咸阳市2021届英语八上期末模拟检测试题(四)
咸阳市2021届英语八上期末模拟检测试题（四）一、选择题1．It ‘s getting dark,please the light.A．turn up B．turn off C．turn on D．turn down2．—What do you think of this cartoon movie?—Oh，I can't ________ it.It's long and boring.A．mind B．lose C．stand D．hear3．What is she going to do when she _____the news?A.is going to hearB.hearC.hearsD.will hear4．I will on my father’s farm this summer.A.take outB.help withC.help outD.look out5．—Let’s go camping after exams.--________. We need to relax.A．Not really B．Lucky you C．It’s my pleasure D．That’s for sure6．We should__________a large hole before we plant the tree.A.cleanB.digC.makeD.peel7．--- does your mother go shopping?---She goes shopping once a week.A．How long B．How much C．How often D．How far8．_____ he ______ to the mountains with his friends last Sunday?A.Did; wentB.Does; goC.Does; wentD.Did; go9．（题文）— Do you mind ______ Linda to join us?— Of course not. The more people, the better.A．ask B．to ask C．asking10．（题文）The man ____ a big tree, but he planted three new ones.A．fell down B．turned down C．cut down D．wrote down11．I ______ get to school by bike.A.sometimeB.some timeC.sometimesD.some times12．（题文）Sometimes the resolutions may be _______ difficult _______.A．too; to keep B．so; that I can keepC．such; that I can keep D．enough; to keep13．I don’t know if she tomorrow. If she ,let me know.A．comes；comes B．will come；comesC．comes；will come D．will come；will come14．New Year's Day is coming. The children are looking forward to ________ gifts from their parents.A．receive B．choose C．receiving D．choosing15．Mrs. Smith ___________ on the ground and hurt her arm.A.fell backB.fell intoC.fell downD.fell behind二、单词填空16．（题文）根据句意及汉语提示, 填写恰当的单词1）（小题1）What ________(其他的) should we do?2）（小题2）We should encourage(鼓励)________(青少年) to take part in more social activities.3）（小题3）The first ________(步骤) is very important.4）（小题4）How do you plan and________(组织) the meeting?5）（小题5）He is able to lead a ________(正常的)life.三、句型转换17．1）If we play together，we'll have fun.(改为同义句)If we play together，we'll ______ ______ ______ ______.2）50 percent of the students like watching the game show. (改为同义句)______ ______ the students like watching the game show.3）I want to know how I should go to the party. (改为简单句)I want to know ______ ______ ______ to the party.4）Work hard, or you will not pass the exam.(改为同义句)______ you ______ work hard, you will ______ the exam.5）This video isn't interesting enough for us to watch. (改为同义句)This video is ______ boring for us ______ watch.四、完成句子18．1）我父母想要我成为一名医生，但是我不确定。
My parents want me to be a doctor, but I_______ ________ ________ ______that.2）当你长大后，你想当什么？What do you want to be when you__________ _________?3）并不是每个人都知道自己未来理想的工作是什么。
__________ _________knows what their dream jobs are in the future.4）高中毕业的时候，吉姆想当一名厨师。
Jim is going to be a cook when he__________ __________ ___________.5）没有人能对自己的未来有把握，但是你可以为之而努力。
No one can ___________ __________ __________the future, but you can work hard for it.五、完形填空19．I’m a middle school student. In the past, I had some bad 1 . I hardly had breakfast 2 I usually got up late. I didn’t like exercising. I 3 exercised one to two times a month. In the evening, I always watched TV 4 more than two hours. Then I 5 the Internet. I liked to play computer games. So I often stayed up late. My parents thought staying up late was bad for my health, so they were not 6 with that. One day, I fainted (昏倒) in a P.E. class. When I 7 , I was in the hospital. The doctor said that my classmates sent 8 there. He said that I was in poor (差的) 9 . Then he told me to eat breakfast every day and do more exercise.From then on, I 10 to give up (戒掉) my bad habits. Now I usually get up early and have a good breakfast. After school, I always play sports with my friends. It’s fun and good for my health.1）A．activities B．habits C．programs2）A．although B．if C．because3）A．only B．least C．never4）A．below B．for C．through5）A．waited B．surfed C．bought6）A．happy B．different C．afraid7）A．cut down B．thought of C．woke up8）A．me B．her C．him9）A．percent B．health C．magazine10）A．asked B．decided C．stopped20．Now more and more people get to know the importance of doing sports. It can bring fun 1 us and it can also make us 2 . 3 wha t should we do? First, we shouldchoose(选择) our favorite ones from 4 kinds of sports. The sports must be right for us. Second, we must warm up (热身) before 5 .Third, 6 a beginner, you had better have an exer cise plan. For example, 7 are you going to exercise each time? How many 8 a week? If you're a 9 of sports,it's important to know this. Last but not least, you had better not exercise if you feel 10 , or it will be very dangerous.1）A．for B．to C．with D．on2）A．health B．healthy C．warm D．healthily3）A．Or B．And C．But D．So4）A．same B．the same C．difference D．different5）A．exercise B．exercising C．to exercise D．exercises6）A．as B．than C．then D．so7）A．how often B．how many C．how much D．how long8）A．time s B．time C．people D．players9）A．love B．loving C．loves D．lover10）A．fine B．fit C．ill D．well六、用所给的词正确形式填空21．（题文）1）You can see yourself in the m________.2）It’s n_______ for you to learn English well if you want to work in America in the future.3）There’s only one bedroom, so we'll have to s________.4）Li Ming is nine years old and he studies in a p_______ school.5）We can get much useful i________ on the Internet.七、阅读理解22．Many people like to travel by plane. Because it is the fastest way. But I don’tlike it because an airport is usually f ar from the city. You can’t open the window. You can’t choose the food you like. Planes are fast, but it still takes hours to get out ofthe airport and into the city.I like travelling by train. I think trains are safe. Train stations are usually in cities. You can see many interesting things on your way. You can walk around. But it takes a little more time.I also like cars. You can start your journey whenever you want to, and you don’t need to get to a train station or a bus stop. Also you can carry many things with you in a car. Butsometimes there are too many cars on the road.1）Why do many people like to travel by plane?A.Because it is the fastest.B.Because it is safe.C.Because you can open the window.D.Because you can walk around.2）Which is not the good thing about the train?A.It is safe.B.It takes a little more time.C.You can see interesting things.D.You can walk around.3）If you want to carry many things, you had better go by __________.A.carB.busC.trainD.air4）What is the bad thing about the car?A.You needn’t go to a station.B.You can start your journey whenever you want to.C.Sometimes there are too many cars on the road.D.You can’t see many interesting things on your way.5）According to the passage, which sentence is right?A.The writer thinks it takes a lot of time to get out of the airport and into the city by plane.B.The writer likes to take a train because it takes a little more time.C.The writer likes to take a car because he has a car.D.The writer likes to take a plane because it is fast.23．阅读短文，从各题所给的三个答案中选择最佳答案回答问题或完成句子。
2020-2021学年陕西省咸阳市秦岭中学高二英语下学期期末试卷含解析
