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2009北京高考英语作文Dear Students,Today, I would like to share with you a sample essay that could have been written for the 2009 Beijing Gaokao English composition section. The topic was "The Most Unforgettable Person in My Life." Here is a sample essay that could have been written for this prompt:The Most Unforgettable Person in My LifeThroughout my life, I have encountered numerous individuals who have left a lasting impression on me. However, there is one person who stands out above the rest, and that is my high school English teacher, Mr. Wang. His influence on my life has been profound and unforgettable.Mr. Wang was not just a teacher; he was a mentor, a guide, and a friend. He had a unique way of teaching that made learning English an enjoyable experience. His classes were never dull; they were filled with laughter, engaging discussions, and thought-provoking questions. He believedthat language is not just about grammar and vocabulary but also about culture and communication.One of the most unforgettable lessons I learned from Mr. Wangwas the importance of perseverance. During my junior year, I struggled with English literature. The dense texts and complex themes seemed insurmountable. Mr. Wang noticed my frustration and offered to help me after school. He patiently guided me through the texts, explaining the nuances of the language and the cultural context. His encouragement and support helped me to overcome my difficulties and develop a love for literature.Moreover, Mr. Wang instilled in me the value of hard work and dedication. He often shared stories of his own struggles and how he overcame them through determination and a strong work ethic. These stories inspired me to push myself beyond my limits and strive for excellence in everything I do.Beyond academics, Mr. Wang also taught me valuable life lessons. He emphasized the importance of kindness, empathy, and respect for others. He believed that these qualities are just as important as academic achievements and encouraged us to be good citizens and responsible individuals.In conclusion, Mr. Wang is the most unforgettable person in my life because he has not only taught me the English language but also life skills that I will carry with me forever. His passion for teaching, his dedication to his students, and his unwavering belief in our potential haveleft an indelible mark on my life. I am grateful for the time I spent in his class and the lessons I learned from him.This essay is a personal reflection on the impact one individual had on the writer's life. It highlights the importance of a good teacher and the lasting influence they can have on their students. Remember, when writing an essay like this, it's crucial to provide specific examples and details that illustrate why the person is so memorable and how they impacted your life.。
2009年北京卷高考真题英语试卷-学生用卷一、单项选择（每小题1分，共15分）1、【来源】 2009年高考真题北京卷第21题1分2009年北京高考John plays basketball well,his favorite sport is badminton.A. soB. orC. yetD. for2、【来源】 2009年高考真题北京卷2009年高考真题北京卷第22题1分You may use room as you like you clean it up afterwards.A. so far asB. so long asC. in caseD. even if3、【来源】 2009年高考真题北京卷第23题1分2009年高考真题北京卷Scientists have many theories about how the universe into being.A. cameB. was comingC. had comeD. would come4、【来源】 2009年高考真题北京卷2009年高考真题北京卷第24题1分For breakfast he only drinks juice from fresh fruit on his own farm.A. grownB. being grownC. to be grownD. to grow5、【来源】 2009年高考真题北京卷第25题1分2009年高考真题北京卷2018~2019学年3月上海普陀区上海市曹杨第二中学高一下学期月考第10题One of the few things you say about English people with certainty is that they talk a lot about the weather.A. needB. mustC. shouldD. can6、【来源】 2009年高考真题北京卷第26题1分2019~2020学年广东深圳罗湖区深圳市翠园中学高一上学期单元测试《Unit 5 定语从句》第2题1分—What do you think of teaching, Bob?—I find it fun and challenging. It is a job you are doing something serious but interesting.A. whereB. whichC. whenD. that7、【来源】 2009年高考真题北京卷第27题1分2011~2012学年北京西城区北京师范大学附属中学高一下学期期中2009~2010学年北京西城区高二上学期期中2015~2016学年广东广州高一上学期期中模拟第8题1分The way the guests in the hotel influenced their evaluation of the service.A. treatedB. were treatedC. would treatD. would be treated8、【来源】 2009年高考真题北京卷第28题1分All of them try to use the power of the work station information in a more effective way.A. presentingB. presentedC. being presentedD. to present9、【来源】 2009年高考真题北京卷第29题1分The wine industry in the area has developed in a special way,little foreign ownership.A. byB. ofC. withD. from10、【来源】 2009年高考真题北京卷第30题1分2018~2019学年广东深圳福田区深圳市高级中学高中部高二下学期期中第25题1分When I talked with my grandma on the phone, she sounded weak, but by the timewe up, her voice had been full of life.A. were hangingB. had hungC. hungD. would hang11、【来源】 2009年高考真题北京卷第31题1分At first he hated the new job but decided to give himself a few months tosee it got any better.A. whenB. howC. whyD. if12、【来源】 2009年高考真题北京卷第32题1分John was given the same suitcase his father and grandfather with them to school.A. tookB. had takenC. were takingD. would take13、【来源】 2009年高考真题北京卷第33题1分The biggest whale is blue whale, which grows to be about 29 meters long---the height of9-story building.A. the; theB. a; aC. a; theD. the; a14、【来源】 2009年高考真题北京卷第34题1分2020~2021学年广东广州番禺区广东仲元中学高二上学期期中第53题1分2018~2019学年10月上海宝山区上海交通大学附属中学高一上学期月考第11题twice, the postman refused to deliver our letters unless we chained our dog.A. Being bittenB. BittenC. Having bittenD. To be bitten15、【来源】 2009年高考真题北京卷第35题1分2020~2021学年广东广州越秀区广州大学附属中学高二上学期期末第28题1分(每题1分)Being a parent is not always easy, and being the parent of a child with special needs often carrieswith extra stress.A. itB. themC. oneD. him二、完形填空（每小题1.5分，共30分）16、【来源】 2017~2018学年北京北京市第一零九中学高一上学期期中第36~55题30分2017~2018学年10月陕西西安碑林区西安交通大学附属中学高一上学期月考第26~45题20分2020~2021学年10月天津河西区天津市新华中学高一上学期月考第11~30题30分2018~2019学年北京西城区北京市第四中学高二下学期期末第11~30题James' New BicycleJames shook his money box again. Nothing! He carefully1the coins that lay on the bed. ＄24.52 was all that he had. The bicycle he wanted was at least$90!2on earth was he going to get the3of the money?He knew that his friends all had bicycles. It was4to hang around with people when you were the only one without wheels. He thought about what he could do. There wasno5asking his parents, for he knew they had no moneyto6.There was only one way to get money, and that was to7it. He would have to find a job.8who would hire him and what could he do? He decided to ask Mr. Clay for advice, who usually had9on most things."Well, you can start right here, " said Mr. Clay. "My windows need cleaning and my car needs washing."That was the10of James's odd job（零工）business. For three months he worked every day after finishing his homework. He was amazed bythe11of jobs that people found for him to do. He took dogs and babies for walks, cleared out cupboards, and mended books. He lost count of the12of cars he washed and windows he cleaned, but the13increased and he knew that he would soon have14for the bicycle he longed for.The day15came when James counted his money and found $94.32.He16no time and went down to the shop to pick up the bicycle he wanted. He rode17home, looking forward to showing his new bicycle to his friends. It had been hard18for the money, but James knew that he valued his bicycle far more19he had bought it with his own money. Hehad20what he thought was impossible, and that was worth even more than the