2015年华二自招测验(2)

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(高清打印版)2015上海数学自招复旦分校真题

/////////////////2015年复旦分校自主招生测试题数学试题1、若4,129x y z xy y +=+=+-，求32x y z ++。

2、若抛物线2y ax bx c =++与x 轴交于A B 、，与y 轴交于C ，且三角形ABC 是直角三角形，求ac 。

3、正方形DEFG EHLB NMKL 、、，边长分别为c a b 、、，求a b c 、、，满足的关系式。

4、若不等式组1252x x x a <->⎧⎪⎨-<<⎪⎩或只有整数2-一个解，求a 的取值范围。

5、若2(1)2(1)0a x x a -+-+=的根都是整数，则整数a 的取值范围？6、已知：Rt ABC ∆，3,4,BC AC D ==为AB 上一动点，作DE BC ⊥，求EF的最小值。

7、从1,2,...,100中取两个不同的数，使两数之和大于100，则有______种不同取法。

8、若12...n x x x 、、、只能取2,0,1-，且满足12...17,n x x x ++=-+22212...37,n x x x ++=+则33312..._______n x x x +++=。

9、已知：等腰ABC ∆，两圆外切且都与AB AC 、相切，两圆半径为1和2，求ABC ∆的面积。

10、已知：正五边形1AG =，_____FG JH DC ++=。

11、已知ABC ∆外接于O ，且AO BC ⊥，垂足为D ，且AB BC=(1)证明：ABC ∆是正三角形；(2)若1,=,,AB AE x PE y ==求y 关于x 的解析是及定义域；(3)在(2)的条件下，,PAC EPC αβ∠=∠=，当y 取何值时，22sin sin 1αβ+=。

12、(1)当04x <<，解22[]0x x x --=；(2)求所有实数x ，使3[]43x x =+。

2015年华二自招测验(5)

【自主招生测验5】已知方程230x ax b -+-=有两个不相等的实数根,方程2(6)60xa xb +-+-=有两个相等的实根,方程2(4)50xa xb +-+-=没有实数根,求,a b 的取值范围.在ABC ∆中，AC AB =， 45=∠A 。

AC 的垂直平分线分别交AB 、AC 于D 、E 两点，连结CD ，如果1=AD ，求：BCD ∠tan 的值。

某公司为了扩大经营，决定购买6台机器用于生产活塞。

现有甲、乙两种机器供选择，其中每种机器的价格和每台机器的日生产活塞数量如下表所示。

经过预算，本次购买机器所需的资金不能超过34万元。

甲乙价格（万元／台）7 5 每台日产量（个） 100 60⑴ 按该公司的要求，可以有几种购买方案？⑵ 若该公司购进的6台机器的日生产能力不能低于380个，为了节约资金，应选择哪种购买方案？已知x 、y 均为实数，且满足17=++y x xy ，6622=+xy y x ，求：代数式432234y xy y x y x x ++++的值。

已知关于x 的方程018)13(3)1(22=+---x m x m 有两个正整数根（m 是整数）。

△ABC 的三边a 、b 、c 满足32=c ，0822=-+a m a m ，0822=-+b m b m 。

求：⑴ m 的值；⑵ △ABC 的面积。

如右图，直线OB 是一次函数2y x =的图像，点A 的坐标是（0，2），点C 在直线OB上且△ACO 为等腰三角形，求C 点坐标。

如图所示，已知边长为4的正方形钢板有一个角锈蚀，其中2=AF ，1=BF 。

为了合理利用这块钢板．将在五边形EABCD 内截取一个矩形块MDNP ，使点P 在AB 上，且要求面积最大，求钢板的最大利用率。

解关于x 的一元二次不等式2(1)10ax a x -++<.四个不同的三位整数的首位数字相同，并且它们的和能被它们中的三个数整除，求这些数．如图所示等腰梯形ABCD 中，AB ∥CD ，AD CB =,对角线AC 与BD 交于O ,60ACD ∠= , 点S P Q 、、分别是O D O A B 、、的中点。

华二初中自招培优讲义之初中自招训练

初中自招训练1、一个平行四边形的一边长为3，两条对角线的长分别为4和25，则它的面积为（）。

2、如图，矩形AOBC 的顶点坐标分别为A （0，3），O(0,0)，B(4,0),C(4,3),动点F 在边BC 上（不与B 、C 重合），过点F 的反比例函数y=xk 的图象与边AC 交于点E ，直线EF 分别与②若k=8,则点C 关于直线EF 的对称点在x 轴上； ③满足题设的k 的取值范围是0＜k≤12； ④若DE•EG=1225,则k=1 其中正确的命题的序号是 4、某人投两次骰子，先后得到点数m,n,用来作为一元二次方程02=++n mx x 的系数，则使方程有实根的概率为（）。

5、设m ,,1N m ∈ 以（0，m ）间的整数为分子，以m 为分母组成分数集合1A ，其所有元素和为1a ；以（0，2m ）间的整数（m N m ∈,1 ）为分子，以2m 为分母组成不属于集合1A 的分数集合2A ;……,依次类推以（0，n m ）间的整数（m N m ∈,1 ）为分子，以n m 为分母组成不属于1A ，2A ，…1-n A 的分数集合n A ，其所有元素和为n a ；则1a +2a +…+n a =( )6、对于任意正整数，定义“n 的双阶乘n!!”如下：对于n 是偶数时， n!!=n(n-2)(n-4)…6⨯ 4⨯2;对于n 是奇数时，n!!=n(n-2)(n-4)…5⨯ 3⨯1,现有如下四个命题：①（2013！！）（2014！！）=（2014！）；②2014！！=10072.1007!;③2014!!的各位数字是0；④2015！！的个位数字是5，正确的命题是（）7、已知无穷数列{}n a 具有如下性质：①1a 为正整数； ②对于任意的正整数n,当n a 为偶数时，21n n a a =+；当n a 为奇数时，211+=+n n a a 。