2020-2021学年陕西省咸阳市秦岭中学高二英语下学期期末试卷含解析一、选择题1. -----Let’s go and have a good drink tonight.----- ______. Have you got the first prize in the competition?A. What for?B. Thanks a lotC. Yes, I’d like toD. Why not?参考答案：A2. The better use you can make__________ time, the greater contributions you willmake__________ society.A. in; toB. of; toC. on; forD. up; for参考答案：B3. 24．____ to give up smoking, he threw away his _____ cigarettes．A．Determined; remained B．Determined; remainingC．Determining; remaining D．Determining; remained参考答案：B略4. Many white-collar people, working in big cities, like to live in the country where there is a(n)________ of peace and calm.A. gloryB. atmosphereC. fantasyD. passion参考答案：B7. The shy girl felt___________ and uncomfortable when she could not answer her teache r’s questions. A. amazed B. awkward C. curious D. amused参考答案：B略6. I feel sure that ___ qualification, ability and experience, you are suited to theposition we offered.A．in need of B．in terms of C．in favor of D．in face of参考答案：B7. The technique is being tried in classrooms to ________ what effect it may have.A. accessB. approveC. appealD. assess参考答案：D8. Modern plastics can _______ very high and very low temperatures.A. standB. holdC. carryD. support参考答案：A9. --I’m applying for Fudan University. --_____A. Why not Tsinghua?B. Take it easy.C. Congratulations.D. Good luck参考答案：D10. _______ the final of UEFA Champions League (欧冠) between Manchester United and FC Barcelona, I would not have stayed up after midnight.A. Were there notB. Were not thereC. Had there not beenD. Had not there been参考答案：C2. It snowed for days on end. We thought it would never come to end!A. an;anB. an;/C. /;anD. /;/参考答案：C略12. ––– I hear Tom ______ playing games recently.–––It is no wonder he is easy to feel sleepy in class.A. addicting toB. addictsC. addict toD. addicted to参考答案：D13. She often shares her deepest thoughts and fears with friends through QQ, as ifthey______ chatting in her living room.A. wereB. areC. had beenD. has been 参考答案：A14. It is ________ any wonder that his friend doesn’t like watching television much.A. mostlyB. nearlyC. hardlyD. shortly参考答案：C略15. —Can I come to ask for some advice at any time?— Yes, of course. _______A. If you’re convenient.B. At your convenience.C. Don’t worry.D. Don’t mention it.参考答案：B16. This is the first time that I _________ this famous temple.A. am visitingB. will visitC. have visitedD. visit参考答案：C 略二、完型填空17. In the autumn of 2010, I was working at a top advertising company. I had recently been 41 as a director and it seemed I was living the American Dream. But it really wasn’t. I began to 42 ill. It started with full-body pain. My 43 shifted and I went from being an energetic, happy girl, to a person my husband James didn’t 44 . Something always got on my 45 and I was very confused. I found myself crying all the time without reason.Things went from bad to worse and I was 46 risk of losing my job. James took me to see my long-term doctor, who told me I had to go on medical 47 immediately. I fought him as if crazy, but in the end, he and James 48 .[I was so ashamed. I thought I had lost my mind, which I didn’t want anyone to 49 . I made James promise not to tell our families and it broke his heart to have everything 50 . I still feel awful that I made James 51 that way, but I’m so 52 to have a partner who was willing to make such a sacrifice for me.I began to take days off before I got my diagnosis: fibromyalgia (纤维肌痛) and a sleep disorder —so severe 53 it forced me to wake up over two dozen times an hour! There is no cure, but both can be 54 .I remembered sitting in my doctor’s office, 55 , “why me?” James and I had been married for just two years and 56 this illness was testing both of us, and testing our marriage as well. It all seemed so 57 . Just then, my doctor said, “No one 58 you anything. You are not owed a perfect, healthy life. Anything you get in this world is a gift. 59 , stop thinking you’re miserable just because life has handed you some 60 . Get over these difficulties and find a way to live.”41. A. appointed B. introduced C. judged D. regarded42. A. behave B. fall C. react D. work43. A. ability B. characteristic C. responsibility D. personality44. A. realize B. recommend C. recognize D. compliment45. A. legs B. nerves C. brains D. arms46. A. in B. under C. at D. on47. A. holiday B. vacation C. credit D. leave48. A. defeated B. resigned C. won D. apologized49. A. notice B. absorb C. upset D. care50. A. hide B. hidden C. hiding D. to hide51. A. suffer B. handle C. experience D. argue52. A. stubborn B. intelligent C. determined D. grateful53. A. and B. that C. as D. thus54. A. announced B. discovered C. proved D. treated55. A. saying B. arguing C. quarreling D. complaining56. A. seriously B. deadly C. obviously D. privately57. A. hopeless B. unfair C. different D. impossible58. A. owes B. spares C. keeps D. saves59. A. However B. Otherwise C. Nevertheless D. Therefore60. A. obstacles B. conflicts C. opportunities D. competitions 参考答案：41．A42．B43．D44．C45．B46．C47．D48．C49．A50．B51．A52．D53．B54．D55．D56．C 57．B58．A59．D60．A41．A考查动词。
2023届陕西省咸阳市高考模拟检测(二)语文参考答案
咸阳市2023年高考模拟检测（二）语文试题参考答案及评分标准第I卷（阅读题共71分）一、现代文阅读（36分）（一）论述类文本阅读（本题共3小题，9分）1．C（“宋朝的皇帝和官员们不思强国”错误，原文为“宋朝的多数皇帝”，选项范围扩大。
）2．D（A“都论证了＊强军强国＇这一主观点”错误，第①段引用词句是为了由电影《满江红》引出下文的论述；B第④段并未对第一个问题做出回答，也未指出宋朝军事落后的根本原因；C文章并未围绕“强军强国”这一观点集中分析论证，而是逐层论述，最后得出结论。
）3．D（D陈述的内容与强军强国无关。
）（二）实用类文本阅读（本题共3小题，12分）4．B（“优于西方人”理解错误，曲解文意。
材料一第一段重在阐述中西方时间标示之“异”，无优劣评价。
）5．A（B“农历已没有太大使用价值”妄加推断且观点错误；C强加因果；D“显示出他们只关注自身的认知局限”于文无据。
）6．①二十四节气将一年的时间划分为二十四个节点，体现了时间的整体性和统观性；②二十四节气依据日月的运行来确定，并以此指导农事活动，体现了天道和时间的交融；③二十四节气对应了季节更替、万物荣枯，蕴含了人对于自身的生命形态和年华盛衰的体验，体现了时间观念的生命感（或二十四节气蕴含了“天人合一”的哲学精神，体现了时间的仪式感和典重感）。
（每点2分，共6分，结合材料并言之成理即可给分）（三）文学类文本阅读（本题共3小题，15分）7．C（“情节一波三折”错误，原文描写老人捻蚕丝织绸布，替爱美的孙女们种指甲草，叙写乡土生活的质朴与琐碎，流露出乡村大地寻常百姓家老人和儿子孙女的款款温情，情节舒缓有致。
并未一波三折。
）8．①热爱质朴、琐碎而又快乐的乡土生活。
②赞美温暖、真挚而又亲密的祖孙之情。
③感恩大奶奶在日常小事和平淡话语中所寄寓的人生智慧。
（每点2分，共6分，言之成理、切合文本即可给分）9．①地方化：小说中多处使用方言，充满浓郁的乡土气息。
如称凤仙花为“指甲草”，以及“捻蚕丝”“衣服哐里哐当”等，亲切自然。
2020-2021学年陕西省咸阳市泾阳县泾干镇泾华学校高二英语上学期期末试题含解析
2020-2021学年陕西省咸阳市泾阳县泾干镇泾华学校高二英语上学期期末试题含解析一、选择题1. The decision ________ for almost a year because of the local residents’ complaint that the newly-built factory would disturb the quietness of the place.A. has been delayedB. was delayedC. would be delayedD. will be delayed参考答案：B2. No one ______ this building without the permission of the police .A is leavingB is to leaveC has leftD will be leaving参考答案：B略3. You job here is only ______, for you will be removed from it when we have a proper post for youA. originalB. accurateC. initialD. temporary参考答案：D略4. many times, he finally understood it.A．Being told B．Telling C．Having told D．Having been told参考答案：D略30. Haiku is a Japanese form of poetry that ______ 17 syllables.A. consist ofB. consisting ofC. makes upD. Is made up of参考答案：D略6. _________the piano, someone suddenly knocked at the door.A. PlayingB. When I was playingC. RepairingD. Examining参考答案：B7. These machines can help recycle the waste which would ______be thrown away.A. otherwiseB. thereforeC. howeverD. though参考答案：A略8. South Korean star PSY's “Gangnam style” video,________, has set the Guinness World Record for the most “liked” video in YouTube history.A. broke recordsB. breaking recordsC. to break recordsD. breaks records参考答案：B9. People have planted a great many trees in order to _______ wind and sand in the desert.A. hold downB. hold upC. hold backD. hold out参考答案：C10. He ______ the local accent while working in Tibet.A. picked upB. made upC. took upD. put up参考答案：A11. Only by imagining and creating ____ a difference. （）A．can make B．you can make C．make D．can you make参考答案：9.【答案】D12. —Do you agree with what he said? —Yes, _____! I t can’t be ____.A. approximately; betterB. ly; betterC. ly; worseD. fortunately; worse参考答案：B13. Dina, for months to find a job as a waitress, finally took a position at a local advertising agency.A. strugglingB. struggledC. having struggledD. to struggle参考答案：C14. You don’t have to worry about the journey because I have made a ______ on the Indian Pacific train to Perth.A. reserveB. bookC. preparationD. reservation参考答案：D15. The hairdresser showed me a book with so many diverse hair styles that I got and didn't know which one to choose.A.confusedB.to confuseC.confusingD.to be confused参考答案：A16. When working out multiple-choices in an English exam, students may find some words or phrase , but of course there exist certain differences.A. familiarB. parallelC. optionalD. relevant参考答案：B17. —I don’t believe we _______ before.—Well, I saw you once at a party, but we ______ introduced.A. have met; weren’tB. met; weren’tC. have met; haven’t beenD. met; haven’t been参考答案：A18. --- You have made the same mistake.--- ______ It’s none of your business.A. Never mind.B. No problem.C. So what?D. Why not?参考答案：C二、短文改错19. 假定英语课上老师要求同桌之间交换修改作文，请你修改你同桌写的以下作文。
