上海市复旦附中2016年高一上学期开学考英语试题
上海交通大学附属中学2021-2022学年高一上学期期中考试英语试卷
A.has been provedB.is provedC.has provedD.proven
1.With the newly bought knife, even the frozen meat ________. You will get what you pay for.
A.is easily cutB.is easy to be cutC.cuts easyD.cuts easily
A.withB.inC.ofD.by
23.They gave me some help in the beginning, but it didn’t ________to much.
A.addB.amountC.equalD.refer
24.The high cost is a major ________ of the new heating system.
C.haven't noticed; have been thinkingD.wasn't noticing; have been thinking
5.We were very interested in the topic our teacher assigned us today.________ even when the bell rang.
26.Don’t open the window! It’s ________cold outsபைடு நூலகம்de.
(通用版)高考英语一轮专题巡练 考点05 代词考点专练(含解析)-人教版高三全册英语试题
考点05 代词人称、物主、反身、指示和疑问代词1．Being a parent is not always easy, and being the parent of a child with special needs often carries with _______ extra stress.A．it B．themC．one D．him2．-- There is still a copy of the book in the library. Will you go and borrow ________ ?---No, I’d rather buy __________ on the internet.A．it; one B．one; oneC．one; it D．it; it3．Isn’t it amazing how the human body heals ____ after an injury?A．himself B．himC．itself D．it4．Those who smoke heavily should remind __________of health, the bad smell and the feelings of other people.A．theirs B．themC．themselves D．oneself5．The quality of education in this small school is better than ______ in some larger schools. A．that B．oneC．it D．this6．The information on the Internet gets around much more rapidly than in the newspaper.A．it B．thoseC．one D．that7．--- Silly me! I forget what my luggage looks like.--- What do you think of_____over there?A．the one B．it C．that D．this8．The English spoken in the United States is only slightly different fromspoken in England. A．which B．what C．that D．the one9．Professional skills are tools like a knife or a hammer. You need ______ to complete the things you want to do.A．it B．that C．them D．ones不定代词10．It was hard for him to learn English in a family, in which _____ of the parents spoke the language.A．none B．neither C．both D．each11．Jim sold most of his things.He has hardly___left in the house.A．anything B．everythingC．nothing D．something12．My brother would like to buy a good watch but _______ was available from that shop. A．nothing B．noneC．no one D．neither13．A study shows the students who are engaged in after-school activities are happier than_________who are not.A．ones B．thoseC．these D．them14．I think Mrs. Stark could be _______ between 50 and 60 years of age.A．anywhere B．anybodyC．anyhow D．anythingit用法15．I’d appreciate ________ if you could let me know in advance whether or not you will come. A．it B．youC．one D．this16．No matter where he is, he makes ________ a rule to go for a walk before breakfast. A．him B．thisC．that D．it17．How would you like ________ if you were watching your favorite TV program and someone came into the room and just shut it off without asking you?A．them B．oneC．those D．it18．This is not my story, nor ______ the whole story. My story plays out differently.A．is there B．there isC．is it D．it is19．I like this house with a beautiful garden in front, but I don’t have enough money tobuy__________.A．one B．it C．this D．that20．It has been proved______eating vegetables in childhood helps to protect you against serious illnesses in later life.A．if B．because C．when D．that21．The employment rate has continued to rise in big cities thanks to the efforts of the local govenments to increase.A．then B．those C．it D．that22．—Who’s that at the door?—is the milkman.A．He B．It C．This D．That1.【2020·全国新课标I】Data about the moon’s composition, such as how much ice and other treasures it contains, could help China decide whether 70． (it) plans for a future lunar (月球的) base are practical.2.【2020·全国新课标I改错】＂Not that way，＂ my mom tried to stop us but failed.3.【2020·全国新课标2改错】Best of luck with yours learning kung fu in China.4.【2020·全国新课标3改错】But he insists on us eating healthy food.5．〔2015·重庆高考真题〕The meeting will be held in September, but ______ knows the date for sure. A．everybody B．nobodyC．anybody D．somebody6．〔2012·四川省高考真题〕New technologies have made ____ possible to turn out new products faster and at a lower cost.A．that B．this C．one D．it7．〔2012·山东省高考真题〕It doesn’t matter ________ you pay by cash or credit card in this store. A．how B．whetherC．what D．why8．〔2012·全国高考真题〕Larry asks Bill and Peter to go on a picnic with him, butof them wants to, because they have work to do.A．either B．anyC．neither D．none9．〔2012·福建省高考真题〕一 Have you figured out how much the trip will cost?—$4,000, or like that.A．anything B．everything C．something D．nothing10．〔2014·山东〕Susan made______ clear to me that she wished to make a new life for herself.A．that B．thisC．it D．her11．〔2015·福建〕The research group produced two reports based on the survey, but _________ contained any useful suggestions.A．all B．noneC．either D．neither12．〔2011·陕西省高考真题〕-Would you get me a bar of chocolate from the kitchen,dear?-one?A．Other B．Every C．Another D．More13．〔2016·浙江省高考真题〕In many ways, the education system in the US is not very different from ________ in the UK.A．that B．thisC．one D．it1．〔2020·某某耀华中学高一月考〕The goods ________ from the Internet are cheaper than ________ we buy in shops.A．be bought; thatB．bought; thoseC．bought; thatD．were bought; the ones2．〔2020·江苏省扬州中学高三月考〕--- The Captain is definitely a touching movie, ______ adapted from a true story.--- Exactly. Captain Liu puts the safety of passengers in the first place.A．one that B．which C．the one D．one3．〔2020·某某理工大学附属中学高一月考〕We thought _______ strange that Tony Stark should come. A．that B．it C．this D．what4．〔2020·江苏省苏州实验中学高三月考〕In public places, improved child-care facilities will benefit ________ genders, not just women.A．both B．all C．either D．other5．〔2020·某某高三二模〕Though reasons for learning Chinese vary, a common was enthusiasm for the country’s rich culture.A．one B．it C．that D．this6．