第4周晚练2必修1第三章

2021年高一英语下学期第4周当堂训练（满分100分，训练时间45分钟）班级_________ 学号_________ 姓名_________ 得分________一单项选择21．The girl devoted all her spare time she had others.A．to help B．helped C．to helping D．help22．I think a cold drink can you after your long hot journey.A．recover B．reward Crelieve D．refresh23．He was robbed last night, but he didn’t have much on him then.A．fortunately B．eventually CcertainlyD．surprisingly24．Was it when he was taking the driving test he ran into a pig in the street?A．where B．when C．that D．how25．The latest survey, by qq, found more than 12 percent of teenagers want to work as pilots.A．to be carried out B．being carried outC．carried out D．having been carried out26． you have got your license, why not drive the car yourself?A．Every since B．Now that C．Even though DAslong as27．With home – schooling growing quickly in the United States, nobody knows exactly how many American children at home.A．will teach B．taughtCare being taughtD．are teaching28．Canada has vast areas of wildness, from the Arctic north, average winter temperatures are usually minus 20 degrees centigrade,to the 8892 – kilometer – long borderthe USA in the southA．where; with B．which; /Cwhere;/ D．which; with29．The police suspected him of carrying drugs so they his bag, but found nothing.A．looked up B．turned in C．searched for Dwent through30．In the past the respect for teachers was driven by the belief they were reliable sources of knowledge.A．what Bthat Cwhy D．whether31．I would have helped you, but I really occupied the whole week.A．was B．am C．had been D．were32．We went to the Lincoln Memorial yesterday, , not surprisingly, was crowded with visitors from all over the world.A．where B．which Cthat D．when33．I waited ages for you –you have said that you weren’t ing!A．could B．would C．might D．must34．It is that she helped others with their lessons after school.A．worth to be praised B．worthy of being praisedC．worth praised D．worthy to praise35．The teacher spoke highly of such as honesty, courage and faithfulness shown by his students.A．deeds B．characteristics Cskills D．virtues二.完形填空In both xx and xx, Li Yong was elected as the Most Valuable TV Show Host. He has been at the 36 of the list for two years. He is considered “ 37 and intelligent(机智的)”. He not only laughs most but also laughs rather wildly in front of the camera. He moves freely between the stage and the 38 whenever he wants. He talks while he walks. He never falters (支吾地说) and he always says, “Don’t leave and we’ll be back 39 the advertisement”, at the 40 .41 a long face, a big mouth and small eyes, Li Yong is a sort of ugly. Someone said that the audience wouldn’t like him when he first appeared on the screen. But now he bees China’s most popular TV show host.“I wouldn’t have bee popular without CCTV. I don’t 42 myselfa celebrity(名人). I am just a host or a public 43 . I don’t fe el like a famous person. I don’t believe that my 44 will always be there.I know 45 I will fall on the stage, then I will walk away. I won’t have others 46 me away,’ he said.Maybe you don’t believe that Li Yong didn’t like to talk with others when he was young. He was born in Xinjiang. In his childhood, he loved 47 . He always went around the street with his palette(调色板) and painted everything he 48 . After 49 senior middle school, he became more and more active. He didn’t 50 hi s teachers’ orders. He always made much trouble. Teachers couldn’t believe he would entera college. But he made up his mind to 51 in entering municationUniversity of China. Now all the teachers 52 taught him say to their students proudly, “Li Yon g－my 53 .”Do you have a dream to be a TV host? Don’t worry about your 54 .Just try your best. Maybe one day you will also stand on the 55 of CCTV.36. A. foot B. top C.beginning D. end37. A. puzzled B. humorous C.handsome D. funny38. A. listeners B. readers C. viewer D. audience39. A. until B. after C. soon after D. behind40. A. camera B. performers C. actors D.other host41. A. Having B. growing C. Because of D. With42. A. think B. consider C. think about D. look43. A. figure B. person C. relation D. pressure44. A. popularity B. position C.impression D. voice45. A. one day B. all day C. another day D. the other day46. A. take B. push C. to take D. to pull47. A. singing B. dancing C. watching TV D. drawing48. A. heard B. touched C. saw D. observed49. A. leaving B. graduating C. entering D. admitting50. A. listen B. obey C. hear D. follow51. A. try B. succeed C. manage D. attempt52. A. what B. which C. who D. whom53. A. friend B. workmate C. proud D. student54. A. weight B. face C. height D. appearance55. A. studio B. stage C. cinema D. theatre24942 616E 慮35742 8B9E 讞@36828 8FDC 远21562 543A 吺h24241 5EB1 庱21285 5325 匥33533 82FD 苽36659 8F33 輳22986 59CA 姊s'Z21732 54E4 哤。

数学必修第二册课时作业(一)【原卷版】1．给出下列物理量：①质量；②速度；③位移；④力；⑤路程；⑥功；⑦加速度．其中是向量的有()A．4个B．5个C．6个D．7个2．【多选题】下列结论正确的是()A．若|a|＝|b|，则a＝b或a＝－bB．非零向量a与b平行，则a与b的方向相同或相反C．起点不同，但方向相同且模相等的向量是相等向量D．与非零向量a平行的单位向量有1个3．设O是△ABC的外心，则AO→，BO→，CO→是()A．相等向量B．模相等的向量C．平行向量D．起点相同的向量4．【多选题】如图，O是正方形ABCD的中心，则下列结论正确的是()A.AO→＝OC→B.AO→∥AC→C.AB→与CD→共线D.AO→＝BO→5．在△ABC中，AB＝AC，D，E分别是AB，AC的中点，则()A.AB→与AC→共线B.DE→与CB→共线C.AD→与AE→相等D.AD→与BD→相等6．四边形ABCD中，AB→＝2DC→，则四边形ABCD为()A．平行四边形B．矩形C．梯形D．菱形7．在坐标平面上，把所有单位向量的起点平移到坐标系的原点，则它们的终点所构成的图形是________．8．已知在边长为2的菱形ABCD中，∠ABC＝60°，则|BD→|＝________．9．某人向正东方向行进100m后，再向正南方向行进1003m，则此人位移的方向是________．10.如图，若四边形ABCD 为正方形，△BCE 为等腰直角三角形，则：(1)图中与AB →共线的向量有________________________________________________________________；(2)图中与AB →相等的向量有________；(3)图中与AB →的模相等的向量有_______________________________________；(4)图中与EC →相等的向量有________．11.如图，在平行四边形ABCD 中，E ，F 分别是AD ，BC 的中点，则以A ，B ，C ，D ，E ，F 这六个点中任意两点分别作为起点和终点的所有向量中，与向量EF →方向相反的向量是________．12．若A 地位于B 地正西方向5km 处，C 地位于A 地正北方向5km 处，则C 地相对于B 地的位移是________．13．一辆消防车从A 地去B 地执行任务，先从A 地向北偏东30°方向行驶2km 到达D 地，然后从D 地沿北偏东60°方向行驶6km 到达C 地，从C 地又向南偏西30°方向行驶2km 才到达B 地．(1)在图中画出向量AD →，DC →，CB →，AB →；(2)描述B 地相对于A 地的位置．14．如图，在四边形ABCD 中，已知M ，N 分别是BC ，AD 的中点，且AB →＝DC →，求证：CN 綉MA .15．中国象棋中规定：马走“日”字．下图是中国象棋的半个棋盘，若马在A 处，可跳到A 1处，也可跳到A 2处，用向量AA 1→或AA 2→表示马走了“一步”．试在图中画出马在B ，C 处走了“一步”的所有情况．1．O 是△ABC 内一点，若|OA →|＝|OB →|＝|OC →|，则O 是△ABC 的()A ．重心B ．内心C ．外心D ．垂心2．【多选题】下列命题中是真命题的是()A ．向量AB →∥CD →(AB →，CD →为非零向量)就是AB →所在的直线平行于CD →所在的直线B ．零向量与任一向量平行C ．相等向量一定是平行向量D ．平行向量一定是相等向量3.如图，已知四边形ABCD 是平行四边形，O 是两条对角线的交点，设点集M ＝{A ，B ，C ，D ，O }，向量集合T ＝{PQ →|P ∈M ，Q ∈M ，且P ，Q 不重合}，求集合T 中元素的个数．4．对于下列各种情况，各向量的终点的集合分别是什么图形？(1)把所有单位向量的起点平行移动到同一点P ；(2)把平行于直线l 的所有向量的起点平移到直线l 上的点P ；(3)把平行于直线l 的所有单位向量的起点平移到直线l 上的点P .5．民间流传的一种智力玩具七巧板是将一块正方形切割为五个等腰直角三角形和一个正方形、一个平行四边形，如图所示．试写出图中与FE →模相等的向量．6．指出下图中的平行向量和相等向量．7.如图所示，在四边形ABCD中，AB→＝DC→，N，M是AD，BC上的点，且CN→＝MA→.