必修5综合训练1

高中英语 Unit1 Great scientists综合检测新人教版必修5（满分100分）Ⅰ．单项填空（共15小题；每小题1分，满分15分）1．Cars do cause us some health problems—in fact far more serious ______ than mobile phones do.A．one B．onesC．it D．those2．I don't think he will pass the exam，______？A．do I B．don't IC．will he D．won't he3．______ the ending，it's a really good film.A．Apart from B．As well asC．Besides D．Instead of4．What do you think of the suggestion ______ by the expert?A．put forward B．put up withC．put off D．put out5．—Can I help you，sir?—Yes，I bought this radio here yesterday，but it ______.A．didn't work B．won't workC．can't work D．doesn't work6．From the ______ look of the boys，I found they were ______.A．satisfying；satisfied B．satisfying；satisfyingC．satisfied；satisfied D．satisfied；satisfying7．The result of this discussion will be ______ tomorrow.A．published B．declaredC．announced D．opened8．Before using the machine，you must ______ carefully to these instructions.A．join B．join inC．take part in D．attend9．With the world ______ fast，we have something new to face every day.A．changed B．changingC．change D．to change10．The bridge being built now will link our town ______ that big city.A．over B．onC．to D．about11．It is widely believed that laughter has ______ effect on physical and mental health.A．positive B．popularC．energetic D．sensitive12．Our monitor is not only strict ______ us，but also cautious ______ expressing her ideas in public.A．with；in B．on；ofC．with；about D．in；about13．It is human being rather than pigs that are ______ for the disastrous H1N1 flu circling the globe.A．to be blamed B．to have blamedC．blaming D．to blame14．No matter how I tried to read it，I just couldn't make ______ of this sentence.A．meaning B．thoughtC．sense D．idea15．Don't stop ______ you come to a word or a phrase you don't know.A．for the first time B．becauseC．every time D．sinceⅡ．完形填空（共20小题；每小题1.5分，满分30分）A successful scientist is generally a good observer.He makes full __16__ of the facts he observes.He doesn't accept ideas which are not __17__ on obvious facts，and therefore refuses to accept authority（权威）as the only __18__ for truth.He always __19__ ideas carefully and makes experiments to prove them.The rise of __20__ science may perhaps be considered to __21__ as far back as the __22__ of Roger Bacon，the wonderful philosopher of Oxford，who lived __23__ the years 1214 and 1292.He was probably the first in the Middle __24__ to suggest that we shoul d learn science __25___ observing and experimenting on the things around us，and he himself __26__ many important discoveries.Galileo，however，who lived more than 300 years later，was the greatest of several great men，__27__ lived in Italy，France，Germany or England，began by __28__ to show how many important __29__ could be discovered by o bservation.Before Galileo，learned men believed that large bodies fell more __30__ towards the earth than small ones，__31__ Aristotle said so.But Galileo，going to the __32__ of the Leaning Tower of Pisa，let fall two __33__ stones and proved Aristotle was wrong.It was Galileo's __34__ of going direct to N ature，and proving our __35__ and theories by experiments，that has led to all the discoveries of modern science.16．A．use B．time C．speed D．trust17．A．worked B．based C．lived D．written18．A．reason B．cause C．advice D．result19．A．thinks B．checks C．has D．learns20．A．natural B．physical C．ancient D．modern21．A．date B．keep C．look D．take22．A．study B．time C．year D．birth23．A．both B．each C．between D．among24．A．Schools B．Ages C．Days D．Countries25．A．in B．with C．on D．by26．A．did B．made C．took D．gave27．A．who B．when C．that D．where28．A．ways B．degrees C．levels D．chance29．A．truths B．problems C．people D．subjects30．A．slowly B．rapidly C．lightly D．heavily31．A．when B．because C．although D．if32．A．place B．foot C．top D．ceiling33．A．big B．small C．equal D．unequal34．A．spirit B．skill C．theory D．discovery35．A．plans B．opinions C．world D．abilityⅢ．阅读理解（共20小题；每小题2分，满分40分）AScientists in the United States have developed a computer program that they say could provide an early warning of major earthquakes.The new system is designed to give warnings seconds before an earthquake strikes.That may not sound like very long.However，it could give people time to get under a table or take shelter someplace safe.The flow of electric power or gas could becut off.Air traffic controllers could warn away pilots.Even so­called “smart buildings” could be de signed to get ready.The system uses a mathematical program to measure information collected by instruments deployed across the Los Angeles area.Southern California has more than 150 seismic（地震的）stations that measure movements within the ground.The information they collect is sent to a high­speed computer.An earthquake shakes the ground with two kinds of waver.The new system uses information from a low­energy wave that moves quickly but causes little damage.Experts call thi s the P­wave.The slower but stronger S­wave is the main threat to life and property.Japan has a system based on P­waves.There，the warnings are used to slow down high­speed b ullet trains.Other countries have warning systems that measure S­waves.The new system is designed to estimate the strength of the earthquake once the shaking starts.Areas directly above the center of the earthquake would have the least warning.Distant places might have as long as forty seconds.The scientists are still testing their proposed system in the Los Angeles area.Professor Allen notes that an early warning system for earthquakes would require a major campaign of public education.36．Which of the following is TRUE about the new earthquake warning system?A．It can give people enough time to run away from earthquakes.B．It can give people seconds to take immediate measures.C．It is developed in Japan.D．It is a system based on S­waves.37．Which of the following can benefit from the new system most?A．Areas far away from the center of the earthquake.B．Areas directly above the center of the earthquake.C．The Los Angeles area.D．High­speed bullet trains.38．From what Professor Allen notes in the last paragraph we can infer that ______.A．it's very important for people to become educatedB．it's very important to warn people before an earthquake startsC．it's very important for people to know about earthquakesD．it's very important for people to know about the new system39．What do we learn from the passage?A．People could cut off electric power or gas immediately after the new system gives a warning.B．A low­energy wave may move slowly and cause little damage.C．Japan has a warning system based on S­waves to slow down bullet trains.D．The new system has been employed in warning of major earthquakes all over the United States.40．What's the BEST title fo r the passage?A．A New Earthquake Warning SystemB．Earthquakes Caused by Different Kinds of WavesC．Warning Systems in Different CountriesD．The Principle of a New Warning SystemBAlbert Einstein had a great effect on science and history，and his achievement was even greater than only a few other great scientists have achieved.An American university president once said that Einstein had made a new outlook，a new view of the universe.It may be some time before the average mind understands fully theidentity（识别）of time and space and so on—but even ordinary men now understand that the universe is something larger than ever thought before.By 1914 young Einstein had been world­famous.He accepted the offer to become a professor at the Prussian Academy of Science in Berlin.He had few duties，little teaching and unlimited chances for study，but soon his peace and quietness were broken by World War Ⅰ.Einstein hated fighting and killing.The great suffering of war affected him deeply，and he sat unhappily in his office doing little.He lost interest in his research.Only when peace came in 1918 was he able to get back to work.In the year following World War Ⅰ honors were increasingly put on him.He became head of the Kaiser Whihem Institute of Theoretical Physics.But he himself refused the effort to put him in a position far above other people.He was well known for his humble（谦逊的）manner.He often said that his success would certainly have been achieved by others if he had never lived.In 1921 he won the Nobel Prize，and he was honored in Germany until the rise of Nazism（纳粹）when he was driven from Germany because he was a Jew.41．The main idea of the first paragraph is ______.A．the difficulty of Einstein's thought to othersB．the feeling of an American university president towards EinsteinC．the difference between science and historyD．the change in human thought produced by Einstein42．It seems to the American university president that ______.A．Einstein achieved more than any other scientist in historyB．everybody understands Einstein's scientific ideasC．the new view of the universe can be quickly learned by everyoneD．our ideas about the universe are different today because of Einstein43．Between 1914 and 1918 Einstein ______.A．received world­wide praiseB．had no chances for studyC．almost did nothing in his officeD．still continued his scientific research44．Einstein did his greatest work ______.A．when Nazism roseB．when he was youngC．during World War ⅠD．after he left EuropeCIn 1901 a German scientist named William Roentgen received the first Nobel Prize in Physics.He had discovered X­rays.He made his discovery when he noticed that in certain experiments a special kind of light was made by the cathode rays（阴极射线）hitting a screen.The other scientists working with him had noticed this light but they did not think it was important.Roentgen was the first person to ask himself questions about this special kind of light.When he examined it，he discover ed a new kind of radiation called X­rays.We often think that a creative idea suddenly comes into our heads from nowhere.This is not usually the case—it comes because we are ready to receive it.When someone asked Issac Newton how he had discovered the laws of gravity（引力），he replied，“I am certain that any mind working on a problem continuously will be prepared for the sudden answer.”Albert Einstein received the Nobel Prize in Physics in 1921 and said that there are discoveries waiting to be made if people continue to look and study all the time.One of the problems facing creative people is that their ideas may be too new for the rest of the world.It often happens that people laugh at the work of artists and scientists.Barbara McClintock is a good example of someone who continued to think she was right although for 30 years other scientists thought that she was wrong.Finally，her work proving that genes（基因）can make sudden changes in heredity （遗传）won her the Nobel Prize in 1983.Many creative people looking for ideas or solutions find them in a visual（幻觉）form first.Elias Howe，the man who invented the sewing machine，had a big problem facing him.He could not understand how to connect the needle to the rest of the machine.Needles had always had a pointed end.One ni ght in a dream he was“attacked”by people with sharp spears（矛）．In the morning he said，“There is the answer staring me in the face！”He made a hole in the sharp end of the needle and then connected it to the sewing machine.45．We can infer from the passage that“creativity”means ______.A．the imagination to find a way of doing thingsB．the plan to make an important discoveryC．the ability to make something differentD．the power to make something new46．The writer's purpose in writing this passage is ______.A．to tell some interesting storiesB．to prove that people are creative in many waysC．to praise some great scientistsD．to explain why some people are clever47．Which of the following statements is BEST supported by the third paragraph?A．Believe in your own ideas.B．Pay attention to your dreams.C．Ask questions about everything around you.D．Collect as much information as you can.48．Which of the following statements is FALSE?A．You can be more creative if you read a lot.B．You can improve your creativity by asking lots of questions.C．You can help your creativity by getting into the habit of thinking about ordinary things.D．Doing a lot of preparation can help you be more creative.DAnthropologists are scientists that study people who often lived hundreds or even thousands of years ago.Items those men，women，and children left behind when they died—everything from clothing to jewelry and tools—tell anthropologists a lot.And sometimes even their words survive.Experts believe that a discovery in Mexico is the oldest example of writing ever found in America.The people who created it probably lived 3,000 years ago，long before Christopher Columbus arrived in the New World.Workers digging in a rock mine happened to notice a stone block with marks carved onto its surface.They found the large piece of stone near the city of Cascajal，Mexico，which was once the capital of an ancient people called the Olmec.“There are signs on the block that seem to show objects used by the Olmec during their religious ceremonies，”says Dr.Stephen H ouston，of Brown University in the USA.“There is a sign that looks very similar to a knife and another that looks like the chair of a King，”he explains.Houston，an expert on the writing systems of ancient cultures，says the discovery is exciting because“it m akes clear that the Olmec could read and write.It is likehearing voices from the past.”But experts don't know exactly what the writing says.Finding other examples of Olmec writing could help decipher the puzzle.“We could have whole sets of ancient writi ng which will basically make ancient people speak to us directly，”says Houston.“There are many puzzles to be solved，and this is just one of them.”49．According to the passage，what items do scientists often use to learn more about the past?A．Clothing and jewelry.B．Books and stones.C．Knives and chairs.D．People and animals.50．What is important about the stone block mentioned in the passage?A．It proves that people lived in this part of the world more than 3,000 years ago.B．It allows scientists to translate the ancient language used in this area.C．It is evidence of the oldest form of writing ever found in America.D．It shows the people of the area have a written language.51．Why are scientists interested in ancient languages?A．It can allow them to better understand ancient societies.B．So they can speak directly with ancient people.C．It will help learn about the origins of modern languages.D．They can learn stories told during that time.EMany parents don't love to read to their children.But there are so many professionals and PhDs doing the opposite thing.And they really promote reading to children，which is so good for children.I have found that this is true with my own children.I am not sure if it is the book or the fact that you are spending quality time with them，undivided time that they will remember for their entire life.Reading to children is one of the most bonding experiences you can have.Children enjoy sitting on your lap while your voice flows over them.As the reader it is fun to watch their little face s as they are moved with the flow of the story.They are laughing one moment and on the next page the story has moved them to tears.As the story teller，I am just as moved as they are；I am caught up in the story as our emotions flow up and down.You can easily lose a whole afternoon in reading.This is an experience that children then try to pass on to younger children，reading books to them.My daughter was so into reading that if she couldn't line up some neighbors' children she could be reading her books to，she would line up her stuffed animals and read to them.Well，she also tried to teach her neighbors and teddy bears math，and “ocean science” as she called it.She would read anything she could to them.And to my astonishment the neighbors' parents told me they really appreciated the fact that their children were doing so much better in school.52．The writer mentions the idea of professionals and PhDs so as to ______.A．introduce his own viewpointB．show the importance of his ideaC．please readersD．attract the attention of readers53．The writer mentions his own daughter in this text for the purpose of ______.A．showing the cleverness of his childrenB．explaining the advantages of reading to daughterC．telling the importance of studying together with parentsD．showing his children were interested in reading54．The writer thinks that ______.A．you should place your children on your lap while reading to themB．children's feelings change while you read books to themC．it is easy to kill time while you are reading books to childrenD．reading to children is one of the most interesting experiences55．According to the text，which of the following is NOT true about the writer's daughter?A．She lined up some neighbors' children and read to them.B．She lined up her stuffed animals and read to them.C．She would teach her teddy bears math and “ocean science”.D．She would read anything she could to her parents.附加题下面文章中有5处（第1～5题）需要添加小标题。