〔2020·江苏省高三二模〕See things in a negative light and you will find faults everywhere and problems where there are really _______.A．none B．no one C．something D．nothing7．〔2020·天水市第一中学高一开学考试〕I would appreciate ______ if you could give me a reply as soon as possible.A．you B．it C．that D．them8．〔2020·河北承德第一中学高一月考〕 ______ has been decided that the 2020 Tokyo Olympic Games will be put off to 2021.A．It B．What C．As D．Which9．〔2020·某某四中高二期中〕The exchange student from Kenya often shares with us the news of his country and ____of his neighbouring countries.A．those B．that C．it D．the one10．〔2020·某某高三月考〕—What do you think of the information from books?— It is easier to forget than ________ from life itself.A．the one B．itC．that D．those11．〔2020·汉中市龙岗学校高一期末〕The old couple preferred a flat in a small town to ______ in so large a city as Chongqing．A．this B．it C．that D．one12．〔2020·某某市静海区大邱庄中学高三月考〕--- Silly me! I forget what my luggage looks like. --- What do you think of_____over there?A．the one B．it C．that D．this13．〔2020·江苏省天一中学高三月考〕What Chinese lawmakers begin to consider is a proposed revision to the country’s Food Safety Law,_______ include s mandatory (强制性) labeling of GM food. A．one B．the one which C．one that D．the one that14．〔2020·江苏省南京师大附中高三一模〕If you see things in a negative light, you will find faults everywhere and problems where there are really ________．A．none B．someC．many D．nothing15．〔2019·江苏省淮阴中学高三月考〕More overseas students are returning home to make contributions to our country, a large proportion of _____ reported to have received first-class education. A．whom B．which C．them D．whoever16．〔2019·黑龙江省哈九中高三月考〕You are really ! I heard marvelous news that you set a new world record.A．anything B．nothing C．something D．everything17．〔2019·黑龙江省哈九中高三月考〕When you go by train, make sure you take an express,___________stops only at big stations.A．one B．one that C．that D．what18．〔2020·某某静海一中高三月考〕“Made in China 2025〞 aims to transform China from a product-making factory into a product-making power, ______________ driven by innovation and emphasizing quality over quantity.A．the one that B．one thatC．one D．the one19．〔2020·江苏省高三开学考试〕“Made in Ch ina 2025〞 initiative aims to transform China from a manufacturing giant into a world manufacturing power, ________ driven by innovation and emphasizes quality over quantity.A．the one that B．one that C．one D．the one20．〔2020·黑龙江省黑龙江实验中学高三开学考试〕He is rich and brave. __________ is no wonder he is popular with girls.A．There B．That C．It D．What21．〔2019·某某曹杨二中高一月考〕I had to buy _____ of these books because I didn’t know which one was the best.A．both B．none C．neither D．all22．〔2020·江西省高一期末〕—I wonder how often you will clean up your room by yourself.—________ other day.A．In B．Every C．For D．Each23．〔2020·黑龙江省哈尔滨三中高二期中〕People were dressed in elegant costumes, all of _____ suitable for a multicultural parade.A．which B．whom C．them D．who24．〔2019·某某市静海区独流中学高二月考〕Praise is like sunlight to the human spirit; without _______ we cannot flower and grow.A．them B．it C．that D．which25．〔2019·某某复旦附中高二期末〕The autobiography(自传) of the happiest man would not be a record of sensational or exciting experiences, but ________ composed of simple and plain incidents or routines.A．it is B．which is C．one that D．one考点练1．A【解析】考查代词的用法。
2022-2023学年人教物理高一上学期分层练习专题5 追击相遇问题含详解
专题5 追击相遇问题一、匀速与变速类追击相遇问题1．甲、乙两个质点沿同一直线运动，其中质点甲以3m/s的速度做匀速直线运动，质点乙做初速度为零的匀变速直线运动，它们的位置x随时间t的变化如图所示。
已知t=3s时，甲、乙图线的斜率相等。
下列判断正确的是（）A．最初的一段时间内，甲、乙的运动方向相反-B．t=3s时，乙的位置坐标为15.5mt=时，两车相遇C．10sD．乙经过原点的速度大小为/s2．（2022·重庆一中高一期末）无线蓝牙耳机可以在一定距离内与手机等设备实现无线连接。
已知无线连接的最远距离为10m，甲和乙两位同学做了一个有趣实验。
甲佩戴无线蓝牙耳机，乙携带手机检测，如图（a）所示，甲、乙同时分别沿两条平行相距6m的直线轨道向同一方向运动，甲做匀速直线运动，乙从静止开始先做匀加速直线运动，再做匀速直线运动，其速度v随时间t的关系如图（b）所示，则在运动过程中，手机检测到蓝牙耳机能被连接的总时间为（）A．4s B．9s C．13s D．17s3．（2022·湖南·高一期中）（多选）无线蓝牙耳机摆脱了线材束缚，可以在一定距离内与手机等设备实现无线连接。
为了研究在运动过程中无线连接的最远距离，甲和乙两位同学做了一个有趣的实验。
乙佩戴无线蓝牙耳机，甲携带手机检测，乙站在甲正前方10m处，二人同时沿同一直线向正前方运动，各自运动的v t-图像如图所示，结果手机检测到蓝牙耳机能被连接的时间为4s。
则下列判断正确的是（）A ．3s 时甲乙相距最远B ．6s 时甲乙相距最近C ．甲减速加速度大小为21m /sD ．5s 时两者距离为7.5m4．（2020·广东·化州市第一中学高一阶段练习）（多选）在平直公路上，自行车与同方向行驶的一辆汽车在t =0时同时经过某一个路标，它们的位移x （m ）随时间t （s ）变化的规律为：汽车为21104x t t =-，自行车为x =6t ，则下列说法正确的是（ ）A ．汽车做匀减速直线运动，且汽车经20s 停止B ．在t =8s 时两车速度相等，自行车第一次追上了汽车C ．在t =0至t =16s 时间内，两车间距离先增大后减小D ．当自行车追上汽车时，它们距路标96m5．甲、乙两车在平直公路上比赛，某一时刻，乙车在甲车前方111m L =处，乙车速度60m/s v =乙，甲车速度50m/s v =甲，此时乙车离终点线尚有2600m L =，如图所示。
上海国际学校招生计划
上海国际学校招生计划国际学校(英语：International School)是提供中等或以下程度的教育，并拥有相当比例的外籍学生，而且实施外国学制的学校。
以下是关于上海国际学校招生计划，希望大家认真阅读!1闵行区民办德闳学校学校性质：面向中国学生的十二年制国际学校学校地址：闵行区曙光路1935号学校概况：学校在实施上海市二期课改课程的基础上，通过国外教育专家的指导，为中国学生提供更多融合了国际先进教育理念和实践经验的国际课程。
招生情况：2017年开始招生2星河湾中学学校性质：完全中学学校概况：学校在区政府、颛桥镇领导和市委的大力支持下建立，旨在建成“主打双语特色，走国际化道路”的完全中学。
招生情况：预计2017年开始招生，预计学生数超过2000人的中学，规划初中24个班(每年级6个，每班40人)、高中24个班(每年级8个，每班45人)3人大附中上海分校学校性质：小学到高中十二年寄宿制学校地址：奉贤海湾旅游区学校概况：中国人民大学附属中学创办于1950年，是一所享誉中外的著名中学。
人大附中上海分校是一所寄宿制民办学校，包括小学、初中和高中，占地规模为130亩，计划于2016年下半年开工建造。
人大附中是“三高”足球训练基地，培养“道德水平高、文化素质高、运动水平高”的足球后备人才。
为此，未来的上海分校也将一并引入“三高”项目，奉贤区提供两片专业足球场用于日常训练，使之成为奉贤区乃至上海市校园足球训练和培训中心。
据悉，上海分校将采用上海教材，但在办学理念和学校管理等方面全面引入人大附中的优质资源。
学校资产归奉贤区政府，由区教育行政部门管理。
招生情况：拟定2018年开始招生，学校拟定为48个班，其中，小学20个班，初中16个班，高中12个班。
面向全市招生，在同等条件下优先招收奉贤区学生。
4位育中学(闵行校区)学校性质：初中学校地址：莘庄工业区(中春路东面，颛盛路南面)招生情况：预计将于2018年竣工，招生计划暂时还未出5嘉定中科院上海实验学校开学时间：2017年/2018年学校概况：中国科学院上海实验学校是经中国科学院批准，隶属于嘉定区教育局的公办学校，学校地处嘉定菊园新区，设一个初中部和一个小学部。
上海师范大学附属中学2022-2023学年高一上学期期中考试英语试卷
