2011年北约华约理工联盟自主招生部分试题汇总

专家建议：平稳心态，提前准备，从物理突破自主招生笔试
从今年的形式可以看出，物理在自主招生考试中有非常重要的地位。

可谓“得失到头皆物理”，面对自主招生考试，学有余力的同学们应该在区分度大，比重大的物理入手，从高二下甚至更早就开始准备。

主要从以下几个方面：
1.做好规划：
（1）集中在高二下之前，解决自己高考范围内的物理知识盲点，比如对于热学部分，对于碰撞问题，动量能量综合问题等。

如果学有余力，可以适当拓展，学习一些奥赛内容，锻炼思维。

（2）高二下申请进入北大清华组织的优秀中学生夏令营。

（3）在高三上学期，学校一轮复习梳理知识的同时，自己多思考一些理论推导，模型的拓展，锻炼自己的推理能力和探究能力。

（4）在12月~1月进行最后的自主招生总复习，适当模考，练就心理承受能力。

2.拓展物理知识：对于没有接触过的热学，物理光学等内容适当的参考教材，和一些奥赛入门书籍，了解基本的模型和思维方法。

2011年综合性大学(北约)自主选拔录取联合考试数学试题请注意:文科考生做1至5题,理科考生做3至7题,每题20分,共100分.1.已知平行四边形的其中两条边长为3和5,一条对角线长为6,求另一条对角线的长.2.求过抛物线2221y x x =--和2523y x x =-++的交点的直线方程.3.在等差数列{}n a 中,3713,3,a a =-=数列{}n a 的前n 项和为n S ,求数列{}n S 的最小项,并指出其值为何.4.在ABC ∆中,2a b c +≥,求证:60C ∠≤.5.是否存在四个正实数,使得它们的两两乘积为2,3,5,6,10,16?6.1C 和2C 是平面上两个不重合的固定圆,C 是平面上的一个动圆,C 与12,C C 都相切,则C 的圆心的轨迹是何种曲线?说明理由.7.求()|1||21||20111|f x x x x =-+-++-的最小值.2012年北约自主招生数学试题1、求x 的取值范围使得12)(-+++=x x x x f 是增函数；2、求1210272611=+-+++-+x x x x 的实数根的个数；3、已知0)2)(2(22=+-+-n x x m x x 的4个根组成首项为41的等差数列，求n m -； 4、如果锐角ABC ∆的外接圆的圆心为O ，求O 到三角形三边的距离之比；5、已知点)0,2(),0,2(B A -，若点C 是圆0222=+-y x x 上的动点，求ABC ∆面积的最小值。

6、在2012,,2,1 中取一组数，使得任意两数之和不能被其差整除，最多能取多少个数？7、求使得a x x x x =-3sin sin 2sin 4sin 在),0[π有唯一解的a ；8、求证：若圆内接五边形的每个角都相等，则它为正五边形；9、求证：对于任意的正整数n ，n )21(+必可表示成1-+s s 的形式，其中+∈N s2013“北约”自主招生试题（时间90分钟，满分120分） 一、选择题（每题8分，共48分）1和1 ） A. 2 B. 3 C. 5 D. 62．66⨯方阵，3个红车，3个黑车，且6个均不在同一行且不在同一列，有（ ）种方法 A. 720 B. 20 C. 518400 D. 14400 3．已知225x y =+，225y x =+，（x y ≠），则32232x x y y -+值为（ ） A. 10- B. 12- C. 14- D. 无法确定 4．在数列{}n a 中，11a =，142n n S a +=+（1n ≥），则2013a 值为（ ） A. 201230192⨯ B. 201330192⨯ C. 201230182⨯ D. 无法确定5．在ABC ∆中，D 为BC 中点，DM 平分ADB ∠交AB 于点M ，DN 平分ADC ∠交AC 于N ，则BM CN +与MN 的关系为（ ）A. BM CN MN +>B. MN CN MN +<C. BM CN MN +=D. 无法确定6．若,,A B C 为三个复数A B C ≠≠，且模全为1，则BC AC ABA B C++++=（ ）A. 12-B. 1C. 2D. 无法确定 二、解答题（每题18分，共72分）7．最多能找多少个两两不相等的正整数使其任意三个数之和为质数，并证明你的结论。