bicycle.A. cleanedB. coveredC. countedD. checkedA. HowB. WhyC. WhoD. WhatA. amountB. partC. sumD. restA. braveB. hardC. smartD. unfairA. pointB. reasonC. resultD. rightA. spellB. spendC. spareD. saveA. borrowB. earnC. raiseD. collectA. OrB. SoC. ForD. ButA. decisionsB. experienceC. opinionsD. knowledgeA. beginningB. introductionC. requirementD. openingA. similarityB. qualityC. suitabilityD. varietyA. brandB. numberC. sizeD. typeA. effortB. pressureC. moneyD. troubleA. allB. enoughC. muchD. someA. finallyB. instantlyC. normallyD. regularlyA. gaveB. leftC. tookD. wastedA. patientlyB. proudlyC. silentlyD. tiredlyA. applyingB. askingC. lookingD. workingA. sinceB. ifC. thanD. thoughA. deservedB. benefitedC. achievedD. learned三、阅读理解（每小题2分，共40分）17、【来源】 2009年高考真题北京卷（A篇）第56~60题10分How I Turned to Be OptimisticI began to grow up that winter night when my parents and I were returning from my aunt's house, and my mother said that we might soon be leaving for America. We were on the bus then. I was crying, and some people on the bus were turning around to look at me. I remember that I could not bear the thought of never hearing again the radio program for school children to which I listened every morning.I do not remember myself crying for this reason again. In fact, I think I cried very little when I was saying goodbye to my friends and relatives. When we were leaving I thought about all the places I wasgoing to see—the strange and magical places I had known only from books and pictures. The country I was leaving never to come back was hardly in my head then.The four years that followed taught me the importance of optimism, but the idea did not come to me at once. For the first two years in New York I was really lost—having to study in three schools as a result of family moves. I did not quite know what I was or what I should be. Mother remarried, and things became even more complex for me. Some time passed before my stepfather and I got used to each other. I was often sad, and saw no end to "the hard times."My responsibilities in the family increased a lot since I knew English better than everyone else at home. I wrote letters, filled out forms, translated at interviews with Immigration officers, took my grandparents to the doctor and translated there, and even discussed telephone bills with company representatives.From my experiences I have learned one important rule: almost all common troubles eventually go away! Something good is certain to happen in the end when you do not give up, and just wait a little! I believe that my life will turn out all right, even though it will not be that easy.(1) How did the author get to know America?A. From her relatives.B. From her mother.C. From books and pictures.D. From radio programs.(2) Upon leaving for America the author felt.A. confusedB. excitedC. worriedD. amazed(3) For the first two years in New York, the author.A. often lost her wayB. did not think about her futureC. studied in three different schoolsD. got on well with her stepfather(4) What can we learn about the author from Paragraph 4?A. She worked as a translator.B. She attended a lot of job interviews.C. She paid telephone bills for her family.D. She helped her family with her English.(5) The author believes that.A. her future will be free from troublesB. it is difficult to learn to become patientC. there are more good things than bad thingsD. good things will happen if one keeps trying18、【来源】 2009年高考真题北京卷（B篇）第61~63题6分2017~2018学年山西太原小店区山西大学附属中学高二上学期期中（A篇）第1~3题6分Dear Friend,The recent success of children's books has made the general public aware that there's a huge market out there.And there's a growing need for new writers trained to create the ＄3 billion worth of children's books bought each year… plus stories and articles needed by over 650 publishers of magazines for children and teenagers.Who are these needed writers? They're ordinary folks like you and me.But am I good enough?I was once where you might be now. My thoughts of writing had been pushed down by self-doubt, and I didn't know where to turn for help.Then, I accepted a free offer from the Institute to test my writing ability, and it turned out to be the inspiration I needed.The promise that paid off.The Institute made the same promise to me that they will make to you, if you show basic writing ability:You will complete at least one manuscript （手稿） suitable to hand in to a publisher by the time you finish our course.I really didn't expect any publication before I finished the course, but that happened. I sold three stories. And I soon discovered that was not unusual at the Institute.Since graduation, I have authored 34 nationally published children's books and over 300 stories and articles.Free test and brochure.We offer a free ability test and will send you a copy of our brochure （小册子） describing our recognized home-study courses on the basis of one-on-one training.Realize your writing dream today. There's nothing sadder than a dream delayed until it fades forever.Sincerely,Kristi Hill, InstructorInstitute of Children's Literature(1) The first two paragraphs tell us that.A. magazines for children and teenagers have drawn public attentionB. there is a growing need for writers of children's booksC. publishers are making $3 billion each yearD. children's books are usually bestsellers(2) You are promised to publish one manuscript when you.A. show basic abilityB. finish the courseC. have sold three storiesD. have passed the test(3) Kristi Hill mentions her own experience mainly to.A. prove she is a qualified instructorB. introduce the home-study coursesC. show she has realized her dreamD. promote the writing program19、【来源】 2009年高考真题北京卷（C篇）第64~67题8分How Room Designs Affect Our Work and FeelingsArchitects have long had the feeling that the places we live in can affect our thoughts, feelings and behaviors. But now scientists are giving this feeling an empirical （经验的，实证的） basis. They are discovering how to design spaces that promote creativity, keep people focused and lead to relaxation.Researches show that aspects of the physical environment can influence creativity. In 2007, Joan Meyers-Levy at the University of Minnesota, reported that the height of a room's ceiling affects how people think. Her research indicates that higher ceilings encourage people to think more freely, which may lead them to make more abstract connections. Low ceilings, on the other hand, may inspire a more detailed outlook.In addition to ceiling height, the view afforded by a building may influence an occupant's ability to concentrate. Nancy Wells and her colleagues at Cornell University found in their study that kids who experienced the greatest increase in greenness as a result of a family move made the most gains on a standard test of attention.Using nature to improve focus of attention ought to pay off academically, and it seems to, according to a study led by C. Kenneth Tanner, head of the School Design & Planning Laboratory at the University of Georgia. Tanner and his team found that students in classrooms with unblocked views of at least 50 feet outside the window had higher scores on tests of vocabulary, language arts and maths than did students whose classrooms primarily overlooked roads and parking lots.Recent study on room lighting design suggests that dim （暗淡的） light helps people to loosen up. If that is true generally, keeping the light low during dinner or at parties could increase relaxation. Researchers of Harvard Medical School also discovered that furniture with rounded edges could help visitors relax.So far scientists have focused mainly on public buildings. "We have a very limited number of studies, so TAL#NBSP we're almost looking at the problem through a straw （吸管）, " architect David Allison says. "How do you take answers to very specific questions and make broad, generalized use of them? That's what we're all struggling with."