上海师范大学附属中学 2022-2023学年高一上学期期中考试英语试卷学校:___________姓名：___________班级：___________考号：___________一、用单词的适当形式完成短文Directions: After reading the passage below, fill in the blanks to make the passage coherent and grammatically correct. For the blanks with a given word, fill in each blank with the proper form of the given word; for the other blanks, use one word that best fits each blank.Charlotte Whitehead was born in England in 1843, and moved to Montreal, Canada at the age of 5 with her family. While ___1___ (take) care of her ill elder sister throughout the years, Charlotte discovered she had an interest in medicine. At 18 she married and started a family. Several years later, Charlotte said she wanted to be a doctor. Her husband supported her decision. Unfortunately, Canadian medical schools did not accept women students at the time. Therefore, Charlotte went to the United States ___2___(study) medicine at the Women’s Medical College in Philadelphia. ___3___ took her five years to earn her medical degree.Upon graduation, Charlotte returned to Montreal and set up a private clinic. Three years later, she moved to Winnipeg, Manitoba, ___4___ she was once again a busy doctor. Many of her patients were from the nearby timber and railway camps. Charlotte found herself operating on damaged limbs and setting broken bones, ___5___ delivering all the babies in the area.But Charlotte had been practising without a license. She had applied for a doctor’s license in both Montreal and Winnipeg, ___6___ was refused. The Manitoba College of Medicines, ___7___ all-male organization, wanted her to complete her studies at a Canadian medical college! Charlotte refused to leave her patients to spend time studying ___8___ she already knew. So in 1887, she appealed to the Manitoba Legislature to issue a license to her but they, too, refused. Charlotte continued to practise ___9___ a license until 1912. She died four years later at the age of 73.In 1993, 77 years after her death, a medical license was issued to Charlotte. This decision ____10____(make) by the Manitoba Legislature to honour “this courageous and pioneering woman”.Directions: After reading the passage below, fill in the blanks to make the passage coherentand grammatically correct. For the blanks with a given word, fill in each blank with the proper form of the given word; for the other blanks, use one word that best fits each blank.My Volunteer Experience in GhanaIt is not quite six in the morning. I am half asleep in a van that in the United States would seat maybe six people. Here in Ghana though, its insides have been replaced with wooden benches that can accommodate twelve of us. I use the term “accommodate” loosely; we are squeezed so tight in together ___11___ sometimes my arm goes numb (麻木的).This goes on for two hours every morning, from Monday to Friday, so that I ___12___ get to the slum town school where I teach.___13___ (arrive) at the destination always fills me with fear. The school originally functioned as a daycare type service free to the children of single mothers. ___14___ the children got older, the coordinator took it upon herself to set up a space to teach them. Into this tiny space they squeeze anywhere between 80 and 120 children under the age of six. It is literally so full of children that you have to move them out of the way by shoving (推开) them with your feet ___15___ (avoid) stepping on them.Within days, I have already learned most of their names. There are the troublemakers who spend their time ___16___(hit) one another and causing chaos in hopes that their negative behavior will attract attention to themselves. All of my child psychology classes tell me to ignore the behavior ___17___ responding to it may reinforce the behavior.Then there are the ones that make your heart bleed. The five-year-old girl who ___18___ (sit) in the front row, is ready to learn whatever knowledge we can throw her way. She learns whatever we teach her, at a pace ___19___ takes my breath away. She far outpaces any Western five-year-old I’ve ever worked with, and she’s learning in English when her family only knows and speaks Twi, a local language.I look at this decrepit (破旧的) classroom and these children in their ragged clothes, and think of ____20____ different this is from the American classrooms I’ve known.二、选用适当的单词或短语补全短文Directions: Complete the following passage by using the words in the box. Each word can only be used once. Note that there is one word more than you need.The digital revolution has had a huge impact on every aspect of our society. For content creators, life looks nothing like it did just a couple of decades ago. Back then, all you had to do is to take the tape off the camera and put it on your ___21___ player. At most, you had to read an article to understand the difference between HS and Betamax. Video was only consumed in two ways: television or cinema. Nowadays, you’ll probably run out of fingers ___22___ how many video formats (格式) there are out there, let alone the number of platforms and devices on which video is played. Understanding what each video format is and does is no child’s play. In fact, it is a “must” if you want your work to look the best regardless of where and how it is played. If, for example, you want to ___23___ a video website, the type of file you’ll need to use will most likely to be different from the one you’ll have to create for your social media accounts. In order to make this whole process a lot easier, we have put together a list of the most common video formats and which ones you should use for each purpose.So what’s the best video format?___24___, there isn’t a cure-all when it comes to video sharing. The good thing is that it’s actually really easy to ___25___ between video formats. If you do so, be sure to always save the original file in order to avoid ___26___ of image quality due to file compression(压缩).Here are three of the most common video uses and the ___27___ formats for each of them:●Social media: MP4 is supported by Facebook, Twitter, Instagram, IGTV, and YouTube. Facebook also___28___ MOV, while YouTube accepts most video formats.●TV and computers: A VI, MOV, and MKV ___29___ the highest quality image which makes them great options for big screen streaming.