七年级上学期期中测试(通用5篇)
七年级数学上学期期中模拟试题 篇1
一、选择 (每题4分,共48分)
1.一个有理数的平方是正数,这个有理数是().
A.正数B.负数C.非零数D.非负数
2.适合a=a3的有理数a有().
A.1个 B.2个 C.3个 D.无数个
3.下列说法正确的是().
A.非负有理数即是正有理数
B.零表示不存在,无实际意义
C.正整数和负整数统称为整数
D.整数和分数统称为有理数
4.稀土元素有独特的性能和广泛的应用,我国稀土资源的总储藏量为1 050 000 000吨,是全世界稀土资源最丰富的国家,将 1 050 000 000吨用科学记数法表示为().
A.1.05×1010吨 B.1.05×109吨
C.10.5×108吨D.0.105×1010吨
5.下列说法错误的是().
A.若x=1,则|x-1|的值为零
B.绝对值最小的数是零
C.绝对值相等的两个数相等
D.若 a<0,则|a|=-a
6.下列说法正确的是().
A.多项式3a-5b+1的项是3a,5b,1
B. 和3a2-ab+b2都是多项式
C. 和 都不是多项式
D.3a2b4-a5+2ab-4是六次四项式
7.若x≠0,y≠0,且 x2y3+kx2y3=0,则k的值为().
A.0B. C.-D.1
8.若a<0,ab<0,计算|b-a+1|-|a-b-5|,结果为().
A.4B.-4 C.-2a+2b+6 D.2a-2b-6
9.下列计算中,正确的是().
A.x2·x4=x8B.(2a2)3=8a6
C.(a+1)2=a2+1D.(±4)2=±16
10.若8a3bm÷28anb2= b2,则m,n的值为().
A.m=4,n=3 B.m=4,n=1
C.m=1,n=3 D.m=2,n=3
11.设a,b都是有理数,则下列命题中,正确的是().
A.若a≠b,则a2≠b2B.若a>|b|,则a2>b2
C.若|a|>|b|,则a>bD.若a2>b2,则a>b
12.已知a,b为实数,且ab=1,设M= + ,N= + ,则M,N的大小关系是().
A.M>NB.M=NC.M 二、填空 (每题4分,共20分) 13.把下列各数用“>”号连接起来:-4 , ,0.7, ,-0.7,-4,2,0. . 14.2.088万是精确到位,它有个有效数字. 15.若|a|=3,|b|=5,且ab>0,则|a+b|=. 16.已知x+y=7,xy=12,那么代数式x2y-x2+xy2+x2的值是. 17.若8÷ 2=2n,则n=. 三、计算 18.(每题6分)计算 + . 19.(每题8分)计算1×- + ×- + ×- +…+ ×- . 20.(本题8分)已知a2x3+ax2-4x3+2x2+x-1是关于x的二次多项式,且多项式4a+b的值是12,求多项式5a2b-6ab-3ab2-2的值. 21.(本题10分)一只球从a米高的地方落下,着地后弹起到 米的地方,第二次落地后又弹起到 × = (米)的地方……当球第八次落地时,这只球经过的路程的和为多少米? 参考答案 一、1.C2.C3.D4.B5.C6.D7.C8.B9.B10.A11.B12.B 二、13.2> >0.7> >0>-0.7>-4>-4 14.十415.816.8417.5 三、18.±2或0.(提示:分情况讨论)19.- .20.略.(提示:先求出a=2,b=4,然后代入求值即可)21. 米. A卷 二、笔试部分(80分)I.单词拼音(将划线部分读音不同的单词的符号填入题号前括号内)(1×10)()1.A.man B.stand C.Kate()2.A.not B.no C.go()3.A.pen B.these C.desk()4.A.this B.sit C.nice()5.A.student B.under C.cup()6.A.sweater B.please C.teacher()7.A.now B.yellow C.How()8.A.they B.three C.mother()9.A.bags B.classes C.maps()10.A.yes B.your C.my Ⅱ.单词分类(将不同类型的单词的符号填入前面括号内)(1×10)()1.A.bus B.car C.cup()2.A.ten B.there C.thirteen()3.A.shoe B.shirt C.bed()4.A.bird B.boy C.girl()5.A.brother B.sister C.family()6.A.are B.in C.under()7.A.China B.American C.Japanese()8.A.teacher B.student C.school()9.A.please B.sit C.stand()10.A.red B.apple C.green Ⅲ.选择填空(将正确选项的符号填入题号前括号内)(1×10)()1.Are you Mrs.Read? Yes, I _______.A.is B.am C.are()2.This is a map Wuhan.A.of B.on C.in()3.These bananas are.A.you B.your C.yours()4.The twins.A.look same B.look the same C.look after()5.It’s time.A.go to bed B.to go to bed C.to go to the bed()6.Who’s that? Is it Lucy Lily? A.or B.and C.but()7.Whose bag is this? I think it’s.A.Jim’s B.Jim C.Jims’()8.That’s his watch,Give it to Please.A.he B.him C.his()9.I have a new bike.bike is under the tree.A.The B.A C.An()10.Are you friends? Yes,.A.I am B.they are C.we are Ⅳ.情景反应(将正确答案的编号填入前括号内)(1×10)()1.一Is everyone here? - _______.A.Yes,I am.B.No.you aren’t.C.Yes,we are all here.()2.-This is a new student,Ann.-.A.What’s your name? B.How do you do? C.Are you Ann?()3.-Good afternoon,class.- , teacher.A.Good morning B.Good afternoon C.Goodbye()4.-Hello,Bill? 一No,.A.this is Sam B.I am Sam C.I’m Sam()5.