清华等五校2011届合作自主选拔通用基础测试英语试题注意事项：1．答卷前，考生务必将自己的姓名、准考证号填写在答题卡上。

2．将答案写在答题卡上，写在本试卷上无效。

3．考试结束后，将本试卷和答题卡一并交回。

Section I ReadingPart ARead the following text．Choose the best word or phrase for each numbered blank．(30 points) Learning theorists emphasize the role of environmental influences in shaping the way a person develops．__1__ their view, child development is guided by both deliberate and __2__ learning experiences in the home, peer group, school, and community．Therefore, childhood growth is significantly __3__ by the efforts of parents, teachers, and others to __4__ children in desirable ways．According to learning theories, the same __5__ that explain how people can use a bicycle or computer also explain how children acquire social skills, emotional self-control, reasoning strategies, and the __6__ skills of walking and running．One kind of learning occurs when a child‟s actions are __7__ by a reward or punishment．A reward increases the probability that behavior will be repeated．For example, a young child may __8__ draw pictures because she receives praise from her parents after __9__ each one．A punishment decreases the probability that behavior will be repeated．For example, a child who touches a hot stove and burns his fingertips is not __10__ to touch the stove again．__11__ kind of learning, classical conditioning, occurs when a person makes a __12__ association between two events．For example, babies begin sucking when they are put in a familiar nursing __13__, children fear dogs whose barking has startled them in the past．A third kind of learning __14__ of imitating the behavior of others．A boy may acquire his father‟s __15__ of talking, his mother‟s tendency to roll her eyes, and his favorite basketball player‟s moves __16__ the court．In doing so, he also acquires __17__ about the consequences of these behaviors．Learning theories provide extremely useful ways of understanding how developmental changes in behavior and thinking __18__ and, for some children, why behavior problems arise．These theories can be studied scientifically and practically applied．Critics point out, __19__, that learning theorists sometimes neglect children‟s __20__ role in their own understanding and development．1．A．For B．From C．In D．With 2．A．unintended B．uninvolved C．undiscovered D．unlimited 3．A．achieved B．created C．developed D．shaped 4．A．moralize B．recognize C．socialize D．standardize 5．A．considerations B．instructions C．principles D．tendencies 6．A．physical B．personal C．original D．technical 7．A．directed B．followed C．pursued D．tracked 8．A．continuously B．immediately C．occasionally D．regularly9．A．completing B．fulfilling C．handling D．obtaining 10．A．liable B．likely C．possible D．ready 11．A．Another B．Next C．One D．Other 12．A．emotional B．intellectual C．mental D．spiritual 13．A．condition B．location C．position D．situation 14．A．contains B．consists C．comprises D．composes 15．A．custom B．means C．type D．style16．A．over B．on C．in D．at 17．A．estimations B．evaluations C．explorations D．expectations 18．A．appear B．emerge C．exist D．occur 19．A．accordingly B．however C．moreover D．therefore 20．A．active B．dominant C．positive D．socialPart BIn the following article, some sentences or paragraphs have been removed．For questions 21 to 25, choose the most suitable one from the list A-F to fit into each of the numbered gaps．There is ONE which does not fit in any of the gaps．(15 points)Horror stories about the food industry have long been with us – ever since 1906, when Upton Sinclair‟s landmark novel The Jungle told some ugly truths about how America produces its meat．In the century that followed, things got much better, and in some ways much worse．The U.S．agricultural industry can now produce unlimited quantities of meat and grains at remarkably cheap prices．But it does so at a high cost to the environment, animals and humans．21．Add to the price tag the acceleration of global warming – our energy-intensive food system uses 19% of U.S．fossil fuels, more than any other sector of the economy．And perhaps worst of all, our food is increasingly bad for us, even dangerous．22． A food system that generates cheap, filling food at the expense of healthier produce is also a major cause of America‟s widespread obesity．At a time when the nation is close to a civil war over health-care reform, obesity adds $147 billion a year to our doctor bills．“The way we farm now is destructive of the soil, the environment and us,” says Doug Gurian-Sherman, a senior scientist with the food and environment program at the Union of Concerned Scientists．23．Film documentaries like the highly critical Food Inc．and the work of investigative journalists like Eric Schlosser and Michael Pollan are repeating Sinclair‟s work, awakening a sleeping public to the uncomfortable realities of how we eat．Change is also coming from the very top．First Lady Michelle Obama‟s White House garden has so far yielded more than 225 lb．of organic produce – and tons of powerful symbolism．24．Sustainable food is also more expensive than conventional food and harder to find．And while large companies like General Mills have opened organic divisions, there is worry that the very definition of sustainability will be used by those companies only for their own good．But we can‟t afford to remain in philosophizing about food．With the exhaustion of the soil, the impact of global warming and the inevitably rising price of oil – which will affect everything from fertilizer to supermarket electricity bills – our industrial style of food production will end sooner or later．25．