 311444a b c =-++；BN 1133b c a =+-． 【解析】【分析】根据向量的加减法计算即可．【详解】解：14BG BM MG BM AM =+=- 131()444BM AB BM BM a =-+=- 3211()4324BC BD a =⨯⨯+- 11()44b ac a a =-+-- 311444a b c =-++； 21()32BN AN AB AC AD AB =-=⨯+- 1133b c a =+-． 【点睛】本题主要考查向量的加减法和几何表示，属于基础题.19. 求过点(2,3)P ，且满足下列条件的直线方程：（1）倾斜角等于直线340x -+=的倾斜角的二倍的直线方程； （2）在两坐标轴上截距相等的直线方程．【答案】（13330x y -+-= ．（2）320x y -=或50x y +-= ．【解析】分析：（1）求出直线的倾斜角，利用点斜式求出直线方程；（2）分类讨论，可得在两坐标轴上截距相等的直线方程.详解：（1） 由题意，可知 3tan α=，所以 30α=， 则 tan2tan603k α=== )332y x --，33230x y -+-=．（2） 当直线过原点时方程为：32y x =，当直线不过原点时方程为：155x y +=． 故所求直线的方程为 320x y -= 或 50x y +-=．点睛：本题考查直线方程，考查分类讨论的数学思想.20. 已知ABC ∆的顶点(2,8)C -，直线AB 的方程为211y x =-+，AC 边上的高BH 所在直线的方程为320x y ++=（1）求顶点A 和B 的坐标；（2）求ABC ∆外接圆的一般方程.【答案】（1）()5,1和()7,3-；（2）2246120x y x y +-+-= 【解析】 【分析】（1）联立直线AB 与直线BH 的方程可得点B 的坐标，由AC BH ⊥，进而设出直线AC 的方程，将C 的坐标代入得方程，再与直线AB 方程联立即可得点A 的坐标；（2）由（1）知A ，B ，C 的坐标，设ABC ∆外接圆的一般方程，代入求解即可.【详解】（1）由211320y x x y =-+⎧⎨++=⎩可得顶点(7,3)B -, 又因AC BH ⊥得，13BH k =- 所以设AC 的方程为3y x b =+，将(2,8)C -代入得14b =-由211314y x y x =-+⎧⎨=-⎩可得顶点为(5,1)A 所以A 和B 的坐标分别为(5,1)和(7,3)-（2）设ABC ∆的外接圆方程为220x y Dx Ey F ++++=，将(5,1)A 、(7,3)B -和(2,8)C -三点的坐标分别代入，得52607358028680D E F D E F D E F +++=⎧⎪-++=⎨⎪-++=⎩，解得4612D E F =-⎧⎪=⎨⎪=-⎩，所以ABC ∆的外接圆的一般方程为2246120x y x y +-+-=.【点睛】本题主要考查两直线交点的求法，待定系数法求圆的方程，属于基础题.21. 已知直线方程为()()221340m x m y m -++++=.（1）证明：直线恒过定点；（2）m 为何值时，点()3,4Q 到直线的距离最大，最大值为多少？（3）若直线分别与x 轴，y 轴的负半轴交于,A B 两点，求AOB 面积的最小值及此时直线的方程.【答案】（1）证明见解析（2）47=m ；213（3）最小值为4；此时直线的方程240x y ++= 【解析】【分析】（1）证明：利用直线是直线系求出直线恒过定点，即可；（2）点(3,4)Q 到直线的距离最大，转化为两点间的距离，求出距离就是最大值．（3）若直线分别与x 轴，y 轴的负半轴交于A ．B 两点，设出直线的方程，求出A ，B ，然后求出AOB ∆面积，利用基本不等式求出的最小值及此时直线的方程．【详解】（1）证明：直线方程为()()221340m x m y m -++++=，可化为()()24230x y m x y +++-++=，对任意m 都成立，所以230240x y x y -++=⎧⎨++=⎩，解得12x y =-⎧⎨=-⎩，所以直线恒过定点()1,2--；（2）解：点()3,4Q 到直线的距离最大，可知点Q 与定点()1,2P --的连线的距离就是所求最大值， ()()223142213+++=423312PQ k +==+， ()()221340m x m y m -++++=的斜率为23-， 可得22321m m --=-+，解得47=m . （3）解：若直线分别与x 轴，y 轴的负半轴交于,A B 两点，直线方程为()21y k x +=+，k 0<，则21,0A k ⎛⎫- ⎪⎝⎭，()0,2B k -， ()121222121222242222AOB k k S k k k k k k --⎛⎫⎛⎫=--=--=++≥+⋅= ⎪ ⎪--⎝⎭⎝⎭△，当且仅当2k =-时取等号，面积的最小值为4.