(1)求证：DN→＝MB→；(2)试写出图中与向量DN→共线的向量．数学必修第二册课时作业(一)【解析版】1．给出下列物理量：①质量；②速度；③位移；④力；⑤路程；⑥功；⑦加速度．其中是向量的有()A ．4个B ．5个C ．6个D ．7个答案A解析速度、位移、力、加速度这4个物理量是向量，它们都有大小和方向．2．【多选题】下列结论正确的是()A ．若|a |＝|b |，则a ＝b 或a ＝－bB ．非零向量a 与b 平行，则a 与b 的方向相同或相反C ．起点不同，但方向相同且模相等的向量是相等向量D ．与非零向量a 平行的单位向量有1个答案BC解析A 中两个向量未必共线；D 中与非零向量a 平行的单位向量有2个．3．设O 是△ABC 的外心，则AO →，BO →，CO →是()A ．相等向量B ．模相等的向量C ．平行向量D ．起点相同的向量答案B4．【多选题】如图，O 是正方形ABCD 的中心，则下列结论正确的是()A.AO →＝OC →B.AO →∥AC →C.AB →与CD →共线 D.AO →＝BO→答案ABC解析根据正方形的特征，结合相等向量，平行向量作出判断，只有D 是错误的，AO →与BO →只是模相等，由于方向不相同，所以不是相等向量．5．在△ABC 中，AB ＝AC ，D ，E 分别是AB ，AC 的中点，则()A.AB →与AC →共线 B.DE →与CB →共线C.AD →与AE →相等D.AD →与BD →相等答案B解析如图，因为D ，E 分别是AB ，AC 的中点，所以由三角形的中位线定理可得DE ∥BC .所以DE →与CB →共线．6．四边形ABCD 中，AB →＝2DC →，则四边形ABCD 为()A ．平行四边形B ．矩形C ．梯形D ．菱形答案C解析∵AB →＝2DC →，∴AB ∥DC 且AB ≠DC .∴四边形ABCD 为梯形．7．在坐标平面上，把所有单位向量的起点平移到坐标系的原点，则它们的终点所构成的图形是________．答案单位圆8．已知在边长为2的菱形ABCD 中，∠ABC ＝60°，则|BD →|＝________．答案23解析由题意知AC ⊥BD ，且∠ABD ＝30°，设AC ，BD 交点为O ，在Rt △ABO 中，|BO →|＝|AB →|·cos 30°＝2×32＝3，∴|BD →|＝2|BO →|＝23.9．某人向正东方向行进100m 后，再向正南方向行进1003m ，则此人位移的方向是________．答案南偏东30°解析如图所示，此人从点A 出发，经点B ，到达点C ，则tan ∠BAC ＝BC BA ＝1003100＝3，∵∠BAC 是三角形的内角，∴∠BAC ＝60°，即位移的方向是南偏东30°.10.如图，若四边形ABCD 为正方形，△BCE 为等腰直角三角形，则：(1)图中与AB →共线的向量有________________________________________________________________；(2)图中与AB →相等的向量有________；(3)图中与AB →的模相等的向量有_______________________________________；(4)图中与EC →相等的向量有________．答案(1)DC →，BE →，BA →，CD →，EB →，AE →，EA →(2)DC →，BE→(3)BA →，BE →，EB →，DC →，CD →，AD →，DA →，BC →，CB →(4)BD→11.如图，在平行四边形ABCD 中，E ，F 分别是AD ，BC 的中点，则以A ，B ，C ，D ，E ，F 这六个点中任意两点分别作为起点和终点的所有向量中，与向量EF →方向相反的向量是________．答案BA →，FE →，CD→解析由平行四边形的性质，可知AB 綉EF 綉DC .则与向量EF →方向相反的向量有BA →，FE →，CD →.12．若A 地位于B 地正西方向5km 处，C 地位于A 地正北方向5km 处，则C 地相对于B 地的位移是________．答案西北方向52km解析根据题意画出图形如图所示，由图可知|BC →|＝52km ，且∠ABC ＝45°，故C 地相对于B 地的位移是西北方向52km.13．一辆消防车从A 地去B 地执行任务，先从A 地向北偏东30°方向行驶2km 到达D 地，然后从D 地沿北偏东60°方向行驶6km 到达C 地，从C 地又向南偏西30°方向行驶2km才到达B 地．(1)在图中画出向量AD →，DC →，CB →，AB →；(2)描述B 地相对于A 地的位置．解析(1)作向量AD →，DC →，CB →，AB →如图所示．(2)由题意知AD →＝BC →，所以四边形ABCD 为平行四边形，所以AB →＝DC →，所以B 地相对于A 地的位置为“北偏东60°，相距6km ”．14．如图，在四边形ABCD 中，已知M ，N 分别是BC ，AD 的中点，且AB →＝DC →，求证：CN 綉MA .证明因为AB →＝DC →，所以AB ＝DC ，且AB ∥DC .所以四边形ABCD 是平行四边形．所以AD →＝BC →.又因为M ，N 分别是BC ，AD 的中点，所以AN ＝MC ，且AN ∥MC .所以四边形AMCN 是平行四边形．所以CN 綉MA .15．中国象棋中规定：马走“日”字．下图是中国象棋的半个棋盘，若马在A 处，可跳到A 1处，也可跳到A 2处，用向量AA 1→或AA 2→表示马走了“一步”．试在图中画出马在B ，C 处走了“一步”的所有情况．解析根据规则，作出符合要求的所有向量，如图．1．O 是△ABC 内一点，若|OA →|＝|OB →|＝|OC →|，则O 是△ABC 的()A ．重心B ．内心C ．外心D ．垂心答案C解析由条件知点O 到△ABC 三个顶点的距离相等，所以O 是△ABC 的外心．2．【多选题】下列命题中是真命题的是()A ．向量AB →∥CD →(AB →，CD →为非零向量)就是AB →所在的直线平行于CD →所在的直线B ．零向量与任一向量平行C ．相等向量一定是平行向量D ．平行向量一定是相等向量答案BC解析向量AB →∥CD →包含AB →所在的直线与CD →所在的直线平行和重合两种情况，故A 是假命题；零向量的方向是任意的，因此与任一向量平行，故B 是真命题；相等向量的方向相同，因此相等向量一定是平行向量，故C 是真命题；平行向量的长度不一定相同，即使长度相同，方向也有可能相反，因此平行向量不一定是相等向量，故D 是假命题．3.如图，已知四边形ABCD 是平行四边形，O 是两条对角线的交点，设点集M ＝{A ，B ，C ，D ，O }，向量集合T ＝{PQ →|P ∈M ，Q ∈M ，且P ，Q 不重合}，求集合T 中元素的个数．解析从模和方向两个角度考虑，以下向量是互不相等的向量：AB →，BA →，AD →，DA →，AO →，OA →，AC →，CA →，BO →，OB →，BD →，DB →，其他向量都与它们中的某一个相等，故集合T 中有12个元素．4．对于下列各种情况，各向量的终点的集合分别是什么图形？(1)把所有单位向量的起点平行移动到同一点P ；(2)把平行于直线l 的所有向量的起点平移到直线l 上的点P ；(3)把平行于直线l 的所有单位向量的起点平移到直线l 上的点P .解析(1)是以P 点为圆心，以1个单位长度为半径的圆．(2)是直线l .(3)是直线l 上与点P 的距离为1个单位长度的两个点．5．民间流传的一种智力玩具七巧板是将一块正方形切割为五个等腰直角三角形和一个正方形、一个平行四边形，如图所示．试写出图中与FE →模相等的向量．解析与FE →模相等的向量有EF →，DO →，OD →，GH →，HG →，OB →，BO →，AO →，OA →，共9个．6．指出下图中的平行向量和相等向量．解析平行向量有CD →∥AB →∥IJ →∥MN →，EF →∥GH →∥KL →.相等向量有CD →＝MN →，AB →＝IJ →.7.如图所示，在四边形ABCD 中，AB →＝DC →，N ，M 是AD ，BC 上的点，且CN →＝MA →.(1)求证：DN →＝MB →；(2)试写出图中与向量DN →共线的向量．解析(1)证明：因为AB →＝DC →，所以|AB →|＝|DC →|，且AB ∥CD .因此四边形ABCD 是平行四边形，所以|DA →|＝|CB →|，且DA ∥CB .同理，由CN →＝MA →，可证四边形CNAM 是平行四边形，所以CM →＝NA →.所以|MB →|＝|DN →|，即DN →与MB →的模相等，又DN →与MB →的方向相同，故DN →＝MB →.(2)图中与向量DN →共线的向量有NA →，AN →，ND →，CM →，MC →，MB →，BM →，CB →，BC →，DA →，AD →.。

一、根据首字母填写单词(单词拼写)1. The country’s o_______________ language is Spanish. (根据首字母单词拼写)2. Cultural e________ is a good way of building bridge between nations. （根据首字母单词拼写）3. There are six o________ languages of the UN: Arabic, Chinese, English, French, Russian and Spanish. (根据首字母单词拼写)二、根据汉语意思填写单词(单词拼写)4. He is surprised to learn that she speaks ________ (流利的) German, though they have only written each other in English.(根据汉语提示单词拼写)5. The price of the house is so high that it is well beyond our _______________(范围).(根据汉语提示单词拼写)6. Life in the ocean __________ (范围) from the tiniest plankton all the way up to giants like sharks and whales.三、根据中英文提示填写单词(单词拼写)7. All five kids, r________(范围是) in age from 10 to 19, were in this wedding. （根据中英文提示填空）8. The price of the house is so high that it is well beyond our r______ (范围). (根据中英文提示填空)四、完成句子9. 我知道中文是一门难学的语言，但我希望在毕业时能说一口流利的汉语。

课时练(三) Discovering Useful Structures基础知识夯实进阶训练第一层Ⅰ.单句语法填空1．At 7：00 tomorrow evening I ________ (watch) the news programme on TV.2．Tom has been dreaming of becoming a banker and I believe he ________ (study) finance in university at the age of 20.3．—I have heard Mr Morgan will be here at 4：00 pm next Thursday.—No, he ________ (board) at that time.4．—How about going sightseeing this Saturday afternoon?—Sorry, I ________ (write) my research report the whole weekend.5．—Is it convenient for you if I call you up at 9 o'clock tomorrow morning?—I'm afraid not. I ________ (attend) a meeting then.6．—Six o'clock in the morning?—No, don't call me that early in the morning. I ________ (sleep)．7．—The president will come to attend your lecture at 10：00 tomorrow.—I'm sorry, by then my lecture will have just ended and I ________ (meet) my fans in my office.8．—Can I pay a visit to you this Friday afternoon, manager?—Sorry, I ________ (meet) some American business partners the whole afternoon.9．—Shall we meet at the school gate at 7：30 tomorrow morning?—Sorry, I ________ (have) my breakfast then. What about 8：00?10．—Hey, look where you are going!—Oh, New Year is coming, and I am wondering what we ________ (do) this time next Sunday.Ⅱ.根据汉语完成句子1．经理已经告诉小王去江苏省徐州市出差，当他的火车进站时，在那里将会有人等他。