●Website: WEBM and HTML5 are the video formats meant for your account. However, using tools such as Wix Video will allow you to work with almost any file ____30____, as well as offer you much more control over your work.Directions: Complete the following passage by using the words in the box. Each word canonly be used once. Note that there is one word more than you need.The value of upcyclingRecycling is a well-known idea that refers to reusing waste materials in any way possible. But what about “upcycling”? It’s a new word, even though it’s something that has been going on since human ____31____ began. It means reusing waste materials so that they have greater value. Throughout history, people have always done creative things with “trash”. For example, they’ve used straw and dead leaves to make roofs, skin from dead animals to make leather goods, and wood from ____32____ trees to make boats. So why is there a new word for it now?One answer to this question is that we reuse fewer and fewer things, and so have become a (n) “____33____” society. This has raised huge questions about waste: Where can we____34____ it all? Will it pollute the environment? Could it be ____35____ to our health? The evidence is everywhere—even in the Pacific Ocean, where billions of bits of broken____36____ float near the surface. Fish eat them, and then we eat the fish.So upcyclers have adopted this new word to focus people’s attention on how waste cannot simply be reused, but be reused ____37____. In fact, upcyclers don’t like the idea of waste and prefer to call it an “asset”, something of value. Nowadays, there are lots of organizations that ____38____ products with upcycled material. Some artists and designers have upcycled things like denim from old jeans to make rugs, and wood from old houses to make furniture. Others have even used candy wrappers to make handbags! Sometimes they’ll add a(n) ____39____ element to their products, such as a beautiful mosaic (马赛克) made with broken dishes. With a (n) _____40_____ supply of “assets”, it seems that upcycling has a great future.三、完形填空Deliverance(解救) for Delivery Drivers?ZOOOOOOOOM A gust of air, the hum of spinning wheels, and a flash of color whizzing past. I’m sure that we have all stepped into the crosswalk, at one time or another, and experienced this sudden, heart-stopping ___41___. I am, of course, referring to deliverydrivers ___42___ through crossroads to deliver takeaway food.Over the past few years, delivery drivers have been criticized, fined, and even ___43___ for breaking traffic laws across China. Delivery drivers often ___44___ red lights, disregard speed limits, and pay little attention to pedestrian crossings. But they also often pay the___45___ price — each year, it seems like fatal injuries involving delivery drivers are increasing. ___46___, since delivery drivers are usually poorly paid, how much blame do they really deserve for running against the clock?Well, it ___47___ to be a complicated issue. Recently an article titled “Delivery Drivers, Stuck in the System” was shared widely on social media. According to the article, ___48___ between delivery services Ele.me and Meituan has caused each app to take increasingly extreme measures to out-do(超过) the other. This has included the shortening of delivery time windows. It has been ___49___, in part, through a combination of rewards for early delivery and severe punishments for late delivery. Over time, the platforms have only increased these ___50___ on drivers. As a result, drivers find themselves in a dilemma(左右两难的困境)：break traffic laws or, in some cases, lose hundreds of yuan in earnings for late deliveries.According to surveys and reactions online, ___51___ care much whether their deliveries are a few minutes late, while most believe that it is not worth risking someone’s life for. In response, Ele.me and Meituan have added a combination of features to ___52___ delivery times for slow elevators or bad weather conditions.Although these new ___53___ cannot solve the underlying problems of intense competition or the inherent dangers of driving, they should, in some measure, start reducing some of the pressure on drivers. In the meantime, as ___54___, we can help to make life for delivery riders safer and easier. ___55___, be a considerate person and don’t leave them waiting for too long. If you see a food delivery guy rushing toward you on an escalator or into a door, consider standing back and letting him go first. After all, they are putting their lives on the line just to deliver our food.41．A．fright B．delight C．pain D．sorrow 42．A．running B．walking C．rushing D．wandering 43．A．arrested B．fired C．blamed D．praised 44．A．destroy B．ignore C．observe D．interrupt 45．A．everlasting B．dramatic C．slight D．final 46．A．However B．Namely C．Indeed D．Surprisingly47．A．decides B．refuses C．turns out D．takes turns 48．A．comparison B．contrast C．competition D．cooperation 49．A．adapted B．achieved C．adopted D．assumed 50．A．motivations B．advantages C．pressures D．skills 51．A．some B．few C．many D．we 52．A．reduce B．expand C．narrow D．extend 53．A．actions B．behaviors C．features D．conditions 54．A．humans B．consumers C．friends D．taxpayers 55．A．In one word B．As a result C．For example D．On the contrary四、阅读理解Who is a genius? This question has greatly interested humankind for centuries.Let’s state clearly: Einstein was a genius. His face is almost the international symbol for genius. But we want to go beyond one man and explore the nature of genius itself. Why is it that some people are so much more intelligent or creative than the rest of us? And who are they?In the sciences and arts, those praised as geniuses were most often white men, of European origin. Perhaps this is not a surprise. It’s said that history is written by the victors, and those victors set the standards for admission to the genius club. When contributions were made by geniuses outside the club—women, or people of a different color or belief—they were unacknowledged and rejected by others.A study recently published by Science found that as young as age six, girls are less likely than boys to say that members of their gender(性别)are “really, really smart.” Even worse, the study found that girls act on that belief: Around age six they start to avoid activities said to be for children who are “really, really smart.” Can our planet afford to have any great thinkers become discouraged and give up? It doesn’t take a genius to know the answer: absolutely not.Here’s the good news. In a wired world with constant global communication, we’re all positioned to see flashes of genius wherever they appear. And the more we look, the more we will see that social factors(因素)like gender, race, and class do not determine the appearance of genius. As a writer says, future geniuses come from those with “intelligence, creativity, perseverance(毅力), and simple good fortune, who are able to change the world.”56．