在数列{}n a 中，若当n k ≥时，n a =1.当1k n ≤时，n a 1.（N k k ∈≥,2），则1a 可取数字的个数为（）（用k 表示）7.如果*12...()n S n n N =+++∈，*3223...(2,)111n n n S S S T n n N S S S =⨯⨯⨯≥∈---，则下列各数中与2010T 最接近的数是（）（A ）2.9 (B) 3.0 (C) 3.1 (D) 3.28.已知12,x x 是关于x 的一元二次方程222(1)50x m x m -+++=的两实数根.（1）若12(1)(1)28x x --=.求m 的值；（2）已知等腰ABC ∆的一边长为7，若12,x x 恰好是ABC ∆另外两边的长，求这个三角形的周长.9. 如图，已知一次函数b x y +=211的图象l 与二次函数b mx x y ++-=22的图象'C 都经过点B （0,1）和点C ，且图象'C 过点A （52-，0）.（1）求二次函数的最大值；（2）设使12y y >成立的x 取值的所有整数和为s ，若s 是关于x 的方程033)111(=-+-+x x a 的根，求a 的值；（3）若点F 、G 在图象'C 上，长度为5的线段DE 在线段BC 上移动，EF 与DG 始终平行于y 轴，当四边形DEFG 的面积最大时，在x 轴上求点P ，使PD+PE 最小，求出点P 的坐标.10. 如图，在直角梯形ABCD 中，AB ∥CD ，AD ⊥AB ，∠B=60°，AB=10，BC=4，点P 沿线段AB 从点A 向点B 运动，设AP=x ．（1）求AD 的长；（2）点P 在运动过程中，是否存在以A 、P 、D 为顶点的三角形与以P 、C 、B 为顶点的三角形相似？若存在，求出x 的值；若不存在，请说明理由；（3）设△ADP 与△PCB 的外接圆的面积分别为S 1、S 2，若S=S 1+S 2，求S 的最小值．11. 已知：如图，在Rt △ABC 中，C90∠=，4BC =，12tan CAB ∠=，点O 在边AC 上，以点O 为圆心的圆过A 、B 两点，点P 为AB 上一动点.（1）求⊙O 的半径；（2）联结AP 并延长，交边CB 延长线于点D ，设AP x =，BD y =，求y 关于x 的函数解析式，并写出定义域；（3）联结BP ，当点P 是AB 的中点时，求△ABP 的面积与△ABD 的面积比ABP ABDS S △△的值．第11题图备用图B（方涛、蔡桂英编辑）。

华二初中自招培优讲义之自主招生考试数学试题

自主招生考试数学试题一、选择题（每小题3分）1、已知81cos sin =⋅αα，且︒<<︒9045α，则ααsin cos -的值为（） A. 23 B. 23- C. 43 D. 23± 2、若c b a ，，为正数，已知关于x 的一元二次方程02=++c bx ax 有两个相等的实根，则方程()()01212=+++++c x b x a 的根的情况是（）A.没有实根B.有两个相等的实根C.有两个不等的实根D.根的情况不确定3、已知半径为1和2的两个圆外切与点P ，则点P 到两圆外公切线的距离为（）A. 43B. 34C. 23 D. 3 4、下图的长方体是由A ，B ，C ，D 四个选项中所示的四个几何体拼接而成的，而且这四个几何体都是由4个同样大小的小正方体组成的，那么长方体中，第四部分所对应的几何体应是（）二、填空题（每小题4分）5、某次数学测验共有20题，每题答对得5分，不答得0分，答错得－2分，若小丽这次测验得分是质数，则小丽这次最多答对题6、已知⊙O 的直径AB=20，弦CD 交AB 于G ，AG>BG ，CD=16，AE ⊥CD 于E ，BF ⊥CD 于F ，则 AE －BF=7、如图，两个反比例函数x k y 1=和xk y 2=在第一象限内的图像依次是1C 和2C ，设点P 在1C 上，PC ⊥x 轴于点C ，交2C 于点A ，PD ⊥x 轴于点D ，交2C 于点B ，则四边形PAOB 的面积为8、若二次方程组⎪⎩⎪⎨⎧+-==-1)2(122x k y y x 有唯一解，则k 的所有可能取值为 9、设正△ABC 的边长为2，M 是AB 中点，P 是BC 边上任意一点，PA+PM 的最大值和最小值分别为s 和t ，则22t s -=10、在△ABC 中， AC=2011，BC=2010，AB=20112010+，则C A cos sin ⋅=11、已知c b a ，，为实数，且514131=+=+=+c a ac c b bc b a ab ，，，则=++cabc ab abc 12、已知Rt △ABC 的三个顶点A ，B ，C 均在抛物线2x y =上，且斜边AB 平行于x 轴，设斜边上的高为h ，则h 的取值为13、方程xx x 222=-的正根个数为 14、已知，124=+=+ab n b a ，，若221914919b ab a ++的值为2011，则n=15、任意选择一个三位正整数，其中恰好为2的幂的概率为16、勾股定理有着悠久的历史，它曾引起很多人的兴趣。

【自招】2004-2015年上海自主招生数学试题

6
2011 年华师二附自主招生数学试题及答案
一、填空题（每题 4 分） 1. 已知关于 x 的多项式 ax7 bx5 x2 x 12 （ a 、 b 为常数），且当 x 2 时，该多项式的值为 8 ，则当 x 2 时，该多项式的值为__________．
2. 已知关于 x 的方程 x2 a 2 x a 1 0 的两实根 x1 、 x2 满足 x12 x22 4 ，则实
15. 某养鱼户为了估计鱼塘内鱼的条数和重量，先网出 100 条鱼，做上标记后全部放回
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约有_________条．
16. 如图，四边形 ABCD 中， ADC 和 ABC 都是直角，
D
DE 垂直于 AB ， AD 边与 CD 边长度相等．已知四边
C 形 ABCD 的面积为 16 ，那么线段 DE 的长度是
_________． A
17. 在 ABC 中， D 为 BC 的中点， E 为 AD 的中点，如
果延长 BE 交 AC 于 F ，那么 AF : FC ________．
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B
第 16 题
18. 如图， ABC 中，已知 AB AC ， DEF 是 ABC 的内接正三角形， BDF ，
圆括号内），一律得零分．
1
1.
计算
3
4
3
9 2
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，得（）
A． 11 9
B． 1
C． 5 9
D． 1 9
2. 如果 a b ，那么下列结论正确的是（）