-This is your eraser.Here you are.-Thank you,Mr.Hu.-.A.Thank you,too B.That’s all right C.All right()6.- , where is the toilet? -Over there.A.Excuse me B.Sorry C.How are you()7.-Sit down and have a cup of tea.-.A.I sit down B.I have a cup of tea C.Thank you()8.一Can I see your watch? -.A.That’s OK B.OK C.OK.It’s not here()9.-What’s the time? 一.A.Yes,it’s a clock B.It’s the time C.It’s eight o’clock()10.-.-It’s an egg.A.What’s this in English? B.Is this your egg? C.Can you speak English? Ⅴ.选错(将错误选项的符号填入题号前括号内)(1×10)()1.They are in the same classes.A B C()2.Can you look the broom behind the door? A B C()3.We aren’t know his name.A B C()4.Who is this blouse? It’s mine.A B C()5.What colour is her coats? A B C()6.Class three has a map of China.A B C()7.Can he sing a song? No,he isn’t.A B C()8.This is my a classroom.A B C()9.These are your books.Please put they there.A B C()10.He isn’t a old policeman.A B C Ⅵ.句型转换(按括号内要求完成镇空,一空一问)(1×10)1.Are you in Row One,Kate?(肯定回答)Yes,.2.They are in Grade Two.(改成否定句)in Grade Two.3.She can spell her name.(改成一般疑问句)spell her name? 4.Their bags are black.(对划线部分提问)are their bags? 5.This is a boat.(变复数)are.Ⅶ.看图填空完成句子(一空一词)(1×10) 1.this a bus a car? a.2.Where the ? They’re the.3.Mr.White’s is under the.Ⅷ.完型填空(将选项的符号填入题号前面括号内)(1×10)Look ① the picture,please.We can see Green’s ② in it.They are ③.Mr.④ Mrs.Green are here.Kate ⑤ there.⑥ is Jim? Oh,⑦ is behind the chair.What’s that? It ⑧ like a hat.But it’s a cat.It’s on ⑨ floor.Do you know ⑩ name?()1.A.at B.like C.after()2.A.family B.father C.mother()3.A.English girls B.English C.English boys()4.A.or B.and C.of()5.A.am B.is C.are()6.A.Who B.How C.Where()7.A.he B.He C.His()8.A.look B.looks C.is looking()9.A.the B.a C.an()10.A.it’s B.is C.its B卷 Ⅰ.选择填空(将正确答案的代号填入题前括号内)(2×10)()1.are the man and the woman the bike? A.Whose,on B.What,in C.Who,on D.Who’s,in()2.It’s not shirt is in the bag.A.her,Her B.mine,My C.your,Your D.his,His()3.The brothers don’t look the same.But they are eleven.A.twins,all B.twin’s,both C.twin,both D.twin,all()4.Your sister is very.You must her at home.A small,look like B.young,look after C.short,look at D.nice,look()5.Do you know the name your English teacher? I think Mr Wang.A.of, it’s B.of, hers C.in, She’s D.of, She’s()6.Can you the school bag to Lucy? She to take it.A.put,forgets B.give,forgets C.give,puts D.forget,has()7.That’s the bell.It’s time to classroom.A.to go,a B.go,the C.to go,the D.go,a()8.I can see books.books are on the table.A.a,The B.some,The C.the,Some D.the,The()9.He his pen in his bag,but he can’t it now.A.puts,find B.looks,see C.has,finds D.looks,find()10.The coat is.Don’t.A.your father, put on it B.Jim’s, put them on C.your father’s,put it on D.Jim,put it on Ⅱ.补全对话(选用所给词语补全对话,一空一词,每词只能使用一次)(2×10)A: 1 skirts are these 2 the desk? Are 3 yours? B: Let me see,Oh,no,they 4 mine.Mine 5 blue,and those are red.I think they are 6.A: Whose? B: Lily’s.A: 7 Lily!Are those red 8 yours? B: Oh,9,they are mine.A: Put 10 in your bag,please.B: OK.Thanks very much.1.2.3.4.5..6.7.8.9.10..