Unit 3 ComputersPeriod One Warming Up & ReadingⅠ.单句填空1.It is such a hard problem that it is impossible for me (solve).2.Many people think highly of him, but (person), I think he is actually dishonest.3.Time passed quickly and three weeks went by we knew it.4. Music is a (universe) language which can bring people pleasure.5.The house needed (total) rebuilding, so we decided against buying it.6. You need to set goals in order to turn your dreams into (real).7.With the (apply) of the new technology,the output of steel almost doubled what it was last year.8.English in this story (simplify) several times to make it easier for children to understand.9. time goes by, we come to understand why the teachers are so strict with us.10.She was so weak she was out of breath after walking for only a few minutes.1.It taught me a lesson. I made a determination that I would never put off anything important until the next day.2.Tom hurt his leg when he was running. , he has to leave school for a few days.3.Small as the factory is, its products are .4.I am disagreement with you as to the value of your plan.5.We parents should make our children learn to toys each other.6.He never wasted any money, so as the years , he saved quite a bit of money.7.Although we have met by chance, we do .8.In theory, this problem should not arise, but it does.9.She had a worker for five years before she became a teacher.10.All I thought was that the programme was interesting and could me the knowledge I wanted. Ⅲ.句型训练1.Much to my disappointment, by the time I got to the stadium, the match .(finish)令我非常失望的是,当我到达体育馆时,比赛已经结束了。

课时作业(二) Discovering Useful Structures & Listeningand TalkingⅠ.单词拼写1．It is very important to know first aid kit in case of e________．2．The government promised the public to increase the ________ (供应量) of green products in the market.3．Having a good sense of smell, dogs are often used to search for those s________ in an earthquake.4．People were swimming in the ocean despite the ________ (飓风) warning.5．When seeing we were trapped, one of the rescue workers came to our a ________．6．The judge finally blew the w________， which meant the game came to a stop.7．The ________ (火山) erupted last year killing about 600 people.8．Sara is usually a c________ (沉着的) girl, but she lost it after being laughed at in public.Ⅱ.单句语法填空1．With the more strict competition regulations, there is almost not ________ black whistle in the event.2．The workers went ________ strike, asking their boss for more pay and better working conditions.3．There's no doubt that China is on the way to becoming a ________ (power) country in the world.4．It is reported that the volcano in Japan is showing signs of ________ (erupt) in recent years.5．Because of the recent frequent earthquakes, people have to sleep ________ the open air.6．To our great joy, a rescue worker heard the last survivor ________ (tap) his fingers on the pipe.7．One of the water ________ (pipe) burst and the entire apartment was flooded.Ⅲ.单元语法练习用适当的关系代词填入下列各句1．The old man ________ we visited yesterday is a famous artist.2．Miss Wang is taking care of the child ________ parents have gone to Beijing .3．I live in the room ________ windows face south.4．I'll never forget the days ________ I spent in the countryside.5．The man ________ my father shook hands with just now is our headmaster.6．This is one of the most interesting films ________ I have ever seen.7．He is the only one of the students who ________ (be) a winner of scholarship for three years.8．—Why does she always ask you for help?—There is no one else ________ she can turn to, is there?9．Will you please show me the way to the only tall building that ________ (stand) near the post office?10．The comrade I want to learn from is the one ________ studies hard and works hard.Ⅳ.完成句子1．那个人的汽车抛锚了，他们都跑过去帮忙。

北师大版高中数学必修2全册课时练习第一章《立体几何初步》简单旋转体1．给出以下说法：①圆台的上底面缩小为一点时(下底面不变)，圆台就变成了圆锥；②球面就是球；③过空间四点总能作一个球．其中正确说法的个数是( )A．0 B．1 C．2 D．3答案 B解析根据圆锥和圆台的形状之间的联系可知①正确；球面是曲面，球是球体的简称，是实心的几何体，故②不正确；当空间四点在同一条直线上时，过这四点不能作球，故③不正确．2．如图阴影部分，绕中间轴旋转一周，形成的几何体形状为( )A．一个球体B．一个球体中间挖去一个圆柱C．一个圆柱D．一个球体中间挖去一个棱柱答案 B解析按旋转体的定义得到几何体B.3．有下列三个命题：①圆柱是将矩形旋转一周所得的几何体；②圆台的任意两条母线的延长线，可能相交也可能不相交；③圆锥的轴截面是等腰三角形．其中错误命题的个数是( )A．0 B．1 C．2 D．3答案 C解析①将矩形的一边作为旋转轴旋转一周得到的几何体是圆柱．②圆台的两条母线的延长线必相交，故①②错误，③是正确的．4．如图所示的几何体由一个圆柱挖去一个以圆柱的上底面为底面，下底面圆心为顶点的圆锥而得到的．现用一个竖直的平面去截这个几何体，则所截得的图形可能是( )A．(1)(2) B．(1)(3) C．(1)(4) D．(1)(5)答案 D解析轴截面为(1)，平行于圆锥轴截面的截面是(5)．5．下列命题中，错误的是( )A．圆柱的轴截面是过母线的截面中面积最大的一个B．圆锥的轴截面是所有过顶点的截面中面积最大的一个C．圆台的所有平行于底面的截面都是圆面D．圆锥所有的轴截面都是全等的等腰三角形答案 B解析当圆锥的截面顶角大于90°时，面积不是最大．6．圆锥被平行于底面的平面所截，若截面面积与底面面积之比为1∶2，则此圆锥的高被分成的两段之比为( )A．1∶2 B．1∶4C．1∶(2＋1) D．1∶(2－1)答案 D解析根据相似性，若截面面积与底面面积之比为1∶2，则对应小圆锥与原圆锥高之比为1∶2，那么圆锥的高被截面分成的两段之比为1∶(2－1)．7．一个正方体内有一个内切球，作正方体的对角面，所得截面图形是下图中的( )答案 B解析由组合体的结构特征知，球只与正方体的六个面相切，而与两侧棱相离，故正确答案为B.8．将等边三角形绕它的一条中线旋转180°，形成的几何体是________．答案圆锥解析 由旋转体的概念可知，得到的几何体是圆锥．9．圆台两底面半径分别是2 cm 和5 cm ，母线长是310 cm ，则它的轴截面的面积是________．答案 63 cm 2解析 画出轴截面，如图，过A 作AM ⊥BC 于M ，则BM ＝5－2＝3(cm)，AM ＝AB 2－BM 2＝9(cm)，∴S 四边形ABCD ＝＋2＝63(cm 2)．10．如图所示的四个几何体中，哪些是圆柱与圆锥，哪些不是，并指出圆柱与圆锥的结构名称．解 ②是圆锥，圆面AOB 是圆锥的底面，SO 是圆锥的高，SA ，SB 是圆锥的母线． ③是圆柱，圆面A ′O ′B ′和圆面AOB 分别为上、下底面，O ′O 为圆柱的高，A ′A 与B ′B 为圆柱的母线．①不是圆柱，④不是圆锥．简单多面体1．五棱柱中，不同在任何侧面且不同在任何底面的两顶点的连线称为它的对角线，那么一个五棱柱对角线的条数共有( )A ．20B ．15C ．12D ．10 答案 D解析 如图，在五棱柱ABCDE －A 1B 1C 1D 1E 1中，从顶点A 出发的对角线有两条：AC 1，AD 1，同理从B ，C ，D ，E 点出发的对角线均有两条，共2×5＝10(条)．2．有两个面平行的多面体不可能是( )A．棱柱 B．棱锥C．棱台 D．长方体答案 B解析棱锥的各面都相交，故有两个面平行的多面体不可能是棱锥．3．如图，将装有水的长方体水槽固定底面一边后倾斜一个小角度，则倾斜后水槽中的水形成的几何体是( )A．棱柱B．棱台C．棱柱与棱锥的组合体D．不能确定答案 A解析形成的几何体前后两个面互相平行，其余各面都是四边形，并且每相邻两个四边形的公共边都互相平行，符合棱柱的定义．4．若棱台上、下底面的对应边之比为1∶2，则上、下底面的面积之比是( )A．1∶2 B．1∶4C．2∶1 D．4∶1答案 B解析由棱台的结构特征知，棱台上、下底面是相似多边形，面积比为对应边之比的平方，故选B.5．某同学制作了一个对面图案相同的正方体礼品盒(如下图1)，则这个正方体礼品盒的表面展开图应该为( )答案 A 解析 两个不能相邻，B 、D 错误；两个不能相邻，C 错误，故选A.也可通过制作模型来判断．6．如下图所示，在三棱台A ′B ′C ′－ABC 中，截去三棱锥A ′－ABC 后，剩余部分是( )A ．三棱锥B ．四棱锥C ．三棱柱D ．三棱台 答案 B解析 剩余部分是四棱锥A ′－BB ′C ′C .7．若一个正棱锥有6个顶点，所有侧棱长的和为20 cm ，则每条侧棱的长为________cm. 答案 4解析 依题意，正棱锥有6个顶点，则该正棱锥为正五棱锥，所以每条侧棱长为205＝4 cm.8．在下面的四个平面图形中，属于侧棱都相等的四面体的展开图的是________(填序号)．答案①②解析③④中的图不能组成四面体，只有①②行．9．一个正方体内接于一个球，过球心作一截面，则截面的可能图形有________．答案①②③解析当截面平行于正方体的一个侧面时得③，当截面过正方体的对角线时得②，当截面不平行于任何侧面也不过对角线时得①，但无论如何都不能截出④.10．已知长方体ABCD－A1B1C1D1(如下图所示)．(1)这个长方体是棱柱吗？如果是，是几棱柱？为什么？(2)用截面BCFE把这个长方体分成两部分后，各部分形成的几何体还是棱柱吗？如果是，是几棱柱？如果不是，请说明理由．解(1)是棱柱，并且是四棱柱．因为以长方体相对的两个面作底面，这两个面都是四边形且平行，其余各面都是矩形，当然是平行四边形，并且四条侧棱互相平行．(2)截面BCFE上方部分是棱柱，且是三棱柱BEB1－CFC1，其中△BEB1和△CFC1是底面．截面BCFE下方部分也是棱柱，且是四棱柱ABEA1－DCFD1，其中四边形ABEA1和四边形DCFD1是底面．2 直观图1．关于斜二测画法的叙述，其中正确的个数为( ) (1)两条相交直线的直观图可能是平行直线； (2)两条互相垂直的直线的直观图仍然垂直； (3)正方形的直观图可能是梯形； (4)平行四边形的直观图是平行四边形； (5)相等线段的直观图仍然相等． A ．1 B ．2 C ．3 D ．4 答案 A解析 由于斜二测画法保共点性，所以(1)错；保平行性，所以(3)错，(4)对；原来垂直的两线段，在直观图中夹角为45°，所以(2)错；与y 轴平行的线段长度变为原来的一半，所以(5)错．2．如下图建立坐标系，得到的正三角形ABC 的直观图不是全等三角形的一组是( )答案 C解析 在A 、B 、D 中，三角形ABC 的直观图的底面边长和高均相等，它们是全等的，只有C 不全等．3．已知正三角形ABC 的边长为a ，那么△ABC 的平面直观图△A ′B ′C ′的面积为( ) A.34a 2 B.38a 2 C.68a 2 D.616a 2 答案 D解析 先根据题意，画出直观图，然后根据直观图△A ′B ′C ′的边长及夹角求解．图(2)所示为实际图形的直观图，由(2)可知，A ′B ′＝AB ＝a ，O ′C ′＝12OC ＝34a ，在图(2)中作C ′D ′⊥A ′B ′于D ′，则C ′D ′＝22O ′C ′＝68a . ∴S △A ′B ′C ′＝12A ′B ′·C ′D ′＝12×a ×68a ＝616a 2.4．如下图，用斜二测画法画一个水平放置的平面图形的直观图为一个正方形，则原来图形的形状是( )答案 A解析 直观图边长为1，对角线为2，则原图形中对应的对角线为2 2.故选A.5．如图所示是水平放置的正方形ABCO ，在平面直角坐标系xOy 中，点B 的坐标为(4,4)，则由斜二测画法画出的正方形的直观图中，顶点B ′到x ′轴的距离为( )A. 2B.22C ．2 2D ．2 答案 A解析 由斜二测画法规则画出直观图如图所示，作B′E⊥x′轴于点E，在Rt△B′C′E中，B′C′＝2，∠B′C′E＝45°，B′E＝B′C′sin45°＝2×22＝ 2.6．