(1) What does Joan Meyers-Levy focus on in her research?A. Light.B. Ceilings.C. Windows.D. Furniture.(2) The passage tells us that.A. the shape of furniture may affect people's feelingsB. lower ceilings may help improve students' creativityC. children in a dim classroom may improve their gradesD. students in rooms with unblocked views may feel relaxed(3) The underlined sentence in the last paragraph probably means that.A. the problem is not approached step by stepB. the researches so far have faults in themselvesC. the problem is too difficult for researchers to detectD. research in this area is not enough to make generalized patterns(4) Which of the following shows the organization of the passage? CP: Central Point P: Point SP: Sub-point（次要点） C: ConclusionA.B.C.D.20、【来源】 2009年高考真题北京卷（D）篇第68~71题8分DWhen students and parents are asked to rate subjects according to their importance, the arts are unavoidably at the bottom of the list. Music is nice，people seem to say, but not important. Too often it is viewed as mere entertainment, but certainly not an education priority （优先）. This view is shortsighted. In fact, music education is beneficial and important for all students.Music tells us who we are. Because music is an expression of the beings who create it, it reflects their thinking and values, as well as the social environment it came from. Rock music represents a lifestyle just as surely as does a Schubert song. The jazz influence that George Gershwin and other musicians introduced into their music is obviously American because it came from American musical traditions. Music expresses our character and values. It gives us identity as a society.Music provides a kind of perception （感知）that cannot be acquired any other way. Science can explain how the sun rises and sets. The arts explore emotive （情感的） meaning of the same phenomenon. We need every possible way to discover and respond to our world for one simple but powerful reason: No one way can get it all.The arts are forms of thought as powerful in what they communicate as mathematical and scientific symbols. They are ways we human beings "talk" to each other. They are the language of civilization through which we express our fears, our curiosities, our hungers, our discoveries, our hopes. The arts are ways we give form to our ideas and imagination so that they can be shared with others. When we do not give children access to an important way of expressing themselves such as music, we take away from them the meanings that music expresses.Science and technology do not tell us what it means to be human. The arts do. Music is an important way we express human suffering, celebration, the meaning and value of peace and love. So music education is far more necessary than people to realize.(1) According to Paragraph 1, students.A. regard music as a way of entertainmentB. disagree with their parents on educationC. view music as an overlooked subject .D. prefer the arts to science(2) In Paragraph 2, the author uses jazz as an example to.A. compare it with rock musicB. show music identifies a societyC. introduce American musical traditionsD. prove music influences people's lifestyles(3) According to the passage, the arts and science .A. approach the world from different anglesB. explore different phenomena of the worldC. express people’s feelings in different waysD. explain what it means to be human differently.(4) What is the main idea of the passage?A. Music education deserves more attention.B. Music should be of top education priority.C. Music is an effective communication tool.D. Music education makes students more imaginative.21、【来源】 2009年高考真题北京卷（E篇）第72~75题8分2016~2017学年北京西城区北京师范大学附属中学高一下学期期中（C篇）第39~42题8分Camping wild is a wonderful way to experience the natural world and, at it best, it makes little environmental influence. But with increasing numbers of people wanting to escape into the wilderness. It is becoming more and more important to camp unobtrusively（不引人注目的）and leave no mark.Wild camping is not permitted in many places, particularly in crowded lowland Britain, Wherever you are, find out about organizations responsible for managing wild spaces, and contact them to find out their policy on camping and shelter building. For example, it is fine to camp wild in remote parts of Scotland, but in England you must ask the landowner's permission, except in national parks.Camping is about having relaxation, sleeping outdoors, experiencing bad weather, and making do without modern conveniences. A busy, fully-equipped campsite seems to go against this, so seek out smaller, more remote places with easy access to open spaces and perhaps beaches. Better still, find a campsite with no road access: Walking in makes a real adventure.Finding the right spot to camp is the first step to guaranteeing a good night's sleep. Choose a campsite with privacy and minimum influence on others and the environment. Try to use an area where people have obviously camped before rather than creating a new spot. When camping in woodland, avoid standing dead trees, which may fall on a windy night. Avoid animal runs and caves, and possible homes of biting insects. Make sure you have most protection on the windward side. If you make a fire, do so downwind of your shelter. Always consider what influence you might have on the nature word. Avoid damaging plants. A good campsite is found, not made-changing it should be unnecessary.(1) You needn't ask for permission when camping in.A. national parks in EnglandB. most parts of ScotlandC. crowded lowland BritainD. most parts of England(2) The author thinks that a good campsite is one.A. with easy accessB. used previouslyC. with modern conveniencesD. far away from beaches(3) The last paragraph mainly deals with.A. protecting animalsB. building a campfireC. camping in woodlandD. finding a campsite with privacy(4) The passage is mainly about.A. the protection of campsitesB. the importance of wild campingC. the human influence on campsitesD. the dos and don'ts of wild camping四、情景作文（共20分）22、【来源】 2009年高考真题北京卷第76题20分假设你是红星中学高三（一）班的学生李华，为响应绿化祖国的号召，你班四月十二日去郊区植树，请根据以下四幅图的先后顺序，介绍植树活动的全过程，给某英文杂志的 "绿色行动" 专栏写一篇以 "Green Action in Our Class" 为题的英文稿件。