2011-2012学年下期高中数学必修综合测试题（四）一、选择题：本大题共10小题，每小题5分，满分50分. 在每小题给出的四个选项中，只有一项是符合题目要求的.1．函数1y x =-的定义域为（ ）A ．(),1-∞B ．(],1-∞C ．()1,+∞D ．[)1,+∞ 2．直线30x y -=的倾斜角为（ ） A ．6π B ．3π C ．23π D ．56π3．已知全集{}1,2,3,4,5,6,7,8U =，集合{}2,4,6,8A =，{}1,2,3,6,7B =，则()U A B = ð（ ） A ．{}2,4,6,8 B ．{}1,3,7 C ．{}4,8 D ．{}2,64．某赛季，甲、乙两名篮球运动员都参加了7场比赛，他们所有比赛得分的情况用如图1所示的茎叶图表示，则甲、乙两名运动员得分的平均数分别为（ ） A ．14、12 B ．13、12 C ．14、13 D ．12、145．在边长为1的正方形ABCD 内随机取一点P ，则点P 到点A 的距离小于1的概率为（ ）A ．4π B ．14π- C ．8π D ．18π- 6．已知向量a 与b 的夹角为120，且1==a b ，则－a b 等于（ ） A ．1 B ．3 C ．2D ．37．有一个几何体的三视图及其尺寸如图2所示（单位：cm ），则该几何体的表面积．．．为（ ） A ．212cm π B. 215cm πC. 224c m π D. 236cm π 8．若23x <<，12xP ⎛⎫= ⎪⎝⎭，2log Q x =，R x =，则P ，Q ，R 的大小关系是（ ） A ．Q P R << B ．Q R P << C ．P R Q << D ．P Q R <<9．已知函数()2sin()f x x ωϕ=+0,2πωϕ⎛⎫>< ⎪⎝⎭的图像如图3所示，则函数)(x f 的解析式是（ ） A ．10()2sin 116f x x π⎛⎫=+ ⎪⎝⎭B ．10()2sin 116f x x π⎛⎫=- ⎪⎝⎭C ．()2sin 26f x x π⎛⎫=+ ⎪⎝⎭ D ．()2sin 26f x x π⎛⎫=- ⎪⎝⎭ 10．一个三角形同时满足：①三边是连续的三个自然数；②最大角是 最小角的2倍，则这个三角形最小角的余弦值为（ ）A ．378 B ．34 C ．74D ．18二、填空题：本大题共4小题，每小题5分，满分20分.11．圆心为点()0,2-，且过点()14，的圆的方程为 ．12．如图4，函数()2x f x =，()2g x x =，若输入的x 值为3， 则输出的()h x 的值为 . 13．设不等式组0,02036x y x y x y -+-⎧⎪-+⎨⎪⎩≤≥≥，表示的平面区域为D ，若直线0kx y k -+=上存在区域D 上的点，则k 的取值范围是 ．14．若函数()()()2213f x a x a x =-+-+是偶函数，则函数()f x 的单调递减区间为 ．三、解答题：本大题共6小题，满分80分. 解答须写出文字说明、证明过程和演算步骤. 15．（本小题满分12分）在△ABC 中，角A ，B ，C 成等差数列．（1）求角B 的大小；（2）若()2sin 2A B +=，求sin A 的值．0 1 2 1 3 55 8 7 5 9 9 7 5 4 86 甲 乙图165主视图65侧视图俯视图图21 Oxy 1112π图3 否是开始 ()()h x f x = ()()f xg x >输出()h x 输入x结束 ()()h x g x =图416．（本小题满分12分）某校在高二年级开设了A ，B ，C 三个兴趣小组，为了对兴趣小组活动的开展情况进行调查，用分层抽样方法从A ，B ，C 三个兴趣小组的人员中，抽取若干人组成调查小组，有关数据见下表（单位：人） （1）求x ，y 的值；（2）若从A ，B 两个兴趣小组抽取的人中选2人作专题发言，求这2人都来自兴趣小组B 的概率． 17．（本小题满分14分）如图5，在四棱锥P ABCD -中，底面ABCD 为正方形，PA ⊥平面ABCD ，PA AB =，点E 是PD 的中点．（1）求证：PB 平面ACE ；（2）若四面体E ACD -的体积为23，求AB 的长．18．（本小题满分14分）已知数列{}n a 是首项为1，公比为2的等比数列，数列{}n b 的前n 项和2n S n =． （1）求数列{}n a 与{}n b 的通项公式；（2）求数列n n b a ⎧⎫⎨⎬⎩⎭的前n 项和． 19．（本小题满分14分）直线y kx b =+与圆224x y +=交于A 、B 两点，记△AOB 的面积为S （其中O 为坐标原点）．（1）当0k =，02b <<时，求S 的最大值； （2）当2b =，1S =时，求实数k 的值． 20．（本小题满分14分）已知函数()213f x ax x a =+-+()a ∈R 在区间[]1,1-上有零点，求实数a 的取值范围．兴趣小组 小组人数 抽取人数A 24 xB 363C48y ABCDPE图52010学年度广州市高中二年级学生学业水平测试数学试题参考答案及评分标准一、选择题：本大题主要考查基本知识和基本运算．共10小题，每小题5分，满分50分．题号 1 23 4 56 78 9 10答案D B C A A B C D C B二、填空题：本大题主要考查基本知识和基本运算．共4小题，每小题5分，满分20分．11．()22225x y ++=（或224210x y y ++-=） 12．913．()0,+∞（或[)0,+∞） 14．122⎡⎤⎢⎥⎣⎦，三、解答题15．本小题主要考查解三角形、三角恒等变换等基础知识，考查运算求解能力．满分12分． 解：（1）在△ABC 中，A B C π++=，由角A ，B ，C 成等差数列，得2B A C =+．解得3B π=．（2）方法1：由()2sin 2A B +=，即()2sin 2C π-=，得2sin 2C =．所以4C π=或34C π=． 由（1）知3B π=，所以4C π=，即512A π=． 所以5sin sinsin 1246A πππ⎛⎫==+ ⎪⎝⎭sincoscossin4646ππππ=+23212222=⨯+⨯ 264+=． 方法2：因为A ，B 是△ABC 的内角，且()2sin 2A B +=， 所以4A B π+=或34A B π+=．由（1）知3B π=，所以34A B π+=，即512A π=．以下同方法1． 方法3：由（1）知3B π=，所以2sin 32A π⎛⎫+= ⎪⎝⎭． 即2sin coscos sin332A A ππ+=． 即132sin cos 222A A +=． 即3cos 2sin A A =-．即223cos 222sin sin A A A =-+． 因为22cos 1sin A A =-，所以()2231sin 222sin sin A A A -=-+．即24sin 22sin 10A A --=．解得26sin 4A ±=． 因为角A 是△ABC 的内角，所以sin 0A >．故26sin 4A +=．16．本小题主要考查统计与概率等基础知识，考查数据处理能力．满分12分．解：（1）由题意可得，3243648x y==， 解得2x =，4y =．（2）记从兴趣小组A 中抽取的2人为1a ，2a ，从兴趣小组B 中抽取的3人为1b ，2b ，3b ，则从兴趣小组A ，B 抽取的5人中选2人作专题发言的基本事件有()12,a a ，()11,a b ，()12,a b ，()13,a b ，()21,a b ，()22,a b ，()23,a b ，()12,b b ，()13,b b ，()23,b b 共10种．设选中的2人都来自兴趣小组B 的事件为X ，则X 包含的基本事件有()12,b b ，()13,b b ，()23,b b 共3种．所以()310P X =． 故选中的2人都来自兴趣小组B 的概率为310．17．本小题主要考查直线与平面的位置关系、体积等基础知识，考查空间想象能力、推理论证能力和运算求解能力．满分14分．（1）证明：连接BD 交AC 于点O ，连接EO ，因为ABCD 是正方形，所以点O 是BD 的中点． 因为点E 是PD 的中点，所以EO 是△DPB 的中位线．所以PB EO ．因为EO ⊂平面ACE ，PB ⊄平面ACE ， 所以PB 平面ACE ．（2）解：取AD 的中点H ，连接EH ， 因为点E 是PD 的中点，所以EH PA ． 因为PA ⊥平面ABCD ，所以EH ⊥平面ABCD ． 设AB x =，则PA AD CD x ===，且1122EH PA x ==． 所以13E ACD ACD V S EH -∆=⨯ 1132AD CD EH =⨯⨯⨯⨯ 3111262123x x x x === ． 解得2x =．故AB 的长为2．18．本小题主要考查等差数列、等比数列等基础知识，考查运算求解能力和推理论证能力．满分14分． 解：（1）因为数列{}n a 是首项为1，公比为2的等比数列，所以数列{}n a 的通项公式为12n n a -=． 因为数列{}n b 的前n 项和2n S n =．所以当2n ≥时，1n n n b S S -=-()22121n n n =--=-，当1n =时，111211b S ===⨯-， 所以数列{}n b 的通项公式为21n b n =-． （2）由（1）可知，1212n n n b n a --=． 设数列n n b a ⎧⎫⎨⎬⎩⎭的前n 项和为n T ， 则 213572321124822n n n n n T ----=++++++ ， ① 即 111357232122481622n n n n n T ---=++++++ ， ② ①－②，得2111112111224822n n nn T --=++++++- 11121211212n nn -⎛⎫- ⎪-⎝⎭=+-- ABC DP E OH2332nn +=-， 所以12362n n n T -+=-． 故数列n n b a ⎧⎫⎨⎬⎩⎭的前n 项和为12362n n -+-．19．本小题主要考查直线与圆、基本不等式等基础知识，考查运算求解能力．满分14分． 解：（1）当0k =时，直线方程为y b =，设点A 的坐标为1()x b ，，点B 的坐标为2()x b ，，由224x b +=，解得2124x b =±-，，所以22124AB x x b =-=-．所以12S AB b =24b b =- 22422b b +-=≤．当且仅当24b b =-，即2b =时，S 取得最大值2．（2）设圆心O 到直线2y kx =+的距离为d ，则221d k =+．因为圆的半径为2R =，所以2222244211AB k R d k k =-=-=++． 于是222241212111k k S AB d k k k =⨯=⨯==+++，即2410k k -+=，解得23k =±．故实数k 的值为23+，23-，23-+，23--．20．本小题主要考查二次函数、函数的零点等基础知识，考查运算求解能力，以及分类讨论的数学思想方法．满分14分． 解法1：当0a =时，()1f x x =-，令()0f x =，得1x =，是区间[]1,1-上的零点． 当0a ≠时，函数()f x 在区间[]1,1-上有零点分为三种情况： ①方程()0f x =在区间[]1,1-上有重根， 令()14130a a ∆=--+=，解得16a =-或12a =． 当16a =-时，令()0f x =，得3x =，不是区间[]1,1-上的零点． 当12a =时，令()0f x =，得1x =-，是区间[]1,1-上的零点． ②若函数()y f x =在区间[]1,1-上只有一个零点，但不是()0f x =的重根， 令()()()114420f f a a -=-≤，解得102a <≤． ③若函数()y f x =在区间[]1,1-上有两个零点，则()()⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧≥≥<-<->++-=∆>.01-,01,1211,01412,02f f a a a a 或()()⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧≤≤<-<->++-=∆<.01-,01,1211,01412,02f f a a a a 解得a ∈∅．综上可知，实数a 的取值范围为10,2⎡⎤⎢⎥⎣⎦．解法2：当0a =时，()1f x x =-，令()0f x =，得1x =，是区间[]1,1-上的零点．当0a ≠时，()213f x ax x a =+-+在区间[]1,1-上有零点⇔()231x a x +=-在区间[]1,1-上有解⇔213xa x -=+在区间[]1,1-上有解． 问题转化为求函数213xy x -=+在区间[]1,1-上的值域．设1t x =-，由[]1,1x ∈-，得[]0,2t ∈．且()2013ty t =≥-+．而()214132ty t t t==-++-． 设()4g t t t=+，可以证明当(]0,2t ∈时，()g t 单调递减． 事实上，设1202t t <<≤， 则()()()()121212121212444t t t t g t g t t t t t t t --⎛⎫⎛⎫-=+-+= ⎪ ⎪⎝⎭⎝⎭， 由1202t t <<≤，得120t t -<，1204t t <<，即()()120g t g t ->． 所以()g t 在(]0,2t ∈上单调递减． 故()()24g t g ≥=． 所以()1122y g t =≤-．故实数a 的取值范围为10,2⎡⎤⎢⎥⎣⎦．。