Unless Americans radically rethink the way they grow and consume food, they face a future of damaged farmland, emptied-out countryside, terrifying germs, higher health costs –and tasteless diet．Sustainable food has an élitist reputation, but each of us depends on the soil, animals and plants –and as every farmer knows, if you don‟t take care of your land, it can‟t take careA．Some Americans are paying attention to such warnings and working to transform the way the country eats –farmers who are raising sustainable food in ways that don‟t bankrupt the earth．B．A series of recalls involving polluted foods this year –including a case of salmonella poisoning from tainted peanuts that killed at least eight people and sickened 600 – has consumers rightly worried about the safety of their meals．C．But, despite increasing public awareness, sustainable agriculture remains a tiny enterprise: according to the most recent data from the U.S．Department of Agriculture, less than 1% of American cropland is farmed organically．D．And when the rains come, the excess fertilizer that helped to take so much corn out of the ground will be washed into the Mississippi River and down into the Gulf of Mexico, where it will help kill fish for miles and miles around．E．Those hidden prices are the gradual damage of our rich farmland, cages for egg-laying chickens so packed that the birds can‟t even raise their wings and the horrible rise of antibiotic-resistant bacteria among farm animals．F．As the developing world grows richer, hundreds of millions of people will want to shift to the same calorie-heavy, protein-rich diet that has made Americans so unhealthy – demand for meat and poultry worldwide is set to rise 25% by 2015 –but the earth can no longer deliver．26．Read the following text and summarize it in Chinese．You should write approximately 200 characters．(20 points)Word travels quickly in the small fishing village of Port Washington, Wisconsin．So when Mardy McGarry wanted to build a playground for kids with special needs, she knew it wouldn‟t take long to create interest in the project．But she never expected that 2,800 people – a third of the town – would roll up their sleeves and use their vacation days to bring her vision to life．“A lot of learning comes through play,” says McGarry, 52, a special education teacher for 28 years．But her students were too often left out．She‟d seen the wood chips and sand of traditional playgrounds stop wheelchairs dead in their tracks．McGarry started researching play equipment and contacting design firms．When a piece of land became available, the city council agreed to give a portion for a playground if she would build it．McGarry asked classrooms of kids for their wish list．“They all said pirate ships,” she says．She also asked physical, occupational, and speech specialists for their input．And she brought on board her friend Sue Mayer, whose eight-year-old son, Sam, is disabled．“Neither of us is good at math, which is why $450,000 didn‟t sound like a lot of money,” McGarry says of the initial estimate．Her Kiwanis Club came through with $7,000, and that‟s when the grass-roots movement really got started．One woman gave $25,000 and had her company match it．Soon smaller businesses were joining in．Families bought wood blocks for the fence at $30 apiece．Bricks for the walkways went for $50 to $750 each．There were silent money-raising activities, T-shirt sales, a coin drive, and a run-walk-and-roll-a-thon．The local Pieper Family Foundation offered to donate half of the remaining $170,000 balance if McGarry could raise the rest．The $450,000 covered materials, but the actual construction would cost an additional $900,000．Out of the question．But the community could build it．The design firm they‟d selected, Leathers & Associates, had sent a 164-page how-to binder．All McGarry needed now was 500 volunteers to work six 12-hour days．On September 16, 2008, the first day of construction, they came．Two women heard about the project from a friend on the way to work and took the day off to help．A couple in their 80s manned the tool trailer．Ten-year-olds sanded surfaces and cleared up wood pieces．“It was truly an amazing week,” says McGarry．“The site looked like an anthill．So many people can take ownership of this playground.”Only three “build captains,” sent by Leathers & Associates, were paid．Volunteers with “build experience” became coordinators, those who could operate power tools formed a separate group, and so on –down to th e “runners.” One team served meals donated from local restaurants and churches, and another organized activities for the children of volunteers．Today, Possibility Playground is one of the most popular destinations in Ozaukee County．The finished wonderland, the length of a football field, sits on a slope overlooking Lake Michigan．There‟s a giant rocking pirate ship, a lighthouse, a rock-climbing wall, high and low rings, monkey bars, palm drums, sandboxes, swings, slides, bridges, and slip roads．All children play shoulder to shoulder．“It‟s neat to see Tori fitting in,” says Charlene Landing of her five-year-old wheelchair-bound daughter．“Some playgrounds have special equipment in a different section．Here, you see all the kids on the same playground having f un.”It‟s exactly what McGarry wished for．“People used to ask, …Why do you want to build a playground just for children with disabilities?‟” she says．“They didn‟t get it．It‟s only when you build a playground for children with disabilities that you build o ne for all children.”Section II WritingPart A27．Translate the following text into English．(10 points)世界上最容易的事情，莫过于踩着别人的脚印走。