2024年咸阳市一模(文)试题
咸阳市2024年高考模拟检测(一)数学(文科)试题注意事项：1．本试卷共6页，满分150分，时间120分钟；2．答卷前，考生须准确填写自己的姓名、准考证号，并认真核准条形码上的姓名、准考证号；3．第Ⅰ卷选择题必须使用2B铅笔填涂，第Ⅱ卷非选择题必须使用0.5毫米黑色墨水签字笔书写，涂写要工整、清晰；4．考试结束，监考员将试题卷、答题卡一并收回．第Ⅰ卷一、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.若集合=U2−2−3<0，集合=∈U<2，则集合∩=（）0,1−1,0,1u第一象限u第二象限u第三象限u第四象限3.已知 =(1,−1), =(s2)，若( + )|| ,则2 ⋅ =（）u−8u−7u7u84.已知数列的前n项和为S n，且等比数列满足=l2,若362=4，则9=（）u3u4u5u65.著名的本福特定律:以数字1开头的数字在各个领域中出现的频率似乎要高于其他数，也称为“第一位数定律”或者“首位数现象”.意指在一堆从实际生活中得到的十进制数据中，一个数的首位数字是o=1,2,3,⋯9)的概率为B(1+1).以此判断,一个数的首位数字是1的概率与首位数字是9的概率之比约为多少？（）（参考数据：B2≈0.301，B3≈0.477）u2.9u3.8u4.5u6.56.直线++=0与圆G+12+−12=5有公共点的一个充分不必要条件是（）u∈−10，10u∈[−10,+∞)u∈−10，10u∈−10,+∞7.某同学寒假期间想到咸阳市的6个旅游景点乾陵、茂陵、汉阳陵、旬邑马栏革命旧址、长武亭口活动旧址、泾阳安吴青训班中的2个景点进行旅游，其中旬邑马栏革命旧址、长武亭口活动旧址、泾阳安吴青训班三个景点为红色旅游景点，则他所去的景点中至少包含一个红色旅游景点的概率是（）A.13B.15C.23D.458.将一个棱长为4的正四面体同一侧面上的各棱中点两两连接，得到一多面体，则这个多面体的外接球的表面积为（）u8u839.等差数列中的2,2024是函数op=3−62+4−2024的极值点，则l81013=（）u13u−3u3u−1310.已知函数op=2+2+1,≤0B,>0，若方程=有四个根1,2,3,4,且1<2<3<4，则下列说法错误的是（）u1+2=−2u3+4>2．三、解答题：共70分．解答应写出文字说明、证明过程或演算步骤．第17—21题为必考题，每个试题考生都必须作答．第22、23题为选考题，考生根据要求作答．（一）必考题：共60分．18.（12分）已知三棱柱B−111如图所示，P是11上一动点，点、分别是B、B的中点，B⊥B,B1=B=B=2。
2021学年陕西省咸阳市三原县五年级(下)期末英语试卷有答案
2021学年陕西省咸阳市三原县五年级（下）期末英语试卷五、按要求写单词（10分）1. 按要求写单词（1）________ happily（形容词）（2）________ two（序数词）（3）________ snowman（复数）2. 按要求写单词（1）________ snowy（名词）（2）________． You'd better．（完整形式）（3）________ 十一月（汉译英）3. （1）________potatoes and tomatoes（英译汉）（2）________ speak with mouth full（英译汉）4. 英译汉．（1）________potatoes and tomatoes（2）________ speak with mouth full六、看图，补全句子（10分）You should________． You should have a good habit．Teachers' Day is in________．Dad is watching the________in the living room．I eat many candies． I________ now．七、单项选择（10分）There are many kinds of flowers in the village in_____．（）A.winterB.springC.autumnWe have many classes and much homework． I feel very _____．（）A.tired B.sad C.angryTake your kite with you． You can fly kites on the _____ day．（）A.snowy B.windy C.rainyChristmas Day is on _____ 25th．（）A.DecemberB.NovemberC.JanuaryYou shouldn't make a noise． You should_____．（）A.litter on the floorB.keep quietC.write on treesI have a headache． I will go to ____．（）A.the libraryB.schoolC.see a doctorThe third month of a year is ___﹣．（）We shouldn't have the book so close ______our eyes．A.forB.inC.toDon't talk_____ in class． It's a bad habit．（）A.loudB.loudlyC.louds八、句型转换（10分）There are ________ seasons in a year．（对划线部分提问）________seasons are there in a year？We are very happy．（变感叹句）________happy we________ !It's ________ today．（对划线部分提问）________ is the weather________ today？Will you go to the the Palace Museum？（做否定回答）________ ， we________．You should read in bed．（变否定句）You________ in bed．九、连词成句（注意大小写和标点符号，10分）rest， You'd， a， better， good， have （．）连词成句________tomorrow， do， What， you， will （？）连词成句________lessons， about， worry， your， Don't （．）连词成句________first， New Year， of， the， in， the， is， month， year （．）连词成句________care， trees， We， take， should， of， the （．）连词成句________阅读理解Today is Saturday． It is sunny and hot． Jack goes to play football with his friends in the park． He feels very hot． And he has an ice﹣cream． Now, he has a stomachache． His mother is going to take him to see a doctor． He is worrying about his homework． What should he do?（1）What day is it today？ It's________．A.SeptemberB. SaturdayC. cloudy．（2）How is Jack feeling？ He feels very________．A. hotB. coldC. happy．（3）What's wrong with Jack？ He has a________．A. stomachacheB. toothacheC. cold．（4）Jack will go to a________with his mother．A. cinemaB. supermarketC. hospital．（5）He is worrying about his________．A. homeworkB. lessonsC. dog．参考答案与试题解析2021学年陕西省咸阳市三原县五年级（下）期末英语试卷五、按要求写单词（10分）1.【答案】happy,second,snowmen【考点】词性转换可数名词及其单复数序数词【解析】（1）高兴的，高兴的．（2）第二，两个．（3）雪人，雪人．【解答】（1）happy．考查副词变形容词，happily的形容词是happy，故填：happy．（2）senond．考查基数词变序数词．two的序数词是second，故填：senond．（3）snowmen．考查单数变复数，snowman的复数形式是snowmen，故填：snowmen．2.【答案】snow,You had better,November【考点】词性转换全写和缩写单词、词组【解析】（1）有雪的．（2）你最好．（3）十一月．【解答】（1）snow．考查形容词变名词．snowy是形容词，有雪的，名词是雪花，即snow．故填：snow．（2）You had better．考查全写． You'd better．译为：你最好．完整形式是You had better．故填：You had better．（3）November考查单词翻译．十一月，November．故填：November．故填：November．3.【答案】土豆和西红柿,嘴里有东西的时候说话【考点】翻译能力（1）土豆和西红柿（2）嘴里有东西的时候说话【解答】本题考查短语翻译．（1）土豆和西红柿．potatoes and tomatoes土豆和西红柿．故答案为：土豆和西红柿．（2）嘴里有东西的时候说话．speak with mouth full嘴里有东西的时候说话．故答案为：嘴里有东西的时候说话．4.【答案】土豆和西红柿,嘴里有东西的时候说话【考点】翻译能力【解析】（1）土豆和西红柿（2）嘴里有东西的时候说话【解答】本题考查短语翻译．（1）土豆和西红柿．potatoes and tomatoes土豆和西红柿．故答案为：土豆和西红柿．（2）嘴里有东西的时候说话．speak with mouth full嘴里有东西的时候说话．故答案为：嘴里有东西的时候说话．六、看图，补全句子（10分）【答案】take exercise【考点】看图完成句子【解析】你应该锻炼，你应该有一个好习惯．【解答】该题考查看图完成句子．图示为锻炼，译为take exercise．should的后面用动词原形．故答案为：take exercise．【答案】September【考点】看图完成句子【解析】教师节在九月份．【解答】该题考查看图完成句子．图示为教师节，根据常识，教师节在九月份，九月份September．故答案为：September．【答案】mapofChina【考点】看图完成句子【解析】爸爸正在客厅里看中国地图．该题考查看图完成句子．图示为中国地图，中国地图map of China．故答案为：map of China．【答案】am having a toothache【考点】看图完成句子【解析】我吃了很多糖果．我现在牙疼．【解答】该题考查看图完成句子．根据"I eat many candies"结合图示推测：我现在牙疼．牙疼have a toothache，该句为现在进行时．故答案为：am having a toothache．七、单项选择（10分）【答案】B【考点】名词【解析】在春天在这个村子有很多种花．【解答】考查名词词义辨析．由题，A为"冬天"，B为"春天"，C为"秋天"，按照逻辑句意为"在春天在这个村子有很多种花．"符合，B选项符合．故选：B．【答案】A【考点】形容词【解析】我们有许多课，许多作业．我感觉很累．【解答】本题主要考查形容词的辨析．句子大意为：我们有许多课程许多作业，我感觉_____．A选项tired意思为：累的；B选项sad意思为：难过的；C选项angry意思为：生气的．可知A选项符合句意．故选：A．【答案】B【考点】形容词【解析】带着你的风筝．你可以在刮风的一天放．【解答】本题主要考查形容词的词义辨析．A选项snowy意思为：下雪的；B选项windy意思为：刮风的；C选项rainy意思为：下雨的，可知放风筝应该在刮风的一天．故选：B．【答案】A年、月、日、时、星期的表达法【解析】圣诞节在12月25日．【解答】考查日期的表达法．圣诞节是12月25日，12月用December，25th表示第二十五天．B是十一月，C是一月．故选：A．【答案】B【考点】动词短语【解析】你不应该出声．你应该保持安静．【解答】考查动词短语．由题，A为"在地板上乱扔垃圾"，B为"保持安静"，C为"在树上写"，按照逻辑句意为"你不应该出声．你应该保持安静．"符合，B选项符合．故选：B．【答案】C【考点】动词短语【解析】我头疼．我要去看医生．【解答】考查短语，headache头疼，生病了要去看医生，即see a doctor．故选：C．【答案】C【考点】年、月、日、时、星期的表达法【解析】一年中第三个月份是三月．【解答】考查月份的表达法．The third month of a year表示一年中第三个月份，即三月，March． A是二月，B是四月．故选：C．【答案】C【考点】介词【解析】我们不应该让书离我们的眼睛这么近．【解答】考查介词的辨析．由题，按照逻辑句意为"我们不应该让书离我们的眼睛这么近．"，be close to为固定搭配表示"接近于……，离……很近"．故选：C．B【考点】副词【解析】上课时不要大声说话．它是一个坏习惯．【解答】考查副词的用法．由题判断题干句意是：上课时不要大声说话．它是一个坏习惯．loud可做副词或形容词，没有C的形式，故排除．loud与loudly都可作副词，表示："响亮地、高声地"，放在动词后面．但loud着重于声音大，loudly则含有"喧闹"或"嘈杂"的意思，有令人讨厌的意味．题干是表示"不要在课堂上喧闹"的意思，loudly符合题意．故选：B．八、句型转换（10分）【答案】four,How many【考点】就划线部分提问【解析】一年有四季．一年有多少个季节？【解答】考查特殊疑问句．特殊疑问句的结构：特殊疑问词+一般疑问句．句子的划线部分是four，是数量，询问多少，确定疑问词为How many，句子的剩余部分变成一般疑问句跟在后面，句子中有系动词are，变一般疑问句are提前即可．用How many seasons are there in a year？提问．故答案为：How many．【答案】How,are【考点】感叹句【解析】我们很高兴．我们多高兴啊！【解答】本题考查感叹句的表达．What相当于一个形容词，修饰的是一个名词结构；How相当于是一个副词，修饰形容词/副词；What+形容词+可数名词复数！