What does the author think of victors’ standards for joining the genius club? A．They’re unfair.B．They’re conservative.C．They’re objective.D．They’re strict.57．What can we infer about girls from the study in Science?A．They think themselves smart.B．They look up to great thinkers.C．They see gender differences earlier than boys.D．They are likely to be influenced by social beliefs58．Why are more geniuses known to the public?A．Improved global communication.B．Less discrimination against women.C．Acceptance of victors’ concepts.D．Changes in people’s social positions.59．What is the best title for the text?A．Geniuses Think Alike B．Genius Takes Many Forms C．Genius and Intelligence D．Genius and LuckFor many travelers these days, finding a way to make travel more meaningful has become a top priority. They are increasingly pursuing journeys that include time to volunteer, work on a conservation project, protect endangered animals or to do other activities that allow for giving back while exploring a new part of the world. Here are some opportunities that combine travel with doing good deeds.The AzoresThe trips offer the opportunity to work on carefully selected projects with famous scientists while also enjoying meetings with some of the world’s most attractive sea creatures in water. It allows participants to join a research team in the Azores that’s studying the animals to support future protection of the species. Advanced diving experience is required.Red Sea, EgyptCombining an opportunity for world-class diving with being part of an important research effort conducted by Hurghada Environmental Protection and Conservation Association. The seven-night trip includes learning how to gather water data with the latest scientific equipment and conducting onboard and underwater experiments.AustraliaExplore Queensland, Australia, while also supporting a registered charity that rescues farm animals from abuse, neglect, and situations where they’re in danger. Participants help with daily cleaning of barns (牲口圈) and cages, feeding, health checks, light repair, monitoring animals’ movements, and intensive care of ill, old and baby animals. The experience includes free accommodations with WiFi and a swimming pool.RomaniaWork with bears at Romania’s largest bear sanctuary (庇护所), a place that was created as a heaven for bears that have been treated in a cruel way. Located in the Carpathian Mountains, the sanctuary needs volunteers to help with preparing food, monitoring animal health and leading conservation tours. The sanctuary is home to 84 bears, with 20 more due to be rescued.60．What do the first two trips have in common?A．Both involve underwater activities.B．Both provide free accommodations.C．Both require advanced diving experience.D．Both teach how to use scientific equipment.61．Which place best suits a girl who likes horses?A．The Azores.B．Red Sea.C．Australia.D．Romania. 62．Why are the bears adopted by the sanctuary in Romania?A．They have no food.B．They are badly treated.C．They should be monitored.D．They need to live in mountains. 63．Where does the passage most probably appear?A．In a fashion magazine.B．In a shopping guide.C．In a travel brochure.D．In a biography.My son, Toby, 17 months old, has just tested positive for COVID-19. He is the first in our household to test positive, and all the information provided for people in this situation is designed for adults who are(or should be) concerned about protecting their families. Toby, obviously, cannot comprehend such advice. He cannot keep a distance from everybody else, or eat and wash in a separate room. We called 911, and the medic we spoke to agreed that the official guidance didn’t really apply to our situation.Indeed. Not much we can do. So, here we are, trapped within our four walls with a little baby infected with COVID-19, who continues to climb on us, spit in our faces, love us and hug us. It seems inevitable that we, and our elder son, are going to get COVID now. Not much we can do.My first reaction, after absorbing his diagnosis, was to cook sausages. I didn’t even know that was my comfort food of choice. Perhaps an afternoon crisis would have caused a different cooking desire? Anyway, a full stomach helped me take stock. Yes, it may now be inevitable that we’re going to contract the very virus we’ve spent a year avoiding, but as long as we don’t contract it at the same time, then, hopefully, one other of us will be available to look after our kids (thus answering my five-year-old’s most pressing concern: “…but who will make the pudding?”).So, to minimise transmission, we’ve opened all the windows. And we’ve decided to wear face coverings whenever we are with Toby. This decision has been insignificant to him—I imagine he literally cannot remember life before masks—but for me, it’s distressing. I’ve become accustomed to wearing masks in supermarkets and coffee shops, of course, but to actually walk around with half my face covered in my own house is quite another matter—it is telling me that my home has been infected, That it’s no longer a safe space.And so it was, when I sat down to write this column, that I ended up writing about COVID, which is probably the last thing you wanted to read. Sorry about that. COVID has coloured my thoughts today, even though I know that my wife and children will be fine, and that really we should just be grateful we haven’t passed it on to my grandmother.I predict there will be more sausages in the morning.64．