1. 元素与集合的关系U x A x C A ∈⇔∉,U x C A x A ∈⇔∉. 2.德摩根公式();()U U U U U U C A B C A C B C A B C A C B == .3.包含关系A B A A B B =⇔= U U A B C B C A ⇔⊆⇔⊆U A C B ⇔=Φ U C A B R ⇔=4.容斥原理()()card A B cardA cardB card A B =+-()()card A B C cardA cardB cardC card A B =++-()()()()card A B card B C card C A card A B C ---+ .5．集合12{,,,}n a a a 的子集个数共有2n 个；真子集有2n –1个；非空子集有2n –1个；非空的真子集有2n –2个.6.二次函数的解析式的三种形式(1)一般式2()(0)f x ax bx c a =++≠;(2)顶点式2()()(0)f x a x h k a =-+≠; (3)零点式12()()()(0)f x a x x x x a =--≠. 7.解连不等式()N f x M <<常有以下转化形式()N f x M <<⇔[()][()]0f x M f x N --< ⇔|()|22M N M N f x +--<⇔()0()f x N M f x ->-⇔11()f x NM N>--.8.方程0)(=x f 在),(21k k 上有且只有一个实根,与0)()(21<k f k f 不等价,前者是后者的一个必要而不是充分条件.特别地, 方程)0(02≠=++a c bx ax 有且只有一个实根在),(21k k 内,等价于0)()(21<k f k f ,或0)(1=k f 且22211k k ab k +<-<,或0)(2=k f 且22122k ab k k <-<+.9.闭区间上的二次函数的最值二次函数)0()(2≠++=a c bx ax x f 在闭区间[]q p ,上的最值只能在ab x 2-=处及区间的两端点处取得，具体如下：(1)当a>0时，若[]q p ab x ,2∈-=，则{}m in m ax m ax()(),()(),()2b f x f f x f p f q a=-=；[]q p ab x ,2∉-=，{}max max()(),()f x f p f q =，{}min min ()(),()f x f p f q =.(2)当a<0时，若[]q p ab x ,2∈-=，则{}m i n ()m i n (),()f x f p f q=，若[]q p ab x ,2∉-=，则{}m a x ()m a x (),()f x f p f q=，{}min ()min (),()f x f p f q =. 10.一元二次方程的实根分布依据：若()()0f m f n <，则方程0)(=x f 在区间(,)m n 内至少有一个实根 . 设q px x x f ++=2)(，则（1）方程0)(=x f 在区间),(+∞m 内有根的充要条件为0)(=m f 或2402p q p m ⎧-≥⎪⎨->⎪⎩；（2）方程0)(=x f 在区间(,)m n 内有根的充要条件为()()0f m f n <或2()0()0402f m f n p q p m n>⎧⎪>⎪⎪⎨-≥⎪⎪<-<⎪⎩或()0()0f m af n =⎧⎨>⎩或()0()0f n af m =⎧⎨>⎩； （3）方程0)(=x f 在区间(,)n -∞内有根的充要条件为()0f m <或2402p q p m ⎧-≥⎪⎨-<⎪⎩ .11.定区间上含参数的二次不等式恒成立的条件依据(1)在给定区间),(+∞-∞的子区间L （形如[]βα,，(]β,∞-，[)+∞,α不同）上含参数的二次不等式(,)0f x t ≥(t 为参数)恒成立的充要条件是m in (,)0()f x t x L ≥∉.(2)在给定区间),(+∞-∞的子区间上含参数的二次不等式(,)0f x t ≥(t 为参数)恒成立的充要条件是(,)0()man f x t x L ≤∉.(3)0)(24>++=c bx ax x f 恒成立的充要条件是000a b c ≥⎧⎪≥⎨⎪>⎩或2040a b ac <⎧⎨-<⎩.12.13.14.四种命题的相互关系15.充要条件（1）充分条件：若p q ⇒，则p 是q 充分条件.（2）必要条件：若q p ⇒，则p 是q 必要条件.（3）充要条件：若p q ⇒，且q p ⇒，则p 是q 充要条件. 注：如果甲是乙的充分条件，则乙是甲的必要条件；反之亦然. 16.函数的单调性(1)设[]2121,,x x b a x x ≠∈⋅那么 []1212()()()0x x f x f x -->⇔[]b a x f x x x f x f ,)(0)()(2121在⇔>--上是增函数； []1212()()()0x x f x f x --<⇔[]b a x f x x x f x f ,)(0)()(2121在⇔<--上是减函数.(2)设函数)(x f y =在某个区间内可导，如果0)(>'x f ，则)(x f 为增函数；如果0)(<'x f ，则)(x f 为减函数.17.如果函数)(x f 和)(x g 都是减函数,则在公共定义域内,和函数)()(x g x f +也是减函数; 如果函数)(u f y =和)(x g u =在其对应的定义域上都是减函数,则复合函数)]([x g f y =是增函数.18．奇偶函数的图象特征 奇函数的图象关于原点对称，偶函数的图象关于y 轴对称;反过来，如果一个函数的图象关于原点对称，那么这个函数是奇函数；如果一个函数的图象关于y 轴对称，那么这个函数是偶函数．19.若函数)(x f y =是偶函数，则)()(a x f a x f --=+；若函数)(a x f y +=是偶函数，则)()(a x f a x f +-=+.20.对于函数)(x f y =(R x ∈),)()(x b f a x f -=+恒成立,则函数)(x f 的对称轴是函数2b a x +=;两个函数)(a x f y +=与)(x b f y -= 的图象关于直线2b a x +=对称.21.若)()(a x f x f +--=,则函数)(x f y =的图象关于点)0,2(a 对称; 若)()(a x f x f +-=,则函数)(x f y =为周期为a 2的周期函数.22．多项式函数110()n n n n P x a x a x a --=+++ 的奇偶性多项式函数()P x 是奇函数⇔()P x 的偶次项(即奇数项)的系数全为零. 多项式函数()P x 是偶函数⇔()P x 的奇次项(即偶数项)的系数全为零. 23.函数()y f x =的图象的对称性(1)函数()y f x =的图象关于直线x a =对称()()f a x f a x ⇔+=- (2)()f a x f x ⇔-=.(2)函数()y f x =的图象关于直线2a b x +=对称()()f a mx f b mx ⇔+=-()()f a b mx f mx ⇔+-=.24.两个函数图象的对称性(1)函数()y f x =与函数()y f x =-的图象关于直线0x =(即y 轴)对称. (2)函数()y f mx a =-与函数()y f b mx =-的图象关于直线2a b x m+=对称.(3)函数)(x f y =和)(1x f y -=的图象关于直线y=x 对称.25.若将函数)(x f y =的图象右移a 、上移b 个单位，得到函数b a x f y +-=)(的图象；若将曲线0),(=y x f 的图象右移a 、上移b 个单位，得到曲线0),(=--b y a x f 的图象.26．互为反函数的两个函数的关系a b fb a f =⇔=-)()(1.27.若函数)(b kx f y +=存在反函数,则其反函数为])([11b x fk y -=-,并不是)([1b kx fy +=-,而函数)([1b kx fy +=-是])([1b x f ky -=的反函数.28.几个常见的函数方程(1)正比例函数()f x cx =,()()(),(1)f x y f x f y f c +=+=.(2)指数函数()x f x a =,()()(),(1)0f x y f x f y f a +==≠.(3)对数函数()log a f x x =,()()(),()1(0,1)f xy f x f y f a a a =+=>≠.(4)幂函数()f x x α=,'()()(),(1)f xy f x f y f α==.(5)余弦函数()cos f x x =,正弦函数()sin g x x =，()()()()()f x y f x f y g x g y -=+，()(0)1,lim1x g x f x→==.29.几个函数方程的周期(约定a>0)（1）)()(a x f x f +=，则)(x f 的周期T=a ； （2）0)()(=+=a x f x f ， 或)0)(()(1)(≠=+x f x f a x f ，或1()()f x a f x +=-(()0)f x ≠,或[]1(),(()0,1)2f x a f x +=+∈,则)(x f 的周期T=2a ；(3))0)(()(11)(≠+-=x f a x f x f ，则)(x f 的周期T=3a ；(4))()(1)()()(212121x f x f x f x f x x f -+=+且1212()1(()()1,0||2)f a f x f x x x a =⋅≠<-<，则)(x f 的周期T=4a ；(5)()()(2)(3)(4)f x f x a f x a f x a f x a +++++++()()(2)(3)(4)f x f x a f x a f x a f x a =++++,则)(x f 的周期T=5a ；(6))()()(a x f x f a x f +-=+，则)(x f 的周期T=6a. 30.分数指数幂(1)mn a =0,,a m n N *>∈，且1n >）.(2)1m nmnaa -=（0,,a m n N *>∈，且1n >）.31．根式的性质（1）na =.（2）当n为奇数时，a =； 当n,0||,0a a a a a ≥⎧==⎨-<⎩.32．有理指数幂的运算性质(1) (0,,)r s r sa a a a r s Q +⋅=>∈. (2) ()(0,,)r s rsa a a r s Q =>∈. (3)()(0,0,)r r rab a b a b r Q =>>∈.注： 若a ＞0，p 是一个无理数，则a p 表示一个确定的实数．上述有理指数幂的运算性质，对于无理数指数幂都适用.33.指数式与对数式的互化式log ba Nb a N =⇔=(0,1,0)a a N >≠>.34.对数的换底公式log log log m a m N N a=(0a >,且1a ≠,0m >,且1m ≠, 0N >).推论 log log mn a a n b b m=(0a >,且1a >,,0m n >,且1m ≠,1n ≠, 0N >).35．对数的四则运算法则若a ＞0，a ≠1，M ＞0，N ＞0，则 (1)log ()log log a a a M N M N =+; (2) log log log aa a M M N N=-; (3)log log ()na a Mn M n R =∈.36.设函数)0)((log )(2≠++=a c bx axx f m,记ac b 42-=∆.若)(x f 的定义域为R ,则0>a ，且0<∆;若)(x f 的值域为R ,则0>a ，且0≥∆.对于0=a 的情形,需要单独检验.37. 对数换底不等式及其推广 若0a >,0b >,0x >,1x a ≠,则函数log ()ax y bx =(1)当a b >时,在1(0,)a 和1(,)a +∞上log ()ax y bx =为增函数. ，(2)当a b <时,在1(0,)a和1(,)a+∞上log ()ax y bx =为减函数.推论:设1n m >>，0p >，0a >，且1a ≠，则 （1）log ()log m p m n p n ++<. （2）2log log log 2a a am n m n +<.38. 平均增长率的问题如果原来产值的基础数为N ，平均增长率为p ，则对于时间x 的总产值y ，有(1)x y N p =+. 39.数列的同项公式与前n 项的和的关系11,1,2n n n s n a s s n -=⎧=⎨-≥⎩( 数列{}n a 的前n 项的和为12n n s a a a =+++ ).40.等差数列的通项公式*11(1)()n a a n d dn a d n N =+-=+-∈；其前n 项和公式为1()2n n n a a s +=1(1)2n n na d -=+211()22d n a d n =+-.41.等比数列的通项公式1*11()n n n a a a q q n N q-==⋅∈；其前n 项的和公式为 11(1),11,1n n a q q s q na q ⎧-≠⎪=-⎨⎪=⎩或11,11,1n n a a qq q s na q -⎧≠⎪-=⎨⎪=⎩.42.等比差数列{}n a :11,(0)n n a qa d a b q +=+=≠的通项公式为 1(1),1(),11n n n b n d q a bq d b q d q q -+-=⎧⎪=+--⎨≠⎪-⎩；其前n 项和公式为(1),(1)1(),(1)111nn nb n n d q s d q db n q q q q +-=⎧⎪=-⎨-+≠⎪---⎩. 43.分期付款(按揭贷款)每次还款(1)(1)1nnab b x b +=+-元(贷款a 元,n 次还清,每期利率为b ).44．常见三角不等式（1）若(0,)2x π∈，则sin tan x x x <<.(2) 若(0,)2x π∈，则1sin cos x x <+≤(3) |sin ||cos |1x x +≥.45.同角三角函数的基本关系式22sin cos 1θθ+=，tan θ=θθcos sin ，tan 1cot θθ⋅=. 46.正弦、余弦的诱导公式212(1)sin ,sin()2(1)s ,nn n co απαα-⎧-⎪+=⎨⎪-⎩ 212(1)s ,s()2(1)sin ,nn co n co απαα+⎧-⎪+=⎨⎪-⎩47.和角与差角公式sin()sin cos cos sin αβαβαβ±=±;cos()cos cos sin sin αβαβαβ±= ;tan tan tan()1tan tan αβαβαβ±±=.22sin()sin()sin sin αβαβαβ+-=-(平方正弦公式); 22cos()cos()cos sin αβαβαβ+-=-. sin cos a b αα+)αϕ+(辅助角ϕ所在象限由点(,)a b 的象限决定,tan b aϕ=).48.二倍角公式sin 2sin cos ααα=.2222cos 2cos sin 2cos 112sin ααααα=-=-=-.22tan tan 21tan ααα=-. 49. 三倍角公式3sin 33sin 4sin 4sin sin()sin()33ππθθθθθθ=-=-+.3cos 34cos 3cos 4cos cos()cos()33ππθθθθθθ=-=-+.323tan tan tan 3tan tan()tan()13tan 33θθππθθθθθ-==-+-.50.三角函数的周期公式函数sin()y x ωϕ=+，x ∈R 及函数cos()y x ωϕ=+，x ∈R(A,ω,ϕ为常数，且A ≠0，ω＞0)的周期2T πω=；函数tan()y x ωϕ=+，,2x k k Z ππ≠+∈(A,ω,ϕ为常数，且A ≠0，ω＞0)的周期T πω=.51.正弦定理2sin sin sin a b c R ABC===.52.余弦定理2222cos a b c bc A =+-; 2222cos b c a ca B =+-; 2222cos c a b ab C =+-.53.面积定理 （1）111222a b c S ah bh ch ===（a b c h h h 、、分别表示a 、b 、c 边上的高）.（2）111sin sin sin 222S ab C bc A ca B ===.(3)O A B S ∆=54.三角形内角和定理在△ABC 中，有()A B C C A B ππ++=⇔=-+222C A B π+⇔=-222()C A B π⇔=-+.55. 简单的三角方程的通解sin (1)arcsin (,||1)k x a x k a k Z a π=⇔=+-∈≤. s 2arccos (,||1)co x a x k a k Z a π=⇔=±∈≤.tan arctan (,)x a x k a k Z a R π=⇒=+∈∈.特别地,有sin sin (1)()kk k Z αβαπβ=⇔=+-∈.s cos 2()co k k Z αβαπβ=⇔=±∈.tan tan ()k k Z αβαπβ=⇒=+∈.56.最简单的三角不等式及其解集sin (||1)(2arcsin ,2arcsin ),x a a x k a k a k Z πππ>≤⇔∈++-∈.sin (||1)(2arcsin ,2arcsin ),x a a x k a k a k Z πππ<≤⇔∈--+∈.cos (||1)(2arccos ,2arccos ),x a a x k a k a k Z ππ>≤⇔∈-+∈.cos (||1)(2arccos ,22arccos ),x a a x k a k a k Z πππ<≤⇔∈++-∈. tan ()(arctan ,),2x a a R x k a k k Z πππ>∈⇒∈++∈.tan ()(,arctan ),2x a a R x k k a k Z πππ<∈⇒∈-+∈.57.实数与向量的积的运算律 设λ、μ为实数，那么(1) 结合律：λ(μa )=(λμ)a ;(2)第一分配律：(λ+μ)a =λa +μa; (3)第二分配律：λ(a +b )=λa +λb . 58.向量的数量积的运算律： (1) a ·b= b ·a （交换律）; (2)（λa ）·b= λ（a ·b ）=λa ·b = a ·（λb ）;(3)（a+b）·c= a·c +b·c.59.平面向量基本定理如果e1、e2是同一平面内的两个不共线向量，那么对于这一平面内的任一向量，有且只有一对实数λ1、λ2，使得a=λ1e1+λ2e2．不共线的向量e1、e2叫做表示这一平面内所有向量的一组基底．60．向量平行的坐标表示设a=11(,)x y,b=22(,)x y，且b≠0，则a b(b≠0)1221x y x y⇔-=.53. a与b的数量积(或内积)a·b=|a||b|cosθ．61. a·b的几何意义数量积a·b等于a的长度|a|与b在a的方向上的投影|b|cosθ的乘积．62.平面向量的坐标运算(1)设a=11(,)x y,b=22(,)x y，则a+b=1212(,)x x y y++.(2)设a=11(,)x y,b=22(,)x y，则a-b=1212(,)x x y y--.(3)设A11(,)x y，B22(,)x y,则2121(,)AB OB OA x x y y=-=--.(4)设a=(,),x y Rλ∈，则λa=(,)x yλλ.