8. 某生产队想筑一面积为 144 m2 的长方形围栏， 围栏一边靠墙 . 现有铁丝网 50 m ，筑成这样 的围栏最少要多少铁丝网？已有的墙最多利用多长？最少利用多长？
9. 在正方形 ABCD 中，过一个顶点 D 作对角线 CA 的平行线 DE ，若 CE CA ，且 CE 交边 DA 于点 F . 求证： AE AF .
12. 以 A 为圆心， 以 2cos (0
) 为半径的圆外有一点 B . 已知 AB 2sin ，设过 B 且与 2
圆 A 外切于点 C 的圆的圆心为 M .
（1）当 取某个值时，说明点 M 的轨迹 P 是什么曲线？
（ 2）点 M 是轨迹 P 上的动点，点 N 是圆 A 上的动点，记 MN 的最小值为 f . 求 f 的
2011 年清华等五校联考（华约）自主招生数学试卷
1. 设 n N * ， n 15 . 集合 A 、 B 都是 I 1,2, , n 的真子集， A B 集合 A 或 B 中，必有两个不同的数，它们的和为完全平方数 .
， A B I . 证明：
2. 设 f x ax2 bx x(a 0) ，方程 f x 0 t x1 . 试比较 f t 与 x1 的大小 .
x 的两个根是 x1 和 x2 ，且 x1 0 ， x2 x1 1 ，又 a
3. 求函数 f x max x 1 , x2 5 的最小值，并求出相应的 x 的值 .
4. 已 知 f x 是 定 义 在 R 上 的 不 恒 为 0 的 函 数 ， 且 对 于 任 意 的 a, b R ， 有
f a b a f b b f. a
取值范围 .
13. 设数列 an 的前 n 项和为 Sn ，点 (n, Sn )( n N * ) 均在函数 y 3x 2 的图像上 . n