华二初中自招培优讲义之自招真题试卷

自主招生数学试题5一、选择题（每小题6分，共计36分） 1、方程2681x x -+=实根的个数为（）A 、1个B 、2个C 、3个D 、4个2、某班进行一次标准化测试，试卷由25道选择题组成，每题答对得4分，不答得0分，答错扣1分，那么下列分数中不可能的是（）A 、95B 、89C 、79D 、75 3、已知二次函数2y ax bx c =++的图象如图所示，则下列6个代数式：ab 、ac 、a b c ++、a b c -+、2a b +、2a b -，其值为正的式子的个数是（）A 、2个B 、3个C 、4个D 、5个4、两个相同的瓶子装满酒精溶液，在一个瓶子中酒精与水的容积之比是:1p ，而在另一个瓶子中是:1q ，若把两瓶溶液混合一起，混合液中的酒精与水的容积之比是（）A 、2p q +B 、22p q p q++ C 、2pq p q + D 、22p q pq p q ++++ 5、已知直角三角形有一条直角边的长是质数n ，另外两条边长是两个自然数，那么它的周长是（）A 、21n +B 、21n -C 、2n n +D 、2n n - 6、如图，ABC ∆中，AB AC =，40A ∠=，延长AC 到D ，使CD B C =，点P 是ABD∆的内心，则BPC ∠=（）A 、145B 、135C 、120D 、105二、填空题（共6小题，每题6分，共36分）7、设直线(1)10kx k y ++-=与坐标轴所构成的直角三角形的面积是k S ，则122008...___________S S S +++=。

8、已知方程121011x x x x m -+-+-+-=无解，则实数m 的取值范围___________。

9、已知11x x -=，则242____________20071x x x =++。

10、如图，电路中有4个电阻和一个电流表A ，若没有电流通过电流表A ，问电阻器断路的可能情况共有______________种。

上海中考自招真题26套及其答案

四校八大历年自招真题答案目录2013年上中自招试卷2014年上中自招试卷2015年上中自招试卷2011年华二自招试卷2012年华二自招试卷2014年华二自招试卷2013年华二冬令营数学试卷2015年年华二自招试卷2017年年华二自招试卷2013年复附自招试题2014年复附自招试题一2014年复附自招试题二2015年复附自招试题一2015年复附自招试题二2012年交附自招试题2013年交附自招试题2014年交附自招试题2015年交附自招试题2016年交附自招试题2014年七宝自招试题2016年七宝自招试题2016年南模自招试题2016年建平自招试题2017年建平自招试题建平数学培训资料试卷2015年控江自招试题2013年华二冬令营数学试卷1、“帽子函数”的图像如图所示：(1)求此函数的解析式；(2)若有抛物线23(),4y x a a =-+<求它与“帽子函数”图像的交点个数； (3)请试写出一个抛物线解析式，使它与“帽子函数”图像有且只有2个交点，横坐标分别为5722，．【解析】：⑴1,211,12x k x k y x k k x k ⎧≤<+⎪⎪=⎨⎪-+++≤<+⎪⎩⑵0a <时，无交点0a =时，一个交点304a <<时，两个交点 ⑶考虑到34a =时，抛物线234y x =-+与帽子函数交于11,22⎛⎫- ⎪⎝⎭、11,22⎛⎫ ⎪⎝⎭两点，所以可以将234y x =-+向右平移3个单位，即满足条件该抛物线解析式为()2334y x =--+2、在一个8×8的正方形方格纸中，一个角剪去一个2×2的小正方形，问其余部分可否剪成15块“L ”型(如图)纸片，若能剪，给出剪切方法，若不能剪，请说明理由。