(5)设a=11(,)x y,b=22(,)x y，则a·b=1212()x x y y+.63.两向量的夹角公式cosx x y yθ+=(a=11(,)x y,b=22(,)x y).64.平面两点间的距离公式,A Bd=||AB==11(,)x y，B22(,)x y).65.向量的平行与垂直设a=11(,)x y,b=22(,)x y，且b≠0，则A||b⇔b=λa1221x y x y⇔-=.a⊥b(a≠0)⇔a·b=01212x x y y⇔+=.66.线段的定比分公式设111(,)P x y，222(,)P x y，(,)P x y是线段12P P的分点,λ是实数，且12P P PPλ=，则121211x xxy yyλλλλ+⎧=⎪⎪+⎨+⎪=⎪+⎩⇔121O P O PO Pλλ+=+⇔12(1)OP tOP t OP=+-（11tλ=+）.67.三角形的重心坐标公式△ABC三个顶点的坐标分别为11A(x,y)、22B(x,y)、33C(x,y),则△ABC的重心的坐标是123123(,)33x x x y y yG++++.68.点的平移公式''''x x h x x hy y k y y k⎧⎧=+=-⎪⎪⇔⎨⎨=+=-⎪⎪⎩⎩''O P O P P P⇔=+.注:图形F上的任意一点P(x，y)在平移后图形'F上的对应点为'''(,)P x y，且'PP的坐标为(,)h k.69.“按向量平移”的几个结论（1）点(,)P x y按向量a=(,)h k平移后得到点'(,)P x h y k++.(2) 函数()y f x=的图象C按向量a=(,)h k平移后得到图象'C,则'C的函数解析式为()y f x h k =-+.(3) 图象'C 按向量a =(,)h k 平移后得到图象C ,若C 的解析式()y f x =,则'C 的函数解析式为()y f x h k =+-.(4)曲线C :(,)0f x y =按向量a =(,)h k 平移后得到图象'C ,则'C 的方程为(,)0f x h y k --=. (5) 向量m =(,)x y 按向量a =(,)h k 平移后得到的向量仍然为m =(,)x y . 70. 三角形五“心”向量形式的充要条件设O 为A B C ∆所在平面上一点，角,,A B C 所对边长分别为,,a b c ，则 （1）O 为A B C ∆的外心222O A O B O C ⇔== .（2）O 为A B C ∆的重心0OA OB OC ⇔++=.（3）O 为A B C ∆的垂心OA OB OB OC OC OA ⇔⋅=⋅=⋅.（4）O 为A B C ∆的内心0aOA bOB cOC ⇔++=.（5）O 为A B C ∆的A ∠的旁心aOA bOB cOC ⇔=+. 71.常用不等式：（1）,a b R ∈⇒222a b ab +≥(当且仅当a ＝b 时取“=”号)． （2）,a b R +∈⇒2a b +≥(当且仅当a ＝b 时取“=”号)． （3）3333(0,0,0).a b c abc a b c ++≥>>>（4）柯西不等式22222()()(),,,,.a b c d ac bd a b c d R ++≥+∈（5）b a b a b a +≤+≤-. 72.极值定理已知y x ,都是正数，则有（1）若积xy 是定值p ，则当y x =时和y x +有最小值p 2；（2）若和y x +是定值s ，则当y x =时积xy 有最大值241s .推广 已知R y x ∈,，则有xy y x y x 2)()(22+-=+ （1）若积xy 是定值,则当||y x -最大时,||y x +最大； 当||y x -最小时,||y x +最小.（2）若和||y x +是定值,则当||y x -最大时, ||xy 最小； 当||y x -最小时, ||xy 最大.73.一元二次不等式20(0)ax bx c ++><或2(0,40)a b ac ≠∆=->，如果a 与2ax bx c ++同号，则其解集在两根之外；如果a 与2ax bx c ++异号，则其解集在两根之间.简言之：同号两根之外，异号两根之间.121212()()0()x x x x x x x x x <<⇔--<<；121212,()()0()x x x x x x x x x x <>⇔--><或.74.含有绝对值的不等式 当a> 0时，有22x a x aa x a <⇔<⇔-<<.22x a x a x a >⇔>⇔>或x a <-.75.无理不等式 （1()0()0()()f x g x f x g x ≥⎧⎪>⇔≥⎨⎪>⎩.（22()0()0()()0()0()[()]f x f x g x g x g x f x g x ≥⎧≥⎧⎪>⇔≥⎨⎨<⎩⎪>⎩或. （32()0()()0()[()]f x g x g x f x g x ≥⎧⎪<⇔>⎨⎪<⎩.76.指数不等式与对数不等式 (1)当1a >时,()()()()f x g x aaf xg x >⇔>;()0log ()log ()()0()()a a f x f x g x g x f x g x >⎧⎪>⇔>⎨⎪>⎩.(2)当01a <<时,()()()()f x g x aaf xg x >⇔<;()0log ()log ()()0()()a a f x f x g x g x f x g x >⎧⎪>⇔>⎨⎪<⎩77.斜率公式2121y y k x x -=-（111(,)P x y 、222(,)P x y ）. 78.直线的五种方程（1）点斜式 11()y y k x x -=- (直线l 过点111(,)P x y ，且斜率为k )． （2）斜截式 y kx b =+(b 为直线l 在y 轴上的截距). （3）两点式 112121y y x x y y x x --=--(12y y ≠)(111(,)P x y 、222(,)P x y (12x x ≠)).(4)截距式1x ya b+=(a b 、分别为直线的横、纵截距，0a b ≠、)（5）一般式 0Ax By C ++=(其中A 、B 不同时为0).79.两条直线的平行和垂直(1)若111:l y k x b =+，222:l y k x b =+ ①121212||,l l k k b b ⇔=≠; ②12121l l k k ⊥⇔=-.(2)若1111:0l A x B y C ++=,2222:0l A x B y C ++=,且A 1、A 2、B 1、B 2都不为零, ①11112222||A B C l l A B C ⇔=≠；②1212120l l A A B B ⊥⇔+=； 80.夹角公式 (1)2121tan ||1k k k k α-=+.(111:l y k x b =+，222:l y k x b =+,121k k ≠-) (2)12211212tan ||A B A B A A B B α-=+.(1111:0l A x B y C ++=,2222:0l A x B y C ++=,12120A A B B +≠). 直线12l l ⊥时，直线l 1与l 2的夹角是2π.81. 1l 到2l 的角公式 (1)2121tan 1k k k k α-=+.(111:l y k x b =+，222:l y k x b =+,121k k ≠-) (2)12211212tan A B A B A A B B α-=+.(1111:0l A x B y C ++=,2222:0l A x B y C ++=,12120A A B B +≠). 直线12l l ⊥时，直线l 1到l 2的角是2π.82．四种常用直线系方程(1)定点直线系方程：经过定点000(,)P x y 的直线系方程为00()y y k x x -=-(除直线0x x =),其中k 是待定的系数; 经过定点000(,)P x y 的直线系方程为00()()0A x x B y y -+-=,其中,A B 是待定的系数．(2)共点直线系方程：经过两直线1111:0l A x B y C ++=,2222:0l A x B y C ++=的交点的直线系方程为111222()()0A x B y C A x B y C λ+++++=(除2l )，其中λ是待定的系数．(3)平行直线系方程：直线y kx b =+中当斜率k 一定而b 变动时，表示平行直线系方程．与直线0Ax By C ++=平行的直线系方程是0Ax By λ++=(0λ≠)，λ是参变量．(4)垂直直线系方程：与直线0Ax By C ++= (A ≠0，B ≠0)垂直的直线系方程是0Bx Ay λ-+=,λ是参变量．83.点到直线的距离d =点00(,)P x y ,直线l ：0Ax By C ++=).84. 0Ax By C ++>或0<所表示的平面区域设直线:0l Ax By C ++=，则0Ax By C ++>或0<所表示的平面区域是：若0B ≠，当B 与Ax By C ++同号时，表示直线l 的上方的区域；当B 与Ax By C ++异号时，表示直线l 的下方的区域.简言之,同号在上,异号在下.若0B =，当A 与Ax By C ++同号时，表示直线l 的右方的区域；当A 与Ax By C ++异号时，表示直线l 的左方的区域. 简言之,同号在右,异号在左.85. 111222()()0A x B y C A x B y C ++++>或0<所表示的平面区域设曲线111222:()()0C A x B y C A x B y C ++++=（12120A A B B ≠），则111222()()0A x B y C A x B y C ++++>或0<所表示的平面区域是： 111222()()0A x B y C A x B y C ++++>所表示的平面区域上下两部分； 111222()()0A x B y C A x B y C ++++<所表示的平面区域上下两部分.86. 圆的四种方程（1）圆的标准方程 222()()x a y b r -+-=.（2）圆的一般方程 220x y Dx Ey F ++++=(224D E F +-＞0). （3）圆的参数方程 cos sin x a r y b r θθ=+⎧⎨=+⎩.（4）圆的直径式方程 1212()()()()0x x x x y y y y --+--=(圆的直径的端点是11(,)A x y 、22(,)B x y ).87. 圆系方程(1)过点11(,)A x y ,22(,)B x y 的圆系方程是1212112112()()()()[()()()()]0x x x x y y y y x x y y y y x x λ--+--+-----=1212()()()()()0x x x x y y y y ax by c λ⇔--+--+++=,其中0ax by c ++=是直线A B 的方程,λ是待定的系数．(2)过直线l :0Ax By C ++=与圆C :220x y Dx Ey F ++++=的交点的圆系方程是22()0x y Dx Ey F Ax By C λ+++++++=,λ是待定的系数．(3) 过圆1C :221110x y D x E y F ++++=与圆2C :222220x y D x E y F ++++=的交点的圆系方程是2222111222()0x y D x E y F x y D x E y F λ+++++++++=,λ是待定的系数．88.点与圆的位置关系点00(,)P x y 与圆222)()(r b y a x =-+-的位置关系有三种若d =d r >⇔点P 在圆外;d r =⇔点P 在圆上;d r <⇔点P 在圆内.89.直线与圆的位置关系直线0=++C By Ax 与圆222)()(r b y a x =-+-的位置关系有三种: 0<∆⇔⇔>相离r d ; 0=∆⇔⇔=相切r d ; 0>∆⇔⇔<相交r d .其中22BA CBb Aa d +++=.90.两圆位置关系的判定方法设两圆圆心分别为O 1，O 2，半径分别为r 1，r 2，d O O =21 条公切线外离421⇔⇔+>r r d ; 条公切线外切321⇔⇔+=r r d ;条公切线相交22121⇔⇔+<<-r r d r r ; 条公切线内切121⇔⇔-=r r d ; 无公切线内含⇔⇔-<<210r r d .91.圆的切线方程(1)已知圆220x y Dx Ey F ++++=．①若已知切点00(,)x y 在圆上，则切线只有一条，其方程是 0000()()022D x xE y y x x y yF ++++++=.当00(,)x y 圆外时, 0000()()022D x xE y y x x y yF ++++++=表示过两个切点的切点弦方程．②过圆外一点的切线方程可设为00()y y k x x -=-，再利用相切条件求k ，这时必有两条切线，注意不要漏掉平行于y 轴的切线．③斜率为k 的切线方程可设为y kx b =+，再利用相切条件求b ，必有两条切线．(2)已知圆222x y r +=．①过圆上的000(,)P x y 点的切线方程为200x x y y r +=;②斜率为k的圆的切线方程为y kx =±. 92.椭圆22221(0)x y a b a b +=>>的参数方程是cos sin x a y b θθ=⎧⎨=⎩.93.椭圆22221(0)x y a b ab +=>>焦半径公式)(21c ax e PF +=，)(22x cae PF -=.94．椭圆的的内外部 （1）点00(,)P x y 在椭圆22221(0)x y a b ab+=>>的内部2200221x y ab⇔+<.（2）点00(,)P x y 在椭圆22221(0)x y a b ab+=>>的外部2200221x y ab⇔+>.95. 椭圆的切线方程 (1)椭圆22221(0)x y a b ab+=>>上一点00(,)P x y 处的切线方程是00221x x y y ab+=.（2）过椭圆22221(0)x y a b ab+=>>外一点00(,)P x y 所引两条切线的切点弦方程是00221x x y y ab+=.（3）椭圆22221(0)x y a b ab +=>>与直线0Ax By C ++=相切的条件是22222A aB b c +=.96.双曲线22221(0,0)x y a b ab-=>>的焦半径公式21|()|aPF e x c=+，22|()|aPF e x c=-.97.双曲线的内外部(1)点00(,)P x y 在双曲线22221(0,0)x y a b a b -=>>的内部2200221x y a b ⇔->. (2)点00(,)P x y 在双曲线22221(0,0)xya b a b-=>>的外部2200221x y ab ⇔-<.98.双曲线的方程与渐近线方程的关系(1）若双曲线方程为12222=-by ax ⇒渐近线方程：22220x y ab-=⇔x ab y ±=.(2)若渐近线方程为x ab y ±=⇔0=±by a x ⇒双曲线可设为λ=-2222by ax .(3)若双曲线与12222=-by ax 有公共渐近线，可设为λ=-2222by ax （0>λ，焦点在x 轴上，0<λ，焦点在y 轴上）.99. 双曲线的切线方程(1)双曲线22221(0,0)x y a b ab-=>>上一点00(,)P x y 处的切线方程是00221x x y y ab-=.（2）过双曲线22221(0,0)x y a b ab-=>>外一点00(,)P x y 所引两条切线的切点弦方程是00221x x y y ab-=.（3）双曲线22221(0,0)x y a b ab-=>>与直线0Ax By C ++=相切的条件是22222A aB b c -=.100. 抛物线px y 22=的焦半径公式抛物线22(0)y px p =>焦半径02p C F x =+.过焦点弦长p x x p x p x CD ++=+++=212122.101.抛物线px y 22=上的动点可设为P ),2(2y py 或或)2,2(2pt pt P P (,)x y ，其中 22y px = .102.二次函数2224()24b ac b y ax bx c a x aa-=++=++(0)a ≠的图象是抛物线：（1）顶点坐标为24(,)24bac ba a--；（2）焦点的坐标为241(,)24b ac b aa-+-；（3）准线方程是2414ac b y a--=.103.抛物线的内外部(1)点00(,)P x y 在抛物线22(0)y px p =>的内部22(0)y px p ⇔<>. 点00(,)P x y 在抛物线22(0)y px p =>的外部22(0)y px p ⇔>>. (2)点00(,)P x y 在抛物线22(0)y px p =->的内部22(0)y px p ⇔<->. 点00(,)P x y 在抛物线22(0)y px p =->的外部22(0)y px p ⇔>->. (3)点00(,)P x y 在抛物线22(0)x py p =>的内部22(0)x py p ⇔<>. 点00(,)P x y 在抛物线22(0)x py p =>的外部22(0)x py p ⇔>>. (4) 点00(,)P x y 在抛物线22(0)x py p =>的内部22(0)x py p ⇔<>. 点00(,)P x y 在抛物线22(0)x py p =->的外部22(0)x py p ⇔>->. 104. 