Ⅲ.单词拼写(根据句子意思写出完整的词,其首字母已给出,拼写时不得重复)(2×10)1.Li Lei isn’t at school today.Not e is here.2.Is that 8287645? No.That’s a w number.3.Can you g the name of the girl in the red hat? 4.Mr.and Mrs.Smith are in China now.But their d,Rose,is in England.She is too young.5.These are the s books.They are on the desk.6.This book is y.You must look after it.7.Now it’s 4 in the afternoon.It’s time to go and play g.8.What colour is your kite? It’s o.9.W father is a teacher? Li Lei’s.10.The girl in a red coat is Lucy’s s.Ⅳ.连词成句(选择能连成正确句子的答案)(2×1O)()1.A.is B.my C.That D.girl E.friend A.BACED+句号 B.CDABE+句号 C.CABDE+句号 ()2.A.behind B.is C.the one D.the tree E.bike F.My A.FEBCAD+句号 B.FEBADC+句号 C.FEADBC+句号 ()3.A.What B.car C.the D.is E.colour A.AECBD+问号 B.ABDCE+问号 C.AEDCB+问号 ()4.A.are B.These C.long D.theirs E.pencils A.BACDE+句号 B.BCEAD+句号 C.BADCE+句号 ()5.A.Han Meimei’s B.is C.it D.desk A.BCAD+问号 B.BADC+问号 C.BDCA+问号 ()6.A.Grade B.am C.I D.not E.Three F.in A.CBDFAE+句号 B.CDBFAE+句号 C.CBDFEA+句号 ()7.A.green B.isn’t C.yours D.Mine E.is F.but A.DEAFBC+句号 B.DBAFCE+句号 C.DECFBA+句号 ()8.A.Which B.Who’s C.man D.the man A.AD+问号BC+问号 B.BC+问号 AD+问号 C.BD+问号AC+问号 ()9.A.know B.you C.the time D.I E.think A.DABEC+句号 B.DEBAC+句号 C.DEABC+句号 ()10.A.can’t B.watch C.I D.find E.his A.CABDE+句号 B.CDAEB+句号 C.CADEB+句号 Ⅴ.阅读理解(根据短文内容选择正确答案,并将其代号填写在题前的括号内)(2×5)My name is Li Lei.I’m Chinese.I’m twelve.This is a picture of my family at home.You can see my father.My mother,my sister一Li Hua and me.I have a nice bedroom.There is a photo on the wall.It’s old.Who is that one behind the doll? Is it my sister? No,it’s me.I look young.How old am I in the photo? I think one.My mother is a worker.My father is a teacher of English.He has an American friend,Mr Green.His family are in China.His son, Jims is a student.We are in the same school,but different classes.We are good friends,too.()1.I have.A.one sister B.one brother C.two friends.()2.The baby in the old picture is.A.Li Hua B.Li Lei C.Jim()3.Mr.Green is good friend.A.my B.my mother’s C.my father’s()4.My father is.A.a worker B.an English teacher C.a teacher of Chinese()5.Jim and I are not in different.A.schools B.classes C.grades Ⅵ.阅读填词(补全空白处单词,使短文意思完整正确,单词的第一个字母已给出)(2×5) Tom and Mary are brother and sister.They are n students in Jim’s class.Jim l after them at school.They are his good f.Today Jim is on duty.It’s his t to clean the b and the classroom.1.n 2.l 3.f 4.t 5.b.参考答案 A卷 一、略 二、笔试部分 Ⅰ.1-5 CABCA 6-10 ABBCC Ⅱ.1-5 CBCAC 6-10 AACAB Ⅲ.1—5 BACBB 6-10 AABAC Ⅳ.1-5 CBBCB 6-10 ACBCA Ⅴ.1-5 CBAAC 6-10 ACBCB Ⅵ.1.I am 2.They aren’t 3.Can she 4.What colour 5.These,boats Ⅶ.1.Is,or,It’s,bus 2.are,students in,classroom 3.bike,tree.Ⅷ.1-5 AABBB 6-10 CBBAC B卷 Ⅰ.下列各组单词中,有一个单词画线部分的读音与其他三个单词画线部分的读音不同,请选出,What a good, kind girl! 单元验收。