【解析】（一道基础的染色问题）如图进行黑白相间染色，那么L 型放入方格纸中，必定可以盖住1个黑格子和3个白格子，或者3个黑格子和1个白格子。

上海华东师范大学第二附属中学高一自招模拟英语试题含答案

Lecture FourI. Words and phrases:1. Choice:1. Some of the passengers told the reporters about their in the burning train.(details,trips,event,experiences)2. After the president made an official announcement, she expressed her opinion.(personal, private,single,individual)3. He run towards the seaside and into the water.(dived, sank, headed, bathed)4. These football players had no strict until they joined our club.(Practice, training, education, exercise)5. Michael be a policeman, for he is much too short.(needn’t, should, can’t, may)6. I don’t know the restaurant, but it’s to be quite a good one.(told, said, spoken, talked)7. You can take as many as you like because they are free of .(charge, pay, money, fare)8. You look nice in green. Green you.(fits, suits, satisfies, matches)9. The new research team was led by the engineer.(chief, primary, main, major)10. A will be offered for information leading to the arrest of the bank robber.(profit, reward, fund, price)11. Electricity, like other forms of , has greatly increased in price.(power, force, strength, energy)12. The director is very busy, so he is not at home on Sundays.(most, mostly, hardly, already)13. Bob has made up his mind to leave here, so it won’t be easy to him to stay here.(suggest, prevent, persuade, make)14. You may take this kind of medicine without an ill .（effect，affect, aim, action）15. With doc tor’s effort, the patient is certain to before long.(go through, pull through, get through, breakthrough)16. We tried to the nervous old lady that flying was safe.(assure, promise, believe, be sure)17.”your brother’s letter when you have time,” I said, but he asked “Must Ito his letter? ” (answer...reply, reply...answer, answer...answer, reply...reply)18. The girl red when she . (Went...was lying, y, broke...lied, id)19. It is that some mistakes will happen. (Certainly, surely, certain,sure)20. The thief walked around the room , afraid of making any noise. (On his tiptoe, on tiptoe, on his tiptoes, on tiptoes)21. She me a present for my birthday last month. (agreed, approved, promised, meant)22. How did you yourselves during your holidays? (Like, enjoy, admire, appreciate)23. It’s very sad when husbands and wives with each other. (debate, discuss, quarrel, argue)24. New scientific discoveries constantly us. (Wonder, surprise, amaze, astonish)25. The fox the chicken and run away immediately.(held, seized, grasped, snatched)26. The city stands on the right of the river.(beach, bank, coast, shore)27. We must stick to education in all schools. (quality, quantity, character, personality)28. Would you show us some of your work? (models, examples, guides, samples)29. Of victories keeps pouring. (information, news, advertisement, data)30.It wasn’t an accident. He did it on .(intention, reason, purpose, carelessness)31. Two rivers connected to form one waterway.(constant, steady, continual, continuous)32. She is so that she doesn’t even know where Beijing is.(foolish, silly, stupid, ignorant)33. My father was with his new study.(pleasing, pleasant, pleased, pleasure)34. You shouldn’t your opinion on others.(force, impress, make, strike)35. His mother told him not to talk at breakfast time.(proudly, nervously, constantly, properly)36. The children like to watch the planes and land from far away.(take out, take away, take up, take off)37. Your carelessness will you your life.(waste, use, cost, spend)38. She told the dentist that the toothache had kept her the night before.(wake, woke, walking, awake)39. Human history is often compare a long river.(for, to, with, along)40. We saw a good film on TV last night.(beautiful, handsome, lovely, pretty)2. Phrases1. be familiar with2. Be fed up with3. be fit for4. be free from5. be grateful/thankful to sb. or sth6. be ignorant of7. be in favor of8. be independent of9. be infected with10. be inferior to11. be involved in12. be located in13. be lost in thought14. be of good/poor quality15. be on a diet16. be opposed to doing17. be particular about18. be popular with/among19. be present at20. be qualified for3. Understanding the following sentences and pay attention to the underlined parts.1. You must have once been here in Shanghai. You are very familiar with the roads here.2. The name sounds familiar to me.3. He is so talkative that I am a little fed up with him.4. I am fed up with the same food every day.5. I don’t think the job is fit for you.6. If we go on polluting the world, it won’t be fit for us to live in.7. I’m looking forward to living a life free from any worry.8. Even if he could be free from punishment, how can he face his children in the future?9. We are grateful to you for your suggestions.10. But choosing not to be religious is different from choosing to be ignorant of religion.11. For a cultivated man to be ignorant of foreign languages is a great inconvenience.12. Are you in favor of or against our plan?13. The Indian Prime Minister is said to be in favor of trying to build bridges with Pakistan.14. Well they definitely have to be independent of political pressure.15. Judges must be independent of political pressure.16. More than 600,000 people were estimated to be infected with HIV, about one-third of them expected to die from TB.17. Horses are the only species of domestic animal that can be naturally infected with Hendra virus.18. I’m inferior to you in math.19. The professor is so easy going that he never makes others inferior to him.20. I don’t want to be involved in your argument.21. Let your child be involved in the process of choosing books.22. Our school is located in Jading.23. He is sitting in front of the window, lost in thought.24. Lost in thought for a moment, mom replied, “then I would rather buy you a gift.”25. I assure you that our products are of good quality.26. If you want to lose weight you must be on a diet.27. But he said he wouldn’t be opposed to covering something local if it meant he’d get to stay a journalist.28. I’m opposed to rebuilding the bridge.29. You shouldn’t be too particular about food.30. His novel is popular with young people.31. You should be present at our meeting tomorrow.32. We can’t accept you as you are not qualified for the job.Part twoReading1.Reading skills:2.归纳论证归纳论证议论文是一种由个别到一般的论证方法。

华中师大一附中2015年自主招生考试数学试题(word版附答案)