.绝密★启用前高中数学必修五综合考试卷第I 卷（选择题）一、单选题1．数列的一个通项公式是（ ）0,23,45,67⋯A ．B ． a n =n -1n +1（n ∈N *)a n =n -12n +1（n ∈N *)C ．D ．a n =2（n -1）2n -1（n ∈N *)a n =2n2n +1（n ∈N *)2．不等式的解集是（ ）x -12-x ≥0A ．B ．C ．D ． [1,2](-∞,1]∪[2,+∞)[1,2)(-∞,1]∪(2,+∞)3．若变量满足 ，则的最小值是（ ）x,y {x +y ≥0x -y +1≥00≤x ≤1x -3y A ．B ．C ．D ． 4-5-314．在实数等比数列{a n }中，a 2，a 6是方程x 2－34x ＋64＝0的两根，则a 4等于( )A ． 8B ． －8C ． ±8D ． 以上都不对5．己知数列为正项等比数列，且，则（ ）{a n }a 1a 3+2a 3a 5+a 5a 7=4a 2+a 6=A ． 1B ． 2C ． 3D ． 46．数列前项的和为（ ）11111,2,3,4,24816n A ． B ． C ．D ．2122nn n ++21122n n n +-++2122n n n +-+21122n n n +--+7．若的三边长成公差为的 等差数列，最大角的正弦值为ΔABC a,b,c 232的面积为（ ）A ．B ．C ．D ．1541534213435348．在△ABC 中，已知,则B 等于( )a =2,b =2,A =450A ． 30°B ． 60°C ． 30°或150°D ． 60°或120°9．下列命题中正确的是( )A ． a ＞b ⇒ac 2＞bc 2B ． a ＞b ⇒a 2＞b 2C ． a ＞b ⇒a 3＞b 3D ． a 2＞b 2⇒a ＞b.10．满足条件，的的个数是 ( )a =4,b =32,A =45∘A ． 1个B ． 2个C ． 无数个D ． 不存在11．已知函数满足：则应满足（ ）f(x)=ax 2-c -4≤f(1)≤-1,-1≤f(2)≤5.f(3)A ．B ．C ．D ．-7≤f(3)≤26-4≤f(3)≤15-1≤f(3)≤20-283≤f(3)≤35312．已知数列{a n }是公差为2的等差数列，且成等比数列，则为（ ）a 1,a 2,a 5a2A ． -2B ． -3C ． 2D ． 313．等差数列的前10项和，则等于（）{a n }S 10=15a 4+a 7A ． 3B ． 6C ． 9D ． 1014．等差数列的前项和分别为，若，则的值为（ ）{a n },{b n }n S n ,T nS nT n=2n3n +1a 3b 3A ．B ．C ．D ． 3547581219第II 卷（非选择题）二、填空题15．已知为等差数列，且－2＝－1,＝0,则公差={a n }a 7a 4a3d 16．在中，，，面积为，则边长=_________.△ABC A =60∘b =13c 17．已知中，，， ，则面积为_________.ΔABC c =3a =1acosB =bcosA ΔABC 18．若数列的前n 项和，则的通项公式____________{a n }S n =23a n +13{a n }19．直线下方的平面区域用不等式表示为________________．x -4y +9=020．函数的最小值是 _____________．y =x +4x -1(x >1)21．已知，且，则的最小值是______．x ，y ∈R +4x +y =11x +1y三、解答题22．解一元二次不等式（1） （2）-x 2-2x +3>0x 2-3x +5>0.（1）求边上的中线的长；BC AD （2）求△的面积。