如下图，矩形O′A′B′C′是水平放置的一个平面图形的直观图，其中O′A′＝6 cm，O′C′＝2 cm，则原图形是( )A．正方形B．矩形C．菱形D．一般的平行四边形答案 C解析如图，在原图形OABC中，OD＝2O′D′＝2×22＝4 2 cm，CD＝C′D′＝2 cm.∴OC＝OD2＋CD2＝22＋22＝6 cm，∴OA＝OC，故四边形OABC是菱形．7．如图，正方形O′A′B′C′的边长为1 cm，它是水平放置的一个平面图形的直观图，则原图的周长是( )A．8 cm B．6 cmC．2(1＋3) cm D．2(1＋2) cm答案 A解析根据直观图的画法，原几何图形如图所示，四边形OABC为平行四边形，OB＝22，OA＝1，AB＝3，从而原图周长为8 cm.8．若一个三角形，采用斜二测画法作出其直观图，则其直观图的面积是原三角形面积的________倍．答案24解析 从这个三角形的一边所在的直线为x 轴建立坐标系，则在直观图中，该边边长不变，高变为原来的24倍． 9．如图所示，四边形ABCD 是一平面图形的水平放置的斜二测直观图．在斜二测直观图中，ABCD 是一直角梯形，AB ∥CD ，AD ⊥CD ，且BC 与y ′轴平行．若AB ＝6，CD ＝4，AD ＝2，则这个平面图形的实际面积是________．答案 20 2解析 由斜二测直观图作图规则知，该平面图形是梯形，且AB 与CD 的长度不变，仍为6和4，高为42，故面积为20 2.10．已知直角梯形ABCD 中，AD ＝22，AB ＝3，CD ＝1，用斜二测画法画出其直观图如图所示，求直观图中的梯形A ′B ′C ′D ′的周长．解 由斜二测画法可知，A ′D ′＝12AD ＝2，A ′B ′＝AB ＝3，C ′D ′＝CD ＝1.在直观图中，如图，过D ′作D ′E ′⊥A ′B ′于E ′， 过C ′作C ′F ′⊥A ′B ′于F ′.∵∠D ′A ′E ′＝45°，∴C′F′＝D′E′＝A′E′＝2×sin45°＝2×22＝1，∴F′B′＝3－1－1＝1，∴B′C′＝12＋12＝2，故梯形A′B′C′D′的周长为4＋2 2.三视图1．以下说法错误的是( )A．三视图相同的几何体只有球B．直立圆锥的主视图与左视图都是等腰三角形，俯视图是圆和圆心C．直立圆柱的主视图与左视图都是矩形，俯视图是圆D．长方体的三视图都是矩形，正方体的三视图都是正方形(有一面正对观察者)答案 A解析选项A中错在“只有”这两个字上，例如正方体的三视图可以都为正方形；根据圆锥、圆柱、长方体、正方体的几何特征易知B、C、D均正确．故选A.2．下列选项是正六棱柱的三视图，其中画法正确的是( )答案 A解析主视图的矩形中应有两条实线，左视图应为两个全等的矩形且中间为实线．故选A.3．如图所示，下列几何体各自的三视图(阴影面为主视面)中，有且仅有两个视图相同的是( )A．①② B．①③ C．①④ D．②④答案 D解析在各自的三视图中，①正方体的三个视图都相同；②圆锥有两个视图相同；③三棱台的三个视图都不同；④正四棱锥有两个视图相同．4．请根据图中三视图，想象物体的形状，用小正方块搭出这个物体，并数一数有多少个小正方块( )A．7 B．6 C．8或10 D．9或10答案 D解析物体的立体图如图所示，由9个或10个小正方块搭成．5．已知三棱锥的俯视图与左视图如下图所示，俯视图是边长为2的正三角形，左视图是有一条直角边为2的直角三角形，则该三棱锥的主视图可能为( )答案 C解析由题设条件知，该三棱锥的直观图可能如图所示，其底面ABC为正三角形，侧棱PC垂直于底面，在主视图中，PA的投影是虚线．故选C.6．若一个正三棱柱(底面为正三角形，侧面为矩形的棱柱)的三视图如下图所示，则这个正三棱柱的侧棱长和底面边长分别为( )A．2,2 3 B．22， 2C．4,2 D．2,4答案 D解析从三视图可以看出，底面三角形的高为23，侧棱长为2，∴底面边长为4.7．某几何体的主视图与左视图均为边长为1的正方形，则下面四个图形中，可能是该几何体俯视图的个数为( )A．1 B．2 C．3 D．4答案 C解析俯视图从左到右依次记为：如果几何体为棱长为1的正方体，则俯视图如图①；如果几何体为圆柱，它的底面直径为1，高为1，则俯视图如图④；如果几何体为从棱长为1的正方体中挖去直径为2，高为1的圆柱的14，则俯视图如图②；以图③为俯视图的几何体的正视图不是正方形．故选C.8．如图所示，正三棱柱ABC －A 1B 1C 1的主视图是边长为4的正方形，则此正三棱柱的左视图的面积为________．答案 8 3解析 由主视图可知三棱柱的高为4，底面边长为4，所以底面正三角形的高为23，所以左视图的面积为4×23＝8 3.9．如图，正方体ABCD －A 1B 1C 1D 1中，E 、F 分别是AA 1、D 1C 1的中点，G 是正方形BCC 1B 1的中心，则空间四边形AEFG 在该正方体各面上的正投影不可能是下图中的________．答案 (2)解析 四边形在面ABCD 与面A 1B 1C 1D 1的投影为(1)；在面AA 1B 1B 与面DD 1C 1C 的投影为(3)；在面ADD 1A 1与面BCC 1B 1的投影为(4)．10．如图，物体的三视图有无错误？如果有，请指出并改正．解主视图正确，左视图和俯视图错误，正确的画法如图所示．空间图形基本关系的认识空间图形的公理(一)1．如果空间四点A，B，C，D不共面，那么下列判断中正确的是( )A．A，B，C，D四点中必有三点共线B．A，B，C，D四点中不存在三点共线C．直线AB与CD相交D．直线AB与CD平行答案 B解析若A，B，C，D四点中有三点共线，则A，B，C，D四点共面；若AB与CD相交(或平行)，则AB与CD共面，即得A，B，C，D四点共面．故选B.2．若点A∈平面α，点B∈平面α，点C∈直线AB，则( )A．C∈αB．C∉αC．AB⊆/αD．AB∩α＝C答案 A解析因为点A∈平面α，点B∈平面α，所以ABα.又点C∈直线AB，所以C∈α.3．如图所示，用符号语言可表示为( )A．α∩β＝m，nα，m∩n＝AB．α∩β＝m，n∈α，m∩n＝AC．α∩β＝m，nα，A m，A nD．α∩β＝m，n∈α，A∈m，A∈n答案 A解析很明显，α与β交于m，n在α内，m与n交于A，故选A.4．如图，平面α∩平面β＝l，点A∈α，点B∈α，且点C∈β，点C∉l.又AB∩l＝R，设A，B，C三点确定的平面为γ，则β∩γ是( )A．直线AC B．直线BCC．直线CR D．直线AR答案 C解析∵C∈平面ABC，AB平面ABC，而R∈AB，∴R∈平面ABC，而C∈β，lβ，R ∈l，∴R∈β，∴点C，点R为两平面ABC与β的公共点，∴β∩γ＝CR.5．在四面体ABCD的棱AB，BC，CD，DA上分别取E，F，G，H四点，如果EF与HG交于点M，则( )A．M一定在直线AC上B．M一定在直线BD上C．M可能在AC上，也可能在BD上D．M不在AC上，也不在BD上答案 A解析因为E，F，G，H分别是四面体ABCD的棱AB，BC，CD，DA上的点，EF与HG交于点M，所以点M为平面ABC与平面ACD的公共点，而两个平面的交线为AC，所以M一定在直线AC上．6．在正方体ABCD－A1B1C1D1中，E、F分别为棱AA1，CC1的中点，则在空间中与三条直线A1D1、EF、CD都相交的直线( )A．不存在B．有且只有两条C．有且只有三条D．有无数条答案 D解析如下图：在直线CD上任取一点H，则直线A1D1与点H确定一平面A1D1HG.显然EF与平面A1D1HG有公共点O且A1D1∥HG.又O∉HG.连接HO并延长，则一定与直线A1D1相交．由于点H有无数个，所以与A1D1、EF、CD都相交的直线有无数条．7．如图，在这个正方体中，①BM与ED平行；②CN与BM是异面直线；③CN与BE是异面直线；④DN与BM是异面直线．以上四个命题中，正确命题的序号是________．答案②④解析观察图形可知①③错误，②④正确．8．有下面几个说法：①如果一条线段的中点在一个平面内，那么它的两个端点也在这个平面内；②两组对边分别相等的四边形是平行四边形；③两组对边分别平行的四边形是平行四边形；④四边形有三条边在同一平面内，则第四条边也在这个平面内；⑤点A在平面α外，点A和平面α内的任意一条直线都不共面．其中正确的序号是________(把你认为正确的序号都填上)．答案③④解析①中线段可与平面α相交；②中的四边形可以是空间四边形；③中平行的对边能确定平面，所以是平行四边形；④中三边在同一平面内，可推知第四条边的两个端点也在这个平面内，所以第四条边在这个平面内；⑤中点A与α内的任意直线都能确定一个平面．9．已知α，β为两个不同的平面，A，B，M，N为四个不同的点，a为直线，下列推理错误的是________(填序号)．①A ∈a ，B ∈a ，A ∈β，B ∈β⇒a β； ②M ∈α，M ∈β，N ∈α，N ∈β⇒α∩β＝MN ； ③A ∈α，A ∈β⇒α∩β＝A . 答案 ③解析 ∵A ∈α，A ∈β，∴A ∈α∩β，由公理3知α∩β为经过点A 的一条直线而不是一个点A ，故③错误．故填③.10．如下图，四面体ABCD 中，E 、G 分别为BC 、AB 的中点，F 在CD 上，H 在AD 上，且有DF ∶FC ＝2∶3，DH ∶HA ＝2∶3.求证：EF 、GH 、BD 交于一点．证明 如图所示，连接GE 、HF ，∵E 、G 分别为BC 、AB 的中点， ∴GE ∥AC ，GE ＝12AC .又∵DF ∶FC ＝2∶3，DH ∶HA ＝2∶3， ∴HF ∥AC ，HF ＝25AC ，∴GE ∥HF ，GE >HF . ∴G 、E 、F 、H 四点共面． ∴EF 与GH 相交，设交点为O .则O ∈平面ABD ∩平面BCD ，而平面ABD ∩平面BCD ＝BD ， ∴O ∈BD .即EF 、GH 、BD 交于一点．空间图形的公理(二)1．若直线a∥b，b∩c＝A，则a与c的位置关系是( )A．异面 B．相交C．平行 D．异面或相交答案 D解析a与c不可能平行，若a∥c，又因为a∥b，所以b∥c，这与b∩c＝A矛盾，而a与c异面、相交都有可能．2．如图所示，在三棱锥P－ABC的六条棱所在的直线中，异面直线共有( )A．2对 B．3对C．4对 D．6对答案 B解析据异面直线的定义可知共有3对．AP与BC，CP与AB，BP与AC.3．如图所示，在长方体木块ABCD－A1B1C1D1中，E，F分别是B1O和C1O的中点，则长方体的各棱中与EF平行的有( )A．3条 B．4条 C．5条 D．6条答案 B解析由于E、F分别是B1O、C1O的中点，故EF∥B1C1，因为和棱B1C1平行的棱还有3条：AD、BC、A1D1，所以共有4条．4．异面直线a，b，有aα，bβ且α∩β＝c，则直线c与a，b的关系是( ) A．c与a，b都相交B．c与a，b都不相交C．c至多与a，b中的一条相交D．c至少与a，b中的一条相交答案 D解析若c与a、b都不相交，∵c与a在α内，∴a∥c.又c 与b 都在β内，∴b ∥c .由基本性质4，可知a ∥b ，与已知条件矛盾． 如图，只有以下三种情况．故直线c 至少与a ，b 中的一条相交．5．已知E ，F ，G ，H 分别为空间四边形ABCD 的各边AB ，BC ，CD ，DA 的中点，若对角线BD ＝2，AC ＝4，则EG 2＋HF 2的值是(平行四边形的对角线的平方和等于四条边的平方和)( )A ．5B ．10C ．12D ．不能确定 答案 B解析 如图所示，由三角形中位线的性质可得EH 綊12BD ，FG 綊12BD ，再根据公理4可得四边形EFGH 是平行四边形，那么所求的是平行四边形的对角线的平方和，所以EG 2＋HF 2＝2×(12＋22)＝10.6．如图所示的是正三棱锥的展开图(D ，E 分别为PB ，PA 的中点)，则在正三棱锥中，下列说法正确的是( )A ．直线DE 与直线AF 相交成60°角B ．直线DE 与直线AC 相交 C ．直线DE 与直线AB 异面D ．直线AF 与直线BC 平行 答案 A解析 将题中的展开图还原成正三棱锥，如图所示，点F 与点P 重合，易知在△PDE 中，PD ＝PE ＝DE ，△PDE 是等边三角形，故∠PED ＝60°，即直线DE 与AF 相交成60°角，A 项正确．由图易知其余选项均错误．7．如图所示，在三棱锥A －BCD 中，M ，N 分别为AB ，CD 的中点，则下列结论正确的是( )A ．MN ≥12(AC ＋BD )B ．MN ≤12(AC ＋BD )C ．MN ＝12(AC ＋BD )D ．MN <12(AC ＋BD )答案 D解析 如图所示，取BC 的中点E ，连接ME ，NE ，则ME ＝12AC ，NE ＝12BD ，所以ME ＋NE ＝12(AC ＋BD )．在△MNE 中，有ME ＋NE >MN ，所以MN <12(AC ＋BD )．8．如图，在正方体ABCD －A 1B 1C 1D 1中，BD 和B 1D 1是正方形ABCD 和A 1B 1C 1D 1的对角线，(1)∠DBC 的两边与________的两边分别平行且方向相同； (2)∠DBC 的两边与________的两边分别平行且方向相反． 答案 (1)∠D 1B 1C 1 (2)∠B 1D 1A 1解析 (1)B 1D 1∥BD ，B 1C 1∥BC 并且方向相同，所以∠DBC 的两边与∠D 1B 1C 1的两边分别平行且方向相同；(2)B 1D 1∥BD ，D 1A 1∥BC 且方向相反，所以∠DBC 的两边与∠B 1D 1A 1的两边分别平行且方向相反．9．如图，正方体ABCD －A 1B 1C 1D 1中，M ，N 分别是棱C 1D 1，C 1C 的中点．有以下四个结论：①直线AM 与CC 1是相交直线 ②直线AM 与BN 是平行直线 ③直线BN 与MB 1是异面直线 ④直线AM 与DD 1是异面直线其中正确的结论为________(注：把你认为正确结论的序号都填上)． 答案 ③④解析 由异面直线的定义知③④正确．10．如图，设E ，F ，G ，H 依次是空间四边形ABCD 的边AB ，BC ，CD ，DA 上的点，且AEAB＝AH AD ＝λ，CF CB ＝CGCD＝μ.(1)当λ＝μ时，求证：四边形EFGH 是平行四边形；(2)当λ≠μ时，求证：①四边形EFGH 是梯形；②三条直线EF ，HG ，AC 交于一点． 证明 在△ABD 中，AE AB ＝AH AD＝λ， 故EH 綊λBD .同理FG 綊μBD . 由公理4得EH ∥FG ，又可得FG ＝μλEH .(1)若λ＝μ，则FG ＝EH ，故EFGH 是平行四边形． (2)①若λ≠μ，则EH ≠FG ，故EFGH 是梯形．②若λ≠μ，则EH ≠FG ，则在平面EFGH 中EF 、HG 不平行，必然相交． 不妨设λ>μ，EF ∩HG ＝O ，如图所示． 由O ∈EF ，EF 平面ABC ，得O ∈平面ABC . 同理有O ∈HG 平面ACD .而平面ABC ∩平面ACD ＝AC ，所以O ∈AC ， 即EF 、HG 、AC 交于点O .平行关系的判定1．已知两条相交直线a ，b ，a ∥α，则b 与平面α的位置关系是( ) A ．b ∥α B ．b 与α相交 C ．b α D ．b ∥α或b 与α相交答案 D解析 ∵a ，b 相交，∴a ，b 确定一个平面β，如果β∥α，则b ∥α，如果β不平行于α，则b 与α相交．2．不同直线m 、n 和不同平面α、β，给出下列命题：其中错误的有( )A．0个 B．1个 C．2个 D．3个答案 D解析由面面平行与线面平行的定义知：①是正确的．