(6%) ( )1.A.wife B.give C.kite D.kind ( )2.A.full B.put C.sugar D.husband ( )3.A.traffic B.apple C.same D.matter ( )4.A.ready B.head C.clean D.weather ( )5.A.front B.strong C.some D.monkey ( )6.A.careful B.ice C.city D.nice Ⅱ.根据汉语提示,写出空白处所缺单词的正确形式。(6%) 1.What about _____(去)out for a walk after supper? 2.Mike bought some pencils on _____(他)way home. 3.Half an hour _____(以后)they finished the work. 4.The _____(格林一家)are having supper now. 5.I did my homework more _____(认真)than Tom. 6.When did you _____(到达)the farm? Ⅲ.根据句意,用括号中所给单词的适当形式填空。(6%) 1.Miss Black is one of the most popular _____ in our school. (teacher) 2.It took _____ two hours to finish the work. (we) 3.The book on the desk is _____. (I) 4.I usually do some _____ on Sundays. (shop) 5.Look! How _____ it is raining now!(heavy) 6.The Picture looks the most beautiful of all. (three) Ⅳ.英汉词组互译。(10%) 1.生病住院 _______ 2.在马路的尽头 ________ 3.半小时 _______ 4.等待…… _________ 5.在某人回家的`路上 ______ 6.at the side of the road ______ 7.on the left __________ 8.look around ___________ 9.go and work on the farm ____________ 10.twenty minutes later ___________ Ⅴ.选择填空。(16%) ( )1.We often play ____ football after school. A.an B.a C.the D./ ( )2.Ask that policeman. He _____ know the way to the station. A.need B.may C.maybe D.can ( )3.It _____ me about half an hour to wash the clothes. A.took B.used C.caught D.made ( )4.Its very cold. Youd better _____ your coat. A.to put on B.to take off C.put on D.take off ( )5.There is a hospital _____ the end of the street. A.in B.on C.to D.at ( )6._____ cold day it is! A.What a B.What C.How a D.How ( )7.She was born ____ October, 1980. A.on B.in C.at D.for ( )8.Dont worry. There is _____ with your baby. A.something wrong B.nothing wrong C.wrong something D.wrong nothing ( )9.Shes supper at half past six every day, _____? A.isnt she B.is she C.doesnt she D.does she ( )10.Ive five good friends. One comes from America, _____ come from China. A.other B.the others C.the other D.others ( )11.Bring me three _____, please. A.cups of milk B.cup of milks C.cup of milk D.cups of milks ( )12.He carefully, but _____ nothing. A.listened to, heard B.heard, listened to C.listened, heard D.heard, listened ( )13.Peter asked me _____ him with his Chinese. A.help B.helping C.to help D.helped ( )14.There is not _____ tea in the cup. A.some B.many C.a hot D.much ( )15.“Can you speak English?” “Yes, only _____. ” A.a few B.few C.a little D.little ( )16.“Is it going to rain this evening?” “Yes, I think _____. ” A.it B.so C.that D.is Ⅵ.根据句意,选用所给的词语,并用其适当形式填空。(10%) had better, how long, may, Wait for, take, how far, reach, find, what about, keep 1._____ I borrow your bike, please? 2.Go on until you _____ the end of the road, and you will see it. 3.“What are you doing here?” “I _____ a bus. ” 4.“ _____ is it from here?” “Its about five kilometres away. ” 5.Its far from here. You _____ catch a bus. 6.It _____ you about two hours to finish the work, wont it? 7.“ _____ did you work in Shanghai?” “For about five years. ” 8.