＝ How+形容词+可数名词复数+动词！根据句子信息推知，happy是一个形容词，用how引导，we后面跟are．故答案为：How，are．【答案】rainy,What,like【考点】就划线部分提问【解析】今天是有雨的．今天天气怎么样？考查特殊疑问句．特殊疑问句的结构：特殊疑问词+一般疑问句．句子的划线部分是rainy，是天气状况，询问天气怎么样，用What is the weather like…？提问．today放在句末．故答案为：What，like．【答案】No,won't【考点】一般疑问句【解析】你会去故宫吗？不，我们不会的．【解答】本题考查一般疑问句和肯定回答的结构．用Will引导的一般疑问句的肯定回答的结构为：Yes，主语（人称代词）+will．否定回答是．No，主语（人称代词）+won't．故答案为：No，won't．【答案】shouldn'tread【考点】肯定句转否定句【解析】你应该在床上读书．你不应该在床上读书．【解答】考查肯定句转否定句．分析原句，句子中有情态动词should，变否定句在should后加not即可，should not可缩写为shouldn't，原句转变成的否定句为：You shouldn't readin bed．故答案为：shouldn't，read．九、连词成句（注意大小写和标点符号，10分）【答案】You'd better have a good rest．【考点】连词成句【解析】你最好好好休息．【解答】考查连词成句，首先根据所给词汇及标点，可确定组合的句子为陈述句，组合的句意：你最好好好休息．You'd better 你最好，放于句首；have a good rest好好休息，放于句末，结合整句及英语表达的习惯作答．故答案为：You'd better have a good rest．【答案】What will you do tomorrow？【考点】连词成句【解析】你明天要做什么？【解答】问句，组合的句意：你明天要做什么？What什么，特殊疑问词，提出疑问，放于句首；will 助动词，引导一般疑问句，放在特殊疑问词后； you 你，做主语，放于助动词后，主谓倒置；do做，是谓语动词，放于主语后；tomorrow明天，时间副词，常放于句末，结合整句及英语表达的习惯作答．故答案为：What will you do tomorrow？【答案】Don'tworryaboutyourlessons【考点】连词成句【解析】不要担心你的课．【解答】考查连词成句，首先根据所给词汇及标点，可确定组合的句子为否定祈使句，组合的句意：不要担心你的课．Don't 不要，否定词，放于句首；worry about 担心，动词原形，放于否定词后；your lessons你的课，做宾语，放于句末，结合整句及英语表达的习惯作答．故答案为：Don't worry about your lessons．【答案】New Year is in the first month of the year．【考点】连词成句【解析】新年在一年中的第一个月．【解答】本题主要考查句法结构连词成句．句号表示陈述句．New Year意思为：新年，其后接be动词is，意思为：是，in the first month of…意思为：在…的第一个月；the year意思为：年，结合要求完成句子．故答案为：New Year is in the first month of the year．【答案】We should take care of the trees．【考点】连词成句【解析】我们应该照顾树木．【解答】本题主要考查句法结构连词成句．句号表示陈述句．we意思为：我们，其后接should，意思为：应该，take care of意思为：照顾，the trees意思为：树木．结合要求完成句子．故答案为：We should take care of the trees．十、阅读理解（10分）【答案】BAACA【考点】文中细节【解析】今天是星期六．晴朗且很热．Jack和他的朋友去公园踢球．他感觉很热．并且他吃了一个冰激凌．现在，他肚子疼．他妈妈将带他去看医生．他正在担心他的作业．他应该做什么？【解答】（1）B．由文中Today is Saturday．可知今天是星期六．故选：B．（2）A．由文中He feels very hot．可知Jack觉得很热．故选：A．（3）A．由文中Now， he has a stomachache．可知Jack肚子疼．故选：A．（4）C．由文中His mother is going to take him to see a doctor．可知Jack将和妈妈去医院看医生．故选：C．（5）A．由文中He is worrying about his homework．可知Jack在担心他的作业．故选：A．。
2021-2022学年-有答案-陕西省咸阳市某校八年级(上)第二次月考地理试卷
2021-2022学年陕西省咸阳市某校八年级（上）第二次月考地理试卷一、选择题（本大题共30个小题，每小题4分，共60分）1. 如图，完成第（1）～（2）题。
（1）四河流中属于人工运河的是（）A.④B.③C.②D.①（2）四河流冬季无结冰现象的是（）A.①B.②C.③D.④2. 下列河流中，由北向南排列正确的是（）A.黑龙江、黄河、珠江、淮河B.黄河、辽河、淮河、长江C.额尔齐斯河、塔里木河、雅鲁藏布江D.淮河、黄河、长江、珠江3. 下面四条河流中，汛期由长到短排列正确的是（）A.①②③④B.④③②①C.④③①②D.②③④①4. 下列河流的干流与流入的海洋组合正确的是（）A. B.C. D.5. “滚滚长江向东流，流的都是煤石油”，这句话说明了长江流域（）A.煤炭、石油资源丰富且得到利用B.森林资源丰富，砍伐严重C.有色金属矿产丰富D.水能资源丰富，但未得到全面地开发6. 关于长江的叙述，正确的是（）A.是我国汛期最长的河流B.下游江阔水深，有“黄金水道”之称C.中游水土流失严重，河水含沙量大D.受气候影响，结冰期长7. 读漫画“长江与黄河的对话”，下列叙述正确的是（）A.长江与黄河都结冰B.长江与黄河的流量都不大C.长江是名副其实的“黄金水道”D.长江流域生态破坏严重，水土流失导致含沙量增大8. 下列长江的中国河流之最有（）①我国最长的河流②水量最大的河流③流域面积最广的河流④内河货运量最大的河流⑤水能蕴藏量最多的河流⑥干流流经省区最多的河流。
A.①②③B.①②③④C.①②③④⑤D.①②③④⑤⑥9. 长江干流流经的直辖市共有（）A.2个B.3个C.4个D.5个10. 下列河流中跨地势三级阶梯，并穿越三大高原的是（）A.①B.②C.③D.④11. 下列对黄河特征的叙述，不正确的是（）A.“黄河之水天上来”﹣﹣黄河发源于地势高峻的巴颜喀拉山B.“一桶水，半桶泥”﹣﹣黄河泥沙含量大C.“地上河”﹣﹣黄河下游泥沙沉积，河床高出两岸D.“黄金水道”﹣﹣黄河沟通东西交通，运输量大12. 读某省区略图，甲河为流经该省区的最大河流，有关甲河水文特征的叙述正确的是（）A.该河下游河段水流湍急，水能丰富B.该河流经黄土高原，中下游河段含沙量大C.该河为我国第二长河，是我国流量最大的河流D.该河流经暖温带地区，无结冰期13. 图中字母与地形区名称对应错误的是（）A.A青藏高原B.B内蒙古高原C.C四川盆地D.D华北平原14. 俗话说“跳进黄河洗不清”，说明黄河含沙量十分严重．黄河的泥沙主要来自（）A.华北平原B.青藏高原C.内蒙古高原D.黄土高原15. 山脉是我国地形的“骨架”。
2021届高考英 Unit 5 The power of nature课时作业选修6(1)
课时作业(三十)选修6Unit 5The power of nature Ⅰ.单词拼写1．When the train began to move quickly，Helen excitedly________(挥手) at her friends on the platform.2．Most researchers believe that，in theory，mobile phones are________(绝对的) safe.3．My time is________(宝贵的)；I can only give you a few minutes.4．When he came out of the water，he was________(哆嗦) with cold5．Sh e got into a________(惶恐) when she found she had lost her necklace.6．Interest is the best teacher，and a lack of interest is a________(保证) of failure.7．A number of________(可能的) buyers have expressed interest in the company and placed orders for some of its products.8．I think you can come any time in the afternoon，if that________(适合) you.Ⅱ.单项填空1．(2021·四川省南充市高三第一次高考适应性考试)—It’s been a wonderful evening.That’s very kind of you.—________.A．It’s my pleasure B．With pleasureC．Go ahead D．Not really2．(2021·山西高三年级第三次四校联考)No matter how much money you possess，it can’t________a healthy body.A．compare B．suitC．defeat D．match3．(2021·河南豫北三校高三下学期联合考试)—Amy，please tell the people seated and________for their turns to come to my office five minutes later.—Yes，sir.A．wait B．waitingC．waited D．to wait4．(2021·湖南十二校第二次联考)—Don’t throw the waste on the ground.—Oh，I’m very sorry.I________the dustbin there.A．didn’t see B．am not seeingC．don’t see D．haven’t seen5．(2021·安徽江南十校联考)Why waste time going to the bank________you can do it all over the Internet at home？It is really convenient.A．when B．becauseC．though D．unless6．(2021·江西赣州适应性考试)He arrived in the US in 1988，________，five years later，he became a successful businessman.A．that B．whichC．where D．when7．(2021·兰州质检)There are________approaches to English teaching.But not all of them are equally efficient for our Chinese students.A．variety B．accurateC．original D．diverse8．The games are designed to make maths fun and________children’s confidence.A．bring up B．take upC．build up D．break up9．(2021·北京东城期末)Last Monday，I was asked to give the new students a test________their English level.A．evaluating B．to have evaluatedC．to evaluate D．having evaluated10．(2021·浙江卷)The research lacks________evidence，and therefore，its conclusions are doubtful.A．solid B．fierceC．severe D．potential11．Nowadays food safety is a big problem and nobody can guarantee all foods are secure to us，but food produced in the normal companies is________better.A．strictly B．absolutelyC．relatively D．exactly12．He had a talent for music and was________conductor of Shanghai Symphony Orchestra at the age of 30.A．assigned B．appointedC．named D．honored13．(2021·江苏泰州检测)—Will you be able to finish the report before Friday?—I can’t________anything，but I’ll try my best.A．guarantee B．transformC．prepare D．say14．(2021·吉林长春外国语学校月考)In the botanical garden we can find a(n)________of plants that range from tall trees to small flowers.A．species B．groupsC．amount D．variety15．(2021·湖南省六校联考)________the blind wandering through the traffic，motorists would have to step on their brakes at once.A．Seeing B．Having seenC．Seen D．To have seenⅢ.阅读明白得(2021·河南洛阳统一考试)Do you remember we agreed to circulate our ideas for our school project this year？Well，here is our suggestion and we in our school hope you will join us to develop it.The idea came to us in the autumn of 2004.We were sitting in our class when our head teacher made an announcement.“There’ll be a new boy starting school tomorrow.He comes from the circus and he’ll be with us for a while.”Immediately there was whispering around the classroom.I heard remarks like “Those are dirty people”；“They eat raw meat”，and “Watch out for your bags”．The teacher must have heard them because she sto pped the talking and said in a very serious voice，“Jake is our guest for as long as he stays here.Make sure you treat him well.I don’t want to hear any complaints about your behaviour to him.”I was curious because it was the first time I had met someone from a circus.Fortunately for me，the next morning，after being introduced to the class he sat in the only spare seat next to me.Of course I didn’t want to be nosy (好管闲事的) but I couldn’t help looking at him sideways as I began my mathematics task.I noticed Jake was smaller than me and had dirty finger nails.His shirt was clean but had been repaired in many places and his trousers were helped up by a wide，leather belt.As he seemed to have no handkerchief and was sniffing loudly，I passed him one of mine.Immediately the boys began to whisper “What did you do that for？