华中师大一附中2015年高中招生考试数学试题考试时间：80分钟卷面满分：150分说明：所有答案一律书写在答题卡上，写在试卷上作答无效．一、选择题 (本大题共6小题，每小题6分，共36分．在每小题给出的四个选项中，有且只有一项是正确的．)1．如果实数a ，b ，c a b +可以化简为( )....A a b c B a b cC a b cD a b c--+-----+-2．反比例函数4y x=-的图象与直线y kx b =-+交于 ()()1,,,1A m B n -两点，则△OAB 的面积为( )111513..4..222A B C D 3．设12,x x 是一元二次方程230x x +-=的两根，则3212415x x -+等于( )A ．－4B ．8C ．6D ．04．已知a ，b ，c 分别是△ABC 的三边长，且满足44422222222a b c a c b c ++-+=0，则△ABC 是( )A ．等腰三角形B ．等腰直角三角形C ．直角三角形D ．等腰三角形或直角三角形5．在一节数学实践活动课上，老师拿出三个边长都为40mm 的正方形硬纸板，他向同学们提出了这样一个问题：若将三个正方形纸板不重叠地放在桌面上，用一个圆形硬纸板将其盖住，这样的圆形硬纸板的最小直径为（单位：mm ）( ).100A B C D6．如图，△ABC 内接于圆O ，BC =36，∠A =60°，点D 为弧BC 上一动点，BE ⊥直线OD 于E ，当点D 由B 点沿弧BC 运动到点C 时，点E 经过的路线长为( ).54A B C D二、填空题(本大题共7小题，每小题7分，共49分) 7．方程()31641x x x +=+的所有根的和为8．在5瓶饮料中，有2瓶已过了保质期，随机地从这5瓶饮料中取2瓶，取到至少有1瓶过保质期饮料的概率为 9．关于x 的方程211aa x =--无解，则a 的值是10．一辆快车从甲地开往乙地，一辆慢车从乙地开往甲地，两车同时出发，分别以各自的速度在甲乙两地间匀速行驶，1小时后，快车司机发现有重要文件遗忘在出发地，便立即返回拿上文件（取文件时间不计）后再从甲地开往乙地，结果快车先到达乙地．慢车继续行驶到甲地．设慢车行驶速度为x（h ），两车之间的距离为y （km ），y 与x 的函数图象如图所示，则a = 11．已知24,13,234a x y x ax ≥≤≤=-+当时函数的最小值为－23，则a = 12．如图，在单位为1的正方形的网格纸上，123345567A A A ,A A A ,A A A ,,∆∆∆都是斜边在x 轴上，且斜边长分别为2，4，6，…的等腰直角三角形．若123A A A ∆的顶点分别为()()()123A 2,0,A 1,-1,A 0,0,则依图中的规律，2015A 的坐标为13．有一张矩形风景画，长为90cm ，宽为60cm ，现对该风景画进行装裱，得到一个新的矩形，要求其长，宽之比与原风景画的长，宽之比相同，且面积比原风景画的面积大44%．若装裱后的上，下边衬的宽都为a cm ，左，右边衬都为b cm ，那么ab = 三、解答题(本大题共4小题，共66分．解答应写出文字说明、证明过程和演算步骤) 14．（14分）已知m ，n 是方程2310x x ++=的两根，（1）求162102553m m m m m -⎛⎫+-⋅- ⎪--⎝⎭的值；（215．（15分）如图，△ABC 中，AC =BC ，I 为△ABC 的内心，O 为BC 上一点，过B ，I 两点的圆O 交BC 于D 点，1tan ,6,3CBI AB ∠== （1）求线段BD 的长；（2）求线段BC 的长．16．（18分）如图，四边形ABCD 中，AD ∥BC ，∠BCD =90°，AD =6，BC =3，DE ⊥AB 于E ，AC 交DE 于F ，（1）求AE ·AB 的值；（2）若CD =4，求AFFC的值；（3）若CD =6，过A 点作AM ∥CD ，交CE 的延长线于M ，求MEEC的值．17．（18分）二次函数242y x mx n =-+的图象与x 轴交于()()()1212,0,,A x B x o x x <两点，与y 轴交于c 点．（1）若AB =2，且抛物线的顶点在直线y =－x －2上，试确定m ，n 的值；（2）在（1）中，若点P 为直线BC 下方抛物线上一点，当△PBC 的面积最大时，求P 点坐标；（3）是否存在整数m ，n ，使得1212,12,x x <<<<同时成立?请证明你的结论．参考答案一．选择题（6分×6=36分） 1．【解析】由图知0b c a <<<，(),a a a b a b c a a c ==-+=-+=-=-a b +()()a a b a c a b c =-+++-=+-，选D ．2．【解析】（补形）()4.1,:-4,4;xy A m m m =--=-∴=代入(),1:4B n n =-代入．故有A （－1，4），B （－4，1）．作AE ⊥y 轴于E ，BD ⊥x 轴于D ．可知： △AOE ≌△BOD ．且11422AOE BOD S S ∆∆==⨯⨯=．延长EA ，DB 交于C ，则四边形CDOE 是边长为4的正方形，且2416,CDOES==△ABC 是腰长为3的等腰直角三角形，且219322ABC S ∆=⨯=．于是△OAB 的面积为915162222ABC S ∆=-⨯-= 3．【解析】（降次）由韦达定理：221212112211.3,3x x x x x x x x +=-⇒=--+=+=()()()2322212111111141534115341215x x x x x x x x x ∴-+=----+=--+++()2115344x x =-+--=-，故选A ．4．【解析】由条件得： 4442222442440,a b c a c b c ++--=()()222222222222220,22,a c b c c a b a b a b c -+-=∴===+=即或且．故△ABC 是等腰直角三角形，选B ． 5．【解析】当3个正方形按如图排列时，其外接圆直径最小．显然，这个圆是等腰梯形ABCD 的外接圆O ，这里AB ∥CD 且CD =40，AB =80．设此等腰梯形的对称轴交AB 于M ，交CD 于N ，则MN =80． ∵AB >CD ，∴OM <ON ．设OM =40－x ，ON =40+x ，圆半径为r ．△AOM 中，()()22240401r x =+-； △DON 中，()()22220402r x =++（1）－（2）：1512001600,2x x -=∴=，代入（2）290251062562517400,444r r ⨯=+==∴=故所求最小圆的直径为2r =C ．6．【解析】（轨迹法）如解图，连结OB ，分别在弧BC 上取123,,,,,B D D D C 其中2OD BC ⊥，则相应的动点依次为123,,,,.B E E E N 12390BE O BE O BE O BNO ∠=∠=∠=∠=︒．故点E 的轨迹是OB 为直径的优弧BE 2N ．已知BC =36，∴2218.BE BOE =∆是含30°角的直角三角形，∴OB =．设M 为OB 的中点（优弧圆心），连MN ．则圆M 的半径MB=注意到∠BOC =120°，∴∠BON =60°，∠BMN =120°，优弧BE 2N 之长为圆M周长的222,2.33BE N l π∴=⨯⋅=，故选B ．二．填空题（7×7=49分）7．【解析1】3244160x x x --+=．根据广义韦达定理，此方程3根之和为4．即123,1,4b x x x a b a ⎛⎫++=-==- ⎪⎝⎭这里【解析2】由原方程得：()()()124220,4,2,x x x x x -+-=∴==-31232.4x x x x =++=． 