抛物线的切线方程(1)抛物线px y 22=上一点00(,)P x y 处的切线方程是00()y y p x x =+.（2）过抛物线px y 22=外一点00(,)P x y 所引两条切线的切点弦方程是00()y y p x x =+. （3）抛物线22(0)y px p =>与直线0Ax By C ++=相切的条件是22pB AC =.105.两个常见的曲线系方程(1)过曲线1(,)0f x y =,2(,)0f x y =的交点的曲线系方程是12(,)(,)0f x y f x y λ+=(λ为参数).(2)共焦点的有心圆锥曲线系方程22221xya kb k+=--,其中22max{,}k a b <.当22m in{,}k a b >时,表示椭圆; 当2222m in{,}m ax{,}a b k a b <<时,表示双曲线.106.直线与圆锥曲线相交的弦长公式 AB =1212||||AB x x y y ==-=-（弦端点A ),(),,(2211y xB y x ，由方程⎩⎨⎧=+=0)y ,x (F bkx y 消去y 得到02=++c bx ax ，0∆>,α为直线A B 的倾斜角，k 为直线的斜率）.107.圆锥曲线的两类对称问题（1）曲线(,)0F x y =关于点00(,)P x y 成中心对称的曲线是00(2-,2)0F x x y y -=. （2）曲线(,)0F x y =关于直线0Ax By C ++=成轴对称的曲线是22222()2()(,)0A Ax By C B Ax By C F x y A BA B++++--=++.108.“四线”一方程对于一般的二次曲线220Ax Bxy C y D x Ey F +++++=，用0x x 代2x ，用0y y 代2y ，用002x y xy +代xy ，用02x x+代x ，用02y y +代y 即得方程0000000222x y xy x x y y A x x B C y y D E F ++++⋅++⋅+⋅+=，曲线的切线，切点弦，中点弦，弦中点方程均是此方程得到.109．证明直线与直线的平行的思考途径 （1）转化为判定共面二直线无交点；（2）转化为二直线同与第三条直线平行； （3）转化为线面平行； （4）转化为线面垂直； （5）转化为面面平行.110．证明直线与平面的平行的思考途径（1）转化为直线与平面无公共点； （2）转化为线线平行； （3）转化为面面平行.111．证明平面与平面平行的思考途径 （1）转化为判定二平面无公共点； （2）转化为线面平行； （3）转化为线面垂直.112．证明直线与直线的垂直的思考途径 （1）转化为相交垂直； （2）转化为线面垂直；（3）转化为线与另一线的射影垂直； （4）转化为线与形成射影的斜线垂直. 113．证明直线与平面垂直的思考途径（1）转化为该直线与平面内任一直线垂直； （2）转化为该直线与平面内相交二直线垂直； （3）转化为该直线与平面的一条垂线平行； （4）转化为该直线垂直于另一个平行平面； （5）转化为该直线与两个垂直平面的交线垂直. 114．证明平面与平面的垂直的思考途径 （1）转化为判断二面角是直二面角； （2）转化为线面垂直.115.空间向量的加法与数乘向量运算的运算律 (1)加法交换律：a ＋b =b ＋a ．(2)加法结合律：(a ＋b )＋c =a ＋(b ＋c )． (3)数乘分配律：λ(a ＋b )=λa ＋λb ．116.平面向量加法的平行四边形法则向空间的推广始点相同且不在同一个平面内的三个向量之和，等于以这三个向量为棱的平行六面体的以公共始点为始点的对角线所表示的向量.117.共线向量定理对空间任意两个向量a 、b (b ≠0 )，a ∥b ⇔存在实数λ使a =λb ．P A B 、、三点共线⇔||AP AB ⇔AP t AB = ⇔(1)O P t O A tO B =-+.||AB CD ⇔AB、CD 共线且A B C D 、不共线⇔AB tCD = 且A B C D 、不共线. 118.共面向量定理向量p 与两个不共线的向量a 、b 共面的⇔存在实数对,x y ,使p ax by =+．推论 空间一点P 位于平面MAB 内的⇔存在有序实数对,x y ,使M P x M A y M B =+，或对空间任一定点O ，有序实数对,x y ，使O P O M x M A y M B =++.119.对空间任一点O 和不共线的三点A 、B 、C ，满足O P xO A yO B zO C =++（x y z k ++=），则当1k =时，对于空间任一点O ，总有P 、A 、B 、C 四点共面；当1k ≠时，若O ∈平面ABC ，则P 、A 、B 、C 四点共面；若O ∉平面ABC ，则P 、A 、B 、C 四点不共面． C A B 、、、D 四点共面⇔AD 与AB、A C共面⇔A D x A B y A C =+ ⇔ (1)O D x y O A xO B yO C =--++（O ∉平面ABC ）.120.空间向量基本定理如果三个向量a 、b 、c 不共面，那么对空间任一向量p ，存在一个唯一的有序实数组x ，y ，z ，使p ＝x a ＋y b ＋z c ．推论 设O 、A 、B 、C 是不共面的四点，则对空间任一点P ，都存在唯一的三个有序实数x ，y ，z ，使O P xO A y O B z O C =++ .121.射影公式 已知向量AB=a 和轴l ，e 是l 上与l 同方向的单位向量.作A 点在l 上的射影'A ，作B 点在l 上的射影'B ，则''||cos A B AB =〈a ，e 〉=a ·e122.向量的直角坐标运算设a ＝123(,,)a a a ，b ＝123(,,)b b b 则 (1)a ＋b ＝112233(,,)a b a b a b +++； (2)a －b ＝112233(,,)a b a b a b ---； (3)λa ＝123(,,)a a a λλλ (λ∈R)； (4)a ·b ＝112233a b a b a b ++；123.设A 111(,,)x y z ，B 222(,,)x y z ，则 AB OB OA =-= 212121(,,)x x y y z z ---. 124．空间的线线平行或垂直 设111(,,)a x y z =r ，222(,,)b x y z =r，则 a b r r P ⇔(0)a b b λ=≠r r r r ⇔121212x x y y z zλλλ=⎧⎪=⎨⎪=⎩；a b ⊥r r ⇔0a b ⋅=r r⇔1212120x x y y z z ++=.125.夹角公式设a ＝123(,,)a a a ，b ＝123(,,)b b b ，则 cos 〈a ，b 〉.推论 2222222112233123123()()()a b a b a b a a a b b b ++≤++++，此即三维柯西不等式.126. 四面体的对棱所成的角四面体A B C D 中, A C 与B D 所成的角为θ,则 2222|()()|cos 2AB CD BC DA AC BDθ+-+=⋅.127．异面直线所成角cos |cos ,|a b θ=r r=||||||a b a b ⋅=⋅r rr r （其中θ（090θ<≤oo）为异面直线a b ,所成角，,a b r分别表示异面直线a b ,的方向向量）128.直线A B 与平面所成角sin ||||AB m arc AB m β⋅=(m为平面α的法向量). 129.若A B C ∆所在平面若β与过若A B 的平面α成的角θ,另两边A C ,B C 与平面α成的角分别是1θ、2θ,A B 、为A B C ∆的两个内角，则2222212sin sin (sin sin )sin A B θθθ+=+.特别地,当90ACB ∠=时,有22212sin sin sin θθθ+=.130.若A B C ∆所在平面若β与过若A B 的平面α成的角θ,另两边A C ,B C 与平面α成的角分别是1θ、2θ,''A B 、为A B O ∆的两个内角，则222'2'212tan tan (sin sin )tan A B θθθ+=+.特别地,当90AOB ∠=时,有22212sin sin sin θθθ+=. 131.二面角l αβ--的平面角cos ||||m n arc m n θ⋅= 或cos ||||m narc m n π⋅-（m ，n 为平面α，β的法向量）.132.三余弦定理设AC 是α内的任一条直线，且BC ⊥AC ，垂足为C ，又设AO 与AB 所成的角为1θ，AB 与AC 所成的角为2θ，AO 与AC 所成的角为θ．则12cos cos cos θθθ=.133. 三射线定理若夹在平面角为ϕ的二面角间的线段与二面角的两个半平面所成的角是1θ,2θ,与二面角的棱所成的角是θ，则有22221212sin sin sin sin 2sin sin cos ϕθθθθθϕ=+- ;1212||180()θθϕθθ-≤≤-+(当且仅当90θ=时等号成立).134.空间两点间的距离公式若A 111(,,)x y z ，B 222(,,)x y z ，则,A B d =||AB ==.135.点Q 到直线l 距离h =点P 在直线l 上，直线l 的方向向量a =PA，向量b =P Q ).136.异面直线间的距离||||C D n d n ⋅=(12,l l 是两异面直线，其公垂向量为n ，C D 、分别是12,l l 上任一点，d 为12,l l 间的距离). 137.点B 到平面α的距离||||A B n d n ⋅=（n 为平面α的法向量，A B 是经过面α的一条斜线，A α∈）.138.异面直线上两点距离公式d =.d =d =（'E AAF ϕ=--）.(两条异面直线a 、b 所成的角为θ，其公垂线段'AA 的长度为h.在直线a 、b 上分别取两点E 、F ，'A E m =,A F n =,E F d =). 139.三个向量和的平方公式2222()222a b c a b c a b b c c a ++=+++⋅+⋅+⋅2222||||cos ,2||||cos ,2||||cos ,a b c a b a b b c b c c a c a =+++⋅+⋅+⋅140. 长度为l 的线段在三条两两互相垂直的直线上的射影长分别为123l l l 、、，夹角分别为123θθθ、、,则有2222123l l l l =++222123cos cos cos 1θθθ⇔++=222123sin sin sin 2θθθ⇔++=.（立体几何中长方体对角线长的公式是其特例）. 141. 面积射影定理'cos SS θ=.(平面多边形及其射影的面积分别是S 、'S ，它们所在平面所成锐二面角的为θ). 142. 斜棱柱的直截面已知斜棱柱的侧棱长是l ,侧面积和体积分别是S 斜棱柱侧和V 斜棱柱,它的直截面的周长和面积分别是1c 和1S ,则①1S c l =斜棱柱侧. ②1V S l =斜棱柱.143．作截面的依据三个平面两两相交，有三条交线，则这三条交线交于一点或互相平行. 144．棱锥的平行截面的性质如果棱锥被平行于底面的平面所截，那么所得的截面与底面相似，截面面积与底面面积的比等于顶点到截面距离与棱锥高的平方比（对应角相等，对应边对应成比例的多边形是相似多边形，相似多边形面积的比等于对应边的比的平方）；相应小棱锥与小棱锥的侧面积的比等于顶点到截面距离与棱锥高的平方比．145.欧拉定理(欧拉公式)2V F E +-=(简单多面体的顶点数V 、棱数E 和面数F).（1）E =各面多边形边数和的一半.特别地,若每个面的边数为n 的多边形，则面数F 与棱数E 的关系：12E nF =；（2）若每个顶点引出的棱数为m ，则顶点数V 与棱数E 的关系：12E m V =.146.球的半径是R ，则 其体积343V R π=,其表面积24S R π=． 147.球的组合体(1)球与长方体的组合体:长方体的外接球的直径是长方体的体对角线长. (2)球与正方体的组合体:正方体的内切球的直径是正方体的棱长, 正方体的棱切球的直径是正方体的面对角线长, 正方体的外接球的直径是正方体的体对角线长. (3) 球与正四面体的组合体:棱长为a12,4.148．柱体、锥体的体积13V Sh =柱体（S 是柱体的底面积、h 是柱体的高）. 13V Sh =锥体（S 是锥体的底面积、h 是锥体的高）.149.分类计数原理（加法原理） 12n N m m m =+++ . 150.分步计数原理（乘法原理） 12n N m m m =⨯⨯⨯ . 151.排列数公式mn A =)1()1(+--m n n n =！！)(m n n -.(n ，m ∈N *，且m n ≤)．注:规定1!0=. 152.排列恒等式(1）1(1)m m n n A n m A -=-+; （2）1mmn n n A A n m-=-;（3）11m m n n A nA --=; （4）11n n nn n n nA A A ++=-; （5）11m m m n n n A A m A -+=+.(6) 1!22!33!!(1)!1n n n +⋅+⋅++⋅=+- . 153.组合数公式m nC=mn m mA A=mm n n n ⨯⨯⨯+-- 21)1()1(=！！！)(m n m n -⋅(n ∈N *，m N ∈，且m n ≤).154.组合数的两个性质 (1)m n C =m n n C - ; (2) m n C +1-m n C =m n C 1+. 注:规定10=n C . 155.组合恒等式 （1）11m m n nn m C C m --+=;（2）1m mn n n C C n m -=-;（3）11m m n n n C C m--=;（4）∑=nr r n C 0=n 2;（5）1121++++=++++r n r n r r r r r r C C C C C . (6)nn n r n n nn C C C C C 2210=++++++ . (7)14205312-+++=+++n n n n n n n C C C C C C .(8)1321232-=++++n n n n n n n nC C C C .(9)rn m r n r m n r m n r m C C C C C C C +-=+++0110 .(10)nn n n n n n C C C C C 22222120)()()()(=++++ .156.排列数与组合数的关系mmn n A m C =⋅！ .157．单条件排列以下各条的大前提是从n 个元素中取m 个元素的排列. （1）“在位”与“不在位”①某（特）元必在某位有11--m n A 种；②某（特）元不在某位有11---m n m n A A （补集思想）1111---=m n n A A （着眼位置）11111----+=m n m m n A A A （着眼元素）种.（2）紧贴与插空（即相邻与不相邻）①定位紧贴：)(n m k k ≤≤个元在固定位的排列有km k n k k A A --种.②浮动紧贴：n 个元素的全排列把k 个元排在一起的排法有kk k n k n A A 11+-+-种.注：此类问题常用捆绑法； ③插空：两组元素分别有k 、h 个（1+≤h k ），把它们合在一起来作全排列，k 个的一组互不能挨近的所有排列数有kh h h A A 1+种.（3）两组元素各相同的插空m 个大球n 个小球排成一列，小球必分开，问有多少种排法？ 当1+>m n 时，无解；当1+≤m n 时，有nm n nnm C A A 11++=种排法.（4）两组相同元素的排列：两组元素有m 个和n 个，各组元素分别相同的排列数为nn m C +.158．分配问题（1）(平均分组有归属问题)将相异的m 、n 个物件等分给m 个人，各得n 件，其分配方法数共有mnn nn nn mn nn mn nmn n mn C C C C C N )!()!(22=⋅⋅⋅⋅⋅=-- .（2）(平均分组无归属问题)将相异的m ·n个物体等分为无记号或无顺序的m 堆，其分配方法数共有。
2009年普通高校招生统一考试北京卷（英语）(满分150分，时间120分钟)第I卷（选择题，共115分）第一部分听力理解（共两节，30分）第一节（共5小题；每小题1.5分，共7.5分）听下面五段对话。
每段对话后有一道小题。
从每题所给的A、B、C三个选项中选出最佳选项。
听完每段对话后，你将有10秒钟时间来回答有关小题和阅读下一小题。
每段对话你将听一遍。
例：What is the man going to read?A. A newspaper.B. A magazine.C. A book.答案是A。
1. How fast can the woman type?A.15 words a minute.B.45 words a minute.C.80 words a minute.2. Which program does the man like most?A. Sports.B. History.C. News.3. Which language does Mr. White speak well?A. French.B. Spanish.C, Japanese.4. What was in the woman’s missing bag?5. Where does the conversation probably take place?A. In a classroom.B. In a library.C. In a bookstore.第二节（共15小题；每小题1.5分，共22.5分）听下面6段对话或独白。