By repeating the sentence “Not much we can do” in paragraph 2, the writer implies that ________.A．he feels inspired B．he is really helplessC．he wants professional support D．he needs to keep a distance from Toby 65．According to the writer, what did sausages bring to him?A．Appetite.B．Concern.C．Relief.D．Sadness. 66．Why did wearing masks at home distress the writer?A．It discourages him from writing about COVID．B．It brings him back to the days before COVID-19C．It makes him feel at a loss for how to help his son.D．It is a reminder of what is happening to his family.67．What can be concluded from the passage?A．The diagnosis leaves the writer’s family in a tough spot.B．Being infected with COVID-19 leaves Toby in a bad mood.C．The writer is sure that his other kids won’t contract the virus.D．Sausages have long been considered by many to be comfort foodHong Kong isn’t the first place you’d expect to find a peaceful island. But in a quiet corner of Sai Kung district (西贡区), a tiny green island offers a rare glimpse into the past.Nicknamed “Ghost Island,” Yim Tin Tsai (盐田仔) was once home to a flourishing Hakka (客家) community. They settled on the empty island and built salt pans (盐田) to earn a living. When the salt pans were shut down more than 100 years ago, most residents turned to farming and fishing.In the 1960s, more and more families moved to other places. The last of the villagers on the island had moved abroad by the 1990s, and, in the following years, Yim Tin Tsai lay empty.If travelers had visited Yim Tin Tsai a decade ago, they would have found nothing but overgrown weeds (野草) and crumbling houses. That’s exactly what villager Colin Chan saw when he returned to the island after 40 years. But, for him, the island represented something special — a unique side of Hong Kong’s history and culture that shouldn’t be forgotten.“I came back here hoping to recover something I felt I had lost,” Colin said. “I found the island was in ruins and I was very sad. This is the place I grew up. This is my father’s home and his father’s home.”In 1999, Colin was elected as village representative and began what would become a lifelong mission: to make the island come alive. Together with about 10 former villagers, Colin raised money to build a visitor center for travelers. In 2004, a charitable foundation donated funds to renovate the island’s historical chapel (小教堂). In 2005, UNESCO (联合国教科文组织) Asia-Pacific Heritage Awards for Cultural Heritage Conservation honored the chapel with an award of merit (优异奖).Motivated by this, the villagers organized a regular ferry schedule to enable travelers to reach the island, and set out to showcase the village’s unique history. They built a heritage trail, repaired Hakka ancestral homes, created a museum, and even started an organic farmthat sits at the foot of the church.68．According to the article, the Hakka community________.A．all moved to other countries in the 1990sB．settled on Yim Tin Tsai about 100 years agoC．made a living by salt mining, farming and fishing on Yim Tin TsaiD．found Yim Tin Tsai to be a fertile place when they had first arrived there69．The word “crumbling” in paragraph 4 could probably be replaced by “______.”A．frightened B．broken C．disappearing D．decreasing 70．Which of the following is NOT a reason why Colin Chan came back to Yim Tin Tsai? A．Because he wanted to recover the special feeling of the place where he grew up. B．Because as village representative, he had the duty to bring the village back to life. C．Because he didn’t want to see the hometown of his father and grandfather in ruins. D．Because he wanted to preserve Hong Kong’s history and culture mirrored in the island. 71．We can learn from the article that _______.A．UNESCO honored Yim Tin Tsai with an award of merit in 2005.B．If travelers visit Yim Tin Tsai today, they will find it a lush, green island full of life. C．Colin Chan raised money to build a museum to showcase the village’s unique history. D．the villagers on Yim Tin Tsai want to restore the salt pans which were closed more than 100 years ago.五、六选四A selection from the trolleyThe trolley problem used to be a popular question in philosophical ethics(伦理学).It runs as follows: a trolley, or a train, is speeding down a track towards a junction(三叉路口). Some evildoer has tied five people to the track ahead, and another person to the branch line. You are standing next to a lever(操纵杆) that controls the junction. Do nothing, and the five people will be killed. Pull the lever, and only one person dies. What is the ethical course of action?_____72_____ A truly self-driving car, after all, will have to be given ethical instructions of some sort by its human programmers. That has led to a boom for the world’s professional ethicists, who suddenly find themselves in hot demand.In a paper just published in Nature, a team of psychologists and computer scientists, led by Edmond Awad of the Massachusetts Institute of Technology (MIT), describe a different approach. _____73_____They created a website which presents visitors with a series of choices about whom to save and whom to kill. The website proved a hit. In the end it gathered nearly 40m decisions.The strongest preferences, expressed by respondents from all over the world, were for saving human lives over animal ones, preferring to save many rather than few and prioritising children over the old. _____74_____Iyad Rahwan, a computer scientist at MIT and one of the paper’s authors, says that many people dismiss the trolley problem as pointless as it is unlikely to arise in real life. He is unconvinced. The specific situations described by the website may hardly ever occur, he says. _____75_____ He gives the example of overtaking cyclists: “If you stay relatively near to the cycle lane, you’re increasing the chance of hitting a cyclist, but reducing the chance of hitting another car in the next lane over,” he says.A．