必修1、3、4、5综合练习题一、 选择题：1、集合U ＝{1,2,3,4,5,6}，S ＝{1,4,5}，T ＝{2,3,4}，则S ∩(∁U T )等于( )A ．{1,4,5,6}B ．{1,5}C ．{4}D ．{1,2,3,4,5} 2、已知U ＝{y |y ＝log 2x ，x >1}，P ＝⎩⎨⎧⎭⎬⎫y |y ＝1x ，x >2，则∁U P ＝( )A.⎣⎡⎭⎫12，＋∞B.⎝⎛⎭⎫0，12C.()0，＋∞D.(]－∞，0∪⎣⎡⎭⎫12，＋∞ 3、已知函数f (x )＝⎩⎪⎨⎪⎧2x，x >0，x ＋1，x ≤0.若f (a )＋f (1)＝0，则实数a 的值等于( )A ．－3B ．－1C ．1D ．34、已知角θ的顶点与原点重合，始边与x 轴的正半轴重合，终边在直线y ＝2x 上，则cos2θ＝( )A ．－45B ．－35 C.35 D.455、设变量x ，y 满足⎩⎪⎨⎪⎧x ＋y ≤1，x －y ≤1，x ≥0，则x ＋2y 的最大值和最小值分别为( )A ．1，－1B ．2，－2C ．1，－2D ．2，－1 6、若函数f (x )＝x ＋1x －2(x >2)在x ＝a 处取最小值，则a ＝( )A ．1＋ 2B ．1＋ 3C ．3D ．47、如图1－1，矩形ABCD 中，点E 为边CD 的中点．若在矩形ABCD 内部随机取一个点Q ，则点Q 取自△ABE 内部的概率等于( )图1－1A.14B.13C.12D.238、有一个容量为66的样本，数据的分组及各组的频数如下：[11.5,15.5) 2 [15.5,19.5) 4 [19.5,23.5) 9 [23.5,27.5) 18 [27.5,31.5) 11 [31.5,35.5) 12 [35.5,39.5) 7 [39.5,43.5) 3根据样本的频率分布估计，数据落在[31.5,43.5)的概率约是( ) A.16 B.13 C.12 D.239、执行如图1－1所示的程序框图，如果输入的N 是6，那么输出的p 是( )A ．120B ．720C ．1440D ．504010、函数()cos f x x =在[0,)+∞内（A ）没有零点 （B ）有且仅有一个零点 （C ）有且仅有两一个零点（D ）有无穷个零点 11、10名工人某天生产同一零件，生产的件数是15,17,14,10,15,17,17,16,14,12,设其平均数为a ,中位数为b ,众数为c ，则有( )A c b a >>B a c b >>C b a c >>D a b c >>12、已知向量)sin ,(cos θθ=,向量)1,3(-=则|2|-的最大值，最小值分别是（ ）A 0,24B 24,4C 16,0D 4,0二、填空题；13、已知log 2a ＋log 2b ≥1，则3a ＋9b 的最小值为________．14、已知向量a ，b 满足(a ＋2b )·(a －b )＝－6，且|a |＝1，|b |＝2，则a 与b 的夹角为________． 15、从1,2,3,4这四个数中一次随机地取两个数，则其中一个数是另一个数的两倍的概率是________． 16、某市有大型超市200家、中型超市400家、小型超市1400家，为掌握各类超市的营业情况，现按分层抽样方法抽取一个容量为100的样本，应抽取中型超市________家．17、《九章算术》“竹九节”问题：现有一根9节的竹子，自上而下各节的容积成等差数列，上面4节的容积共3升，下面3节的容积共4升，则第5节的容积为________升．18、若22xx ≥，则x 的取值范围是____________三、解答题：19、已知等比数列{a n }的公比q ＝3，前3项和S 3＝133.(1)求数列{a n }的通项公式；(2)若函数f (x )＝A sin(2x ＋φ)(A >0,0<φ<π)在x ＝π6处取得最大值，且最大值为a 3，求函数f (x )的解析式．20、△ABC 的内角A 、B 、C 的对边分别为a 、b 、c .已知A －C ＝90°，a ＋c ＝2b ，求C .21、等比数列{a n }的各项均为正数，且2a 1＋3a 2＝1，a 23＝9a 2a 6.(1)求数列{a n }的通项公式；(2)设b n ＝log 3a 1＋log 3a 2＋…＋log 3a n ，求数列⎩⎨⎧⎭⎬⎫1b n 的前n 项和．22、已知12)(),cos 21,2sin 1(),sin 23,2sin 1(+⋅=-=+=b a x f x x b x x a (I)求函数()f x 的最小正周期和最大值;(II)该函数的图象可由sin ()y x x R =∈的图象经过怎样的平移和伸缩变换得到?参考答案：一、选择题1、 B 【解析】 S ∩(∁U T )＝{1,4,5} ∩{1,5,6}＝{1,5}2、 A 【解析】 因为U ＝{y |y ＝log 2x ，x >1}＝{y |y >0}，P ＝⎩⎨⎧⎭⎬⎫y |y ＝1x ，x >2＝⎩⎨⎧⎭⎬⎫y ⎪⎪0<y <12，所以∁U P ＝⎩⎨⎧⎭⎬⎫y ⎪⎪y ≥12＝⎣⎡⎭⎫12，＋∞3、A 【解析】 由已知，得f (1)＝2；又当x >0时，f (x )＝2x >1，而f (a )＋f (1)＝0，∴f (a )＝－2，且a <0，∴a ＋1＝－2，解得a ＝－3，故选A4、B 【解析】 解法1：在角θ终边上任取一点P (a,2a )(a ≠0)，则r 2＝||OP 2＝a 2＋(2a )2＝5a 2，∴cos 2θ＝a 25a 2＝15，∴cos2θ＝2cos 2θ－1＝25－1＝－35.解法2：tan θ＝2a a ＝2，cos2θ＝cos 2θ－sin 2θcos 2θ＋sin 2θ＝1－tan 2θ1＋tan 2θ＝－35. 5、B 【解析】 画出可行域(如图所示阴影部分)．可知当直线u ＝x ＋2y 经过A (0,1)，C (0，－1)时分别对应u 的最大值和最小值．故u max ＝2，u min ＝－2.6、C 【解析】 ∵x ＞2，∴f (x )＝x ＋1x －2＝(x －2)＋1x －2＋2≥2(x －2)·1x －2＋2＝4，当且仅当x －2＝1x －2，即x ＝3时取等号．7、C 【解析】 因为S △ABE ＝12|AB |·|BC |，S 矩形＝|AB |·|BC |，则点Q 取自△ABE 内部的概率p ＝S △ABE S 矩形＝129、【解析】 k ＝1时，p ＝1；k ＝2时，p ＝1×2＝2； k ＝3时，p ＝2×3＝6； k ＝4时，p ＝6×4＝24； k ＝5时，p ＝24×5＝120； k ＝6时，p ＝120×6＝720.8、B 【解析】 根据样本中的频率分布可得：数据落在[31.5,43.5)的概率约是12＋7＋366＝2266＝13.10、B 【解析】：令1y =2cos y x =，则它们的图像如图故选B11、D 总和为147,14.7a =；样本数据17分布最广，即频率最大，为众数，17c =；从小到大排列，中间一位，或中间二位的平均数，即15b =12、6 D 2(2cos 2sin 1),|2|a b a b θθ-=+-===4，最小值为0二、填空题；13、18 【解析】 ∵log 2a ＋log 2b ＝log 2ab ≥1，∴ab ≥2，∴3a ＋9b ＝3a ＋32b ≥23a ·32b ＝23a＋2b≥2322ab＝18.14、π3 【解析】 设a 与b 的夹角为θ，依题意有(a ＋2b )·(a －b )＝a 2＋a ·b －2b 2＝－7＋2cos θ＝－6，所以cos θ＝12.因为0≤θ≤π，故θ＝π3.15、13 【解析】 一次随机抽取两个数共有1,2；1,3；1,4；2,3；2,4；3,4，一个数是另一个数的2倍的有2种，故所求概率为13.16、20 【解析】 由题意，样本容量为200＋400＋1400＝2000, 抽样比例为1002000＝120，所以中型超市应抽120×400＝20家． 17、6766 【解析】 设所构成的等差数列{}a n 的首项为a 1，公差为d ，由⎩⎪⎨⎪⎧a 1＋a 2＋a 3＋a 4＝3，a 7＋a 8＋a 9＝4，得⎩⎪⎨⎪⎧4a 1＋6d ＝3，3a 1＋21d ＝4，解得⎩⎨⎧a 1＝1322，d ＝766，所以a 5＝a 1＋4d ＝6766.18、 [2,4] 在同一坐标系中画出函数2y x =与2xy =的图象，可以观察得出 三、解答题：19、【解答】 (1)由q ＝3，S 3＝133得a 1(1－33)1－3＝133，解得a 1＝13.所以a n ＝13×3n －1＝3n －2.(2)由(1)可知a n ＝3n －2，所以a 3＝3.因为函数f (x )的最大值为3，所以A ＝3； 因为当x ＝π6时f (x )取得最大值，所以sin ⎝⎛⎭⎫2×π6＋φ＝1. 又0<φ<π，故φ＝π6.所以函数f (x )的解析式为f (x )＝3sin ⎝⎛⎭⎫2x ＋π6. 20、【解答】 由a ＋c ＝2b 及正弦定理可得sin A ＋sin C ＝2sin B .又由于A －C ＝90°，B ＝180°－(A ＋C )，故 cos C ＋sin C ＝2sin(A ＋C ) ＝2sin(90°＋2C ) ＝2cos2C . 故22cos C ＋22sin C ＝cos2C ， cos(45°－C )＝cos2C . 因为0°<C <90°，所以2C ＝45°－C ，C ＝15°.21、【解答】 (1)设数列{a n }的公比为q ，由a 23＝9a 2a 6得a 23＝9a 24，所以q 2＝19. 由条件可知q ＞0，故q ＝13.由2a 1＋3a 2＝1得2a 1＋3a 1q ＝1，所以a 1＝13.故数列{a n }的通项公式为a n ＝13n .(2)b n ＝log 3a 1＋log 3a 2＋…＋log 3a n ＝－(1＋2＋…＋n ) ＝－n (n ＋1)2.故1b n ＝－2n (n ＋1)＝－2⎝⎛⎭⎫1n －1n ＋1， 1b 1＋1b 2＋…＋1b n ＝－2⎝⎛⎭⎫1－12＋⎝⎛⎭⎫12－13＋…＋⎝⎛⎭⎫1n －1n ＋1＝－2nn ＋1. 所以数列⎩⎨⎧⎭⎬⎫1b n 的前n 项和为－2nn ＋1.22、21sin ()2(cos )14x f x x x -=++=15sin(2)264x π++. (I) f(x)的最小正周期为T=22ππ=. ∵sin(2)[1,1]6x π+∈-, ∴f(x)的最大值为74. (II) 将函数sin ()y x x R =∈的图象向左平移6π个单位,再将横坐标与纵坐标均缩小到原来的12倍,最后将图象向上平移54个单位,即可得到.。