对于②，n可能在平面β内．对于③，在正方体ABCD－A1B1C1D1中，如图，AA1平面ADD1A1，CC1平面CDD1C1，而AA1∥C1C，从而A1A与CC1可确定一个平面AA1C1C.即AA1，C1C可以共面．对于④，m可能在平面β内．故②③④错，选D.3．如图，在四面体ABCD中，若M，N，P分别为线段AB，BC，CD的中点，则直线BD 与平面MNP的位置关系为( )A．平行B．可能相交C．相交或BD平面MNP D．以上都不对答案 A解析因为N，P分别为BC，CD的中点．∴NP∥BD.又NP平面MNP，BD⊆/平面MNP，∴BD∥平面MNP.4．平面α与△ABC的两边AB，AC分别交于点D，E，且AD∶DB＝AE∶EC，如图所示，则BC与α的位置关系是( )A ．平行B ．相交C ．异面D ．BC α 答案 A解析 在△ABC 中，AD DB ＝AEEC，∴DE ∥BC . ∵DE α，BC ⊆/ α，∴BC ∥平面α.5．直线l ∥平面α，直线m ∥平面α，直线l 与m 相交于点P ，且l 与m 确定的平面为β，则α与β的位置关系是( )A ．相交B ．平行C ．异面D ．不确定 答案 B解析 因为l ∩m ＝P ，所以过l 与m 确定一个平面β.又因l ∥α，m ∥α，l ∩m ＝P ，所以β∥α.6．一条直线l 上有相异三个点A 、B 、C 到平面α的距离相等，那么直线l 与平面α的位置关系是( )A ．l ∥αB ．l ⊥αC ．l 与α相交但不垂直D ．l ∥α或l α答案 D解析 l ∥α时，直线l 上任意点到α的距离都相等，l α时，直线l 上所有的点到α的距离都是0；l ⊥α时，直线l 上有两个点到α的距离相等；l 与α斜交时，也只能有两点到α的距离相等．7．已知不重合的直线a ，b 和平面α.给出下列命题： ①若a ∥α，b α，则a ∥b ； ②若a ∥α，b ∥α，则a ∥b ； ③若a ∥b ，b α，则a ∥α； ④若a ∥b ，a ∥α，则b ∥α或b α. 其中正确的是________(填序号)． 答案 ④解析 ①若a ∥α，b α，则a ，b 平行或异面； ②若a ∥α，b ∥α，则a ，b 平行或相交或异面；③若a ∥b ，b α，则a ∥α或a α. ④正确．8．对于平面α与平面β，有下列条件：①α，β都平行于平面γ；②α内不共线的三点到β的距离相等；③l ，m 为两条平行直线，且l ∥α，m ∥β；④l ，m 是异面直线，且l ∥α，m ∥α，l ∥β，m ∥β.则可判定平面α与平面β平行的条件是________(填序号)．答案 ①④解析 由面面平行的传递性可知①能得出α∥β.对于④，l ，m 是异面直线，则分别在α，β内作l ′∥l ，m ′∥m 及l ″∥l ，m ″∥m ，则l ′与m ′，l ″与m ″都分别相交，故α∥β.对于②③，平面α与平面β可能相交．9．在四面体ABCD 中，M ，N 分别是△ACD ，△BCD 的重心，则四面体的四个面中与MN 平行的是________．答案 平面ABC 、平面ABD解析 如图，连接AM 并延长交CD 于点E ，连接BN 并延长交CD 于点F .由重心的定义及性质可知，E ，F 重合为一点，设为E ，且该点为CD 的中点，由EM MA ＝EN NB ＝12，得MN ∥AB ， 因此，MN ∥平面ABC 且MN ∥平面ABD .10．如图所示，在三棱锥S －ABC 中，D ，E ，F 分别是棱AC ，BC ，SC 的中点，求证：平面DEF ∥平面SAB .证明 因为D ，E 分别是棱AC ，BC 的中点，所以DE 是△ABC 的中位线，DE ∥AB . 因为DE ⊆/ 平面SAB ，AB 平面SAB ，所以DE ∥平面SAB ， 同理可证：DF ∥平面SAB ，又因为DE ∩DF ＝D ，DE 平面DEF ，DF 平面DEF ，所以平面DEF∥平面SAB.平行关系的性质1．a∥α，b∥β，α∥β，则a与b位置关系是( )A．平行 B．异面C．相交 D．平行或异面或相交答案 D解析如图(1)，(2)，(3)所示，a与b的关系分别是平行、异面或相交．2．三棱锥S－ABC中，E、F分别是SB、SC上的点，且EF∥平面ABC，则( ) A．EF与BC相交 B．EF与BC平行C．EF与BC异面 D．以上均有可能答案 B解析由线面平行的性质定理可知EF∥BC.3．如图，四棱锥P－ABCD中，M，N分别为AC，PC上的点，且MN∥平面PAD，则( )A．MN∥PDB．MN∥PAC．MN∥ADD．以上均有可能答案 B解析∵MN∥平面PAD，MN平面PAC，平面PAD∩平面PAC＝PA，∴MN∥PA.4．下列说法正确的个数是( )①两个平面平行，夹在两个平面间的平行线段相等；②两个平面平行，夹在两个平面间的相等线段平行；③如果一条直线和两个平行平面中的一个平行，那么它和另一个也平行；④平行于同一条直线的两个平面平行．A．1 B．2 C．3 D．4答案 A解析只有①正确．②中的两线段还可能相交或异面；③中的直线可能在另一个平面内；④中的两个平面可能相交．5．平面α截一个三棱锥，如果截面是梯形，那么平面α必定和这个三棱锥的( ) A．一个侧面平行 B．底面平行C．仅一条棱平行 D．某两条相对的棱都平行答案 C解析当平面α∥平面ABC时，如下图(1)所示，截面是三角形，不是梯形，所以A、B 不正确；当平面α∥SA时，如上图(2)所示，此时截面是四边形DEFG.又SA平面SAB，平面SAB∩α＝DG，所以SA∥DG.同理，SA∥EF，所以EF∥DG.同理，当平面α∥BC时，GF∥DE，但是截面是梯形，则四边形DEFG中仅有一组对边平行，所以平面α仅与一条棱平行．所以D不正确，C正确．6．下列说法正确的是( )A．平行于同一条直线的两个平面平行B．平行于同一个平面的两个平面平行C．一个平面内有三个不共线的点到另一个平面的距离相等，则这两个平面平行D．若三直线a，b，c两两平行，则在过直线a的平面中，有且只有一个平面与b，c 均平行答案 B解析平行于同一条直线的两个平面可以平行也可以相交，所以A错；B显然正确；C 中没有指明这三个点在平面的同侧还是异侧，不正确；D不正确，因为过直线a的平面中，只要b ，c 不在其平面内，则与b ，c 均平行．7．设m 、n 是平面α外的两条直线，给出三个论断：①m ∥n ；②m ∥α；③n ∥α.以其中的两个为条件，余下的一个为结论，构成三个命题，写出你认为正确的一个命题：________.(用序号表示)答案 ①②⇒③(或①③⇒②) 解析 ①②⇒③设过m 的平面β与α交于l .∵m ∥α，∴m ∥l ，∵m ∥n ，∴n ∥l ，∵n ⊆/ α，l α，∴n ∥α.8．如图，正方体ABCD －A 1B 1C 1D 1中，AB ＝2，点E 为AD 的中点，点F 在CD 上．若EF ∥平面AB 1C ，则线段EF 的长度等于________．答案2解析 因为直线EF ∥平面AB 1C ，EF 平面ABCD ，且平面AB 1C ∩平面ABCD ＝AC ，所以EF ∥AC ，又因为E 是DA 的中点，所以F 是DC 的中点，由中位线定理可得：EF ＝12AC ，又因为在正方体ABCD －A 1B 1C 1D 1中，AB ＝2，所以AC ＝22，所以EF ＝ 2.9．在棱长为a 的正方体ABCD －A 1B 1C 1D 1中，M ，N 分别是棱A 1B 1，B 1C 1的中点，P 是棱AD 上一点，AP ＝a3，过P ，M ，N 的平面与棱CD 交于Q ，则PQ ＝________.答案22a3解析 ∵MN ∥平面AC ，PQ ＝平面PMN ∩平面ABCD ， ∴MN ∥PQ ，易知DP ＝DQ ＝2a3，故PQ ＝PD 2＋DQ 2＝2DP ＝22a 3.10．如图，在正方体ABCD －A 1B 1C 1D 1中，点N 在BD 上，点M 在B 1C 上，且CM ＝DN .求证：MN ∥平面AA 1B 1B .证明 如图，作MP ∥BB 1交BC 于点P ，连接NP ，∵MP ∥BB 1，∴CM MB 1＝CP PB. ∵BD ＝B 1C ，DN ＝CM ， ∴B 1M ＝BN ，∴CM MB 1＝DN NB ，∴CP PB ＝DN NB， ∴NP ∥CD ∥AB .∵NP ⊆/ 平面AA 1B 1B ，AB 平面AA 1B 1B ， ∴NP ∥平面AA 1B 1B .∵MP ∥BB 1，MP ⊆/ 平面AA 1B 1B ，BB 1平面AA 1B 1B ， ∴MP ∥平面AA 1B 1B .又∵MP 平面MNP ，NP 平面MNP ，MP ∩NP ＝P ， ∴平面MNP ∥平面AA 1B 1B .∵MN 平面MNP ，∴MN ∥平面AA 1B 1B .平面与平面垂直的判定1．下列说法中正确的是( )A ．平面α和β分别过两条互相垂直的直线，则α⊥βB ．若平面α内的一条直线垂直于平面β内的两条平行直线，则α⊥βC ．若平面α内的一条直线垂直于平面β内的两条相交直线，则α⊥β。

2013年高考英语一轮复习系列训练（M2）星期三（U3-M2）第一节单项填空1. The boy was last seen near the West Lake.A. missing; playingB. missing; playC. missed; played D .missed; to play2. It was dangerous for you to swim in that river. You eaten by the fish.A. might have beenB. must have beenC. could have beenD. should have been3. Let’s hold a party to your birthday and at the same time you on your passing thedriving test.A. congratulate; celebrateB. celebrate; celebrateC. celebrate; congratulateD. congratulate; congratulate4. We walk in the moonlight, talking about we were interested in.A. used to; all whatB. were used to; all thatC. used to; whatD. were used to; what5. It is a paradox that in a rich country there should be many poor people.A. such; suchB. such; soC. so; soD. so; such6. The boy wanted to ride his father’s new motorcycle in the crowed street but his father toldhim .A. not to doB. not toC. not do itD. do not so7. The students were told that they at the school gate at 2:00 o’clock.A. metB. will meetC. were to meetD. were met8. I couldn’t find Peter, did I know where he had gone.A. neitherB. norC. soD. and9. __________ , but also they are good friends.A. Not only does he know JimB. Not only he knows JimC．Not does he only know Jim D. Does not only he knows Jim10. During the football match, he hurt his arm as well as his leg.A. breakB. breaksC. breakingD. broke11. He goes to the gymnasium for physical training day.A. every eachB. every oneC. each other;D. every other12. Athletes compete the Olympic Games the honor of winning.A. in; againstB. in; forC. against; inD. for; for13. The German Nazis were considered the Amber Room during World War II.A. to have stolenB. stealingC. having stolenD. to steal14. The weather turned out to be very good, was more than we could expect.A. whatB. whichC. thatD. it15. He had me to come to my birthday party, but he didn’t show up.A. allowedB. promisedC. wantedD. advised16. Which of the following sentences is NOT right?A. There is no doubt that you will succeed.B. We have no doubt that you will succeed.C. We don’t doubt that you will succeed.D. We don’t doubt whether you will succeed17. Take away whatever you.A. belong toB. belongs toC. belonging toD. is belonged to18. The old woman has been ill for several months but is beginning to pick .A. outB. offC. upD. on19. When winter comes, some birds fly to the south and stay.A. anotherB. otherC. othersD. the other20. He told me about the news in a voice.A. amazing; surprisingB. amazed; surprisedC. amazing; surprisedD. amazed; surprising21. The chairman told the speaker that she ___ to speak a little louder so as to make herself _____.A. was expected; heardB. had expected; hear C had hoped; hear D. was hoped; heard22. Do you think that the bridge ______ in a year?A. would be completedB. will be completedC. had been completedD. is being completed23. The window ______, you need not wash it again.A. washedB. is washedC. has been washedD. will be washed24.John had to have his car repaired in a garage because it ____seriously.A. damagedB. was being damagedC. had damagedD. had been damaged25. The police found that the house _______and a lot of things _____A. has broken into; has been stolenB. had broken into; had been stolenC. has been broken into; stolenD. had been broken into; stolen第二节完形填空Dear Classmates,As young people, we don’t always want to think about the past. 26 we often hear our grandfathers and grandmothers talk about cultural relics. They say we 27 protect some of these relics because they are important to our culture. 