“I cant _____ my ticket. ”“It may be in your pocket. ” 9.You must _____ yourselves healthy. 10._____ taking a walk after supper? Ⅶ.按要求改写下列各句,每空一词,缩写词算一词,英语试题《What a good, kind girl! 单元验收》(10%) 1.There are some apples in the basket. (改为单数形式) There _____ _____ _____ in the basket. 2.Classes begin at eight every day, _____ _____?(完成反意疑问句) 3.My father had lunch at school yesterday. (改为否定句) My father _____ _____ lunch at school yesterday. 4.There is some milk in the bottle. (改为一般疑问句) _____ _____ _____ milk in the bottle? Ⅷ.根据中文,完成句子,每空一词,缩写词算一词。(16%) 1.昨天我为我弟弟修理自行车花了大约两个小时的.时间。 It _____ me _____ two hours _____ _____ the bike for my brother yesterday. 2.“今天很冷。”“的确如此。” “It is very cold today. ”“_____ _____ _____. ” 3.我父亲下个月可能去巴黎。 My father _____ _____ to Paris next month. 4.也许李先生知道去邮局的路。 ____ Mr. Li knows the _____ _____ the post office. 5.你最好现在别看电视了。 _____ _____ _____ _____ TV now. Ⅸ.补全对话,每空一词,缩写词算一词。(15%) A: 1 can I 2 for you, sir? B:could you tell me the 3 to the nearest hospital, please? A:I am 4 I 5 know. Please ask that policeman over there. She 6 know, I think. B: 7 you all the 8 .(He runs to the policeman.) Excuse me, sir. 9 there a hospital 10 here? A:Let me 11 . Go along this road, and 12 the first turning on the right. Go 13 until you reach the second traffic lights, and then 14 left. You can find 15 in front of you. Ⅹ.阅读理解。(5%)阅读下面的短文,根据短文内容,选择正确答案。 Mr. White had a farm in England. He and his wife grew a lot of things on the farm. They worked very hard. One day Mr. White said to his wife, “Dear, lets go to London next Sunday. We can have a good lunch there, and then we can go to the cinema. ”His wife was very happy when she heard this, because she and her husband always stayed at home, and she didnt like cooking three times everyday. They went to London by train and walked for two hours. When it was twelve oclock, they wanted to have a meal. There were a few restaurants(饭馆)along the street. In one of them there was a blackboard outside. On the blackboard it said, “Lunch 12:30 to 2:30 &1.50. ” “Well thats good, ”Mrs. White said, “ We eat for two hours for 1.50 here? This is the place for us. ” ( )1.Mr. White took his wife to a very _____ restaurant. A.dear B.cheap C.new D.old ( )2.Mrs. White was happy because _____. A.she didnt have to cook B.Mr. White ate too much C.she couldnt eat much D.they worked hard ( )3.Mr. and Mrs. White wanted to see a film _____. A.in London B.in the morning C.in the afternoon D.in the restaurant ( )4.The words on the blackboard told people that _____. A.they must eat a very slow lunch B.they must eat for two hours C.they could have lunch anytime between 12:30 and 2:30 D.they must be in the restaurant at 12:30 and leave at 2:30 ( )5.Mrs. White wanted to go into the restaurant because she thought ______. A.they could eat quickly B.they could eat for a long time C.she was in London D.she wanted to go to see her friends 参考答案 Ⅰ.1.B 2.D 3.C 4.C 5.B 6.A Ⅱ.1.going 2.his 3.later 4.Greens 5.carefully 6.reach Ⅲ.1.teachers 2.us 3.mine 4.shopping 5.heavily 6.third Ⅳ.1.be ill in hospital 2.at the end of the road 3.half an hour 4.wait for 5.on ones way home 6.