此时直线的方程240x y ++=.【点睛】本题考查直线系过定点，零点的距离公式，基本不等式的应用，考查计算能力，转化思想，属于中档题．22. 如图所示的几何体P ABCDE -中，ABP △和AEP △均为以A 为直角顶点的等腰直角三角形，AB AE ⊥，//AB CE ，//AE CD ，24CD CE AB ===，M 为PD 的中点.（1）求证：CE PE ⊥；（2）求二面角M CE D --的大小；（3）设N 为线段PE 上的动点，使得平面//ABN 平面MCE ，求线段AN 的长.【答案】（1）证明见解析；（2）45︒；（32【解析】【分析】（1）根据题意，得出PA AB ⊥，PA AE ⊥，根据线面垂直的判定定理得出PA ⊥平面ABCDE ，则AB AE ⊥，建立以A 为原点，AB ，AE ，AP 为x ，y ，z 轴的空间直角坐标系，利用向量法能证明CE PE ⊥；（2）求出平面MEC 的法向量和平面DEC 的一个法向量，利用向量法能求出二面角M CE D --的大小；（3）设PN PE λ→→=，[[0λ∈，1])，求出(0N ，2λ，22)λ-，令AN n →→⊥，则0AN n →→=，解得N 为PE 的中点，利用向量法能求出线段AN 的长．【详解】解：依题意得，ABP △和AEP △均为以A 为直角顶点的等腰直角三角形， 则PA AB ⊥，PA AE ⊥，所以PA ⊥面ABCDE ，又AB AE ⊥，可以建立以A 为原点，分别以AB →，AE →，AP →的方向为x 轴，y 轴，z 轴正方向的空间直角坐标系（如图）， 可得()0,0,0A ，()2,0,0B ，()4,2,0C ，()4,6,0D ，()0,2,0E ，()002P ,,，()2,3,1M ，（1）证明：由题意，()4,0,0CE →=-，()0,2,2PE →=-，因为0CE PE →→⋅=，所以CE PE ⊥.（2）解：()2,1,1ME →=---，()2,1,1MC →=--，设(),,n x y z →=为平面MEC 的法向量，则 00n ME n MC ⎧⋅=⎨⋅=⎩，即2020x y z x y z ---=⎧⎨--=⎩， 不妨令1y =，可得()0,1,1n →=-，平面DEC 的一个法向量()0,0,2AP →=，因此有2cos ,2n AP n AP n AP →→→→→→⋅==-，由图可得二面角M CE D --为锐二面角，所以二面角M CE D --的大小为45︒.（3）解：（方法一）设[]()0,1PN PE λλ→→=∈，(),,N x y z ，所以()(),,20,2,2x y z λ-=-，因此()0,2,22N λλ-， 令AN n →→⊥，即0AN n →→⋅=，解得12λ=，即N 为PE 的中点， 因为//AB 平面MCE ，//AN 平面MCE ，AB AN A =，所以当N 为PE 的中点时，平面//ABN 平面MCE ，此时即()0,1,1N ，2220112AN →=++=所以线段AN 2.（方法二）设[]()0,1PN PE λλ→→=∈，(),,N x y z ， 所以()(),,20,2,2x y z λ-=-，因此()0,2,22N λλ-，设(),,m x y z →=为平面ABN 的法向量，则00m AB m AN ⎧⋅=⎨⋅=⎩，即()402220x y z λλ=⎧⎨+-=⎩， 不妨令1y λ=-，可得()0,1,m λλ→=-，因为平面//ABN 平面MCE ，所以//m n →→，解得：12λ=， 此时即()0,1,1N ，2220112AN →=++=所以线段AN 2. 【点睛】本题考查利用空间向量法证明线线垂直，以及利用空间向量法求出二面角和线段长，还涉及空间中线面的判定定理和性质，考查运算求解能力以及化归与转化思想，是中档题．13、2 14.22(1)(2)10x y +++=15：22(4)(2)10x y -+-=（去掉（3,5），（5，-1）两点）。