1-1 Warming up & PrereadingⅠ.根据句意及首字母提示补全单词1．Who invented the first steam e________？2．Who used peas to show how physical c________ are passed from parents to their children?3．Who discovered r________？4．A________the result，and you can get a conclusion.5．Do you know how to prove a new idea in s________ research?答案：1.engine 2.characteristics 3.radium 4.Analyse5．scientificⅡ.根据汉语提示完成句子1．Which scientist discovered that objects in water are ______ (举起；提高；提升)by a force that helps them float?2．Who________ (提出)a theory about black holes?3．________(想出)a method to solve the problem.4．We can safely________(得出结论)from our discussion.5．________ (提问题)if you don't understand.答案：1.lifted up 2.put forward 3.Think of 4.draw a conclusion 5.Make a questionⅢ.用所给词的适当形式填空。

1．There are lots of ________(scientific) in history.2．Discuss in small groups the stages in ________(examination) a new scientific idea.3．Before drawing a ________(conclude)，you should do lots of research.4．The students should________ (analysis)their examinationresults.5．Sorry，I can't hear you clearly. Please________(repetition) it.答案：1.scientists 2.examing 3.conclusion 4.analyse5．repeatⅣ.单项填空1．Thomas Edison________ the electric lamp and Madame Curie ________ radium.A．invented；invented B．discovered；discoveredC．discovered；invented D．invented；discovered答案：D invent“发明”；discover“发现”。

必修5Unit 1(时间：40分钟满分：50分)Ⅰ.单项填空(共15小题；每小题1分，满分15分)1. Considering your salary，you should be able to________at least twenty dollars a week. A．put forward B．put upC．put out D．put aside答案D[句意：就你的工资而言，每周你应至少存20美元。

put forward提出；put up张贴；put out扑灭；put aside储存。

]2．The________，I think，he drew from his simple experiment is not scientific.A．attention B．conclusion C．promise D．relation答案B[句意：我认为他从简单的实验中得出的结论是不科学的。

draw a conclusion“得出结论”；draw one's attention“引起某人注意”。

]3．The more one is ________the English-speaking environment，the better he or she will learn the language.A．exposed to B．filled in C．caught on D．kept up答案A[根据句子的意思判断，此处应该使用be exposed to...，表示“被暴露在……”的意思。

]4．—I was surprised to see wild flowers in Alaska.—Many people think there is nothing there ________ice and snow.A．except for B．rather thanC．apart from D．regardless of答案C[此处表示“除了冰雪以外没有别的东西”，可以使用except，因为ice and snow和nothing属于同类，而except for后的名词和前面的名词不属同类。

综合质量评估(Unit 1～5)本试卷分第I卷(选择题)和第II卷(非选择题)两部分。

满分150分，考试时间120分钟。

I.听力（共20小题，每小题1.5分，满分30分）II.单项填空(共15小题，每小题1分，满分15分)21. In China, ______ pressure to pass _____ exams tends to cause lack of students‟ creativity.【杭州高二检测】A /; theB a; the C. the; / D. / ; the【解析】选C。