But all sorts of choices made by the firms producing self-driving cars will affect who lives and who dies in indirect ways.B．Rather than asking philosophers for their thoughts, they decided instead to ask the public. C．That seems to conflict with most people’s moral preferences.D．The preference for saving women, for instance, was stronger in places with higher levels of gender equality.E．There were weaker preferences for saving women over men, pedestrians over passengers in the car and for taking action rather than doing nothing.F．The excitement around self-driving cars, though, has made the problem famous.六、汉译英（整句）76．从长远来看，回收包装对环境有积极的影响。
2023学年上海高二上学期数学同步精讲练第01讲 空间直线与平面(核心考点讲与练)
第01讲 空间直线与平面（核心考点讲与练）一、空间间位置关系的集合语言集合的语言来描述点、直线和平面之间的关系：点A 在直线l 上，记作：A l ∈；点A 不在直线l 上，记作A l ∉； 点A 在平面α内，记作：A α∈；点A 不在平面α内，记作A α∉； 直线l 在平面α内（即直线上每一个点都在平面α内），记作l α⊂； 直线l 不在平面α内（即直线上存在不在平面α内的点），记作l α⊄； 直线l 和m 相交于点A ，记作{}lm A =，简记为lm A =；平面α与平面β相交于直线a ，记作a αβ=．二、平面的三个公理及推论1.公理一：如果一条直线上的两点在一个平面内，那么这条直线上所有的点都在这个平面内．图形语言表述：如右图： αBlA符号语言表述：,,,A l B l A B l ααα∈∈∈∈⇒⊂ 用途：证明“点在面内”、“线在面内”.2.公理二：经过不在同一条直线上的三点，有且只有一个平面，也可以简单地说成，不共线的三点确定一个平面．图形语言表述：如右图，αCBA符号语言表述：,,A B C 三点不共线⇒有且只有一个平面α，使,,A B C ααα∈∈∈． 用途：证明“两平面重合”、“多点共面”、“点线共面”.3.公理三：如果不重合的两个平面有一个公共点，那么它们有且只有一条过这个点的公共直线．图形语言表述：如右图： Aαaβ符号语言表述：,A a A a αβαβ∈⇒=∈．考点考向用途：证明“多点共线”、“多线共点”.如果两个平面有一条公共直线，则称这两个平面相交，这条公共直线叫做两个平面的交线．4.推论推论1：经过一条直线和直线外的一点，有且只有一个平面．推论2：经过两条相交直线，有且只有一个平面．推论3：经过两条平行直线，有且只有一个平面．三、空间中的平行关系1.平行直线(1)平行公理过直线外一点有且只有一条直线和已知直线平行.(2)基本性质4(空间平行线的传递性)平行于同一条直线的两条直线互相平行.(3)定理如果一个角的两边与另一个角的两边分别对应平行，并且方向相同，那么这两个角相等. 2.直线与平面平行(1)直线与平面平行的定义直线l与平面α没有公共点，则称直线l与平面α平行.(2)判定定理与性质定理文字语言图形表示符号表示判定定理平面外一条直线与此平面内的一条直线平行，则该直线平行于此平面a⊄α，b⊂α，a∥b⇒a∥α性质定理一条直线和一个平面平行，则过这条直线的任一平面与此平面的交线与该直线平行a∥α，a⊂β，α∩β＝b⇒a∥b3.(1)平面与平面平行的定义没有公共点的两个平面叫做平行平面.(2)判定定理与性质定理文字语言图形表示符号表示判定定理一个平面内的两条相交直线与另一个平面平行，则这两个平面平行a⊂α，b⊂α，a∩b＝P，a∥β，b∥β⇒α∥β性质定理两个平面平行，则其中一个平面内的直线平行于另一个平面α∥β，a⊂α⇒a∥β如果两个平行平面同时和第三个平面相交，那么它们的交线平行α∥β，α∩γ＝a，β∩γ＝b⇒a∥b四、空间中的垂直关系1.直线与平面垂直(1)直线与平面垂直的定义如果一条直线和一个平面相交于点O，并且和这个平面内过交点(O)的任何直线都垂直，就说这条直线和这个平面互相垂直.(2)直线与平面垂直的判定定理及其推论文字语言图形语言符号语言判定定理如果一条直线与平面内的两条相交直线垂直，则这条直线与这个平面垂直⎭⎪⎬⎪⎫a⊂αb⊂αa∩b＝Ol⊥al⊥b⇒l⊥α推论1如果在两条平行直线中，有一条垂直于平面，那么另一条直线也垂直于这个平面⎭⎪⎬⎪⎫a∥ba⊥α⇒b⊥α推论2如果两条直线垂直于同一个平面，那么这两条直线平行⎭⎪⎬⎪⎫a⊥αb⊥α⇒a∥b2.(1)定义：一条斜线和它在平面内的射影所成的角叫做斜线和平面所成的角，一条直线垂直于平面，则它们所成的角是直角；一条直线和平面平行或在平面内，则它们所成的角是0°的角.(2)范围：⎣⎢⎡⎦⎥⎤0，π2.3.二面角(1)定义：从一条直线出发的两个半平面所组成的图形叫做二面角；(2)二面角的平面角：在二面角的棱上任取一点，以该点为垂足，在两个半平面内分别作垂直于棱的两条射线，这两条射线所构成的角叫做二面角的平面角. (3)二面角的范围：[0，π]. 4.平面与平面垂直 (1)平面与平面垂直的定义如果两个相交平面的交线与第三个平面垂直，又这两个平面与第三个平面相交所得两条交线互相垂直，就称这两个平面互相垂直.(2)平面与平面垂直的判定定理与性质定理文字语言图形语言 符号语言判定定理 如果一个平面过另一个平面的一条垂线，则这两个平面互相垂直⎭⎪⎬⎪⎫l ⊥αl ⊂β⇒α⊥β性质定理 如果两个平面互相垂直，那么在一个平面内垂直于它们交线的直线垂直于另一个平面⎭⎪⎬⎪⎫α⊥βl ⊂βα∩β＝a l ⊥a⇒l ⊥α1.异面直线的判定方法2.求异面直线所成的角的三步曲方法技巧3．线面平行的证明方法（1）定义法：一般用反证法；（2）判定定理法：关键是在平面内找(或作)一条直线与已知直线平行，证明时注意用符号语言叙述证明过程；（3）性质判定法：即两平面平行时，其中一个平面内的任何直线都平行于另一个平面．4．构造平行直线的常用方法（1）构建三角形或梯形的中位线：可直接利用线段的中点、等腰三角形三线合一或利用平行四边形对角线的交点找中点，从而构建中位线；（2）构建平行四边形：可以利用已知的平行关系(如梯形的上下底边平行)或构建平行关系(如构造两条直线同时平行于已知直线)，从而构建平行四边形．应用线面平行的性质定理时，关键是过已知直线作辅助平面与已知平面相交，所得交线与已知直线平行，还可以利用交线判断已知平面内的直线与已知直线的位置关系，即在已知平面内所有和交线平行的直线都与已知直线平行，所有和交线相交的直线都与已知直线异面．5.判定平面与平面平行的4种方法（1）面面平行的定义，即证两个平面没有公共点(不常用)；（2）面面平行的判定定理(主要方法)；（3）利用垂直于同一条直线的两个平面平行(客观题可用)；（4）利用平面平行的传递性，两个平面同时平行于第三个平面，那么这两个平面平行(客观题可用)．利用线面平行或面面平行的性质，可以实现与线线平行的转化，尤其在截面图的画法中，常用来确定交线的位置．对于线段长或线段比例问题，常用平行线对应线段成比例或相似三角形来解决．6.证明直线和平面垂直的常用方法：①判定定理；②垂直于平面的传递性(a∥b，a⊥α⇒b⊥α)；③面面平行的性质(a⊥α，α∥β⇒a⊥β)；④面面垂直的性质．7.利用判定定理证明平面与平面垂直的一般方法先从现有的直线中寻找平面的垂线，若这样的垂线存在，则可通过线面垂直来证明面面垂直；若这样的垂线不存在，则需通过作辅助线来证明平面及其基本性质1．（2022·上海民办南模中学高二开学考试）“平面内有一条直线，则这条直线上的一点必在这个平面内”用符号表述是（ ）A ．a A A a αα⊂⎫⇒⊂⎬⊂⎭B ．a A A a αα⊂⎫⇒∈⎬∈⎭C ．a A A a αα∈⎫⇒∈⎬⊂⎭D ．a A A a αα∈⎫⇒⊂⎬∈⎭2．（2022·上海·华师大二附中高二期末）设1234P P P P 、、、为空间中的四个不同点，则“1234P P P P 、、、中有三点在同一条直线上”是“1234P P P P 、、、在同一个平面上”的（ ） A ．充分非必要条件 B ．必要非充分条件 C ．充要条件D ．既非充分又非必要条件3．（2021·上海·复旦附中高二期中）若A ∈面α，B ∉面α，C ∉面α，则平面ABC 与平面α的位置关系_________．4．（2021·上海市大同中学高二阶段练习）已知111BAC B AC ∠=∠，11//AB A B ，则AC 与11A C 的位置关系是__________.直线与直线间的位置关系1．（2022·上海市建平中学高二阶段练习）不同在任何一个平面上的两条直线的位置关系是_________直线与平面间的位置关系1．（2021·上海奉贤区致远高级中学高二阶段练习）设m ，n 是两条不同的直线，α表示平面，下列说法正确的是（ ）A ．若m //α，n ⊂α，则m //nB ．若m //α，m ⊥n ，则n ⊥αC ．若m ⊥α，m ⊥n ，则n //αD ．若m ⊥α，n //α，则m ⊥n2．（2021·上海·高二专题练习）若a 、b 是异面直线，则下列命题中的假命题为（ ） A ．过直线a 可以作一个平面并且只可以作一个平面α与直线b 平行 B ．过直线a 至多可以作一个平面α与直线b 垂直 C ．唯一存在一个平面α与直线a 、b 等距 D ．可能存在平面α与直线a 、b 都垂直3．（2021·上海奉贤区致远高级中学高二阶段练习）Rt ABC △的直角边AB 在平面α内，顶点C 在平面α外，则直角边BC 、斜边AC 在平面α上的射影与直角边AB 组成的图形是（ ）A ．线段或锐角三角形B ．线段或直角三角形C ．线段或钝角三角形D ．线段、锐角三角形、直角三角形或钝角三角形 二、填空题4．（2020·上海市七宝中学高二期末）平面α外的直线a 与平面α所成的角是θ，则θ的取值范围是______. 5．（2021·上海市七宝中学高二期中）在三棱锥P ABC -中，已知PA ⊥平面ABC ，BC AC ⊥，则以P ，A ，B ，C 为顶点的直角三角形的个数为________.6．（2021·上海奉贤区致远高级中学高二阶段练习）若2AB =，线段AB 所在直线和平面α成30°角，且αA ∈，则点B 到平面α的距离=___________7．（2021·上海·位育中学高二阶段练习）如图，在直角ABC 中，2C π=，20BC =，40AB =，现将其放置在平面α的上面，其中点A ，B 在平面α的同一侧，点C ∈平面α，BC 与平面α所成的角为6π，则点A 到平面α的最大距离是___________.5．（2021·上海·华师大二附中高二阶段练习）在120°的二面角α－l －β的两个面内分别有点A ，B ，A ∈α，B ∈β，A ，B 到棱l 的距离AC ，BD 分别是2，4，且线段AB ＝10.(1)求C ，D 间的距离；(2)求直线AB 与平面β所成角的正弦值．平面与平面间的位置关系1．（2022·上海·复旦附中高二期中）已知a ，b 是两条不重合的直线，α，β是两个不重合的平面，给出下列命题：①//a α，////b a b α⇒；②//a b ，////b a αα⇒；③//αβ，a α⊂，//b a b β⊂⇒．其中假命题．．．的序号是______．（写出所有假命题．．．的序号） 2．（2021·上海南汇中学高二阶段练习）异面直线a ，b 成60°角，P 是a ，b 外一定点，若过P 点有且只有两条直线与a ，b 所成的角相等且等于θ，则θ的范围为___________.3．（2021·上海市金山中学高二期中）已知斜坡平面与水平面成30°的二面角，一条公路与坡脚成45°的角，沿公路前进100米，则路基升高了________米．4．（2020·上海师范大学第二附属中学高二期中）将边长为1的正方形ABCD 沿对角线AC 折叠，使得点B 和D 的距离为1，则二面角B AC D --的大小为______.异面直线间的距离1．（2021·上海·复旦附中高二期中）已知菱形ABCD 的边长为a ，3A π∠=．