每段对话或独白后有几道小题，从每小题所给的A、B、C三个选项中选出最佳选项。
听每段对话或独白前，你将有5秒钟的时间阅读每小题。
听完后，每小题将给出5秒钟的做答时间。
每段对话或独白你将听两遍。
听第6段材料，回答第6至7题。
6. What’s wrong with the man?A. He has headaches.B. He has a runny nose.C. He has a temperature.7. When did the problem begin?A. Two weeks ago.B. Two months ago.C. Three months ago.听第7段材料，回答第8至9题。
2009年普通高等学校招生全国统一考试（北京卷）英语试题第I 卷（选择题共115）第一部分：听力理解（共两节， 30分）第一节（共5小题 海小题1.5分，共7.5分）听下面5段对话，每段对话后有一道小题，丛每题所给的A,B,C 三个选项中选出最佳 选项。
听完每段对话后，你将有10秒钟的时间来回答有关小题和阅读下一小题，每段对话 你将听一遍。
例: What is the man going to read ?A. A newspaperB. A magazineC. A book 答案是A .1. How fast can the woman type?2 . Which program does the man like most?3 . Which Ian guage does Mr. White speak well?4 . What was in the woman ' missing bag?5 . Where does the con versatio n probably take place?第二节（共15小题：每小题1.5分，共22.5分） 听下面6段对话或独白，每段对白或独白后有几道小题，从每题所给的A . In a classroom B . In a libraryC . In a bookstoreA . 15words a minute .B . 45 words a minute .C . 80rds a minute .A . Sports .B . History .C . News.A . FrenchB . SpanishC .JapaneseA 、B 、C 三个选项中选出最佳选项。
听每段对白或独白前，你将有 5 秒种的时间阅读每小题。
听完后，每小题将给出 5 秒钟的做答时间。
每段对话或独白你将听两遍。
听每 6 段材料，回答第 6 至7 题。
6．What's wrong with the man?A ．He has headaches.B ．He has a runny nose.C ．He has a temperature.7．When did the problem begin?A ．Two weeks ago.B ．Two months ago.A ．Three months ago.听第7 段材料，回答8 至9 题8．Why did the woman make the phone call?A ．To change her appointment.B ．To discuss a business plan.C ．To arrange an exhibition.9．when are they going to meet?A ．Wednesday B．Thursday听第8 段材料，回答第10—11 题。
2009年高考北京英语试题及答案！BDear Friend,The recent success of children's books has made the general public aware that there's a huge market out there.And there's a growing need for new writers trained to create the $3 billion worth of children's books bought each year ... plus stories and articles needed by over 650 publishers of magazines for children and teenagers.Who are these needed writers They're ordinary folks like you and me.But am I good enoughI was once where you might be now. My thoughts of writing had been pushed down by self-doubt, and I didn't know where to turn for help.Then, I accepted a free offer from the Institute to test my writing aptitude(潜能), and it turned out to be the inspiration I needed.The promise that paid offThe Institute made the same promise to me that they will make to you, if you show basic writing abilityYou will complete at least one manuscript(手稿) suitable to hand in to a publisher by the time you finish our course.I really didn't expect any publication before I finished the course, but that happened. I sold three stories. And I soon discovered that was not unusual at the Institute.Since graduation, I have authored 34 nationally published children's books and over 300 stories and articles.Free test and brochureWe offer a free aptitude test and will send you a copy of our brochure describing our recognized home-study courses on the basis of one-on-one training.Realize your writing dream today. There's nothing sadder than a dream delayed until it fades forever.Sincerely,Kristi Holl, InstructorInstitute of Children's Literature61. From the first three paragraphs, we learn that_____.A. children's books are usually bestsellersB. publishers are making $3 billion each yearC. magazines for teenagers have drawn public attentionD. there is a growing need for writers of children's books62. When finishing the course, you are promised to_____.A. be a successful publisherB. become a confident editorC. finish one work for publicationD. get one story or article published63. Kristi Holl mentions her experience mainly to_____.A. prove she is a good instructorB. promote the writing programC. give her advice on course preparationD. show she sold more stories than articleCHow Room Designs Affect Our Work and FeelingsArchitects have long had the feeling that the places we live in can affect our thoughts, feelings and behaviors. But now scientists are giving this feeling an empirical(经验的，实证的) basis. They are discovering how to design spaces that promote creativity, keep people focused and lead to relaxation.Researches show that aspects of the physical environment can influence creativity. In 2007, Joan Meyers-Levy at the University of Minnesota, reported that the height of a room's ceiling affects how people think. Her research indicates that higher ceilings encourage people to think more freely, which may lead them to make more abstract connections. Low ceilings, on the other hand, may inspire a more detailed outlook.In additions to ceiling height, the view afforded by a building may influence an occupant's ability to concentrate. Nancy Wells and her colleagues at Cornell University found in their study that kids who experienced the greatest increase in greenness as a result of a family move made the most gains on a standard test of attention.Using nature to improve focus of attention ought to pay off academically, and it seems to, according to a study led by C. Kenneth Tanner, head of the School Design & Planning Laboratory at The University of Georgia. Tanner and his team found that students in classrooms with unblocked views of at least 50 feet outside the window had higher scores on tests of vocabulary, language arts and maths than did students whose classrooms primarily overlooked roads and parking lots.Recent study on room lighting design suggests than dim(暗淡的) light helps people to loosen up. If that is true generally, keeping the light low during dinner or at parties could increase relaxation. Researchers of Harvard Medical School also discovered that furniture with rounded edges could help visitors relax.So far scientists have focused mainly on public buildings. We have a very limited number of studies, so we're almost looking at the problem through a straw(吸管), architect David Allison says. How do you take answers to very specific questions and make broad, generalized use of them That's what we're all struggling with.64. What does Joan Meyers-Levy focus on in her researchA. Light.B. Ceilings.C. Windows.D. Furniture.65. The passage tells us that ______.A. the shape of furniture may affect people's feelingsB. lower ceilings may help improve students' creativityC. children in a dim classroom may improve their gradesD. students in rooms with unblocked views may feel relaxed66. The underlined sentence in the last paragraph probably means that ______.A. the problem is not approached step by stepB. the researches so far have faults in themselvesC. the problem is too difficult for researchers to detectD. research in this area is not enough to make generalized patterns67. Which of the following shows the organization of the passageCP Central Point P Point SP Sub-point(次要点) C ConclusionDWhen students and parents are asked to rate subjects according to their importance, the arts are unavoidably at the bottom of the list. Music is nice, people seem to say, but not important. Too often it is viewed as mere entertainment, but certainly not an education priority(优先). This view is shortsighted. In fact, music education is beneficial and important for all students.Music tells us who we are. Because music is an expression of the beings who create it, it reflects their thinking and values, as well ad the social environment it came influence that George Gershwin and other musicians introduced into their music is obviously American because it came from American musical traditions. Music expresses our character