考查冠词。

第一空特指通过考试的压力，用the; 第二空为泛指各种考试，不用加冠词。

22.It can be ____ from the graph that there has been a great decline in birth rates in China in the past five years.A. decidedB. determinedC. concludedD. summarized【解析】选C。

decide 决定= determine; conclude 总结，得出结论；summarize 摘要，概述。

句意：从曲线图可以得出结论，最近5年来中国人口出生率已经大大下降。

23. This liquid is not to be exposed ___ air, for it will soon evaporate.A. with B in C. into D. to【解析】选D。

考查介词。

be exposed to... 暴露于/接触....(的环境).【举一反三】The more we are _____ to advertising the less we notice it。

A. exposedB. linkedC. showedD. appeared【解析】选A。

新课标高二上学期5册Unit 5测试Class: Name: Marks: 满分（120）第一部分阅读理解（共两节，满分40分）第一节（共15小题；每小题2分，满分30分）AIt all started off with five. It was raining, and my dad told me that I had to make five free throws (投球) one after another without a break before I could go home. I was ten years old at that time. So I shot (投球) until I indeed made five free throws. Pretty soon the number of free throws increased until I got to high school and I had to make seventy-five before I could leave the gym.I used to get mad when I reached the required amount and I would miss the next one, but my dad made me start over, saying that I missed it on purpose. So in high school I would usually get to 120 before I stopped. My highest ever was 175. I never expected this would ever bring me some benefit, but one game really changed my idea.Our school team was playing basketball match with Royal High School, and we lost the game by twenty points a week earlier, so this game was all about revenge (报仇). With two minutes left in the game, my team was up by six and their coach told them to foul (犯规). Our coach immediately put forward a perfect plan — giving the ball to me every time. And every time my teammates gave me the ball, the other team would foul me. Then I went to the free throw line and made two free throws successfully. I ended up going ten-for-ten that game all in the last two minutes. The final score was 57-50 and we won.1. We can learn from the first two paragraphs that .A. the a uthor’s father was very strict with himB. the author’s father wanted him to create a recordC. the author had to make five free throws every dayD. the author could make 75 free throws at the age of 102. How many times was the author fouled?A. Ten.B. Seven.C. Six.D. Five.3. What can we learn from the text?A. The author could make fifty free throws within two minutes.B. The author’s hard work brought a good result in the game.C. The author’s coach didn’t know the author had t he talent.D. Royal High School team was stronger than the author’s team.BHaving attended a first-aid-at-work course, Brian Cracknell soon found himself responding to two emergencies in a week.Brian was at work in his bus company office in Weston-super-Mare on May 18 when a colleague called to say there had been an accident at a bus stop nearby. He ran to the scene with his first aid kit and found an old lady sitting at the bus stop. When boarding (上车) the bus, the old lady painfully hurt her left leg, which was bleeding.He introduced himself, and asked some basic first aid questions to get a general picture of her health. Then he put his jacket on the floor, using the lady’s coat as a pillow, and gently laid her down so he could raise her legs and apply some bandages. Bryan said, “At first I was a bit worried about remembering to do all the right things. I made sure she was as comfortable as possible.” When he had nearly finished, he told the lady that she was his first one. The lady said he was doing a great job.Just a week later, Brian got a second call-out to help another lady at a bus stop.Having used his new skills twice within a short time, Brian has decided to sign up as a Red Cross fir st aid volunteer. As he said, “I just thought: what’sthe point in having this knowledge if I’m not going to use it? It felt really good to help those people. I was so proud of myself afterwards, so I’m going to join the Red Cross.”4. Before helping the old lady, Brian ..A. had given first aid on many peopleB. was thought to be a skilled first aid volunteerC. knew nothing about how to handle an emergencyD. had learnt first aid skills but hadn’t put them into practice5. What was wrong with the old lady when Brian arrived?A. Her legs were bleeding after being injured.B. Her left leg was injured and was bleeding.C. She fell onto the ground and hurt her head.D. She was knocked down by a bus and lost her legs.6. What does Paragraph 3 mainly tell us?A. The reason why Brian was asked to help.B. Brian handled two emergencies in a week.C. How Brian gave first aid on the old lady.D. How the old lady reacted to Brian’s performance.7. After using his first aid skills twice, Brian .A. was discouraged and decided to give upB. realized he still had a lot of things to learnC. decided to give up his job for the Red CrossD. decided to help more people with his first aid skillsCA Japanese company has ordered all of its 2,700 employees to get identical (一模一样的) hairstyles. For men, the preferred cut is short on the sides and back but longer on top. For women, the company prefers a bob with a longer fringe (刘海) that can be swept to one side. This unusual request is its effort to help the country save energy.The spokeswoman for the Tokyo-based construction firm Maeda Corporation Chizuru Inoue explained, “Our company is eager to protect the environment and we encourage our staff to carry out many environment-friendly actions.”She added, “We are not sure of the data yet, but we believe if people have short hair, they do not need to use their hair driers for so long and they will use less water. If all our employees do this, then we m ay save a lot of power.” Some staff are confused about which style they must have and have been asking which salons (美容院) give the best cut.The energy-saving move is part of a national campaign to reduce energy consumption (消耗) following the Fukushima nuclear disaster. Japan has been struggling to produce enough electricity since the tsunami ended production at the Fukushima plants. The disaster resulted in a review of the country’s energy policy that now meansless than a quarter of its remaining nuclear plants are in use. The government has talked about a move away from nuclear energy towards more sustainable (能持续的) technologies. Many government departments have taken measures to save power, including a reduced use of air conditioning in offices and schools. Many employees cannot turn down the air conditioner below 27 degrees in the hottest summer months.8. The company gave the order of hairstyles to .A. keep the company offices cleanB. show a good image of the businessC. make the staff look tidy and prettyD. help reduce energy consumption9. After the Fukushima nuclear disaster, Japan .A. is building more nuclear plantsB. has to rethink its energy policyC. limits people’s daily energy useD. decides to close old nuclear plants10. It can be inferred from the text that .A. nuclear power forms 1/4 of Japan’s energyB. nuclear plants will soon disappear in JapanC. the company is not alone in saving energyD. other companies have copied the “hairstyle order”11. What would be the best title for the text?A. Sustainable power is welcome in JapanB. Nuclear disaster brings bad resultsC. Short hairstyles may help save energyD. Japan is lacking in energyDIt turns out that nodding off (睡觉) in class may not be a bad idea, as a new study has shown that going to sleep shortly after learning new materials is the best way to remember them.According to US lead author Jessica Payne, a psychologist at the University of NotreDame in Indiana, nodding off after learning something new is like “telling” the sleeping brain what to remember. Along with colleagues, she studied 207 students who slept for at least six hours per night. Students were randomly (随机地) assigned to study something declarative (陈述的) at 9 am or 9 pm, and returned for testing 30 minutes, 12 hours or 24 hours later.Declarative memory refers to the ability to consciously (有意识地) remember facts and events, and can be divided into memory for events and memory for facts about the world. People use both types of memory every day — remembering where they parked today or learning how a colleague prefers to be called.At the 12-hour retest, memory was better following a night of sleep compared to a day of wakefulness (不眠). At the 24-hour retest, with allstudents having received both a full night of sleep and a full day of wakefulness,students’ memories were greater when sleep happened shortly after learning rather than following a full day of wakefulness.“Our study confirms that sleeping directly after learning something new is bene ficial for memory. What’s special for this study is that we have shown sleep’s influence on declarative memory,” Payne said. “Since we found that sleeping soon after learning benefited both types of memory, this means that it would be a good thing to go over any information you need to remember before going to bed. In some sense, you may be telling the sleeping brain what to consolidate (巩固).”12. What’s the main idea of the text?A. How to improve one’s memory is a difficult problem.B. Nodding off after learning something new helps remember it.C. Students should nod off in class after they learn new materials.D. The best way to remember new materials is to nod off in class.13. We can learn about the study that .A. Jessica Payne and her colleagues were testedB. 207 students were eager to take the testC. it tested students studying something declarativeD. people’s memory can be divided into two types14. Paragraph 4 mainly tells us .A. students’ memories are the best if they sleep for 12 hours each dayB. sleeping for 24 hours contributes to improving students’ memoriesC. a full day of wakefulness can make students’ memories improved greatlyD. students’ memories are better when sleep happens shortly after learning15. What suggestion does Payne give in the last paragraph?A. The more you sleep, the more you remember.B. Go over what you want to remember before going to bed.C. Teachers should allow their students to sleep in class.D. If you want to improve your memory, you should sleep.第二节（共5小题；每小题2分，满分10分）根据短文内容，从短文后的选项中选出能填入空白处的最佳选项。