在马路边 7.在左边 8.向四周看 9.去农场劳动 10.二十分钟以后 Ⅴ.1.D 2.B 3.A 4.C 5.D 6.A 7.B 8.B 9.C 10.B 11.A 12.C 13.C 14.D 15.C 16.B Ⅵ.1.May 2.reach 3.am waiting for 4.How far 5.had better 6.will take 7.How long 8.find 9.keep 10.What about Ⅶ.1.is,an,apple 2.dont,they 3.didnt,have 4.Is,there,any Ⅷ.1.took,about,to,mend 2.So,it,is 3.may,go 4.Maybe,way,to 5.Youd,better,not,Watch Ⅸ.1.What 2.do 3.way 4.sorry 5.dont 6.may 7.Thank 8.same 9.Is 10.near 11.see 12.take 13.on 14.turn 15.it Ⅹ.1.B 2.A 3.A 4.C 题目:上中学以后 要求:(1)字数不少于500字,文体不限,诗歌除外 1.已知集合P={ x | x (x-1)≥0},Q={ x | y=ln(x-1)},则P∩Q=. 2.高三(1)班共有56人,学号依次为1,2,3,…,56,现用系统抽样的办法抽取一个容量为4的样本,已知学号为6,34,48的同学在样本中,那么还有一个同学的学号应为. 3.已知i是虚数单位,m∈R,且2-mi1+i是纯虚数,则(2-mi2+mi)2011=. 4.若直线l过点A(-2,-3),且与直线3x+4y-3=0垂直,则直线l的方程为. 5.设正项等比数列{an}的前n项和为Sn,且210S30+S10=(210+1)S20,则数列{an}的公比. 6.设函数f(x)=x2-3x-4,x∈[-3,6],则对任意x0∈[-3,6],使f(x0)≤0的概率为. 7.下图伪代码运行输出的n的值是. j←1n←0While j≤11j←j+1 If mod(j,4)=0 thenn←n+1 End ifj←j+1 End whilePrint nEnd 8.点A在曲线C:x2+(y+2)2=1上,点M(x,y)在平面区域2x-y+2≥0x+y-2≤02y-1≥0 上,则AM的最小值是. 9.设定义在R上的函数f(x)=1|x-1|,x≠1,1,x=1. 若关于x的方程f2(x)+bf(x)+c=0有3个不同的实数解x1,x2,x3,则x1+x2+x3=. 10.设△ABC的BC边上的高AD=BC,a,b,c分别表示角A,B,C对应的三边,则bc+cb的取值范围是. 11.给出下列命题,其中正确的命题是 (填序号). ①若平面α上的直线m与平面β上的直线n为异面直线,直线l是α与β的交线,那么l至多与m,n中的一条相交; ②若直线m与n异面,直线n与l异面,则直线m与l异面; ③一定存在平面γ同时与异面直线m,n都平行. 12.在△ABC中,AH为BC边上的高,tanC2=12,则过点C,以A,H为焦点的双曲线的离心率为. 13.若不等式a+|x2-1x|≥2|log2x|在x∈(12,2)上恒成立,则实数a的取值范围为. 14.如图放置的等腰直角三角形ABC薄片(∠ACB=90°,AC=2)沿x轴滚动,设顶点A(x,y)的轨迹方程是y=f(x),则f(x)在其相邻两个零点间的图象与x轴所围区域的面积为. 二、填空题:本大题共6小题,共计70分.请在指定区域内作答,解答时应写出文字说明、证明过程或演算步骤. 15.(本小题满分14分) 已知点A(3,0),B(0,3),C(cosα,sinα),α∈(π2,3π2). (1)若|AC|=|BC|,求角α的值; (2)若AC·BC=-1,求2sin2α+sin2α1+tanα的值. 16.(本小题满分14分) 如图,在四棱锥PABCD中,底面ABCD是菱形,∠BAD=60°,AB=2,PA=1,PA⊥平面ABCD,E是PC的中点,F是AB的中点. (1)求证:BE∥平面PDF; (2)求证:平面PDF⊥平面PAB; (3)求三棱锥PDEF的体积. 17.(本小题满分14分) 如图,在边长为10的正三角形纸片ABC的边AB,AC上分别取D,E两点,使沿线段DE折叠三角形纸片后,顶点A正好落在边BC上(设为P),在这种情况下,求AD的最小值. 18.(本小题满分16分) 已知F是椭圆C1:x2a2+y2b2=1的右焦点,点P是椭圆C1上的动点,点Q是圆C2:x2+y2=a2上的动点. (1)试判断以PF为直径的圆与圆C2的位置关系; (2)在x轴上能否找到一定点M,使得QFQM=e (e为椭圆的离心率)?若存在,求出点M的坐标;若不存在,请说明理由. 19.(本小题满分16分) 已知函数f(x)=3xa+3(a-1)x,a≠0且a≠1. (1)试就实数a的不同取值,写出该函数的单调增区间; (2)已知当x>0时,函数在(0,6)上单调递减,在(6,+∞)上单调递增,求a的值并写出函数的解析式; (3)记(2)中的函数图象为曲线C,试问是否存在经过原点的直线l,使得l为曲线C的对称轴?若存在,求出直线l的方程;若不存在,请说明理由. 20.(本小题满分16分) 已知数列{an}满足an+1+an=4n-3(n∈N). (1)若数列{an}是等差数列,求a1的值; (2)当a1=2时,求数列{an}的前n项和Sn; (3)若对任意n∈N,都有a2n+a2n+1an+an+1≥5成立,求a1的取值范围. 附加题 21.【选做题】在下面A、B、C、D四个小题中只能选做两题,每小题10分,共20分. A.选修41:几何证明选讲 如图所示,圆O的两弦AB和CD交于点E,EF∥CB,EF交AD的延长线于点F,FG切圆O于点G. (1)求证:△DEF∽△EFA; (2)如果FG=1,求EF的长. B.选修42:矩阵与变换 设M是把坐标平面上点的横坐标不变、纵坐标沿y方向伸长为原来5倍的伸压变换. (1)求直线4x-10y=1在M作用下的方程; (2)求M的特征值与特征向量. C.选修44:坐标系与参数方程 已知极坐标系的极点与直角坐标系的原点重合,极轴与x轴的正半轴重合.若曲线C1的方程为ρ2=8ρsinθ-15,曲线C2的方程为x=22cosαy=2sinα (α为参数). (1)将C1的方程化为直角坐标方程; (2)若C2上的点Q对应的参数为α=3π4,P为C1上的动点,求PQ的最小值. D.