Leave him be！” I concentrated on my work.From my left a slight movement showed that Jake must have heard the remarks，too.He seemed to be having great difficulty with his work and was struggling to hold his pen correctly.He looked around as if for help.I looked around too.The teacher was busy so I leaned over and offered to explain the problems to him.As he listened，his mouth split into a smile.His eyes lit up as if a light had been turned on and he returned to the exercises with enthusiasm.He finished them quickly and proudly presented them to the teacher.“Well done，Jake，” she approved.Jake smiled and was soon busy sorting out information and adding more of his own.I thought about how Jake’s life would be.I remembered seeing the campsite and thinking how poor the facilities (设施) were.So I decided to help.As it was my turn to suggest a project for our grade，I put forward this idea.Our head teacher accepted it and so we began to prepare our plans to improve the campsite.Hope you will join us.1．Why did the students dislike Jake even before they saw him?A．They must have heard bad stories about people in a circus.B．They always judged a person by what he looked like.C．The teacher hadn’t introduced Jake to them beforehand.D．A newcomer was not welcome at that time.2．What’s the author’s attitude towards Jake?A．He copied the other boys’ behavior.B．He was distant about what Jake’s life was like.C．He couldn’t ignore what other people said about Jake.D．He had sympath y for the poor and disadvantaged.3．The author mentions how Jake did maths to show that________.A．the teacher was helpful and friendlyB．the maths problem was too hard for himC．Jake was courageous and determinedD．the author was better at maths4．What’s the author’s purpose in writing this passage?A．To argue people should be treated fairly.B．To ask the reader to join in their project.C．To call on more people to help Jake.D．To show what is in need for a circus.Ⅳ.信息匹配(2021·陕西省咸阳市高考模拟二)A．Set up a meeting.B．Make a shopping list.C．Think in silence.D．Write to your parents.E．Answer useful emails.F．Do things as planned. ,In these early days of 2021，it’s more important than ever to emphasize just how much of a difference the next 15 minutes can make in your day.Then，you perhaps make your new year a perfect future.1.Write a letter or an email of gratitude to your parents.At first，this might sound a little strange，particularly if you don’t have the best relationship with your parents，but write a letter or an email simply thanking your parents for having you.Even if you’ve lost them，even if you don’t know them，even if they aren’t great parents，you can still write this letter.None of us would be here if our parents had not given birth to us，and that’s a unique bond that should not be ignored.You don’t even have to send it if you don’t want to，and the process will be great.2.Make a list of five immediate actions that need to be done in order for progress to be made on the big project(s) on which you are currently working.They don’t have to be big targets，but what you know to be absolute “musts”．T ake massive action on them without hesitation or fear.3.Answer that email you need to get to.You know which one I’m talking about：that email you know is crying out for a response.Maybe it’s workrelated；maybe it’s personal；it doesn’t matter.Respond immediately and clear it from your mind—your focus is needed elsewhere.4.Schedule a meeting that will move you forward.I’m talking about getting together with someone who has information that will be beneficial to you.This doesn’t have to be formal；in fact，it doesn’t even have to be related to your study at all.It can be dinner with a friend from whom you want some advice，for example.Just get that person on the phone，and then get that meeting on the calendar.5.Do absolutely nothing for 15 minutes straight.Yes，choosing to do nothing is，in fact，doing something.Spend 15 minutes in silence，giving your mind and spirit a chance to refresh and regroup.I promise that when the 15 minutes are up you will be able to return to whatever you’re working on with a new perspective.Ⅴ.短文改错(2021·陕西省重点中学二模)A man in Hamburg，Germany，kept getti ng phone，but when he picked up a receiver there was no one on the other end of the phone.To begin with，he paid the phone company block the number.Therefore，it was expensive just as to stop one call，so he phoned the police.The police soon tracked the calls to a barber shop.Which happened was that every time someone paid for the haircut by credit card，the credit card machine dialed t he wrong number in mistake.It was quite beyond the police why somebody at the barber’s or the credit card company noticed the problem.Fortunately，there had a happy outcome and the barber’s credit card phone was reprogrammed.课时作业(三十)Ⅰ.单词拼写1．waved 2.absolutely 3.precious 4.trembling 5.panic6．guarantee 7.potential 8.suitsⅡ.单项填空1．解析：考查交际用语。
陕西省咸阳市周陵中学2021年高二英语月考试题含解析
陕西省咸阳市周陵中学2021年高二英语月考试题含解析一、选择题1. Although her grandma had been ill for a long time, it still came as a shock when she______ died.A. slightlyB. approximatelyC. eventuallyD. directly参考答案：C2. ---- Are you a teacher?---- No, but I ______, and I ______ in No. 1 Middle School for 30 years.A. used to; have workedB. used to be; workedC. am used to be; workD. was used to; had worked参考答案：B3. He gave himself a new name to hide his _____ when he went to carry out the secret task.A. emotionB. identityC. talentD. treasure参考答案：B4. Only those who had tickets were ________ to the stadium for the concert given by the famous musician.A. allowedB. admittedC. committedD. permitted参考答案：B考查动词词义。
根据语境可知句子表示只有有票的人才被允许进入体育场观看著名音乐家的音乐会。
准许进入是be admitted to A. allowed 允许 C. committed作出保证;使承担义务;约束 D. permitted 允许5. All of us , Jane will make a trip to London next month.A. includeB. includedC. includingD. is included参考答案：B6. Those people are trying to change the law to give dangerous drivers more ________ punishment.A. severeB. roughC. cautiousD. positive参考答案：A7. He just graduated from the university last month, so he_____ working experience.A. lacks ofB. lacksC. is lacking ofD. lacking in参考答案：B8. —I am sorry for having been so rude to you.—That’s OK. I am sure you didn’t .A. mean toB. have toC. hope soD. want so参考答案：A略9. We __________ our homework when someone knocked at the door.A. didB. were doingC. have doneD. do参考答案：B10. Witnesses said the security forces use tear gas to protests（抗议）by about 200 people in Teheran, although this can’t be independently confirmed.A. break outB. break downC. break upD. break through参考答案：C11. --Ok, I’ve had enough of it. I give up--You can’t ____ your responsibilities.A.run off withB.run up againstC.run out ofD.run away from参考答案：D12. People who don’t smoke have less _______ of suffering from lung cancer than those who do.A. coincidenceB. loadC.aspect D. potential参考答案：D13. Had he her promise, she would have made it to Yale University.A．looked up to B．lived up toC．kept up with D．come up with参考答案：B考查动词短语辨析。
陕西省咸阳市2020-2021学年四年级下学期语文期末考试试卷D卷
陕西省咸阳市2020-2021学年四年级下学期语文期末考试试卷D卷姓名:________ 班级:________ 成绩:________亲爱的同学，经过一段时间的学习，你们一定学到不少知识，今天就让我们大显身手吧！一、基础知识（32分） (共11题；共35分)1. （10分）给下列多音字组词。
禁：jìn ________jīn ________抹：mǒ ________mò ________mā ________吵：chǎo ________chāo ________露：lù ________lòu ________2. （8分） (2020五上·即墨期末) 下面句子中成语使用错误的一句是________，请换一个合适的词语写在横线上________。