高二数学练习1.在△ABC 中，若19,2,3===c b a ，则C ∠等于A ．030B ．045C ．090D ．01202设n S 是等差数列{}n a 的前n 项和，5283()S a a =+,则53a a 的值为 A ． 56B ．13C ．35D ．163.“p ∨q 为真”是“⌝p 为假”的A ．充分不必要条件.B ．必要不充分条件C ．充要条件D ．既不充分也不必要条件4.若等比数列{}n a 满足：354321=++++a a a a a ，122524232221=++++a a a a a ，则54321a a a a a +-+-的值是A ． 3B ．3-C ．4D ．25.对于实数,,a b c ，下列命题中正确的个数为①,a b ac bc >>若则；②22,;ac bc a b >>若则③0,;a b c a b c a c b >>>>--若则④11,0a b a b a b>>>>若，则 A. 1个 B. 2个 C.3个 D. 4个 6.一元二次方程2210,(0)ax x a ++=≠有一个正根和一个负根的充分不必要条件是A ． 0a <B ． 0a >C ． 1a <-D ． 1a >7.已知实数,x y 满足121y y x x y m ≥⎧⎪≤-⎨⎪+≤⎩，如果目标函数z x y =-的最小值是1-，那么此目标函数的最大值是A ．1B ．2C ．3D ．58.若ABC ∆的内角,,A B C 所对的边,,a b c 满足22()4a b c +-=，且060C =，则a b +的最小值为A ．43B．8- CD9.已知命题2p :x R,mx 10∃∈+≤,命题2q :x R,x mx 10∀∈++>,若p q ∨为假命题,则实数m 的取值范围为 A ．m 2≥ B ．m 2≤- C ． m 2≤- 或m 2≥ D ．2m 2-≤≤10.等差数列{}n a 的前n 项和记为n S ，若2610a a a ++为一个确定的常数，则下列各数中可以用这个常数表示的是 A.10S B.11S C.12S D.13S 11.两个等差数列}{n a 和}{n b ,其前n 项和分别为n n T S ,,且,327++=n n T S n n 则157202b b a a ++等于 A.49 B. 837 C. 1479 D. 2414912.若不等式2229t t a t t ++≤≤在t ∈(0，2]上恒成立，则实数a 的取值范围是⎥⎦⎤⎢⎣⎡1,61.A ⎥⎦⎤⎢⎣⎡1,132.B ⎥⎦⎤⎢⎣⎡1314,132.C ⎥⎦⎤⎢⎣⎡22,61.D13.在△ABC 中,BC=5，sinC=2sinA,则AB 的长为14.若数列}{n a 为等差数列，且12031581=++a a a ，则1092a a -的值等于 . 15.存在实数x ，使得0342<+-b bx x 成立，则b 的取值范围是______ 16.若)2,0(∈x ，则xx y -+=241的最小值是17.设三角形ABC 的内角,,A B C 的对边分别为,,a b c ，4,4sin a c A B === （1）求b 边的长；（2）求角C 的大小；（3）求三角形ABC 的面积.18.设p :实数x 满足22430x ax a -+<,命题:q 实数x 满足2260,280.x x x x ⎧--≤⎪⎨+->⎪⎩（1）若1,a =且p q ∧为真，求实数x 的取值范围； （2）若p ⌝是⌝q 的充分不必要条件,求实数a 的取值范围.19.设等差数列{}n a 的前n 项和为n S ，等比数列{}n b 的前n 项和为n T ，已知0>n b)(5,,1),(23533211b T S a b a b a N n +==+==∈+．（Ⅰ）求数列{}n a 、{}n b 的通项公式； （Ⅱ）求和：1322211++++n n n T T b T T bT T b ． 20.某企业有员工1000名，平均每人每年创造利润10万元。