28 also say that these relics are important to us because they help us remember the 29 of our ancestor s and respect what they have done. I’m sure you will30 . After all, someday we will be 31 ourselves and will want our own children to protect them. So I have a plan for 32 the painting in the old temple, which is a fine cultural relic 33 in our hometown. It should be protected because it was painted by a 34__ artist of the early Qing Dynasty. My plan is to get students to take a 35 to see it on a Saturday next month and then 36 some important people to join us. 37 , we can also write 38 about it for the town newspaper. Later, when others begin to 39 their help, perhaps we can 40 enough money to help the museum buy it. If you like my plan, please give me a note to your teacher. Thank you.Yours.26. A. However B. Yet C. Therefore D. And27. A. can B. must C. will D. need28. A. Some B. We C. They D. Others29. A. days B. dreams C. lives D. styles30. A. refuse B. allow C. doubt D. agree31. A. greater B. stronger C. richer D. older32. A. surviving B. remaining C. saving D. removing33. A. there B. here C. where D. anywhere34. A. famous B. rare C. gifted D. skillful35. A. trip B. look C. bus D. rest36. A. take B. order C. beg D. ask37. A. In a word B. Besides C. First of all D. By the way38. A. a poem B. a passage C. an article D. a diary39. A. prove B. offer C. supply D. provide40. A. make B. earn C. raise D. give第三节语法填空Over time I have been changed quite a lot. I began 41__________ a calculating machine in France in 1642. 42______________ I was young I could simplify difficult sums. I developed very 43(slow)________ and it took nearly two hundred years 44_______________ I was built as an analytical machine by Charles Babbage. After I was programmed 45____________ an operator who used cards with holes, I could “think”logical and produce 46_____________ answer quicker than any person.At that time 47_____________ was considered a technological revolution and the start of my “artificial intelligence”. In 1936 my r eal father, Alan Turing, wrote a book 48____________ how I could be made to work as a “universal machine”49(solve)________________ any difficult mathematical problem. From then on, I grew rapidly 50_____________ in size and in brainpower. 41____________42_____________43_________________44______________45_____________ 46____________47_____________48_________________49______________50_____________第四节阅读理解AEddie McKay, a once-forgotten pilot, is a subject of great interest to a group of history students in Canada.It all started when Graham Broad, a professor at the University of Western Ontario, found McKay’s name in a footnote in a book about university history. McKay was included in a list of university alumni （校友）who had served during the First World War, but his name was unfamiliar to Broad, a specialist in military history. Out of curiosity, Broad spent hours at the local archives （档案馆）in a fruitless search for information on McKay. Tired and discouraged, he finally gave up. On his way out, Broad’s glance happened to fall on an exhibiting case showing some old newspapers. His eye was drawn to an old picture of a young man in a rugby uniform. As he read the words beside the picture, he experienced a thrilling rea lization. “After looking for him all day, there he was, staring up at me out of the exhibiting case,” said Broad. Excited by the find, Broad asked his students to continue his search. They combed old newspapers and other materials for clues. Gradually, a picture came into view.Captain Alfred Edwin McKay joined the British Royal Flying Corps in 1916. He downed ten enemy planes, outlived his entire squadron （中队）as a WWI flyer, spent some time as a flying instructor in England, then returned to the front, where he was eventually shot down over Belgium and killed in December 1917. But there’s more to his story. “For a brief time in 1916 he was probably the most famous pilot in the world,” says Broad. “He was credited with downing Oswald Boelcke, the most famou s German pilot at the time.” Yet, in a letter home, McKay refused to takecredit, saying that Boelcke had actually crashed into another German plane.McKay’s war records were destroyed during a World War II air bombing on London —an explanation for why he was all but forgotten.But now, thanks to the efforts of Broad and his students, a marker in McKay’s memory was placed on the university grounds in November 2007. “I found my eyes filling with tears as I read the word ‘deceased’ （阵亡）next to his name,” sa id Corey Everrett, a student who found a picture of Mckay in his uniform. “This was such a simple example of the fact that he had been a student just like us, but instead of finishing his time at Western, he chose to fight and die for his country.”51. What made Professor Broad continue his search for more information on McKay?A. A uniform of McKay.B. A footnote about McKay.C. A book on McKay.D. A picture of McKay.52. What did the students find out about McKay?A. He trained pilots for some time.B. He lived longer than other pilots.C. He died in the Second World War.D. He was downed by the pilot Boelcke.53. McKay’s flying documents were destroyed in .A. BelgiumB. GermanyC. CanadaD. England54. We can learn from the last paragraph that McKay .A. preferred fight to his studyB. went to war before graduationC. left a picture for Corey EverrettD. set an example for his fellow students55. What is the text mainly about?A. The research into war history.B. The finding of a forgotten hero.C. The pilots of the two world wars.D. The importance of military studies.BEvery object tells a story. Even the most ordinary objects can present to us powerful images. Sometimes it is the ordinary nature of these objects that actually makes them so extraordinary. Such is the case with an old leather shoe in a museum in