8．【解析】（正繁则反）由于从这5瓶饮料中任取2瓶，没有过期饮料的概率为3,5故取2瓶，取到至少有1瓶过保质期饮料的概率为32155-=， 9．【解析】由原方程得：()()()2111a a x =--，关于x 的方程（1）只有唯一解1x =，代入（1）得0a =，此时原方程无解；又在方程（1）中令1,a =得0a =．矛盾．此时方程（1）无解，从而原方程无解．故若原方程无解，则必01a =或． 10．【解析】慢车12．5小时走完全程，()12.5100080x x km =⇒=设快车速度为t （h ）∵1小时后两车相距800km ，即1小时两车共行200km ，∴t =120km （h ）∵a 小时后两车相遇，此时慢车走80akm ，快车走120（a －1）km ，故有：()()80120110002001120, 5.6a a a a h +-=⇒=∴=11．【解析】原式配方得：2392448y x a a ⎛⎫⎛⎫=-+- ⎪ ⎪⎝⎭⎝⎭，抛物线开口向上且对称轴为34x a =．当34,34a a ≥≥时，故当13x ≤≤时，y 随x 增大而减小．故当x =3时有： 22333423945, 5.a a a ⨯-⋅+=-⇒=∴=12．【解析】注意到点13521,,,,n A A A A +全在x 轴上，设其横坐标依次为1352015,,,,x x x x ．．继续分析．点41A n +都在原点右边，其横坐标取正值，点41A n -都在原点左边（其中3A 为原点），其横坐标取0或负值（其中仅3A 横坐标为0）．∵2015=4×504－1，故2015A 必在原点左边，其横坐标必为负值．易求()()34117421114310,021,0224,x x x x x x ⨯-⨯-⨯-====+-⨯==+-⨯=-()201545041025031006x x ⨯-==+-⨯=-，故所求点A 的坐标为：()20151006,0A -．13．【解析】依题意有：9029032360260222a ab b +==⇒=+（据等比定理）故()231a b =又：()()1449026029060100a b ++=⨯⨯()120180454442a b ab ⇒++=⨯，（1）代入（2）：2260318065444603960.b b b b b ⋅++=⨯⇒+-=解得：()666b b ==-或舍，从而9,54a ab =∴=．三．解答题14．【解析】（1）∵2310,m m ++=故()()()()551625162102255353m m m m m m m m m m m +-+--⎛⎫+-⋅-=⋅- ⎪-----⎝⎭ ()()2229223123203m m m m m m m m-++=--=-+-=-⋅=-．（2）m ，n 是方程2310x x ++=的两根，31m n mn +=-⎧∴⎨=⎩设x =，则33442m n m n x n m mn +=++=+()()()2222222221,2229249mn x m nm n m n mn ⎡⎤=∴=+-+=+-=-=⎣⎦7,x ∴=即=7．15．【解析】（1）如解图，I 为△ABC 内心，故BI 平分∠ABC ．设∠ABI =∠CBI =α．连CI ，并延长交AB 于E ，∵CA =CB ，∴CE ⊥AB ，且AE =BE =3．于是1IE=BE tan 31,3BI α⋅=⨯===连DI ，∵BD 为圆O 的直径，∴∠BID =90°．于是10tan 3DI BI BD α=⋅===．（2）连OI ，∵OI =OB =53，∴∠DOI =2α，故OI ∥AB ， △COI ∽△CBE ，5533,539353OI CO CO CO BE CB CO CO =⇒=⇒=++2525515,121234CO BC ∴==+=． 16．【解析】（1）如解图1，作AG ∥BC ，交 CB 延长线于G ，则四边形AGCD 为矩形． ∴GC =AD =6，但BC =3，∴GB =3．已知DE ⊥AB 于E ，∴△AGB ∽△DEA ．于是18.AB BG AB AE AD BG AD AE=⇒⋅=⋅= （2）延长AB ，DC 交于H ．∵AD ∥BC ，且 AD =2BC ，∴BC 为△AHD 的中位线，故 CH =DC =4．由勾股定理知AH =10，AB =BH =5．沿DE ，CB 交于T ，有△AED ∽△BTE ．Rt △ADH 中，DE ⊥AH ，23618,105AD AE AH ∴=== 187555BE AB AE =-=-=．于是 775,186185BT BE BT AD AE =⇒==7716,3333BT CT ∴==+=由△AFD ∽△CFT ，知691638AF AD FC CT ===．（3）如解图3有AB BH ==AE EH==∴==∵△AEM∽△HEC，1.4ME AEEC EH∴===17．（18分）二次函数242y x mx n=-+的图象与x轴交于()()()1212,0,,A xB x o x x<两点，与y轴交于c点．（1）若AB=2，且抛物线的顶点在直线y=－x－2上，试确定m，n的值；（2）在（1）中，若点P为直线BC下方抛物线上一点，当△PBC的面积最大时，求P点坐标；（3）是否存在整数m，n，使得1212,12,x x<<<<同时成立?请证明你的结论．【解析】（1）()2212121AB=2244x x x x x x⇒-=⇒+-=．由韦达定理：121224mx xnx x⎧+=⎪⎪⎨⎪=⎪⎩，故有：()2414mn-=抛物线的顶点为24,44m n m⎛⎫-⎪⎝⎭，代入y=－x－2：()2242224444n m m m mn-=--⇒=--代入（1）：20,8,4mm-=∴=从而12n=．（2）在（1）的条件下，有：241612y x x=-+此抛物线的顶点为（2，－4），交x轴于A（1，0），B（3，0），交y轴于C(0，12）易求直线BC的解析式为412y x=-+．为使△PBC面积最大，只需点P与直线BC距离最远．设过P且平行于BC的直线解析式为4y x b=-+，代入抛物线解析式；22416124412120.x x x b x x b-+=-+⇒-+-=()14416120912,3b b b ∆=--=⇒=-∴=令．此时有33,43 3.22x y ==-⨯+=-即所求点的坐标为3,32P ⎛⎫-⎪⎝⎭．（3）（反证法）假如存在这样的整数m ，n ，使得方程2420x mx n -+=之2根满足121,2x x <<．那么：()122<<4,4<8,,5,6,7;12mx x m m m +=∴<∴=为整数()121<<4,416,,5,6,7,,15;24nx x n n n =∴<<∴=为整数()224160,34m m n n ∆=->∴<方程之2根为：x ==()22148164244m m m m n n m >⇒->⇒-+>-⇒>-()2228464164165m m n m m n m <⇒-⇒-<-+⇒>-当m =5时，2m －4=6>4m －16=4，根据（3），（4），取2m －4<24m n <，即1664n <<，无整数解，舍去；当m =6时， 2m －4=8=4m －16，根据（3），（4），取2m －4<24m n <，即89,n <<无整数解，舍去；当m =7时， 2m －4=10<4m －16=12．根据（5），（4），取2416,4m m n -<<即112124n <<无整数解，舍去．据上分析，不存在整数m ，n ，使得1212,12,x x <<<<同时成立．。