A．新中国成立70周年的国庆大典展现出一个太平盛世。
B．蔺相如诡计多端，秦王只好完璧归赵。
C．学习时要有探究精神，不能一知半解就轻易满足。
3. （1分）下列每组词语中，不是近义词的一项是（）A . 喧哗——喧闹B . 敏捷——便捷C . 哭笑不得——啼笑皆非D . 恍然大悟——豁然开朗4. （1分） (2020四上·盐城月考) 拼音（1）找出下面每道小题中含有错别字的一项。
（）A . 证据B . 竖强C . 睡眠D . 蝙蝠（2）找出下面每道小题中含有错别字的一项。
A . 既使B . 苍蝇C . 成熟D . 研究5. （1分）划线的字解释正确的是（）。
供养不周A . 绕一圈B . 周围C . 普遍D . 周到6. （1分）成年人体内的含水量大约占体重的（）A . 10%——20%B . 80%——90%C . 60%——70%7. （1分）剪不断的绳子。
【道具准备】①长绳一根，短绳一根；②剪刀一把。
【魔术表演】步骤①：展开长绳，把绳子弯折，左手握住弯折处。
步骤②：右手从左手中抽出绳子。
步骤③：用剪刀把露出的绳子剪断。
专题29 空间向量与立体几何(解答题)(新高考地区专用)(解析版)
专题29 空间向量与立体几何（解答题）1．如图，在三棱锥P ABC -中，平面PAC ⊥平面ABC ，PC AC ⊥，BC AC ⊥，2AC PC ==，4CB =，M 是PA 的中点．（1）求证：PA ⊥平面MBC ；（2）设点N 是PB 的中点，求二面角N MC B --的余弦值．【试题来源】陕西省咸阳市2020-2021学年高三上学期高考模拟检测（一）（理）【答案】（1）证明见解析；（2）3． 【解析】（1）平面PAC ⊥平面ABC ，平面PAC平面ABC =AC ，BC ⊂平面ABC ，BC AC ⊥，所以BC ⊥平面PAC ，因为PA ⊂平面PAC ，所以BC PA ⊥，因为AC PC =，M 是PA 的中点，所以CM PA ⊥， 因为CMBC C =，,CM BC ⊂平面MBC ，所以PA ⊥平面MBC ．（2）因为平面PAC ⊥平面ABC ，平面PAC平面ABC =AC ，PC ⊂平面PAC ，PC AC ⊥，所以PC ⊥平面ABC ，因为BC ⊂平面ABC ，所以PC BC ⊥，以C 为原点，CA ，CB ，CP 为x ，y ，z 轴正方向，建立如图所示的空间直角坐标系，(2,0,0)A ，(0,4,0)B ，(0,0,0)C ，(0,0,2)P ，(1,0,1)M ，(0,2,1)N ，则(1,0,1)CM =，(0,2,1)CN =，(2,0,2)PA =-，由（1）知(2,0,2)PA =-是平面MBC 的一个法向量，设(,,)n x y z =是平面MNC 的法向量，则有00CM n CN n ⎧⋅=⎨⋅=⎩，即020x z y z +=⎧⎨+=⎩，令1y =，则2z =-，2x =，所以(2,1,2)n =-，设二面角N MC B --所成角为θ，由图可得θ为锐角，则2cos cos ,||||PA n PA n PA n θ⋅⨯=<>===【名师点睛】解题的关键是熟练掌握面面垂直的性质定理，线面垂直的判定和性质定理，并灵活应用，处理二面角或点到平面距离时，常用向量法求解，建立适当的坐标系，求得所需点的坐标及向量坐标，求得法向量坐标，代入夹角或距离公式，即可求得答案． 2．在四棱锥P ABCD -中，PAB △为直角三角形，90APB ∠=︒且12PA AB CD ==，四边形ABCD 为直角梯形，//AB CD 且DAB ∠为直角，E 为AB 的中点，F 为PE 的四等分点且14EF EP =，M 为AC 中点且MF PE ⊥．（1）证明：AD ⊥平面ABP ；（2）设二面角A PC E --的大小为α，求α的取值范围． 【试题来源】山东省德州市2020-2021学年高三上学期期末 【答案】（1）证明见解析；（2）,32ππα【解析】（1）取PE 的中点N ，连接AN ，DN ，CE ，如图所示：因为12AE AB =，12AP AB =，所以AP AE =，AN PE ⊥．因为四边形ABCD 为直角梯形，且90DAB ∠=︒，12CD AB =， 所以四边形AECD 为正方形，即M 为DE 的中点． 因为14EF EP =，N 为PE 的中点，所以F 为EN 的中点．所以//MF DN ． 因为MF PE ⊥，所以DN PE ⊥．所以PE DN PE ANPE DN AN N ⊥⎧⎪⊥⇒⊥⎨⎪⋂=⎩平面ADN ． 因为DA ⊂平面ADN ，所以PE DA ⊥．所以DA AB DA PEDA PE AB E ⊥⎧⎪⊥⇒⊥⎨⎪⋂=⎩平面ABP ． （2）以A 为原点，AB ，AD 分别为y ，z 轴，垂直AB 的直线为x 轴，建立空间直角坐标系，如图所示：设AD a =，1PA CD ==，2AB =，则()0,0,0A，1,02P ⎫⎪⎪⎝⎭，()0,1,0E ，()0,1,C a ． 31,02AP ⎛⎫= ⎪ ⎪⎝⎭，()0,1,AC a =，1,02PE ⎛⎫=- ⎪ ⎪⎝⎭，()0,0,CE a =-． 设平面PAC 的法向量()111,,n x y z =，则1111310220AP n x yAC n y az ⎧⋅=+=⎪⎨⎪⋅=+=⎩，令1y =，解得11x =，1z =，故1,3,n⎛=- ⎝⎭． 设平面PEC 的法向量()222,,m x y z =，则222310220PE mx y CE m az ⎧⋅=-+=⎪⎨⎪⋅=-=⎩，令2y =21x =，20z =，故()1,3,0m =．由图知，二面角A PC E --的平面角α为锐角，所以11cos 0,2α-⎛⎫==⎪⎝⎭．故,32ππα．3．如图，在四棱锥P ABCD -中，底面ABCD 为直角梯形，AD BC ∥，112BC AD ==且CD =E 为AD 的中点，F 是棱PA 的中点，2PA =，PE ⊥底面ABCD ．AD CD ⊥（1）证明：//BF平面PCD ； （2）求二面角P BD F --的正弦值；（3）在线段PC （不含端点）上是否存在一点M ，使得直线BM 和平面BDF 所成角的正弦值为13？若存在，求出此时PM 的长；若不存在，说明理由． 【试题来源】天津市滨海七校2020-2021学年高三上学期期末联考 【答案】（1）证明见解析；（2（3）存在，7PM = 【解析】（1）由题意得//BC DE ，=BC DE ，90ADC ∠=︒，所以四边形BCDE 为矩形， 又PE ⊥面ABCD ，如图建立空间直角坐标系E xyz -，则()0,0,0E ，()1,0,0A，()B ，()1,0,0D -，(P ，()C -，1,0,22F ⎛ ⎝⎭，设平面PCD的法向量为(),,m x y z=，()0,DC =，(DP =则00DC m DP m ⎧⋅=⎨⋅=⎩，则0x ==⎪⎩，则0y =，不妨设x =1z =，可得()3,0,1m =-，又1,22BF ⎛⎫= ⎪ ⎪⎝⎭，可得0BF m ⋅=，因为直线BF ⊄平面BCD ，所以//BF 平面BCD ．（2）设平面PBD 的法向量为()1111,,x n y z =，()1,DB =，(0,BP =，则1100DB n BP n ⎧⋅=⎪⎨⋅=⎪⎩，即111100x ⎧+=⎪⎨+=⎪⎩，不妨设x =()13,1,1n =--，设平面BDF 的法向量为()2222,,n xy z =，32DF ⎛= ⎝⎭，则2200DB n DF n ⎧⋅=⎪⎨⋅=⎪⎩，即222203022x x z ⎧+=⎪⎨+=⎪⎩，不妨设2x =，可得()2n =-，因此有121212cos ,65n n n n n n ⋅==-⋅，（注：结果正负取决于法向量方向） 于是21212465sin ,1cos ,n n n n =-=，所以二面角P BD F --．（3）设((),PM PC λλλ==-=-，()0,1λ∈(),BM BP PM λ=+=-，由（2）可知平面BDF 的法向量为()23,1,3n =-，2223cos ,BM n BM n BM n⋅===⋅，有23410λλ-+=，解得1λ=（舍）或13λ=， 可得1,333PM ⎛=-- ⎝⎭，所以73PM =． 4．在四棱锥P ABCD -中，PA ⊥平面ABCD ，PA =//DC AB ，90DAB ∠=︒，3AB =，2AD CD ==，M 是棱PD 的中点．（1）求异面直线DP 与BC 所成的角的余弦值； （2）求AM 与平面PBC 所成的角的大小；（3）在棱PB 上是否存在点Q ，使得平面QAD 与平面ABCD 所成的锐二面角的大小为60°？若存在，求出AQ 的长；若不存在，说明理由．【试题来源】天津市南开中学2020-2021学年高三上学期第四次月考 【答案】（1；（2）45︒；（3）125． 【解析】如图，以,,AD AB AP 所在直线分别为,,x y z 轴建立如图所示空间直角坐标系，则(()()()()(,0,0,0,3,0,0,2,2,0,0,2,0,P A B C D M ，（1）(0,DP =-，()1,2,0BC =-，所以cos,DP BC==，即异面直线DP与BC（2）(AM=，(3,0,PB=-，()1,2,0BC=-设平面PBC的法向量(),,m x y z=，则mPBm BC⎧⋅=⎨⋅=⎩，3020xx y⎧-=⎪⎨-+=⎪⎩，所以可取(m=，设AM与平面PBC所成的角为θ，则sin cos,AM mθ===，所以AM与平面PBC所成的角为45︒；（3）平面ABCD的法向量可取()10,0,1n=，设(()3,0,3,0,PQ PBλλλ==-=-，则()3Qλ，所以()3AQλ=，()0,2,0AD =，设平面QAD的法向量为()2222,,n x y z=，则22nAQn AD⎧⋅=⎪⎨⋅=⎪⎩，()2223020x zyλ⎧+=⎪⎨=⎪⎩，可取()223,0,3nλ=-，因为平面QAD与平面ABCD所成的锐二面角的大小为60°.所以121cos,2n n=，12=，解得25λ=或2λ=-（舍）所以6,0,55AQ⎛=⎝⎭，所以61255AQ⎛==5．如图，在正四面体A BCD-中，点E，F分别是,ABBC的中点，点G，H分别在,CD AD 上，且14DH AD=，14DG CD=．（1）求证：直线,EH FG 必相交于一点，且这个交点在直线BD 上； （2）求直线AB 与平面EFGH 所成角的正弦值．【试题来源】陕西省榆林市2020-2021学年高三上学期第一次高考模拟测试（理） 【答案】（1）证明见解析；（2． 【解析】（1）因为//,//EF AC GH AC ，11=,=24EF AC GH AC ，所以//GH EF 且12GH EF =，故E ，F ，G ，H 四点共面，且直线,EH FG 必相交于一点，设=EH FG M ，因为,∈M EH EH平面ABD ，所以M ∈平面ABD ，同理：M ∈平面BCD ，而平面ABD ⋂平面BCD BD =，故M ∈平面BCD ，即直线,EH FG 必相交于一点，且这个交点在直线BD 上； （2）取BD 的中点O ，则,⊥⊥BD OA BD OC ，所以BD ⊥平面AOC ，不妨设OD =，则BD AC ==12AO CO ==， 所以1441441921cos 212123+-∠==⨯⨯AOC ，以O 为坐标原点建立如图所示的空间直角坐标系，则(0,(12,0,0),(6,--A B C F G ，故=BA ，(=-FG ，(8,0,=-AC ，(4,0,=-EF ，设平面EFGH 的法向量为(,,)n x y z =，由00n EF n FG ⎧⋅=⎨⋅=⎩可得50y x ⎧+=⎪⎨-=⎪⎩，令x =，则(52,=n ，则182cos ,3||||92⋅<>===⨯BA n BA n BA n ，故直线AB 与平面EFGH ． 6．如图，已知四边形ABCD 为菱形，对角线AC 与BD 相交于O ，60BAD ∠=︒，平面ADEF平面BCEF =直线EF ，FO ⊥平面ABCD ，22BC CE DE EF ====（1）求证：//EF DA ；（2）求二面角A EF B --的余弦值．【试题来源】江西省五市九校协作体2021届高三第一次联考 【答案】（1）证明见解析；（2）35． 【解析】（1）因为四边形ABCD 为菱形，所以//AD BC ，AD ⊄平面BCEF ，BC ⊂平面BCEF ，//AD ∴平面BCEF ，因为平面ADEF平面BCEF =直线,EF AD ⊂平面ADEF ，所以//EF AD ；（2）因为四边形ABCD 为菱形，所以AC BD ⊥，因为OF ⊥平面ABCD ，所以以O 为坐标原点、OA ，OB ，OF 为x ，y ，z 轴建立空间直角坐标系，取CD 中点M ，连EM ，OM ，60BAD ︒∠=，21BC OA OC OB OD =∴====，2BC CD CE DE CDE ====∴为正三角形，EM =11//,=,//,=22OM BC OM BC EF BC EF BC，//,=//,=EF OM EF OM OF EM OF EM∴∴，从而1(0,1,0),((0,1,0),(22A B C D E---，设平面ADEF一个法向量为(,,)m x y z=，则m DAm DE⎧⋅=⎨⋅=⎩，即12yx y⎧+=⎪⎨+=⎪⎩，令11,(1,x y z m=∴===-，设平面BCEF一个法向量为(,,)n x y z=，则n BCn EC⎧⋅=⎨⋅=⎩，即122yx y⎧-=⎪⎨-+-=⎪⎩，令11,(1,3,1)x y z n=∴==-=--，3cos,5|||,|m nm nm n⋅∴<>==，因此二面角A EF B--的余弦值为35．7．如图，在四棱锥P ABCD-中，90BAD∠=，//AD BC，PA AD⊥，PA AB⊥，122PA AB BC AD====．（1）求证：//BC平面PAD；（2）求平面PAB与平面PCD所成锐二面角的余弦值．【试题来源】北京房山区2021届高三上学期数学期末试题【答案】（1）证明见解析；（2【解析】（1）解法1．因为//BC AD，BC⊄平面PAD，AD⊂平面PAD，所以//BC平面PAD，解法2．因为PA AD⊥，PA AB⊥，AD AB⊥，所以以A为坐标原点，,,AB AD AP所在直线分别为x轴、y轴、z轴，建立如图所示空间直角坐标系A xyz-，则(0,0,0),(2,0,0),(0,4,0),(0,0,2),(2,2,0)A B D P C ，平面PAD 的法向量为(1,0,0)t ， (0,2,0)BC = ，因为 0120000t BC ⋅=⨯+⨯+⨯= ，BC ⊄平面PAD ，所以//BC 平面PAD ； （2）因为PA AD ⊥，PA AB ⊥AD AB ⊥，所以以A 为坐标原点，,,AB AD AP 所在直线分别为x 轴、y 轴、z 轴，建立如图所示空间直角坐标系A xyz -，则(0,0,0),(2,0,0),(0,4,0),(0,0,2),(2,2,0)A B D P C所以平面PAB 的法向量为(0,1,0)n = ， 设平面PCD 的法向量为(,,)m x y z =， (2,2,2)PC =-，(0,4,2)PD =- ，所以2220042020x y z x y m PC m PC y z z y m PD m PD ⎧⎧+-==⎧⎧⊥⋅=⇒⇒⇒⎨⎨⎨⎨-==⊥⋅=⎩⎩⎩⎩，令1(1,1,2)y m ==得 ，cos ,1n mn m n m ⋅<>===⨯，设平面PAB 与平面PCD 所成角为θθ，为锐角， 所以cos θ=． 8．如图，在四棱锥P ABCD -中，底面ABCD 为菱形，平面PAD ⊥平面ABCD ，PA PD ⊥，PA PD =，3BAD π∠=，E 是线段AD 的中点，连结BE ．（1）求证：BE PA ⊥；（2）求二面角A PD C --的余弦值；（3）在线段PB 上是否存在点F ，使得//EF 平面PCD ？若存在，求出PF PB 的值；若不存在，说明理由．【试题来源】北京市朝阳区2021届高三上学期期末数学质量检测试题【答案】（1）证明见解析；（2）7-；（3）存在；12PF PB =． 【解析】（1）因为四边形ABCD 为菱形，所以AB AD =．因为3BAD π∠=，E 为AD 的中点，所以BE AD ⊥． 因为平面PAD ⊥平面ABCD ，平面PAD平面ABCD AD =，所以BE ⊥平面PAD ． 因为PA ⊂平面PAD ，所以BE PA ⊥．（2）连结PE ．因为PA PD =，E 为AD 的中点，所以PE AD ⊥．由（1）可知BE ⊥平面PAD ，所以BE AD ⊥，PE BE ⊥．设2AD a =，则PE a =．如图，建立空间直角坐标系E xyz -．所以(,0,0),,0),(2,0),(,0,0),(0,0,)A a B C a D a P a --．所以),0(D C a =-，(,0,)D a P a =．因为BE ⊥平面PAD ，所以(0,,0)EB =是平面PAD 的一个法向量．设平面PCD 的法向量为(,,)x y z =n ，则00n DC n DP ⎧⋅=⎨⋅=⎩，即00ax ax az ⎧-+=⎪⎨+=⎪⎩，所以,.x x z ⎧=⎪⎨=-⎪⎩令3x =，则1y =，z =(3,1,n =．所以cos ,||||7n EB n EB n EB ⋅===．