Alaska. At first glance it does not look like much. It is a woman’s shoe of a style popular in the 1890s. But what is unique(独特的) about this shoe is where it was found. It was discovered on the Checkout Pass, the famous trail used by the people seeking gold in Alaska. Who it belonged to or why it was left there is not known. Was it perhaps dropped by accident as the woman climbed up the 1500 stairs carved outface? Or did she throwaway goods that she didn’t need in order to travel lighter?Over 100, 000 people with “gold fever” made this trip hoping to become millionaires. Few of them understood that on their way they would have to cross a harsh wildness. Unprepared for such a dangerous journey, many died of starvation and exposure to the cold weather.The Canadian government finally started requiring the gold seekers to bring one ton of supplies with them. This was thought to be enough for a person to survive for one year. They would carry their supplies in backpacks(背包) each weighing up to fifty pounds; it usually took at least 40 trips to get everything to the top and over the pass. Whoever dropped the shoe must have been a brave and determined woman. Perhaps she was successful and made it to Alaska. Perhaps she had to turn back in defeat. No one will ever know for sure, but what we do know is that she took part in one of the greatest adventures in the 19th century.56. Which of the following is right?A. it was found on a famous trailB. it was an important clue to life in the pastC. it at one time belonged to a VIPD. it was a fashionable shoe at that time57. According to this passage, many people who went to Alaska _______.A. eventually became millionairesB. were not properly equippedC. were very poorD. brought with them many shoes58. The Canadian government made gold seekers bring one year’s supplies with them so that ___.A. they would not die of hunger and coldB. the army would have enough food for fighting a warC. they would change these goods with the EskimosD. the supplies would make Alaska rich59. No matter what happened to the woman who owned the shoe, _______.A. she must have been a brave woman.B. she certainly dropped the shoe on purposeC. her adventurous spirit is definitely admiredD. her other shoes were equally fashionable60. what’s the best title of the passage?A. special shoe whose ower is a woman.B. the sad story about the shoe.C. the old shoe that has a special. story.D. gold seekersCJune 26, 2000 — the Human Genome(基因组) Project, a great $3 billion, 15-year task aimed atdrawing the genetic(遗传的) map of humans, is now more than 90 percent completed. The scientific and medical communities are very excited about the chances genetic research provides for getting rid of diseases and prolonging(延长) human life. But those communities and policymakers also are careful about the scientific door they are opening as the project uncovers the mysteries of life.For the last few years, the genetic advances in the developing field of biotechnology(生物技术) have provides material for all kinds of work, but the developments of modern science in unlocking the secrets of the human genetic code have opened a world of possibilities for human health, as well as for the popular imagination.While European and Japanese researchers are making rapid progress in decoding(解码) human DNA, the leading organization for genetic research is in the United States, which began in 1990, is “unlocking the code” of the human body to learn how to defeat fatal diseases . Already, the Human Genome Project has become widely known and praised for finding the genes connected with terrible diseases as yet, and making progress toward separating the genes that show a sign of breast cancer or AIDS.Once these genes are found and studied, researchers can develop new ways to attack infections and genetic diseases. Medical companies are very interested in mapping the human genome, as they expect to develop a lot of new drugs for these illnesses.61. Why did the scientists work hard at mapping the human genome?A. Because the human genome can help us live longer.B. Because they wanted to be better known than others.C. Because the human genome can provide a lot of moneyD. Because the human genome's completion can help them get rid of many diseases.62. Which of the following is NOT true?A. If the genes can be found, scientists can study many new ways to cure illnesses.B. The scientists have made great progress in connecting the genes with the cancers.C. Many medical companies show great interest in drawing the human genome map.D. The United States began the Genes Study early in the 19th century.63. Which country studied the genes most rapidly in the world?A. Japan. C. The United States .B. British England D. China.64. We can conclude that the Human Genome Project can cause _______.A. the policy makers to feel very happy.B. the scientists to work harderC. many people to live longer.D. a lot of companies to produce many new drugs65. What’ the best title of the passage?A. unlocking genetic codeB. the genes' discoveryC. human genome’s scienceD. the genes and the scientistsDGeorge Markov was a famous writer in Bulgaria. In 1969 he suspected that he was going to be imprisoned or killed because one of his plays was regarded as being an attack on leaders of Bulgaria. Markov managed to reach England and got a job with the BBC, writing something in Bulgaria. Some of the BBC programmes were critical of life in Bulgaria. Perhaps as a result of this, Markov received an anonymous telephone call warning him that he would be killed. In September 1978, Markov stopped his car in London and started to walk to his office. When he was passing a bus line, a man in the line seemed to drop his umbrella accidently. Markov felt a sudden pain in the leg.When Markov reached his office, he spoke about the matter to a friend. A few hours later, he began to feel hot. He was sent to hospital and died four days later. The doctors examined his body, and they were puzzled about the cause of his death. Scientists were asked to help and they found a tiny metal pellet in Markov’s leg. The scientists believed that the two holes in it must contain an unknown poison in them.A few weeks before Markov was “shot” with a poisoned pellet fired from an umbrella, another Bulgarian had the same experience in France. Towards the end of August 1978, Kostov felt a sharp pain in the back when he was leaving a railway station in Paris. He was ill for a few days but became well. When news of Markov’s death became known, Kostov was asked to return to hospital for examination. Doctors found a tiny pellet in his back, but it had stuck in an area from which the poison had not been able to spread.The police in both countries are still searching for the reasons why both men were attacked. They hoped to catch their attackers.66.Which of the statement is right about the underlined word?A．