华二自招模拟训练卷(二)

华二自招训练题〔二〕Part 1 Words and phrasesI. Choice:1. A person’s _________ body temperature is about 37C.(A :ordinary, B:normal, C: common,D:usual)2.When I opened the door, a parcel on the floor___________ my eye.(A: met, B: caught, C:drew,D: attracted)3.Many foreigners_____________the Great Wall as the World’s Seventh Wonder.(A: look at ,B: look for,C: look around,D: look on)4. A piece of ______________music will make you happy. (A:fond ,B: pleased, C: merry,D: glad)5.It has been ten years now since the Labour Party came to____________ in that country.(A:power ,B: control ,C: force,D: charge)6.Have you finished your report yet? No, I'll finish in _________ten minutes. (A:other,B:another, C:more,D: less )7.You are __________your time trying to persuade him; He'll never join us. (A:spending,B: losing, C: missing, D: wasting )8.It's nearly seven o'clock. Jack ___________be here at any moment. (A:need,B: must,C: can,D: s hould )9.What did you think of her speech? She ____________for an hour but didn't ____________much . (A:spoke …say;B: said …speak;C: said …say ;D:spoke …speak )11.He was expected to ___________at the meeting but he failed to come. (A:come across, B:show out,C: turn up,D: present out )12.The information ______________on the mystery of Mary's sudden death. (A:threw light,B: ga ve light,C: made light ,D:too light )13.It's necessary for us to ______________our bad habits. (A:break with,B：break out of, C:break away with, D:break up )14._______________her clothing,the woman must be from an Arabian country. (A:Looking at, B: Seen fron,C: Observe by, D:Judging by )15.As we all know, our living conditions can only be ____________step by step. (A:improved, B: raised, C:turned, D:corrected )16. What he said will ____________.(A:realize, B:complete,C: come true, D: finish )17.It will further strengthen the exchange between ___________and thus lead _____________bett er understanding. (A:peoples ……to;B: people ……into; C:peoples ……/ ;D:people ……/)18..This photo ___________me of what we did together during our holiday. (A:reminds,B: recogn izes,C: remembers D :causes ).19.Students in that area learn how to ______________food and animals besides subjects. (A:feed, B:plant, C:grow, D:raise )20.I went to Hangzhou last summer vacation with my brother and visited many _______________ ______.(A:places of interests, B:places of interest, C:place of interest, D: place of interests )21 .Please ________________this little gift as your souvenir. (A:receive, B:get,C:accept, D:gain ).22.We have ______________some victories despite these difficulties. (A:finished, B:completed,C : achieved, D:accomplished )23. All this ____________me that she was innocent. (convinced, persuaded, advised, taught )24 .A daughter's marriage is quite a(n) _____________for a father. (A:incident, B:accident, C:eve nt,D: thing )25.He ran into the burning house and saved the child, which was really a brave ______________. (A:motion,B: action, C:deed, D:act )26.The girl kept a cat as a __________.(A:animal.B: beast, C:pet,D: creature )27. She has no ___________in him. ( A: belief ;B: faith, C: hope, D: trust )28.I was not ____________to see the difference between them. (A: capable, B: able, C: qualified, D: enabled )29. Gold is ___________ , wealth is abstract. (A: real, B: actual, C: concrete, D: exact )30.The town was __________in the early hours of the morning. (A: ever,B: still, C: yet, D: never )31.Is it __________for you to buy so many dresses at a time? (A: vital,B: necessary, C: essential, D: imperative )32.Her behavior is _________of great praises. (A: worthy, B: worth,C: worthwhile ,D: valuable )33.Are we allowed to take __________in the museum? (A: drawings, B: paintings,C: portraits, D: pictures )34.You'd better ___________the holiday arrangement once more before you leave tomorrow. (A: go round, B: go into, C: go after,D: go over )35.Her outstanding ____________in the competition proved that she was a worthy winner. (A: per formance, B: activity, C: behavior, D: personality )36.___________your step, or you might fall into the poll. (A: Feel, B: See, C: Watch, D: Look )37. As we know, there are four ___________of blood. (A: kinds, B: sorts, C: types, D: shapes )38.Please __________that you won't make such a mistake. (A: make sure, B: make out, C: make u p,D: make in )39.There 'll be some baby birds in the nest, these eggs are ready to ____________.(A: search, B: match, C: clench, D: hatch )40.The close _____________are not allowed to get married. (A:relationships,B： relatives,C: rela tions,D: relatives )II. Phrases1.As a whole2.As far as3.As usual4.Ask a favor of6.Ask after7.Associate A with B8.Assure …of/assure sb.that…9.At a loss / at sea10.At all costs / at any cost11.At first sight12.At one time13.At present14.At the cost of15.At the mercy of16.At the moment17.At the risk of one’s life18.At times19.Attach… to20.Attach importance toIII. Put the following into English or vice versa: 1.你最好从整体来看待这个问题。