选修44:不等式选讲 设函数f(x)=|x-1|+|x+1|,若不等式|a+b|-|2a-b|≤|a|·f(x)对任意a,b∈R且a≠0恒成立,求实数x的范围. 22.(本小题满分10分) 如图,在直三棱柱ABCA1B1C1中,AC=3,BC=4,AB=5,AA1=4. (1)设AD=λAB,异面直线AC1与CD所成角的余弦值为925,求λ的值; (2)若点D是AB的中点,求二面角DCB1B的余弦值. 23.(本小题10分) 在0,1,2,3,……,9这是个自然数中,任取三个不同的数字. (1)求组成的三位数中是3的倍数的有多少个? (2)将取出的三个数字按从小到大的顺序排列,设ξ为三个数字中相邻自然数的组数(例如:若取出的三个数字为0,1,2,则相邻的组为0,1和1,2,此时ξ的值是2),求随机变量ξ的分布列及其数学期望Eξ. nlc202309020852 参考答案 一、填空题 1.(1,+∞)解析:P=(-∞,0]∪[1,+∞),Q=(1,+∞),所以P∩Q=(1,+∞). 2.20解析:采用系统抽样,所抽出的样本成等差数列,故另一个同学的学号应是20. 3.i解析:因为2-mi1+i=(2-mi)(1-i)2=(2-m)-(2+m)i2是纯虚数,所以m=2. 故(2-mi2+mi)2011=(2-2i2+2i)2011=(-i)2011=-i3=i. 4.4x-3y-1=0解析:依题意直线l的斜率为43,由点斜式方程得直线l的方程为4x-3y-1=0. 5.12解析:设数列{an}的公比为q,因为210S30+S10=(210+1)S20,所以210(S30-S20)=(S20-S10),由此可得210(S20-S10)q10=(S20-S10),所以q10=(12)10.又因为{an}是正项等比数列,所以q=12. 6.59解析:函数f(x)=x2-3x-4=(x+1)(x-4),因此当x∈[-1,4]时,f(x)≤0,所以对任意x0∈[-3,6],使f(x0)≤0的概率为4-(-1)6-(-3)=59. 7.3. 8.32解析:曲线C是圆x2+(y+2)2=1;不等式组的可行域如图阴影部分所示,A点为(0,-1),当M为(0,12)时,AM最短,长度是32. 9.3解析:易知f(x)的图象关于直线x=1对称.f2(x)+bf(x)+c=0必有一根使f(x)=1,不妨设为x1,而x2,x3关于直线x=1对称,于是x1+x2+x3=3. 10.[2,5]解析:因为BC边上的高AD=BC=a,.所以S△ABC=12a2=12bcsinA,所以sinA=a2bc.又因为cosA=b2+c2-a22bc=12(bc+cb-a2bc),所以bc+cb=2cosA+sinA≤5,同时bc+cb≥2,所以bc+cb∈[2,5]. 11.③解析:①是错误的,因为l可以与m,n都相交;②是错误的,因为m与l可以异面、相交或平行;③是正确的,因为只要将两异面直线平移成相交直线,两相交直线确定一个平面,此平面就是所求的平面. 12.2解析:如图所示,由tanC2=12,得tanC=2tanC21-tan2C2=43.由题可知AH⊥BC,以A,H为焦点的双曲线的离心率e=AHAC-CH.由于△AHC为直角三角形,且tanC=AHCH=43,可设AH=4a,CH=3a,则AC=5a,所以离心率e=AHAC-CH=4a5a-3a=2. 13.a≥1解析:不等式即为a≥-|x2-1x|+2|log2x|,在x∈(12,2)上恒成立.而函数 f(x)=-|x2-1x|+2|log2x|=x,12 14.2+4π解析:作出点A的轨迹中相邻两个零点间的图象,如图所示.其轨迹为两段圆弧,一段是以C为圆心,CA为半径的四分之一圆弧;一段是以B为圆心,BA为半径,圆心角为3π4的圆弧.其与x轴围成的图形的面积为12×22×π2+12×2×2+12×(22)2×3π4=2+4π. 二、解答题 15.解析:(1)解法1:由题意知AC=(cosα-3,sinα),BC=(cosα,sinα-3).由|AC|=|BC|,化简整理得cosα=sinα.因为α∈(π2,3π2),所以α=5π4. 解法2:因为|AC|=|BC|,所以点C在直线y=x上,则cosα=sinα.因为α∈(π2,3π2),所以α=5π4. (2)由AC·BC=-1,得(cosα-3)cosα+sinα(sinα-3)=-1,即sinα+cosα=23.所以(sinα+cosα)2=1+2sinαcosα=49,即2sinαcosα=-59. 所以2sin2α+sin2α1+tanα=2sinαcosα=-59. 16.解析:(1)取PD的中点为M,连结ME,MF,因为E是PC的中点,所以ME是△PCD的中位线.所以ME∥CD,ME=12CD.又因为F是AB的中点,且由于ABCD是菱形,AB∥CD,AB=CD,所以ME∥FB,且ME=FB.所以四边形MEBF是平行四边形,所以BE∥MF. 连结BD,因为BE平面PDF,MF平面PDF,所以BE∥平面PDF. (2)因为PA⊥平面ABCD,DF平面ABCD,所以DF⊥PA. 连结BD,因为底面ABCD是菱形,∠BAD=60°,所以△DAB为正三角形. 因为F是AB的中点,所以DF⊥AB. 因为PA,AB是平面PAB内的两条相交直线,所以DF⊥平面PAB. 因为DF平面PDF,所以平面PDF⊥平面PAB. (3)因为E是PC的中点,所以点P到平面EFD的距离与点C到平面EFD的距离相等,故VPDEF=VCDEF=VEDFC,又S△DFC=12×2×3=3, E到平面DFC的距离h=12PA=12, 所以VEDFC=13×3×12=36. 17.解析:显然A,P两点关于折线DE对称,连结DP,图(2)中,设∠BAP=θ,∠BDP=2θ. 再设AD=x,所以DP=x,DB=10-x. 在△ABC中,∠APB=180°-∠ABP-∠BAP=120°-θ. 在△BDP中,由正弦定理知BDsin∠BPD=DPsin∠DBP,即10-xsin(120°-2θ) =xsin60°,所 以x=1032sin(120°-2θ)+3. 因为0°≤θ≤60°,所以0°≤120°-2θ≤120°, 所以当120°-2θ=90°, 即θ=15°时,sin(120°-2θ)=1. 此时x取得最小值1032+3=203-30, 且∠ADE=75°. 所以AD的最小值为203-30. 18.解析:(1)取PF的中点记为N,椭圆的左焦点记为F1,连结ON,则ON为△PFF1的中位线,所以ON=12PF1.又由椭圆的定义可知,PF1+PF=2a,从而PF1=2a-PF,故ON=12PF1=12(2a-PF)=a-12PF.所以以PF为直径的圆与圆C2内切. (2)设椭圆的半焦距为c,M (x,0),Q (x0,y0),F (c,0),由QFQM=e,得QF2=e2QM2,即(x0-c)2+y20=e2[(x0-x)+y20].