stranger B．well-known C．friendly D. unknown67．Which of the following was not mentioned in the passage?A．George Markov was working for the BBCB．George Markov wrote many plays at that timeC．kostov was “shot” with a poisoned pelletD．Both kostov and George Markov died at last68．According to the passage, Markov’s suspicions turned out to beA．wrong B．right C．reasonable D．unreasonable 69．Who killed Markov?A．Bulgarian B．not known C．French D．British70. We can infer that _________A. Leaders in Bulgaria disliked George MarkovB. The police didn’t find the person who kil led George MarkovC. George Markov wrote some critical playsD. George Markov lived in Bulgaria all his life答案星期三（U3-M2）第一节单项填空1-5 A A C C B 6-10 B C B A C 11-15 D B A B B 16-20 D B C C C 21-25 A B C D D 第二节完形填空26-30. BBCCD 31-35. DCBAA 36-40. DBCBC第三节语法填空41. as 42. although 43. slowly 44. before 45. by46. an 47. it 48. about 49. to solve 50. both第四节阅读理解51-55 DADBB 56-60 ABACC 61-65 DDCBA 66-70 DBBA。

一、根据首字母填写单词(单词拼写)1. This reference book is easily a________ to beginners.2. You are a ______mistaken! Such an honest person can’t tell a lie to us. (根据首字母单词拼写)3. He should be a________of himself for telling such lies. （根据首字母单词拼写）二、根据汉语意思填写单词(单词拼写)4. Over those years she ________ (逐渐地) built up a reputation as a successful lawyer. (根据汉语提示单词拼写)5. Jack is late again. It is _________ (典型的) of him to keep others waiting. (根据汉语提示单词拼写)6. The epidemic situation (疫情) was ________(令人不安的). (根据汉语提示完成句子)三、根据中英文提示填写单词(单词拼写)7. It now looks e____________ (完全) different from what it was.(根据中英文提示单词拼写)8. This museum is d________ (绝对地) worth visiting again. （根据中英文提示填空）四、完成句子9. 中国是一个有吸引力的国家，有许多神秘的故事和传奇人物。

(where)China is an attractive country ________ ________ ________ ________ ________ mysterious stories and legends.10. 果不其然，走下楼梯，来到站台上就像跳进一座正在喷发的火山一样。

第三章 章末检测(B)(时间：120分钟 满分：150分)一、选择题(本大题共12小题，每小题5分，共60分)1．如图直线l 1，l 2，l 3的倾斜角分别为α1，α2，α3，则有( )A ．α1<α2<α3B ．α1<α3<α2C ．α3<α2<α1D ．α2<α1<α32．直线x ＋2y －5＝0与2x ＋4y ＋a ＝0之间的距离为5，则a 等于( ) A ．0 B ．－20 C ．0或－20 D ．0或－103．若直线l 1：ax ＋3y ＋1＝0与l 2：2x ＋(a ＋1)y ＋1＝0互相平行，则a 的值是( ) A ．－3 B ．2 C ．－3或2 D ．3或－2 4．下列说法正确的是( )A ．经过定点P 0(x 0，y 0)的直线都可以用方程y －y 0＝k (x －x 0)表示B ．经过定点A (0，b )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x a ＋yb＝1表示D ．经过任意两个不同的点P 1(x 1，y 1)、P 2(x 2，y 2)的直线都可以用方程(y －y 1)(x 2－x 1)＝(x －x 1)(y 2－y 1)表示5．点M (4，m )关于点N (n ，－3)的对称点为P (6，－9)，则( ) A ．m ＝－3，n ＝10 B ．m ＝3，n ＝10 C ．m ＝－3，n ＝5 D ．m ＝3，n ＝56．以A (1,3)，B (－5,1)为端点的线段的垂直平分线方程是( ) A ．3x －y －8＝0 B ．3x ＋y ＋4＝0 C ．3x －y ＋6＝0 D ．3x ＋y ＋2＝07．过点M (2,1)的直线与x 轴，y 轴分别交于P ，Q 两点，且|MP |＝|MQ |，则l 的方程是( ) A ．x －2y ＋3＝0 B ．2x －y －3＝0 C ．2x ＋y －5＝0 D ．x ＋2y －4＝08．直线mx －y ＋2m ＋1＝0经过一定点，则该点的坐标是( ) A ．(－2,1) B ．(2,1) C ．(1，－2) D ．(1,2) 9．如果AC <0且BC <0，那么直线Ax ＋By ＋C ＝0不通过( ) A ．第一象限 B ．第二象限 C ．第三象限 D ．第四象限10．直线2x ＋3y －6＝0关于点(1，－1)对称的直线方程是( ) A ．3x －2y ＋2＝0 B ．2x ＋3y ＋7＝0 C ．3x －2y －12＝0 D ．2x ＋3y ＋8＝011．已知点P (a ，b )和Q (b －1，a ＋1)是关于直线l 对称的两点，则直线l 的方程是( ) A ．x ＋y ＝0 B ．x －y ＝0 C ．x ＋y －1＝0 D ．x －y ＋1＝012．设x ＋2y ＝1，x ≥0，y ≥0，则x 2＋y 2的最小值和最大值分别为( )A ．15，1B ．0,1C ．0，15D ．15，2二、填空题(本大题共4小题，每小题5分，共20分)13．不论a 为何实数，直线(a ＋3)x ＋(2a －1)y ＋7＝0恒过第________象限．14．原点O 在直线l 上的射影为点H (－2,1)，则直线l 的方程为______________． 15．经过点(－5,2)且横、纵截距相等的直线方程是____________________．16．与直线3x ＋4y ＋1＝0平行且在两坐标轴上截距之和为73的直线l 的方程为______________．三、解答题(本大题共6小题，共70分)17．(10分)已知直线2x ＋(t －2)y ＋3－2t ＝0，分别根据下列条件，求t 的值：(1)过点(1,1)；(2)直线在y 轴上的截距为－3．18．(12分)直线l 过点(1,4)，且在两坐标轴上的截距的积是18，求此直线的方程．19．(12分)光线从A (－3,4)点出发，到x 轴上的点B 后，被x 轴反射到y 轴上的C 点，又被y 轴反射，这时反射光线恰好过D (－1,6)点，求直线BC 的方程．20．(12分) 如图所示，某县相邻两镇在一平面直角坐标系下的坐标为A (1,2)，B (4,0)，一条河所在的直线方程为l ：x ＋2y －10＝0，若在河边l 上建一座供水站P ，使之到A ，B 两镇的管道最省，那么供水站P 应建在什么地方？21．(12分)已知△ABC的顶点A为(3，－1)，AB边上的中线所在直线方程为6x＋10y－59＝0，∠B的平分线所在直线方程为x－4y＋10＝0，求BC边所在直线的方程．22．(12分)已知直线l过点P(3,1)，且被两平行直线l1：x＋y＋1＝0和l2：x＋y＋6＝0截得的线段长度为5，求直线l的方程．第三章直线与方程(B) 答案1．B2．C3．A4．D[斜率有可能不存在，截距也有可能不存在．]5．D [由对称关系n ＝4＋62，－3＝m ＋(－9)2，可得m ＝3，n ＝5．]6．B [所求直线过线段AB 的中点(－2,2)，且斜率k ＝－3，可得直线方程为3x ＋y ＋4＝0．] 7．D [由题意可知M 为线段PQ 的中点，Q (0,2)，P (4,0)，可求得直线l 的方程x ＋2y －4＝0．] 8．A [将原直线化为点斜式方程为y －1＝m (x ＋2)，可知不论m 取何值直线必过定点(－2,1)．]9．C [将原直线方程化为斜截式为y ＝－A B x －CB ，由AC <0且BC <0，可知AB >0，直线斜率为负，截距为正，故不过第三象限．]10．D [所求直线与已知直线平行，且和点(1，－1)等距，不难求得直线为2x ＋3y ＋8＝0．] 11．D [∵k PQ ＝a ＋1－bb －1－a＝－1，∴k l ＝1．显然x －y ＝0错误，故选D ．] 12．A [x 2＋y 2为线段AB 上的点与原点的距离的平方，由数形结合知，O 到线段AB 的距离的平方为最小值，即d 2＝15，|OB |2＝1为最大值．]13．二解析 直线方程可变形为：(3x －y ＋7)＋a (x ＋2y )＝0．由⎩⎪⎨⎪⎧ 3x －y ＋7＝0x ＋2y ＝0得，⎩⎪⎨⎪⎧x ＝－2y ＝1． ∴直线过定点(－2,1)．因此直线必定过第二象限． 14．2x －y ＋5＝0解析 所求直线应过点(－2,1)且斜率为2，故可求直线为2x －y ＋5＝0．15．y ＝－25x 或x ＋y ＋3＝0解析 不能忽略直线过原点的情况． 16．3x ＋4y －4＝0解析 所求直线可设为3x ＋4y ＋m ＝0，再由－m 3－m 4＝73，可得m ＝－4．17．解 (1)代入点(1,1)， 得2＋(t －2)＋3－2t ＝0，则t ＝3． (2)令x ＝0，得y ＝2t －3t －2＝－3，解得t ＝95．18．解 设直线l 的方程为x a ＋yb＝1，则⎩⎪⎨⎪⎧ab ＝181a ＋4b ＝1，解得⎩⎪⎨⎪⎧a ＝3b ＝6或⎩⎪⎨⎪⎧a ＝32b ＝12则直线l 的方程2x ＋y －6＝0 或8x ＋y －12＝0．19．解如图所示，由题设，点B 在原点O 的左侧，根据物理学知识，直线BC 一定过(－1,6)关于y 轴的对称点(1,6)，直线AB 一定过(1,6)关于x 轴的对称点(1，－6)且k AB ＝k CD ，∴k AB ＝k CD ＝4＋6－3－1＝－52．∴AB 方程为y －4＝－52(x ＋3)．令y ＝0，得x ＝－75，∴B ⎝⎛⎭⎫－75，0． CD 方程为y －6＝－52(x ＋1)．令x ＝0，得y ＝72，∴C ⎝⎛⎭⎫0，72． ∴BC 的方程为x －75＋y72＝1，即5x －2y ＋7＝0． 20．解如图所示，过A 作直线l 的对称点A ′，连接A ′B 交l 于P ， 若P ′(异于P )在直线上，则|AP ′|＋|BP ′|＝|A ′P ′|＋|BP ′|>|A ′B |． 因此，供水站只有在P 点处，才能取得最小值，设A ′(a ，b )， 则AA ′的中点在l 上，且AA ′⊥l ， 即⎩⎪⎨⎪⎧a ＋12＋2×b ＋22－10＝0，b －2a －1·⎝⎛⎭⎫－12＝－1，解得⎩⎪⎨⎪⎧a ＝3，b ＝6，即A ′(3,6)．所以直线A ′B 的方程为6x ＋y －24＝0，解方程组⎩⎪⎨⎪⎧6x ＋y －24＝0，x ＋2y －10＝0，得⎩⎨⎧x ＝3811，y ＝3611，所以P 点的坐标为⎝⎛⎭⎫3811，3611． 故供水站应建在点P ⎝⎛⎭⎫3811，3611处．21．解 设B (4y 1－10，y 1)， 由AB 中点在6x ＋10y －59＝0上， 可得：6·4y 1－72＋10·y 1－12－59＝0，y 1＝5， 所以B (10,5)．设A 点关于x －4y ＋10＝0的对称点为A ′(x ′，y ′)， 则有⎩⎪⎨⎪⎧x ′＋32－4·y ′－12＋10＝0y ′＋1x ′－3·14＝－1⇒A ′(1,7)，∵点A ′(1,7)，B (10,5)在直线BC 上， ∴y －57－5＝x －101－10， 故BC ：2x ＋9y －65＝0．22．解 方法一 若直线l 的斜率不存在，则直线l 的方程为x ＝3，此时与直线l 1，l 2的交点分别为A (3，－4)，B (3，－9)．截得的线段AB 的长为|AB |＝|－4＋9|＝5，符合题意．若直线l 的斜率存在，则设直线l 的方程为y ＝k (x －3)＋1．解方程组⎩⎪⎨⎪⎧y ＝k (x －3)＋1，x ＋y ＋1＝0得⎩⎪⎨⎪⎧x ＝3k －2k ＋1，y ＝－4k －1k ＋1，所以点A 的坐标为⎝ ⎛⎭⎪⎫3k －2k ＋1，－4k －1k ＋1． 解方程组⎩⎪⎨⎪⎧y ＝k (x －3)＋1，x ＋y ＋6＝0得⎩⎪⎨⎪⎧x ＝3k －7k ＋1，y ＝－9k －1k ＋1，所以点B 的坐标为⎝ ⎛⎭⎪⎫3k －7k ＋1，－9k －1k ＋1． 因为|AB |＝5，所以⎝ ⎛⎭⎪⎫3k －2k ＋1－3k －7k ＋12＋⎣⎢⎡⎦⎥⎤⎝ ⎛⎭⎪⎫－4k －1k ＋1－⎝ ⎛⎭⎪⎫－9k －1k ＋12＝25． 解得k ＝0，即所求直线为y ＝1．综上所述，所求直线方程为x ＝3或y ＝1．方法二 设直线l 与直线l 1，l 2的交点分别为A (x 1，y 1)，B (x 2，y 2)， 则x 1＋y 1＋1＝0，x 2＋y 2＋6＝0． 两式相减，得(x 1－x 2)＋(y 1－y 2)＝5．① 因为|AB |＝5，所以(x 1－x 2)2＋(y 1－y 2)2＝25．②由①②可得⎩⎪⎨⎪⎧ x 1－x 2＝5，y 1－y 2＝0，或⎩⎪⎨⎪⎧x 1－x 2＝0，y 1－y 2＝5.所以直线的倾斜角为0°或90°．又P (3,1)在l 上，所以x ＝3或y ＝1．。