章末综合测评(一) 解三角形满分：150分 时间：120分钟一、选择题(本大题共12小题，每小题5分，满分60分．在每小题给出的四个选项中，只有一项是符合题目要求的)1．在△ABC 中，a ＝k ，b ＝3k (k >0)，A ＝45°，则满足条件的三角形有( ) A ．0个 B ．1个 C ．2个D ．无数个A [由正弦定理得a sin A ＝bsin B ，所以sin B ＝b sin A a ＝62>1，即sin B >1，这是不成立的．所以没有满足此条件的三角形．]2．已知三角形三边之比为5∶7∶8，则最大角与最小角的和为( ) A ．90° B ．120° C ．135°D ．150°B [设最小边为5，则三角形的三边分别为5，7，8，设边长为7的边对应的角为θ，则由余弦定理可得49＝25＋64－80cos θ，解得cos θ＝12，∴θ＝60°.则最大角与最小角的和为180°－60°＝120°.]3．在△ABC 中，A ＝π3，BC ＝3，AB ＝6，则C ＝( ) A ．π4或3π4 B ．3π4 C ．π4D ．π6C [由BC sin A ＝AB sin C ，得sin C ＝22. ∵BC ＝3，AB ＝6，∴A >C ， 则C 为锐角，故C ＝π4.]4．在△ABC 中，a ＝15，b ＝20，A ＝30°，则cos B ＝( )A ．±53 B ．23 C ．－53D ．53A [因为a sin A ＝b sinB ，所以15sin 30°＝20sin B ，解得sin B ＝23.因为b >a ，所以B >A ，故B 有两解，所以cos B ＝±53.]5．在△ABC 中，已知(b ＋c )∶(c ＋a )∶(a ＋b )＝4∶5∶6，则sin A ∶sin B ∶sin C 等于( )A ．6∶5∶4B ．7∶5∶3C ．3∶5∶7D ．4∶5∶6B [∵(b ＋c )∶(c ＋a )∶(a ＋b )＝4∶5∶6， ∴b ＋c 4＝c ＋a 5＝a ＋b 6.令b ＋c 4＝c ＋a 5＝a ＋b6＝k (k >0)，则⎩⎨⎧b ＋c ＝4k ，c ＋a ＝5k ，a ＋b ＝6k ，解得⎩⎪⎨⎪⎧a ＝72k ，b ＝52k ，c ＝32k .∴sin A ∶sin B ∶sin C ＝a ∶b ∶c ＝7∶5∶3.]6．在△ABC 中，a ，b ，c 分别为A ，B ，C 的对边，如果2b ＝a ＋c ，B ＝30°，△ABC 的面积为32，那么b 等于( )A ．1＋32B ．1＋ 3C ．2＋22D ．2 3B [∵S △ABC ＝12ac sin B ，∴ac ＝6.又∵b 2＝a 2＋c 2－2ac cos B＝(a ＋c )2－2ac －2ac ·cos 30°＝4b 2－12－63， ∴b 2＝4＋23，∴b ＝1＋ 3.]7．已知△ABC 中，sin A ∶sin B ∶sin C ＝k ∶(k ＋1)∶2k ，则k 的取值范围是( )A ．(2，＋∞)B ．(－∞，0)C ．⎝ ⎛⎭⎪⎫－12，0D ．⎝ ⎛⎭⎪⎫12，＋∞D [由正弦定理得：a ＝mk ，b ＝m (k ＋1)，c ＝2mk ，(m >0)， ∵⎩⎨⎧a ＋b >c ，a ＋c >b ，即⎩⎨⎧m （2k ＋1）>2mk ，3mk >m （k ＋1）， ∴k >12.]8．在△ABC 中，角A ，B ，C 的对边分别为a ，b ，c ，且sin 2A 2＝c －b2c ，则△ABC 的形状为( )A ．等边三角形B ．直角三角形C ．等腰三角形D ．等腰直角三角形B [由已知可得1－cos A 2＝12－b 2c ，即cos A ＝bc ，b ＝c cos A .法一：由余弦定理得cos A ＝b 2＋c 2－a 22bc ，则b ＝c ·b 2＋c 2－a 22bc ， 所以c 2＝a 2＋b 2，由此知△ABC 为直角三角形． 法二：由正弦定理，得sin B ＝sin C cos A . 在△ABC 中，sin B ＝sin(A ＋C )，从而有sin A cos C ＋cos A sin C ＝sin C cos A ， 即sin A cos C ＝0.在△ABC 中，sin A ≠0，所以cos C ＝0.由此得C ＝π2，故△ABC 为直角三角形．]9．已知圆的半径为4，a ，b ，c 为该圆的内接三角形的三边，若abc ＝162，则三角形的面积为( )A ．2 2B ．8 2C ． 2D ．22C [∵a sin A ＝b sin B ＝c sin C＝2R ＝8， ∴sin C ＝c 8，∴S △ABC ＝12ab sin C ＝abc 16＝16216＝ 2.]10．在△ABC 中，三边长分别为a －2，a ，a ＋2，最大角的正弦值为32，则这个三角形的面积为( )A ．154B ．1534C ．2134D ．3534B [∵三边不等，∴最大角大于60°.设最大角为α，故α所对的边长为a ＋2，∵sin α＝32，∴α＝120°.由余弦定理得(a ＋2)2＝(a －2)2＋a 2＋a (a －2)，即a 2＝5a ，故a ＝5，故三边长为3，5，7，S △ABC ＝12×3×5×sin 120°＝1534.]11．如图，海平面上的甲船位于中心O 的南偏西30°，与O 相距15海里的C 处．现甲船以35海里/小时的速度沿直线CB 去营救位于中心O 正东方向25海里的B 处的乙船，则甲船到达B 处需要的时间为( )A ．12小时 B ．1小时 C ．32小时D ．2小时B [在△OBC 中，由余弦定理，得CB 2＝CO 2＋OB 2－2CO ·OB cos 120°＝152＋252＋15×25＝352，因此CB ＝35，3535＝1(小时)，因此甲船到达B 处需要的时间为1小时．]12．如图，在△ABC 中，D 是边AC 上的点，且AB ＝AD ，2AB ＝3BD ，BC ＝2BD ，则sin C 的值为()A ．33B ．36C ．63D ．66D [设BD ＝a ，则BC ＝2a ，AB ＝AD ＝32a . 在△ABD 中，由余弦定理，得cos A ＝AB 2＋AD 2－BD 22AB ·AD ＝⎝ ⎛⎭⎪⎫32a 2＋⎝ ⎛⎭⎪⎫32a 2－a 22×32a ·32a ＝13.又∵A 为△ABC 的内角，∴sin A ＝223. 在△ABC 中，由正弦定理得，BC sin A ＝ABsin C . ∴sin C ＝AB BC ·sin A ＝32a 2a ·223＝66.]二、填空题(本大题共4小题，每小题5分，共20分，把答案填在题中横线上)13．已知△ABC 为钝角三角形，且C 为钝角，则a 2＋b 2与c 2的大小关系为________．a 2＋b 2<c 2[∵cos C ＝a 2＋b 2－c 22ab ，且C 为钝角，∴cos C <0，∴a 2＋b 2－c 2<0，故a 2＋b 2<c 2.]14．设△ABC 的内角A ，B ，C 所对边的长分别为a ，b ，c .若b ＋c ＝2a ，3sin A ＝5sin B ，则角C ＝________．2π3 [由3sin A ＝5sin B ，得3a ＝5b .又因为b ＋c ＝2a ， 所以a ＝53b ，c ＝73b ，所以cos C ＝a 2＋b 2－c 22ab ＝⎝ ⎛⎭⎪⎫53b 2＋b 2－⎝ ⎛⎭⎪⎫73b 22×53b ×b ＝－12.因为C ∈(0，π)，所以C ＝2π3.]15．在锐角△ABC 中，BC ＝1，B ＝2A ，则ACcos A 的值等于________，AC 的取值范围为________．2 (2，3) [设A ＝θ⇒B ＝2θ. 由正弦定理得AC sin 2θ＝BCsin θ， ∴AC 2cos θ＝1⇒ACcos θ＝2.由锐角△ABC 得0°<2θ<90°⇒0°<θ<45°. 又0°<180°－3θ<90°⇒30°<θ<60°， 故30°<θ<45°⇒22<cos θ<32， ∴AC ＝2cos θ∈(2，3)．]16．在锐角三角形ABC 中，角A ，B ，C 的对边分别是a ，b ，c ，若b a ＋ab ＝6cos C ，则tan C tan A ＋tan Ctan B ＝________．4 [∵b a ＋ab ＝6cos C ， ∴a 2＋b 2ab ＝6·a 2＋b 2－c 22ab ， ∴2a 2＋2b 2－2c 2＝c 2，又tan C tan A ＋tan C tan B ＝sin C cos A sin A cos C ＋sin C cos B sin B cos C ＝sin C （sin B cos A ＋cos B sin A ）sin A sin B cos C ＝sin C sin （B ＋A ）sin A sin B cos C ＝sin 2C sin A sin B cos C ＝c 2ab cos C ＝c 2ab a 2＋b 2－c 22ab＝2c 2a 2＋b 2－c 2＝4.]三、解答题(本大题共6小题，共70分．解答应写出文字说明、证明过程或演算步骤)17．(本小题满分10分)△ABC的三个内角A，B，C所对的边分别为a，b，c，a sin A sin B＋b cos2A＝2a.(1)求b a；(2)若c2＝b2＋3a2，求B.[解](1)由正弦定理得，sin2A sin B＋sin B cos2A＝2sin A，即sin B(sin2A＋cos2A)＝2sin A.故sin B＝2sin A，所以ba＝ 2.(2)由余弦定理和c2＝b2＋3a2，得cos B＝（1＋3）a2c.由(1)知b2＝2a2，故c2＝(2＋3)a2.可得cos2B＝12，又cos B>0，故cos B＝22，所以B＝45°.18．(本小题满分12分)已知△ABC的内角A，B，C所对的边分别为a，b，c，且a＝2，cos B＝3 5.(1)若b＝4，求sin A的值；(2)若△ABC的面积S△ABC＝4，求b，c的值．[解](1)∵cos B＝35>0，且0<B<π，∴sin B＝1－cos2B＝4 5.由正弦定理得asin A＝bsin B，sin A＝a sin Bb＝2×454＝25.(2)∵S △ABC ＝12ac sin B ＝4， ∴12×2×c ×45＝4，∴c ＝5.由余弦定理得b 2＝a 2＋c 2－2ac cos B ＝22＋52－2×2×5×35＝17，∴b ＝17. 19．(本小题满分12分)已知A ，B ，C 为△ABC 的三个内角，其所对的边分别为a ，b ，c ，且2cos 2A2＋cos A ＝0.(1)求角A 的值；(2)若a ＝23，b ＝2，求c 的值． [解] (1)∵cos A ＝2cos 2A2－1， ∴2cos 2A2＝cos A ＋1.又2cos 2A2＋cos A ＝0，∴2cos A ＋1＝0， ∴cos A ＝－12，∴A ＝120°.(2)由余弦定理知a 2＝b 2＋c 2－2bc cos A ， 又a ＝23，b ＝2，cos A ＝－12， ∴(23)2＝22＋c 2－2×2×c ×⎝ ⎛⎭⎪⎫－12，化简，得c 2＋2c －8＝0， 解得c ＝2或c ＝－4(舍去)．20．(本小题满分12分)某观测站在城A 南偏西20°方向的C 处，由城A 出发的一条公路，走向是南偏东40°，在C 处测得公路距C 处31千米的B 处有一人正沿公路向城A 走去，走了20千米后到达D 处，此时C 、D 间的距离为21千米，问这人还要走多少千米可到达城A ?[解] 如图所示，设∠ACD ＝α，∠CDB ＝β. 在△CBD 中，由余弦定理得 cos β＝BD 2＋CD 2－CB 22BD ·CD＝202＋212－3122×20×21＝－17，∴sin β＝437.而sin α＝sin(β－60°)＝sin βcos 60°－sin 60°cos β＝437×12＋32×17＝5314.在△ACD 中，21sin 60°＝ADsin α，∴AD ＝21×sin αsin 60°＝15(千米)．所以这人还要再走15千米可到达城A .21．(本小题满分12分)在△ABC 中，角A ，B ，C 所对的边分别为a ，b ，c ，cos 2C ＋22cos C ＋2＝0.(1)求角C 的大小；(2)若b ＝2a ，△ABC 的面积为22sin A sin B ，求sin A 及c 的值． [解] (1)∵cos 2C ＋22cos C ＋2＝0， ∴2cos 2C ＋22cos C ＋1＝0， 即(2cos C ＋1)2＝0， ∴cos C ＝－22. 又C ∈(0，π)，∴C ＝3π4.(2)∵c 2＝a 2＋b 2－2ab cos C ＝3a 2＋2a 2＝5a 2， ∴c ＝5a ，即sin C ＝5sin A ， ∴sin A ＝15sin C ＝1010. ∵S △ABC ＝12ab sin C ，且S △ABC ＝22sin A sin B ， ∴12ab sin C ＝22sin A sin B ，∴absin A sin B sin C ＝2，由正弦定理得 ⎝ ⎛⎭⎪⎫c sin C 2sin C ＝2，解得c ＝1. 22．(本小题满分12分)在△ABC 中，a ，b ，c 分别为内角A ，B ，C 所对的边，且满足sin A ＋3cos A ＝2.(1)求角A 的大小；(2)现给出三个条件：①a ＝2；②B ＝π4；③c ＝3b .试从中选出两个可以确定△ABC 的条件，写出你的方案并以此为依据求△ABC 的面积．(写出一种方案即可)[解] (1)依题意得2sin ⎝ ⎛⎭⎪⎫A ＋π3＝2， 即sin ⎝ ⎛⎭⎪⎫A ＋π3＝1，∵0<A <π，∴π3<A ＋π3<4π3，∴A ＋π3＝π2， ∴A ＝π6.(2)参考方案：选择①②.由正弦定理a sin A ＝b sin B ，得b ＝a sin Bsin A ＝2 2. ∵A ＋B ＋C ＝π，∴sin C ＝sin(A ＋B )＝sin A cos B ＋cos A sin B ＝2＋64，∴S △ABC ＝12ab sin C ＝12×2×22×2＋64＝3＋1.。