把x20+y20=a2代入并化简整理,得2(c-e2x)x0+e2a2+e2x2-a2-c2=0,要此方程对任意的Q (x0,y0)均成立,只要c-e2x=0即可,此时x=ce2=a2c.所以x轴上存在点M,使得QFQM=e,M的坐标为(a2c,0). nlc202309020852 19.解析:(1)①当a<0时,函数f(x)的单调增区间为(-a(a-1),0),(0,a(a-1)); ②当0<a<1时,函数f(x)的单调增区间为(-∞,0),(0,+∞); ③当a>1时,函数f(x)的单调增区间为(-∞,-a(a-1)),(a(a-1),+∞). (2)由题设及(1)中③知a(a-1)=6,且a>1,解得a=3,因此函数解析式为f(x)=3x3+23x(x≠0). (3)假设存在经过原点的直线l为曲线C的对称轴,显然x,y轴不是曲线C的对称轴,故可设l:y=kx(k≠0). 设P(p,q)为曲线C上的任意一点,P′(p′,q′)与P(p,q)关于直线l对称,且p≠p′,q≠q′,则P′也在曲线C上,由此得q+q′2=k·p+p′2,q-q′p-p′=-1k,且q=p3+23p,q′=p′3+23p′, 整理得k-1k=23,解得k=3或k=-33. 所以存在经过原点的直线y=3x及y=-33x为曲线C的对称轴. 20.解析:(1)若数列{an}是等差数列,则an=a1+(n-1)d,an+1=a1+nd. 由an+1+an=4n-3,得(a1+nd)+[a1+(n-1)d]=4n-3,即2d=4,2a1-d=-3,解得d=2,a1=-12. (2)由an+1+an=4n-3(n∈N),得an+2+an+1=4n+1(n∈N). 两式相减,得an+2-an=4. 所以数列{a2n-1}是首项为a1,公差为4的等差数列. 数列{a2n}是首项为a2,公差为4的等差数列. 由a2+a1=1,a1=2,得a2=-1. 所以an=2n,n=2k-12n-5,n=2k (k∈Z). ①当n为奇数时,an=2n,an+1=2n-3. Sn=a1+a2+a3+…+an =(a1+a2)+(a3+a4)+…+(an-2+an-1)+an =1+9+…+(4n-11)+2n =n-12×(1+4n-11)2+2n =2n2-3n+52. ②当n为偶数时,Sn=a1+a2+a3+…+an=(a1+a2)+(a3+a4)+…+(an-1+an)=1+9+…+(4n-7) =2n2-3n2. 所以Sn=2n2-3n+52,n=2k-12n2-3n2,n=2k (k∈Z). (3)由(2)知,an=2n-2+a1,n=2k-12n-3-a1,n=2k (k∈Z). ①当n为奇数时,an=2n-2+a1,an+1=2n-1-a1. 由a2n+a2n+1an+an+1≥5,得a21-a1≥-4n2+16n-10. 令f(n)=-4n2+16n-10=-4(n-2)2+6. 当n=1或n=3时,f(n)max=2,所以a21-a1≥2. 解得a1≥2或a1≤-1. ②当n为偶数时,an=2n-3-a1,an+1=2n+a1. 由a2n+a2n+1an+an+1≥5,得a21+3a1≥-4n2+16n-12. 令g(n)=-4n2+16n-12=-4(n-2)2+4. 当n=2时,g(n)max=4,所以a21+3a1≥4. 解得a1≥1或a1≤-4. 综上所述,a1的取值范围是(-∞,-4]∪[2,+∞). 附加题 21.【选做题】 A.选修41:几何证明选讲 (1)因为EF∥CB,所以∠BCE=∠FED,又∠BAD=∠BCD,所以∠BAD=∠FED, 又∠EFD=∠EFD,所以△DEF∽△EFA. (2)由(1)得,EFFA=FDEF,EF2=FA·FD. 因为FG是切线,所以FG2=FD·FA,所以EF=FG=1. B.选修42:矩阵与变换 (1)M=1005. 设(x′,y′)是所求曲线上的任一点,1005xy=x′y′, 所以x′=x,y′=5y, 所以x=x′,y=15y′, 代入4x-10y=1得,4x′-2y′=1, 所以所求曲线的方程为4x-2y=1. (2)矩阵M的特征多项式f(λ)=λ-100λ-5=(λ-1)(λ-5)=0, 所以M的特征值为λ1=1,λ2=5. 当λ1=1时,由Mα1=λ1α1, 得特征向量α1=10; 当λ2=5时,由Mα2=λ2α2, 得特征向量α2=01. C.选修44:坐标系与参数方程 (1)x2+y2-8y+15=0. (2)当α=3π4时,得Q(-2,1),点Q到C1的圆心的距离为13, 所以PQ的最小值为13-1. D.选修45:不等式选讲 由f(x)≥|a+b|-|2a-b||a|,对任意的a,b∈R,且a≠0恒成立, 而|a+b|-|2a-b||a|≤|a+b+2a-b||a|=3,f(x)≥3,即|x-1|+|x+1|≥3, 解得x≤-32,或x≥32,所以x的范围为{x|x≤-32,或x≥32}. 22.(1)以CA,CB,CC1分别为x,y,z轴建立如图所示空间直角坐标, 因为AC=3,BC=4,AA1=4,所以A(3,0,0), B(0,4,0),C(0,0,0),C1=(0,0,4), 所以AC1=(-3,0,4),因为AD=λAB, 所以点D(-3λ+3,4λ,0),所以CD=(-3λ+3,4λ,0), 因为异面直线AC1与CD所成角的余弦值为925, 所以|cos〈AC1,CD〉|=|9λ-9|5(3-3λ)2+16λ2=925, 解得λ=12. (上接第75页) (2)由(1)得B1(0,4,4),因为 D是AB的中点,所以D(32,2,0), 所以CD=(32,2,0),CB1=(0,4,4), 平面CBB1C1的法向量n1=(1,0,0), 设平面DB1C的一个法向量n2=(x0,y0,z0), 则n1,n2的夹角(或其补角)的大小就是二面角DCB1B的大小, 由n2·CD=0,n2·CB1=0, 得32x0+2y0=0,4y0+4z0=0, 令x0=4,则y0=-3,z0=3, 所以n2 =(4,-3,3), cos〈n1,n2〉=n1·n2|n1|·|n2| =434=23417, 所以二面角DB1CB的余弦值为23417. 23.(1)要想组成的三位数能被3整除,把0,1,2,3,…,9这十个自然数中分为三组:0,3,6,9;1,4,7;2,5,8. 若每组中各取一个数,含0,共有C13C13C12A22=36种; 若每组中各取一个数不含0,共有C13C13C13A33=162种; 若从每组中各取三个数,共有3A33+C23A22A22=30种. 所以组成的三位数能被3整除,共有36+162+30=228种. 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