由题知，二面角A PD C --为钝角，所以其余弦值为- （3）当点F 是线段PB 的中点时，//EF 平面PCD ．理由如下： 因为点E ∈/平面PCD ，所以在线段PB 上存在点F 使得//EF 平面PCD 等价于0EF ⋅=n ．假设线段PB 上存在点F 使得//EF 平面PCD ．设([0,1])PF PBλλ=∈，则PF PB λ=．所以(0,0,),),)EF EP PF EP PB a a a a a λλλ=+=+=+-=-．由)0EF a a a λ⋅=-=n ，得12λ=． 所以当点F 是线段PB 的中点时，//EF 平面PCD ，且12PF PB =． 9．如图，在四棱锥P ABCD -中，PD ⊥平面ABCD ，4PD =，底面ABCD 是边长为2的正方形，E ，F 分别为PB ，PC 的中点．（1）求证：平面ADE ⊥平面PCD ；（2）求直线BF 与平面ADE 所成角的正弦值．【试题来源】北京市东城区2021届高三上学期期末考试【答案】（1）证明见解析；（2）15． 【解析】（1）因为PD ⊥平面ABCD ，所以PD AD ⊥．因为底面ABCD 是正方形，所以AD CD ⊥．因为PD CD D ⋂=，所以AD ⊥平面PCD ．因为AD ⊂平面ADE ，所以平面ADE ⊥平面PCD ．（2）因为PD ⊥底面ABCD ，所以PD AD ⊥，PD CD ⊥．因为底面ABCD 是正方形，所以AD CD ⊥．如图建立空间直角坐标系D xyz -．因为4PD =，底面ABCD 为边长为2的正方形，所以()0,0,4P ，()2,0,0A ，()2,2,0B ，()0,2,0C ，()0,0,0D ，()1,1,2E ，()0,1,2F ． 则()2,0,0DA =，()1,1,2DE =，()2,1,2BF =--．设平面ADE 的法向量(),,m x y z =，由00m DA m DE ⎧⋅=⎨⋅=⎩，可得2020x x y z =⎧⎨++=⎩． 令1z =-，则0x =，2y =．所以()0,2,1m =-．设直线BF 与平面ADE 所成角为θ，则,sincos ,9BF mBF m BF m θ====．所以直线BF 与平面ADE ． 【名师点睛】本题考查了面面垂直的判定，核心是要求面面垂直，先考虑线面垂直；同时也考查了线面角的计算方法，核心是要求正弦值，先求余弦值．10．如图，已知11ABB A 是圆柱1OO 的轴截面，O 、1O 分别是两底面的圆心，C 是弧AB 上的一点，30ABC ∠=，圆柱的体积和侧面积均为4π．（1）求证：平面1ACA ⊥平面1BCB ；（2）求二面角11B A B C --的大小．【试题来源】江西省吉安市2021届高三大联考数学（理）（3-2）试题【答案】（1）证明见解析 ；（2）60 ．【解析】（1）因为1AA 是圆柱的母线，所以1AA ⊥平面ABC ，因为BC ⊂平面ABC ， 所以1AA BC ⊥，又C 是弧AB 上的一点，且AB 是圆O 的直径，所以AC BC ⊥，因为1AA AC A =，所以BC ⊥平面1ACA ，又BC ⊂平面1BCB ，所以平面1ACA ⊥平面1BCB ；（2）设圆柱的底面半径为r ，母线长为l ，因为圆柱的体积和侧面积均为4π，所以2244rl r l ππππ=⎧⎨=⎩，解得，2r ，1l =，即4AB =，11AA =，因为30ABC ∠=，所以2AC =，BC =设圆柱过C 点的母线为CD ，以C 为原点，CA ，CB ，CD 所在直线分别为x 轴､y 轴､z 轴建立空间直角坐标系C xyz -，如图所示；则()0,0,0C ，()B ，()12,0,1A ，()1B ；所以()12,0,1CA =，()10,CB =，()12,BA =-，()10,0,1BB = 设平面11CA B 的法向量为(),,n x y z =，由1120000x z n CA n CB z ⎧+=⎧⋅=⎪⎪⇒⎨⎨⋅=+=⎪⎪⎩⎩，取z =x =1y =-，所以平面11CA B的一个法向量为(3,n =--， 设平面11BA B 的法向量为(),,m a b c=，由1102000m BA a c m BB c ⎧⎧⋅=-+=⎪⎪⇒⎨⎨⋅==⎪⎪⎩⎩， 取1b =，则a =0c ，所以平面11BA B 的一个法向量为()3,1,0m =， 所以1cos ,23n mm n n m ⋅===-+⋅， 由图中可看出二面角11B A B C --是锐角，故二面角11B A B C --的值为60．【名师点睛】证明面面垂直的方法：（1）利用面面垂直的判定定理，先找到其中一个平面的一条垂线，再证明这条垂线在另外一个平面内或与另外一个平面内的一条直线平行即可； （2）利用性质：//,αββγαγ⊥⇒⊥（客观题常用）；（3）面面垂直的定义（不常用）； （4）向量方法：证明两个平面的法向量垂直，即法向量数量积等于0．11．如图1，正方形ABCD ，边长为a，,E F 分别为,AD CD 中点，现将正方形沿对角线AC 折起，折起过程中D 点位置记为T ，如图2．（1）求证：EF TB ⊥；（2）当60TAB ︒∠=时，求平面ABC 与平面BEF 所成二面角的余弦值．【试题来源】安徽省黄山市2020-2021学年高三上学期第一次质量检测（理）【答案】（1）证明见解析；（2． 【解析】（1）取AC 中点O ，连,,OT OB BT ，因为ABCD 为正方形，所以,AC OT AC OB ⊥⊥，又OT OB O ⋂=，所以AC ⊥平面OBT ，而TB ⊂平面OBT ，所以AC TB ⊥． 又,E F 分别为,AD CD 中点，所以//EF AC ，所以EF TB ⊥；（2）因为60TAB ︒∠=，所以TAB △为等边三角形，TB a =，又2OT OB a ==，所以222TB OB OT =+，即OT OB ⊥． 如图建立空间直角坐标系O xyz -，则,0,0,0,,B E F ⎫⎛⎛⎪ ⎝⎭⎝⎭⎝⎭，220,,0,,,2244EF a EB a ⎛⎫⎛⎫==- ⎪ ⎪⎝⎭⎝⎭，平面ABC 法向量(0,0,1)m =设平面BEF 法向量(,,1)x n y =，由00n EF n EB ⎧⋅=⎨⋅=⎩，00244y ay =⎧+-=⎩，012y x =⎧⎪⎨=⎪⎩，1,0,1,cos ,2||||11mn n m n m n ⋅⎛⎫=<>=== ⎪⋅⎝⎭⋅， 记平面ABC 与平面BEF 所成二面角为θ，则θ为锐角，所以cos 5θ=即平面ABC 与平面BEF ． 12．如图所示，四棱柱1111ABCD A B C D -的底面是菱形，侧棱垂直于底面，点E ，F 分别在棱1AA ，1CC 上，且满足113AE AA =，113CF CC =，平面BEF 与平面ABC 的交线为l ．（1）证明：直线l ⊥平面1BDD ；（2）已知2EF =，14BD =，设BF 与平面1BDD 所成的角为θ，求sin θ的取值范围．【试题来源】海南省2021届高三年级第二次模拟考试【答案】（1）证明见解析；（2）35⎫⎪⎪⎝⎭．【解析】（1）如图，连接AC ，与BD 交于点O ．由条件可知//AE CF ，且AE CF =，所以//AC EF ，因为EF ⊂平面BEF ，所以//AC 平面BEF ．因为平面BEF 平面ABC l =，所以//AC l ． 因为四棱柱1111ABCD A B C D -的底面是菱形，且侧棱垂直于底面，所以AC BD ⊥，1AC BB ⊥，又1BD BB B ⋂=，所以AC ⊥平面1BDD ，所以l ⊥平面1BDD ．（2）如图所示，以O 为坐标原点，分别以OB ，OC 的方向为x ，y 轴的正方向建立空间直角坐标系．设2BD a =，因为1BD BD <，所以02a <<．则OB a =，1DD ==所以(,0,0)B a ，(0,1,0)C，F ⎛ ⎝． 由（1）可知(0,1,0)OC =是平面1BDD的一个法向量，而BF a ⎛=- ⎝， 所以sin cos ,OC BF OC BF OC BF θ⋅=<>===当02a <<35<<，即3sin 5θ⎫∈⎪⎪⎝⎭．【名师点睛】求空间角的常用方法：（1）定义法，由异面直线所成角、线面角、二面角的定义，结合图形，作出所求空间角，再结合题中条件，解对应三角形，即可求出结果；（2）向量法：建立适当的空间直角坐标系，通过计算向量夹角（直线方向向量与直线方向向量、直线方向向量与平面法向量，平面法向量与平面法向量）余弦值，即可求出结果．13．在三棱柱111ABC A B C -中，1AB AC ==，1AA =AB AC ⊥，1B C ⊥平面ABC ，E 是1B C 的中点．（1）求证：平面1AB C ⊥平面11ABB A ；（2）求直线AE 与平面11AAC C 所成角的正弦值．【试题来源】浙江省宁波市2020-2021学年高三上学期期末【答案】（1）证明见解析；（2【解析】（1）由1B C ⊥平面ABC ，AB平面ABC ，得1AB B C ⊥， 又AB AC ⊥，1CB AC C =，故AB ⊥平面1AB C ， AB 平面11ABB A ，故平面11ABB A ⊥平面1AB C ．（2）以C 为原点，CA 为x 轴，1CB 为z 轴，建立如图所示空间直角坐标系， 则()0,0,0C ，()1,0,0A ，()1,1,0B，又BC =11BB AA == 故11CB =，()10,0,1B ，10,0,2E ⎛⎫ ⎪⎝⎭，()1,0,0CA =， ()111,1,1AA BB ==--，11,0,2AE ⎛⎫=- ⎪⎝⎭，设平面11AAC C 的一个法向量为(),,n x y z =，则100n CA n AA ⎧⋅=⎪⎨⋅=⎪⎩，即00x x y z =⎧⎨--+=⎩，令1y =，则1z =， ()0,1,1n =， 设直线AE 与平面11AAC C 所成的角为θ，故1sin 102nAEn AE θ⋅===，即直线AE 与平面11AAC C14．如图，在平面四边形PABC 中，PA AC ⊥，AB BC ⊥，PA AB ==，2AC =，现把PAC △沿AC 折起，使P 在平面ABC 上的射影为O ，连接OA 、OB ，且OB//AC ．（1）证明：OB ⊥平面PAO ；（2）求二面角O PB C --的余弦值．【试题来源】安徽省六安市示范高中2020-2021学年高三上学期教学质量检测（理）【答案】（1）证明见解析；（2） 【解析】（1）PO ⊥平面ABC ，AC ⊂平面ABC ，PO AC ∴⊥，又PA AC ⊥，PAPO P =，所以AC ⊥平面PAO ， //OB AC ，所以OB ⊥平面PAO ；（2）在Rt ABC 中，AB =2AC =，则1BC ==，30BAC ∴∠=，在Rt OAB 中，903060OAB ∠=-=，所以12OA AB ==，32OB =，Rt PAO 中，PA =AO =32OP ∴==， 以点O 为坐标原点，OB 、OA 、OP 所在直线分别为x 、y 、z 轴建立空间直角坐标系O xyz -，则0,,02A ⎛⎫ ⎪ ⎪⎝⎭、,02C ⎛⎫ ⎪ ⎪⎝⎭、3,0,02B ⎛⎫ ⎪⎝⎭、30,0,2P ⎛⎫ ⎪⎝⎭，所以33,0,22PB ⎛⎫=- ⎪⎝⎭，32PC ⎛⎫=- ⎪ ⎪⎝⎭，由（1）可知()0,1,0m =为平面POB 的一个法向量，设平面平PBC 的法向量为(),,n x y z =，则有330223202x z x y z ⎧-=⎪⎪⎨⎪-=⎪⎩y x z x ⎧=⎪⇒⎨⎪=⎩，取x =(3,n =-，cos ,717m n m n m n ⋅===-⋅⨯， 由图可知，二面角O PB C --为钝角，所以，二面角O PB C --的余弦值为7-． 15．在四棱锥P ABCD -中，平面PAD ⊥平面ABCD ，底面ABCD 为直角梯形，//,90BC AD ADC ∠=︒，11,2BC CD AD E ===为线段AD 的中点，过BE 的平面与线段,PD PC 分别交于点,G F ．（1）求证：GF ⊥平面PAD ；（2）若PA PD ==G为PD 的中点，求平面PAB 与平面BEGF所成锐二面角的余弦值．【试题来源】安徽省名校2020-2021学年高三上学期期末联考（理）【答案】（1）证明见解析；（2．【解析】证明：（1）因为12BC AD =，且E 为线段AD 的中点，所以BC DE =， 因为//BC AD ，所以四边形BCDE 为平行四边形，所以//BE CD ，因为CD ⊂平面,PCD BE ⊂/平面PCD ，所以//BE 平面PCD ，又平面BEGF ⋂平面PCD GF =，所以//BE GF ，又BE AD ⊥，且平面PAD ⊥平面ABCD ，平面PAD平面ABCD AD =， 所以BE ⊥平面PAD ，所以GF ⊥平面PAD ；（2）因为,PA PD E =为线段AD 的中点，所以PE AD ⊥，‘’因为平面PAD ⊥平面ABCD ，所以PE ⊥平面ABCD ，以E 为坐标原点，EA 的方向为x 轴正方向建立如图所示的空间直角坐标系E xyz -；则11(0,0,1),(1,0,0),(0,1,0),(0,0,0),(1,0,0),,0,22P A B E D G ⎛⎫--⎪⎝⎭， 则11(1,0,1),(0,1,1),(0,1,0),(1,0,1),,0,22PA PB BE DP EG ⎛⎫=-=-=-==- ⎪⎝⎭， 设平面PAB 的法向量为()111,,m x y z =，则0{0PA m PB m ⋅=⋅=，，，即11110,0x z y z -=⎧⎨-=⎩， 不妨令11x =，可得(1,1,1)n =为平面BEGF 的一个法向量，设平面BEGF 的法向量为()222,,n x y z =，则0{0BE n EG n ⋅=⋅=，，，即222011022y x z =⎧⎪⎨-+=⎪⎩，，不妨令21x =，可得(1,0,1)n =为平面BEGF 的一个法向量，设平面PAB 与平面BEGF 所成的锐二面角为α，于是有2cos |cos ,|32m n α=〈〉==； 所以平面PAB 与平面BEGF ．16．如图所示，在四棱锥S ABCD -中，底面ABCD 是正方形，对角线AC 与BD 交于点F ，侧面SBC 是边长为2的等边三角形，点E 在棱BS 上．（1）若//SD 平面AEC ，求SE EB的值； （2）若平面SBC ⊥平面ABCD ，求二面角B AS C --的余弦值．【试题来源】江苏省G4（苏州中学、常州中学、盐城中学、扬州中学）2020-2021学年高三上学期期末联考【答案】（1）1；（2． 【解析】（1）连结EF ，因为//SD 平面AEC ，SD ⊂平面BSD ，平面BSD ⋂平面AEC EF =，所以//SD EF ．因为底面ABCD 是正方形，F 为AC 中点，所以EF 是SD 的中位线，则1SE EB=． （2）取BC 的中点为O ，AD 的中点为M ，连结MO ，则MO BC ⊥， 因为平面SBC ⊥平面ABCD ，平面SBC平面ABCD BC =，OM ⊂平面ABCD ， 所以OM ⊥平面SBC ．又OS BC ⊥，所以O 为坐标原点．以{},,OS OC OM 为正交基底建立空间直角坐标系O xyz -．则()0,1,2A -，()010B -,,，()0,1,0C，)S，1,022E ⎛⎫- ⎪ ⎪⎝⎭，从而()SC =-，()0,2,2AC =-，()0,0,2AB =-，()3,1,2AS =-． 设平面ASC 的法向量为(),,m x y z =， 则0,0.m SC m AC ⎧⋅=⎪⎨⋅=⎪⎩，即0,0.y y z ⎧+=⎪⎨-=⎪⎩取1x =，则y =z = 所以平面ASC的一个法向量为(1,3,m =．设平面ASB 的法向量为(),,n x y z =， 则0,0.n AB n AS ⎧⋅=⎪⎨⋅=⎪⎩，即20,20.z y z -=⎧⎪+-=取y =1x =-，0z =． 所以平面ASB 的一个法向量为()1,3,0n =-．所以7cos ,7m n m n m n ⋅〈〉==． 因为二面角B AS C --的平面角为锐角，所以二面角B AS C --的余弦值为7． 【名师点睛】本题的核心在考查空间向量的应用，需要注意以下问题：（1）求解本题要注意两点：一是两平面的法向量的夹角不一定是所求的二面角，二是利用方程思想进行向量运算，要认真细心，准确计算．（2）设,m n 分别为平面α，β的法向量，则二面角θ与,m n <>互补或相等．求解时一定要注意结合实际图形判断所求角是锐角还是钝角．17．在三棱锥P ABC -中，底面ABC 为正三角形，平面PBC ⊥平面,1,ABC PB PC D ==为AP 上一点，2,AD DP O =为三角形ABC 的中心．（1）求证：AC ⊥平面OBD ；（2）若直线PA 与平面ABC 所成的角为45︒，求二面角A BD O --的余弦值．【试题来源】山东省威海市2020-2021学年高三上学期期末【答案】（1）证明见解析；（2）5． 【解析】（1）证明：连接AO 并延长BC 交于点E ，则E 为BC 中点，连接PE ．如图所示：因为О为正三角形ABC 的中心，所以2,AO OE =又2AD DP =，所以//,DO PE 因为PB PC =，E 为BC 中点，所以,PE BC ⊥ 又平面PBC ⊥平面ABC ，平面PBC 平面ABC BC =，所以PE ⊥平面,ABC 所以DO ⊥平面,ABC AC ⊂平面PBC ，所以,DO AC ⊥又,AC BO DO BO O ⊥⋂=，所以AC ⊥平面OBD ．