and values. It gives us identity as a society.Music provides a kind of perception(感知)that cannot be acquired any other way. Science can explain how the sun rises and sets. The arts explore the emotive(情感的)meaning of the same phenomenon. We need every possible way to discover and respond to our world for one simple but powerful reason No one way can get it all.The arts are forms of thought as powerful in what they communicate as mathematical and scientific symbols. They are ways we human beings “talk”to each other. They are the language of civilization through which we express our fears, our curiosities, our hungers, our discoveries, our hopes. The arts are ways we give form to our ideas and imagination so that they can be shared with others. When we do not give children access to an important way of expressing themselves such as music, we take away from them the meanings that music expresses. Science and technology do not tell us what it means to be human. The arts do. Music is an important way we express human suffering, celebration, the meaning and value of peace and love.So music education is far more necessary than people seem to realize.68. According to Paragraph 1, studentsA. regard music as a way of entertainmentB. disagree with their parents on educationC. view music as an overlooked subjectD. prefer the arts to science69. In Paragraph 2, the author uses jazz an example toA. compare it with rock musicB. show music identifies a societyC. introduce American musical traditionsD. prove music influences people’s lifestyles70. According to the passage, the arts and science .A. approach the world from different anglesB. explore different phenomena of the worldC express people’s feelings in different waysD. explain what it means to be human differently71. What is the main idea of the passageA. Music education deserves more attention.B. Music should be of top education priority.C. Music is an effective communication tool.D. Music education makes students more imaginative.23. Scientists have many theories about how the universe ___ into being.A. cameB. was comingC. had comeD. would come24. For breakfast he only drinks juice from fresh fruit ___ on his own farm.A. grownB. being grownC. to be grownD. to grow25. One of the few things you ___ say about English people with certainty is that they talk a lot about the weather.A. needB. mustC. shouldD. can26. –What do you think of teaching, Bob?–I find it fun and challenging. It is a job ___ you are doing something serious but interesting.A. whereB. WhichC. WhenD. that27. The way the guests ___ in the hotel influenced their evaluation of the service.A. treatedB. were treatedC. would treatD. would be treated28. All of them try to use the power of the workstation ___ information in a more effective way.A. presentingB. presentedC. being presentedD. to present29. The wine industry in the area has developed in a special way, ____ little foreign ownership.A. byB. ofC. withD. from30. When I talked with my grandma on the phone, she sounded weak, but by the time we ___ up, her voice had been full of life.A. were hangingB. had hungC. hungD. would hang31. At first he hated the new job but decided to give himself a few months to see ___ it got any better.A. whenB. howC. whyD. if32. John was given the same suitcase his father and grandfather ___ with them to school.A. tookB. had takenC. were takingD. would take33. The biggest whale is ___ blue whale, which grows to be about 29 meters long—the height of ___ 9-story building.A. the; theB. a; aC. a; theD. the; a34. ____ twice, the postman refused to deliver our letters unless we changed our dog.A. Being bittenB. BittenC. Having bittenD. To be bitten35. Being a parent is not always easy, and being the parent of a child with special needs often carries with ___ extra stress.A. itB. themC. oneD. him第二节完形填空（共20小题，每小题1.5分，共30分）阅读下面短文，掌握其大意，从每题所给的A、B、C、D四个选项中，选出最佳选项。
21. John plays basketball well, ___his favorite sport is badminton.A. soB. orC. yetD. for【答案】C。
【解析】考查连词的用法。
约翰篮球打的好，然而他最喜欢的运动却是羽毛球。
用yet表示转折。
22. You may use the room as you like ___ you clean it up afterwards.A. so far asB. so long asC. in caseD. even if【答案】B.【解析】考查状语从句的用法。
该句话的意思是：只要你后来把房子打扫干净，你就可以按照你喜欢的方式使用这间房子。
as long as =so long as只要。
23. Scientists have many theories about how the universe ___ into being.A. cameB. was comingC. had comeD. would come【答案】A。
【解析】考查时态的用法。
宇宙是如何形成的，因为宇宙的形成这个动作已经发生，所以用一般过去时来表示已经发生的动作。
24. For breakfast he only drinks juice from fresh fruit ___ on his own farm.A. grownB. being grownC. to be grownD. to grow【答案】A。
【解析】考查非谓语的用法。
fresh fruit与grow的关系是被动和完成，所以用过去分词做定语。
25. One of the few things you ___ say about English people with certainty is that they talk a lot about the weather.A. needB. mustC. shouldD. can【答案】D。
2009年普通高等学校招生全国统一考试英语【总卷评析】从测试反馈和卷面总体分析的情况来看，体现了语言能力的考查和高考试题体现的选拔性，突出了考试大纲对语言基本功的测试。
第一卷（选择题）第一部分英语知识运用（共三节，满分50分）第一节语音知识（共5小题；每小题1分，满分5分）【总体评析】观察试题单词，可看出均侧重基础和高频词汇的考查。
所以，总体而言，语音知识这道大题，难易度的设置较为适中，有选拔的梯度设计。
这套试卷选取了字母两个辅音字母“c, x”、一个元音字母及一个元音字母组合“i, ei”和一个半元音字母“y”来进行基本语音考查。
基本上能涵盖完整的元辅音及组合元音字母发音的考查。
从辨音难易度来看，第1、2和4小题读音区别度比较明显，考生不易失分。
第3小题的迷惑度较大，再次就是第5小题具有一定得迷惑度，但相对第3小题来看，要容易得分。
从A、B、C、D四个选项中，找出其划线部分与所给单词的划线部分读音相同的选项，并在答题卡上将该项涂黑。
例：haveA. gaveB. saveC. hatD. made答案是C1. JulyA. diaryB. energyC. replyD. daily【答案】C【解析】该选项y读音/ai/,而，其余发/i/。
区别度较大，考生一般不易失分。
2. medicineA. twiceB. medicalC. perfectD. clinic【答案】A。
【解析】字母“c”发/s/,而其余排除项发音/k/,比较明显，不含糊，考生基本能判别。
3. seizeA. neighbourB. weighC. eightD. receive【答案】D。
【解析】seize中的“ei”读音/i:/。
而其余项读音/ei/。
4. determineA. remindB. ministerC. smileD. tidy【答案】B。
【解析】该字母读音/i/,其余排除项读音/ai/。
尽管读音的区别度大，容易判断，但是对于minister“大臣, 牧师”一词，依然会有相当的学生不熟悉。
2009年北京卷英语高考阅读理解部分翻译解析刘金路老师翻译解析QQ 872969615AHow I Turned to Be Optimistic我如何变得更加乐观I began to grow up that winter night when my parents and I were returning from my aunt's house, and my mother said that we might soon be leaving for America. We were on the bus then. I was crying, and some people on the bus were turning around to look at me. I remember that I could not bear the thought of never hearing again the radio program for school children to which I listened every morning.那年冬天，当我的父母和我往家赶的时候，我感觉自己开始长大了，母亲说，我们不就可能要搬到美国去住。
之后我们上了公交车。
我当时在哭，车上的很多人都看我。
我记着，自己真是无法忍受那样的想法，就是自己再也无法听到学校专门为孩子们播放的节目了。
I do not remember myself crying for this reason again. In fact, I think I cried very little when I was saying goodbye to my friends and relatives. When we were leaving I thought about all the places I was going to see-—the strange and magical places I had known only from books and pictures. The country I was leaving never to come back was hardly in my head then.我记得自己再也没有因为这件事情哭过。
2009年普通高校招生统一考试北京卷（英语）（满分150分，时间120分钟） 第I 卷（选择题，共115分）第一部分 听力理解（共两节，30分）第一节 （共5小题；每小题1.5分，共7.5分）听下面五段对话。
每段对话后有一道小题。
从每题所给的 A 、B 、C 三个选项中选出最佳选项。
听完每段对话后，你将有 10秒钟时间来回答有关小题和阅读下一小题。
每段对话你将听一遍。
例： What is the man going to read?A. A n ewspaper.B. A magaz ine.C. A book. 答案是A 。
1. How fast can the woma n type?A. 15 words a minu te.B. 45 words a mi nu te.C. 80 words a minute.2. Which program does the man like most?A. Sports.B. History.C. News.3. Which Ian guage does Mr. White speak well?A. Fren ch.B. Spani sh. C, Japanese.4. What was in the woma n5. Where does the con versati onprobably take place? A. In a classroom. B. I n a library. C. I n a bookstore.第二节 （共15小题；每小题1.5分，共22.5 分）听下面6段对话或独白。
每段对话或独白后有几道小题，从每小题所给的 个选项中选出最佳选项。
听每段对话或独白前，你将有 5 秒钟的时间阅读每小题。
听完后， 每小题将给出 5秒钟的做答s miss ing bag?A.H.U时间。
每段对话或独白你将听两遍。
听第 6 段材料，回答第 6 至7 题。