上海市华东师范大学第二附属中自主招生英语试题含答案

华二自招试卷（八）Lecture NineWords and phrasesChoice:1.What you said just now________me of that American professor.(mentioned,informed,reminded,memorized)2.Fred is second to none in maths in our class,but believe it or not,he_______passed the last exam.(easily,hardly,actually,successfully)3.She is______to leave as soon as possible.(hurried,worried,anxious,nervous)4.Mary has______a difficult question to me.(put,asked,made up,set)5.Will you_____me a favor,please?(give,make,do,bring)6.Some of the houses on the hillside are________to cars.(impossible,inconvenient,inadequate,inaccessible)7.If you are_______about Australian cities,just read the book written by Dr.Johnson.(interested,curious,anxious,upset)8.Many countries are increasing their use of natural gas,wind and other forms of_________.(source,power,material,energy)9.The cars give off a great deal of waste gas in the streets.Yes.But I’m sure something will be done to _______air pollution.(remove,reduce,warn,collect)10.The thing that_______is not whether you fail or not,but whether you try or not.(cares,considers,matters,minds)11.Although she has taken the medicine for two months,she can’t_______the cold.(take off,do away,get rid of,put away)12.We had a few _______days of rest in the mountain area.(quiet,silent,calm,still)13.Two important secrets of long life are regular exercise and_____from worry.(feelings,motion,freedom,process)14.If you disobey traffic rules,I think it will______you much trouble.(offer,take,give,cause.)15.What does she is often contrary________what she says.(against,for,to,on)16.I would rather stay at home_______swim.(than,to ,for,instead of)17.We all find that Mr.White seems to be_________these days.(more thankful,more considerate,more personal,more surprising)18.In the field of medicine,nowadays,the new______of psychology seems to be more popular.(application,reply,relief,practice)19.The bedroom was in a state of______because of these naughty boys.(dirty,disorder,mind,confidence)20.The special service____all the members of the club.(is needed to,is needed for,is available to,is available with)21.You are so kind to me that I wonder how i can_________you your help.(award,reward,grant,deserve)22.People used to_______that the earth was flat.(trust,believe,believe in, assure)23.Can i ______this novel for three weeks?(take,lend,borrow,keep)24.His brother will be____to work in the countryside.(taken,brought,sent,carried)25.She said she______the countryside to the city.(picked,chose,preferred,selected)26.The Chinese nation_______more than 50 national minorities besides theHans.(includes,holds,contains,involves)27.Perhaps i can not ______any time except on Sunday.(spend,take,cost,spare)28.Her honesty and willingness to work are the_________of her success.29.He gave away several______of old newspapers and magazines.(parcels,bunches,bundles,packages)30.I’m glad to tell you that your______for the job is tomorrow morning.(conversation,dialogue,interview,chat)31.For a reception male guests usually wear black_______and white shirts.(suits,dresses,garments,clothes.)32.The Victorian______of the English history is the time when Victoria was queen.(time,period,ages,day)33His parents felt very happy because he had passed the college_______examination.(gate,door,exit,entrance)34.His novel is perhaps the_______one of his century.(biggest,largest,greatest,hugest)35.She looked________while she was lying there.(quiet,silent,still,peaceful)36.He started his______life as an actor at the age of 20.(experienced,professional,qualified,practiced)37.There are usually at least two_______of looking at every question.(ways,views,means,directions)38.She has searched_______for the ring but can not find it.(hardly,deeply,heavily,thoroughly.)39.His mother often complains________the music he plays because it’s too______(of...aloud,about...loudly,of...loudly,about...loud)40.Wilson tried hard to_______a new world record.(set up,set out,set about,set)Phrases:bine...withe acrosse down withe into existence/beinge oute to(oneself)e into powere to/arrive at a conclusione up withe upment onmit a crimemon sensepare A to Bpare A with Bpete with sb. For sth.17.Concentrate on18.Conduct a survey19.Confuse A with B20.Congratulate sb.on sth.Underlined the following sentences:1.When they are allowed to combine with oxygen,this energy is released as heat and light.2.When you come across something important,write it down.3.What should I do when I come across situations like these again?4.Scientists are really interested in this idea that if you infect people with the virus,everyone will get infected,but only 75%of the people will actually come down with the cold.5.In Australia on Saturday,health officials said a person in Queensland who had been travelling in the U.S. had come down with the disease but had fully recovered.6.The implication is that these are jobs that would not otherwise have come into existence without the bridge.7.New products would be developed,new companies would come into existence and retailers would increase their sales.8.What he said came out to be true.9.The patient came to himself with the help of the doctors.10.Half a year later,no matter how significant their achievements are,2 new successors will take place of them and come into power after the reelection.11.We can’t come to any conclusion before we do any investigation.12.I am going to discuss this topic and strive to give the arguments of both sides.I will then come to a conclusion based on what i have learned.13.You would take things as they came.All you had to do was think a little harder and come up with a new conclusion.14.With theentire team gathered together,each team member must come up with something positive from the past three months and something negative from the past three months.15.You should see all the services come up on the backup machine in under a minute.16.The little boy came up to the singer and presented the flowers to him.17.You’d better keep silent and don’t comment on anything.18.The young who break the law and commit a crime are both vermin and victims19.It’s common sense to dress more warmly when it gets cold20.These are all common sense treatments.21.Nurses are compared to angels in white.22.How does the US compare to Canada,Russia,India-other populous nations?23.Weeds compete with plantings for water and nutrients.24.In spite of this,however,women in many places are still denied the right to compete with men for jobs on equal terms and the right to equal opportunities for promotion.25.He can hardly concentrate himself on his study.26.If you concentrate on the project,you can finish it on time.27.To understand our target customers’ needs better,we would like to conduct a survey.28.In accordance with the standard for students healthy condition,we conduct a survey on 13033 students in Nantong University.29.They do original reporting,they do not confuse fact with opinion or rumor,and they make sound editorial decisions.30.The twins are so much alike that i often confuse one with the other.31.I come here to congratulate on your promotion.32.Congratulate on their new position and please kindly support them for the coming new termas always!Part TwoReading:Reading SkillsII.如何阅读记叙文1.夹叙夹议阅读此类文体时一定要在通读全文的基础上弄清文章的结构。

华二自招英语模拟卷(二)

C. watch
D. to be watching
2. ________he can win the contest depends on how hard he tries.
A. That
B. If
C. Whether
D. What
3. ________some of us have taken part in the sports meet and only a few have won the game.
and changed back into his suit．Alf did this for over two years and his fellow dustmen
kept his 7 ．Alf's wife has never discovered that she married a dustman and she
B. Look up
C. has turned C. Look out
D. had turned D. Look down
II. 用括号内所给单词的适当形式完成下列句子。每空格限填一词。 1. Commonly, Chinese people would like to go back to their________ homes on important festivals for family unions, (national) 2. Don’t________ on your dictionary all the time when you meet with a new word, (independent) 3. Leave the________ on. I want to lay the table for supper, (clothing) 4. She was really unfortunate and many ________ turned her hair white. (sorrow) 5. Tom travelled around the world and formed many new________ over the summer. (friend)

华二自招飞行考五门学科

华二自招飞行考五门学科（原创实用版）目录1.华东师范大学第二附属中学自招飞行考试科目2.五门学科的内容和重要性3.备考建议和注意事项正文华东师范大学第二附属中学（简称华二）自招飞行考试涉及五门学科，分别是数学、物理、化学、英语和语文。

这五门学科在飞行考试中占据重要地位，下面我们来具体了解一下这五门学科的内容和重要性。

首先是数学，数学是飞行考试的基础，涉及到的内容包括代数、几何、三角函数等。

学好数学能够帮助考生在解决飞行问题时更加得心应手。

其次是物理，物理学科主要包括力学、电磁学、光学等内容。

在飞行考试中，物理题目通常较为复杂，需要考生具备较强的分析和解决问题的能力。

接着是化学，化学主要涉及化学反应、物质性质、化学方程式等内容。

学好化学有助于考生更好地理解飞行器燃料和动力系统的原理。

然后是英语，英语主要测试考生的听、说、读、写能力。

在飞行考试中，英语题目通常以阅读理解、翻译等形式出现，考生需要具备扎实的英语基本功。

最后是语文，语文主要测试考生的阅读理解、古文翻译、作文等能力。

学好语文有助于提高考生的表达能力，更好地应对飞行考试中的各类题目。

针对这五门学科的备考，考生可以采取以下建议：一是制定合理的学习计划，确保每个学科都得到充分的复习；二是多做真题和模拟题，总结经验和规律；三是及时查漏补缺，针对自己的薄弱环节进行针对性强化训练。

在备考过程中，考生还需要注意以下几点：一是保持良好的心态，避免焦虑和紧张影响考试发挥；二是注重时间分配，确保每个学科都有足够的复习时间；三是注意休息，保证充足的睡眠和适当的锻炼，以保持良好的身体状态。

总之，华二自招飞行考试涉及五门学科，分别是数学、物理、化学、英语和语文。

学好这五门学科，对于备考飞行考试至关重要。