（2）由PE ⊥平面ABC 知，所以45PAE ∠=︒ ，所以,PE AE =所以,ABE PBE ≌ 所以1AB PB BC AC ====，由（1）知，,,EA EB EP 两两互相垂直，所以分别以,,EA EB EP 的方向为,,x y z 轴正方向，建立如图所示空间直角坐标系，则1,0,,0,0,0,,22263A B P D ⎛⎫⎛⎫⎛⎛⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎝⎭⎝⎭⎝⎭⎝⎭，10,,02C ⎛⎫- ⎪⎝⎭所以31,,0,2231,,623AB BD ⎛⎫-⎛= ⎪ ⎪⎝⎭=-⎝⎭， 设平面ABD 的法向量为(),,n x y z =， 则302302x y nBD z y n AB x ⎧⋅=-=⎪⎪⎨⎪⋅=-+=⎪⎩，令1,x =可得1y z ==，则()1,3,1n =． 由（1）知AC ⊥平面,DBO 故1,02AC ⎛⎫=-- ⎪ ⎪⎝⎭为平面DBO 的法向量， 所以2cos ,5nAC n AC n AC -⋅===-，由图可知二面角A BD O --的为锐二面角，所以二面角A BDO --的余弦值为5． 18．如图，在几何体ABCDEF 中，四边形ABCD 为等腰梯形，且22AB CD ==，60ABC ∠=︒，四边形ACFE 为矩形，且FB =，M ，N 分别为EF ，AB 的中点．（1）求证：//MN 平面FCB；（2）若直线AF 与平面FCB 所成的角为60°，求平面MAB 与平面MAC 所成锐二面角的余弦值．【试题来源】山西省运城市2021届高三上学期期末（理）【答案】（1）证明见解析；（2．【解析】（1）取BC 的中点Q ，连接NQ ，FQ ，则1//2NQ AC ，且12NQ AC =， 又1//2MF AC ，且12MF AC = ，所以//MF NQ 且MF NQ =， 所以四边形MNQF 为平行四边形，所以//MN FQ ，因为FQ ⊂平面FCB ，MN ⊄平面FCB ，所以//MN 平面FCB ；（2）由四边形ABCD 为等腰梯形，且22AB CD ==，60ABC ∠=︒，可得1BC =，AC =90ACB ∠=︒，所以AC BC ⊥．因为四边形ACFE 为矩形，所以AC CF ⊥，所以AC ⊥平面FCB ，所以AFC ∠为直线AF 与平面FCB 所成的角，即60AFC ∠=︒，所以1FC =．因为FB =，所以222FB FC CB =+，所以FC BC ⊥．则可建立如图所示的空间直角坐标系C xyz -，3(3,0,0),(0,1,0),,0,12A B M ⎛⎫ ⎪⎝⎭，所以3,0,1,(3,1,0)2MA AB ⎛⎫=-=- ⎪⎝⎭，设(,,)m x y z =为平面MAB 的法向量，则00MA m AB m ⎧⋅=⎨⋅=⎩，即30230x z x y ⎧-=⎪⎨⎪-+=⎩，取23x =，则(23,6,3)m =为平面MAB 的一个法向量，又(0,1,0)n =为平面MAC的一个法向量， 所以657257cos 571||m n mn m n ⋅〈〉====⨯∣∣， 故平面MAB 与平面MAC 所成锐二面角的余弦值为5719． 19．如图，该多面体由底面为正方形ABCD 的直四棱柱被截面AEFG 所截而成，其中正方形ABCD 的边长为4，H 是线段EF 上（不含端点）的动点，36==FC EB ．（1）若H 为EF 的中点，证明：//GH 平面ABCD ；（2）若14=EH EF ，求直线CH 与平面ACG 所成角的正弦值． 【试题来源】河南省驻马店市2020-2021学年高三上学期期末考试（理） 【答案】（1）证明见解析；（26． 【解析】（1）证明：取BC 的中点M ，连接HM ，DM ．因为该多面体由底面为正方形ABCD 的直四棱柱被截面AEFG 所截而成，所以截面AEFG 是平行四边形，则4=-=DG CF EB ．因为36==FC EB ，所以1(26)42=⨯+=HM ，且DG//HM ，所以四边形DGHM 是平行四边形，所以GH //DM ．因为DM ⊂平面ABCD ，GH ⊄平面ABCD ，所以//GH 平面ABCD ．（2）解：如图，以D 为原点，分别以DA ，DC ，DG 的方向为x 轴、y 轴、z 轴的正方向，建立空间直角坐标系D xyz -，则(4,0,0)A ，(0,4,0)C ，(0,0,4)G ，(3,4,3)H ，(4,4,0)=-AC ，(4,0,4)=-AG ，(3,0,3)=CH ．设平面ACG 的法向量为(,,)n x y z =，则440440AC n x y AG n x z ⎧⋅=-+=⎨⋅=-+=⎩，令1x =，得(1,1,1)n =．因为cos ,3||||32⋅〈〉===⨯CH n C n n CH H ，所以直线CH 与平面ACG 所成角的正弦值为3．【名师点睛】本题考查了立体几何中的线面平行的判定和线面角的求解问题，意在考查学生的空间想象能力和逻辑推理能力；解答本题关键在于能利用直线与直线、直线与平面关系的相互转化，通过严密推理证明线线平行从而得线面平行，同时对于立体几何中角的计算问题，往往可以利用空间向量法，通过求解平面的法向量，利用向量的夹角公式求解．20．如图，已知四边形ABCD 和BCEG 均为直角梯形，//AD BC ，//CE BG ，且2BCD BCE π∠=∠=，120ECD ∠=︒．22BC CD CE AD BG ====．（1）求证：//AG 平面BDE ；（2）求二面角E BD C --的余弦值．【试题来源】安徽省蚌埠市2020-2021学年高三上学期第二次教学质量检查（理）【答案】（1）证明见解析；（2 【解析】（1）证明：在平面BCEG 中，过G 作GN CE ⊥于N ，交BE 于M ，连DM ， 由题意知，MG MN =，////MN BC DA 且12MN AD BC ==， 因为//MG AD ，MG AD =，故四边形ADMG 为平行四边形，所以//AG DM ， 又DM ⊂平面BDE ，AG ⊂/平面BDE ，故//AG 平面BDE ．（2）由题意知BC ⊥平面ECD ，在平面ECD 内过C 点作CF CD ⊥交DE 于F ， 以C 为原点，CD ，CB ，CF 的方向为x ，y ，z 轴的正方向建立空间直角坐标系，不妨设1AD =，则22BC CD CE BG ====．且()0,0,0C ，()2,0,0D ，()0,2,0B ，(E -，设平面EBD 的法向量(),,n x y z =，则由0,0,DE n BD n ⎧⋅=⎨⋅=⎩得30,220,x x y ⎧-=⎪⎨-=⎪⎩ 取1y =，得(1,1,3n =，易知平面BCD 的一个法向量为()0,0,1m =，3cos ,51m nm n m n ⋅==⋅=⋅E BD C --． 21．如图，在四棱锥P ABCD -中，底面ABCD 是边长为2的正方形，M 为PC 的中点．（1）求证：//AP 平面BDM ；（2）若PB PC ==CD PC ⊥，求二面角C DM B --的余弦值．【试题来源】河南省湘豫名校2020-2021学年高三上学期1月月考（理）【答案】（1）证明见解析；（2． 【解析】（1）连接AC 交BD 于E ，连接EM ，则E 为AC 中点，所以EM 为APC △的中位线，所以//EM AP ，因为EM ⊂平面BDM ，AP ⊄平面BDM ，所以//AP 平面BDM ．（2）在PBC 中，因为2224PB PC BC +==，所以PB PC ⊥，取BC 中点O ，AD 中点F ，连接PO ，OF ，则PO BC ⊥，1PO =，因为BC CD ⊥，CD PC ⊥，BC 、PC ⊂平面PBC ，BC PC C ⋂=，所以CD ⊥平面PBC ，因为PO ⊂平面PBC ，所以CD PO ⊥，因为PO BC ⊥，BC CD C ⋂=，BC 、CD ⊂平面ABCD ，所以PO ⊥平面ABCD ，因为OF ⊂平面ABCD ，所以PO OF ⊥，所以PO ，OF ，OB 两两垂直，如图所示，以O 为原点，OF ，OB ，OP 分别为x 轴，y 轴，z 轴建立空间直角坐标系，则(2,1,0)D -，(0,0,1)P ，(0,1,0)B ，(0,1,0)C -，所以110,,22M ⎛⎫- ⎪⎝⎭，可得112,,22DM ⎛⎫=- ⎪⎝⎭，(2,2,0)BD =-，(2,0,0)CD =．设平面BDM 的法向量为()111,,m x y z =， 则0 0m BD m DM ⎧⋅=⎨⋅=⎩，即11111220112022x y x y z -=⎧⎪⎨-++=⎪⎩，取(1,1,3)m =， 设平面CDM 的法向量为()222,,n x y z =，则00n CD n DM ⎧⋅=⎨⋅=⎩，即222220112022x x y z =⎧⎪⎨-++=⎪⎩，取(0,1,1)n =-，所以222cos ,11||||112m n m nm n ⋅〈〉===⋅⨯， 所以二面角C DM B --的余弦值为11．22．如图所示，矩形ABCD 和梯形BEFC 所在平面互相垂直， //BE CF ，BCF CEF ∠=∠=90°，AD =EF =（1）求证：EF ⊥平面DCE（2）当AB 的长为何值时，二面角A EF C --的大小为60°． 【试题来源】山东省菏泽市2020-2021学年高三上学期期末【答案】（1）证明见解析；（2）60°．【解析】（1）因为平面ABCD ⊥平面BEFC ，平面ABCD 平面BEFC BC =，CD BC ⊥，CD ⊂平面ABCD ，所以CD ⊥平面BEFC ，EF ⊂平面BEFC ，从而CD EF ⊥． 因为EF CE ⊥，CD CE C =，,CD CE ⊂平面CDE ，所以EF ⊥平面CDE ．（2）如图所示，以点C 为坐标原点，以CB 、CF 和CD 所在直线分别为x 轴、y 轴和z 轴建立空间直角坐标系．过点E 作EG CF ⊥于点G ．在Rt EFG中，EG AD ==EF =1FG =．因为CE EF ⊥，则90EFC ECF BCE ∠=︒-∠=∠，所以Rt EFG Rt ECB △△，EG GF EF BE BC EC==，所以2,BE CE == 所以2CG =，所以3CF =．设AB a ，则()0,0,0C，)A a，)E ，()0,3,0F ．()0,2,AE a =-，()EF =-，()2,2,0CE =， 设平面AEF 的法向量(),,n x y z =．则00n AE n EF ⎧⋅=⎨⋅=⎩，即200y az y -=⎧⎪⎨+=⎪⎩， 令2z=，得,2n a ⎫=⎪⎭．因为CD ⊥平面EFC ，()0,0,CD a =，所以1cos ,2n CD ==，解得a =所以当AB =A EF C --的大小为60°．【名师点睛】本题考查空间向量法求二面角．求空间角的方法：（1）几何法（定义法）：根据定义作出二面角的平面角并证明，然后解三角形得出结论； （2）空间向量法：建立空间直角坐标系，写出各点为坐标，求出平面的法向量，由两个平面法向量的夹角得二面角（它们相等或互补）．23．如图，四棱锥E ABCD -中，底面ABCD 为直角梯形，其中AB BC ⊥，//CD AB ，面ABE ⊥面ABCD ，且224AB AE BE BC CD =====，点M 在棱AE 上．（1）证明：当2MA EM =时，直线//CE 平面BDM ；（2）当AE ⊥平面MBC 时，求二面角E BD M --的余弦值．【试题来源】内蒙古赤峰市2021届高三模拟考试（理）【答案】（1）证明见解析；（2． 【解析】（1）连结BD 与AC 交于点N ，连结MN ，//AB CD ，24AB CD ==， CND ANB ∴△∽△，12CD CN AB AN ∴==， 12EM MA =，EM CN MA AN∴=，MN //EC ∴， 又MN ⊂面BDM ，CE ⊂面BDM ，//CE ∴平面BDM ．（2）AE 平面MBC ，AE BM ∴⊥，M ∴是AE 的中点，取AB 的中点为O ， OE ∴⊥平面ABCD ，以OD ，OA ，OE 所在的直线为x ，y ，z 轴建立空间直角坐标系O xyz -，则(0,2,0)B-，E ，(2,0,0)D ，(0,2,0)A ，M ，设平面EBD 的法向量为()1111,,x n y z=，则1111112200020x y n BD n BE y ⎧+=⎧⋅=⎪⎪⇒⎨⎨⋅=+=⎪⎪⎩⎩， 令11z =，则1y=1x =1(3,3,1)n ∴=-，设平面BDM 的法向量为()2222,,n x y z =，则2222222200030x y n BD n BM y ⎧+=⎧⋅=⎪⎪⇒⎨⎨⋅==⎪⎪⎩⎩，令2z 21y =-，21x =，1(1,13)n ∴=-， 1212123105cos ,||n n n n n n ⋅∴<>===⋅ ∴二面角E BD M --的余弦值为35． 24．已知正方体1111ABCD A B C D -，棱长为2，M 为棱CD 的中点，N 为面对角线1BC 的中点，如图．（1）求证：ND AN ⊥；（2）求平面1AMD 与平面11AAC C 所成锐二面角的余弦值．【试题来源】安徽省池州市2020-2021学年高三上学期期末（理）【答案】（1）证明见解析；（2 【解析】（1）取BC 的中点分别为F ，连接NF ，DF ，因为N ，F 分别为1BC ，BC 的中点，1111ABCD A B C D -是正方体，易得NF ⊥平面ABCD ，所以NF AM ⊥；因为FC MD =，AD DC =，FCD MDA ∠=∠，所以FCD MDA ≌△△，所以CFD DMA ∠=∠，所以90FDC DMA ∠+∠=︒，所以FD AM ⊥，因为NF FD F =，NF ⊂平面NFD ，FD ⊂平面NFD ，所以AM ⊥平面NFD ， 又DN ⊂平面NFD ，所以ND AM ⊥；（2）以A 为原点，分别以AB 、AD 、1AA 方向为x 轴、y 轴、z 轴正方向，建立如下图所示空间直角坐标系：连接BD ，1C D ，在正方体1111ABCD A B C D -中，易知1BD C D =，且N 为1BC 中点，所以1DN BC ⊥．又11//BC AD ，所以1AD DN ⊥． 因为1AD AM A =，1AD ⊂平面1AMD ，AM ⊂平面1AMD ，所以ND ⊥平面1AMD ，故ND 为平面1AMD 的一个法向量；由1111ABCD A B C D -是正方体，得BD ⊥平面11AAC C ，故BD 为平面11AAC C 的一个法向量，因为()2,0,0B ，()0,2,0D ，()2,1,1N ， 所以()2,1,1ND =--，()2,2,0BD =-， 所以(cos ,ND BDND BD ND BD -⋅<>===⋅则平面1AMD 与平面11AAC C25．如图，正方形ADEF 与梯形ABCD 所在的平面互相垂直，AD CD ⊥，AB ∥CD ，122AB AD CD ===，点M 在线段EC 上．（1）当点M 为EC 中点时，求证：BM ∥平面ADEF ；（2）当平面BDM 与平面ABFM 在线段EC 上的位置．【试题来源】宁夏固原市第五中学2021届高三年级期末考试（理）【答案】（1）证明见解析；（2）点M 为EC 中点．【解析】（1）以直线DA 、DC 、DE 分别为x 轴、y 轴、z 轴建立空间直角坐标系，则(2,0,0)A ，(2,2,0)B ，(0,4,0)C ，(0,0,2)E ，所以(0,2,1)M ．所以(2,0,1)BM =-， 又(0,4,0)DC =是平面ADEF 的一个法向量．因为0BM DC ⋅=即BM DC ⊥，BM ⊄平面ADEF ，所以BM ∥平面ADEF ；（2）设(,,)M x y z ，则(,,2)EM x y z =-，又(0,4,2)EC =-，设()01EM EC λλ=≤≤，则0,4,22x y z λλ===-，即(0,4,22)M λλ-．设111(,,)n x y z =是平面BDM 的一个法向量，则11112204(22)0DB n x y DM n y z λλ⎧⋅=+=⎪⎨⋅=+-=⎪⎩，取11x =得11y =-，此时显然1λ=时不符合，则121z λλ=-，即2(1,1,)1n λλ=--， 又由题设，(2,0,0)DA =是平面ABF 的一个法向量，所以cos ,622DA n DA n DA n ⋅===⋅，解得12λ=，即点M 为EC 中点． 【名师点睛】利用法向量求解空间面面角的关键在于“四破”：第一，破“建系关”，构建恰当的空间直角坐标系；第二，破“求坐标关”，准确求解相关点的坐标；第三，破“求法向量关”，求出平面的法向量；第四，破“应用公式关”．26．如图所示，在多面体ABCDEF 中，//AB CD ，AB BC ⊥，22AB BC CD ==，四边形ADEF 为矩形，平面ADEF ⊥平面ABCD ，AF AB λ=．（1）证明：//DF 平面BCE ；（2）若二面角C BE F --λ的值． 【试题来源】江西宜春市2021届高三上学期数学（理）期末试题【答案】（1）证明见解析；（2）1．【解析】（1）取AB 的中点为M ，连接FM CM DM ，，，因为//AM CD 且AM CD =，四边形AMCD 为平行四边形，所以//AD MC 且AD MC =，因为四边形ADEF 为矩形，所以//FE MC 且=FEMC ，所以四边形EFMC 是平行四边形，所以//FM EC ，且EC ⊂平面BEC ，FM ⊄平面BEC ，。