将菱形ABCD 沿对角线折成二面角θ，若2,33ππθ⎡⎤∈⎢⎥⎣⎦，则异面直线AC 与BD 距离的最大值为（ ）A ．32aB ．34aC ．34aD ．32a 2．（2022·上海·复旦附中高二期中）棱长为1的正方体中，异面直线1A D 与11B C 之间的距离为______． 3．（2016·上海·位育中学高二期末）若RtΔABC 的斜边AB =5，BC =3，BC 在平面α内，A 在平面α内的射影为O ，AO =2，则异面直线AO 与BC 之间的距离为___________．题型一：点、直线、平面之间的位置关系一、单选题1．（2021·上海师范大学第二附属中学高二期中）已知异面直线a 、b 所成角为80︒，P 为空间一定点，则过P 点且与a 、b 所成角都是50︒的直线有且仅有（ ）条． A ．2B ．3C ．4D ．62．（2022·上海市控江中学高二期末）如图，已知正方体1111ABCD A B C D -，点P 是棱1CC 的中点，设直线AB 为a ，直线11A D 为b ．对于下列两个命题：①过点P 有且只有一条直线l 与a 、b 都相交；②过点P 有且只有两条直线l 与a 、b 都成75︒角．以下判断正确的是（ ）能力拓展A ．①为真命题，②为真命题B ．①为真命题，②为假命题C ．①为假命题，②为真命题D ．①为假命题，②为假命题3．（2021·上海·位育中学高二阶段练习）已知菱形ABCD ，60DAB ∠=︒，E 为边AB 上的点（不包括A B ，），将ABD △沿对角线BD 翻折，在翻折过程中，记直线BD 与CE 所成角的最小值为α，最大值为β（ ） A ．αβ，均与E 位置有关B ．α与E 位置有关，β与E 位置无关C ．α与E 位置无关，β与E 位置有关D ．αβ，均与E 位置无关4．（2021·上海·高二专题练习）三条直线两两异面，有几条直线同时与这三条直线相交（ ） A ．一条B ．两条C ．无数条D ．没有5．（2021·上海·高二专题练习）如图两正方形ABCD ，CDFE 所在的平面垂直，将EFC ∆沿着直线FC 旋转一周，则直线EC 与AC 所成角的取值范围是（ ）A ．5,1212ππ⎡⎤⎢⎥⎣⎦B ．7,1212ππ⎡⎤⎢⎥⎣⎦C ．,122ππ⎡⎤⎢⎥⎣⎦D ．,62ππ⎡⎤⎢⎥⎣⎦6．（2021·上海·高二专题练习）设123l l l 、、为空间中三条互相平行且两两间的距离分别为4、5、6的直线，给出下列三个结论：①存在()123i i A l i ∈=，，，使得123A A A △是直角三角形； ②存在()123i i A l i ∈=，，，使得123A A A △是等边三角形； ③三条直线上存在四点()1234i i A l i ∈=，，，，使得四面体1234A A A A 为在一个顶点处的三条棱两两互相垂直的四面体，其中，所有正确结论的个数是 A ．0B ．1C ．2D ．37．（2021·上海·格致中学高二期中）从正方体八个顶点的两两连线中任取两条直线a ，b ，且a ，b 是异面直线，则a ，b 所成角的余弦值的所有可能取值构成的集合是（ ）A ．1320,,,,1232⎧⎫⎪⎪⎨⎬⎪⎪⎩⎭； B ．13260,,,,2323⎧⎫⎪⎪⎨⎬⎪⎪⎩⎭C ．1260,,,223⎧⎫⎪⎪⎨⎬⎪⎪⎩⎭； D ．1320,,,232⎧⎫⎪⎪⎨⎬⎪⎪⎩⎭． 二、填空题8．（2021·上海市复兴高级中学高二期中）已知两条异面直线a 与b 所成角为30，P 是空间一点，若过点P 与a 和b 所成角都是θ的直线有4条，则θ的范围是___________9．（2021·上海市甘泉外国语中学高二期中）若直线a 、b 是平面α上的两条直线，且a 、b 均在平面β外，则“//a β，//b β”是“//αβ”的 ____条件.（填“充分非必要”、“必要非充分”、“充要”、“既非充分又非必要”） 10．（2021·上海·格致中学高二阶段练习）直线m 和平面α所成角为6π，则直线m 和平面α内任意直线所成角的取值范围是_____11．（2022·上海长宁·高二期末）如图是一个边长为2的正方体的平面展开图，在这个正方体中，则下列说法中正确的序号是___________.①直线AF 与直线CN 垂直；②直线BM 与直线CN 相交；③直线ME 与直线CN 平行；④直线AB 与直线CN 异面；12．（2022·上海长宁·高二期末）已知平面α和两条不同的直线,a b ，则下列判断中正确的序号是___________. ① 若a ∥,b α∥α，则a ∥b ； ② 若,a b αα⊥⊥，则a ∥b ； ③ 若,//a b b α⊥，则a α⊥； ④ 若a ∥,b b α⊥，则a α⊥；13．（2021·上海奉贤区致远高级中学高二阶段练习）若两异面直线a 、b 所成的角为60，过空间内一点P 作与直线a 、b 所成角均是60的直线l ，则所作直线l 的条数为_________．14．（2021·上海市大同中学高二阶段练习）空间四边形SABC 中，23AB AC BC ===25SA SB SC ===，E 、F 分别为AC 、SB 的中点，平面α过点A ，//α平面SBC ，α平面=ABC l ，则异面直线l 和EF 所成角的余弦值为_______.15．（2021·上海·华东师范大学第三附属中学高二阶段练习）刍甍，中国古代算数中的一种几何形体，《九章算术》中记载：“刍甍者，下有袤有广，而上有袤无广.刍，草也，甍，屋盖也.”翻译为“底面有长有宽为矩形，顶部只有长没有宽为一条棱.刍甍字面意思为茅草屋顶.”如图为一个刍瓷的五面体，其中四边形ABCD 为矩形，ADE 和BCF △都是等腰三角形，2AE ED BF CF AD ====，//EF AB ，若3AB EF =，且2AD EF =，则异面直线AE 与CF 所成角的大小为______.16．（2021·上海市奉贤区奉城高级中学高二阶段练习）如图∶已知A 是BCD △所在平面外一点，AD BC =，E 、F 分别是AB 、CD 的中点，若异面直线AD 与BC 所成角的大小为θ，AD 与EF 所成角的大小为_______________．17．（2021·上海市进才中学高二阶段练习）空间给定不共面的A ，B ，C ，D 四个点，其中任意两点间的距离都不相同，考虑具有如下性质的平面α：A ，B ，C ，D 中有三个点到的距离相同，另一个点到α的距离是前三个点到α的距离的2倍，这样的平面α的个数是___________个18．（2021·上海·高二专题练习）已知两个不同平面α，β和三条不重合的直线a ，b ，c ，则下列命题： （1）若//a α，b αβ=，则//a b（2）若a ，b 在平面α内，且c a ⊥，c b ⊥，则c α⊥（3）若α，β分别经过两异面直线a ，b ，且⋂=c αβ，则c 必与a 或b 相交（4）若a ，b ，c 是两两互相异面的直线，则存在无数条直线与a ，b ，c 都相交其中正确的命题是________．（请写上正确命题的序号）三、解答题19．（2021·上海市洋泾中学高二期中）如左图所示，在直角梯形ABCD 中，//BC AD ，AD CD ⊥，2BC =，3AD =，3CD =AD 上一点E 满足1DE =.现将ABE △沿BE 折起到1A BE 的位置，使平面1A BE ⊥平面BCDE ，如右图所示.(1)求证：1A C BE ⊥；(2)求异面直线1A C 与BE 的距离；(3)求平面1A BE 与平面1A CD 所成锐二面角的余弦值.20．（2022·上海市控江中学高二期末）如图，在长方体1111ABCD A B C D -中，11,2DD DA AB ===，点E 在棱AB 上运动．(1)证明：11B C D E ⊥；(2)当E 为棱AB 的中点时，求直线AB 与平面1DEC 所成角的正弦值；(3)AE 等于何值时，二面角1A DE C --的大小为150︒？题型二：直线、平面平行的判定与性质一、单选题1．（2021·上海市洋泾中学高二期中）已知m 、n 是两条不同的直线，α、β是两个不同的平面，则以下命题正确的是（ ）A ．若m 、n 平行于同一平面，则m 与n 平行B ．若α、β垂直于同一平面，则α与β平行C ．若m 、n 不平行，则m 与n 不可能垂直于同一平面D ．若α、β不平行，则在α内不存在与β平行的直线2．（2021·上海市吴淞中学高二阶段练习）已知空间直线l 不在平面α内，则“//l α”是“直线l 上有两个点到平面α的距离相等”的（ ）A ．充分非必要条件B ．必要非充分条件C ．充要条件D ．非充分非必要条件 3．（2021·上海市洋泾中学高二阶段练习）如图，在矩形ABCD 中，2AB =，1BC =，E 、N 分别为边AB 、BC 的中点，沿DE 将ADE ∆折起，点A 折至1A 处（1A 与A 不重合），若M 、K 分别为线段1A D 、1A C 的中点，则在ADE ∆折起过程中（ ）A ．DE 可以与1A C 垂直B ．不能同时做到//MN 平面1A BE 且//BK 平面1A DEC ．当1MN AD ⊥时，MN ⊥平面1A DED ．直线1AE 、BK 与平面BCDE 所成角分别为1θ、2θ，1θ、2θ能够同时取得最大值二、填空题4．（2021·上海市吴淞中学高三期中）与不共面的四点等距离的平面有___________个.5．（2021·上海·华师大二附中高二阶段练习）给出下列4个命题：①过平面外一点，与该平面成θ角的直线一定有无穷多条；②一条直线与两个相交平面都平行，则它必与这两个平面的交线平行；③过空间任意一点有且只有一个平面与两条异面直线都平行；④与确定的两条异面直线所成的角相等的平面有无数个．其中正确命题的序号有_____（请把所有正确的序号都填上）．6．（2021·上海·位育中学高二期中）已知四边形ABCD 为矩形, 24AB AD ==,M 为AB 的中点,将ADM ∆沿DM 折起,得到四棱锥1A DMBC -,设1A C 的中点为N ,在翻折过程中,得到如下有三个命题:①//BN 平面1A DM ，且BN 5②三棱锥N DMC -22； ③在翻折过程中，存在某个位置，使得1DM AC ⊥. 其中正确命题的序号为__________．（写出所有正确结论的序号）三、解答题7．（2021·上海市徐汇中学高二期中）如图，几何体ABCDE 中，ABC 是正三角形，AE 和CD 都垂直于平面ABC ，且2EA AB ==， 1,,DC F G =分别为EB 和AB 的中点.(1)求证：FD ∥平面ABC ；(2)求二面角B FC G --的正切值.8．（2022·上海·高三专题练习）如图，将边长为2的正方形ABCD 沿对角线BD 折叠，使得平面ABD 与平面CBD 所成二面角为直角，AE ⊥平面ABD ，且2AE =.（1）求证：直线EC 与平面ABD 没有公共点；（2）求点C 到平面BED 的距离.题型三：直线、平面垂直的判定与性质一、单选题1．（2021·上海市市西中学高二期中）在下列命题中，假命题是（ ）A ．若平面α内的一条直线垂直于平面β内的任一直线，则α⊥βB ．若平面α内任一直线平行于平面β，则α∥βC ．若平面α⊥平面β，任取直线l ⊂α，则必有l ⊥βD ．若平面α∥平面β，任取直线l ⊂α，则必有l ∥β二、填空题2．（2021·上海市南洋模范中学高二期中）在正方体1111ABCD A B C D -中，M 是棱AB 的中点，过1A ，M ，C 三点的平面交棱11C D 于点N ，则11A B 与平面1A MCN 所成角的正弦值是___________.3．（2021·上海师大附中高二期中）如图，已知,,60,1AP BP AP PC ABP ACP BAC PA ⊥⊥∠=∠=∠=︒=，D 是BC 中点，则点B 到平面APD 的距离是___________.4．（2021·上海师范大学第二附属中学高二期中）在长方体1111ABCD A B C D -中，4AB =，3BC =，12BB =，那么BC 到平面11ADD A 的距离为______.5．（2021·上海·华东师大附属枫泾中学高二期中）如图，在正方体1111ABCD A B C D -中，2AB =，则直线1AA 到平面11BB D D 的距离为___________.6．（2022·上海·复旦附中高二期中）在矩形ABCD 中，AB ＝2，23BC =，沿对角线AC将矩形折成直二面角D －AC －B ，则线段BD 的长为______．7．（2021·上海市进才中学高二期中）在Rt ABC 中，已知12BC =，5AC =，D 是斜边AB 上任意一点，如图沿直线CD 将ABC 折成直二面角B CD A --.若折叠后A ，B 两点间的距离为d ，则d 的最小值为___________.三、解答题8．（2022·上海金山·高二期末）如图，在正方体1111ABCD A B C D -中，E 是棱1DD 的中点.(1)试判断直线1BD 与平面AEC 的位置关系，并说明理由；(2)求证：直线1BD ⊥面1AB C .9．（2021·上海市大同中学高二阶段练习）如图，正方体1111ABCD A B C D -中，1AA a =.（1）求证：平面11//AB D 平面1C BD ；（2）求两平面11AB D 与1C BD 之间的距离.10．（2021·上海市南洋模范中学高二期中）已知，如图P 是平面α外一点，PA 是平面α的斜线，交α于点A ，过点P 作平面α的垂线PO ，垂足是O ，直线OA 是PA 在平面α上的投影.求证：对平面α上任一直线a ，a OA ⊥是a PA ⊥的充要条件.一、单选题1．（2021·上海奉贤区致远高级中学高二期中）如图，在下列四个正方体中，A ，B 为正方体的两个顶点，M ，N ，Q 为所在棱的中点，则在这四个正方体中，直线AB 与平面MNQ 不平行的是（ ）．A ．B ．C ． D ． 2．（2022·上海·复旦附中高二期中）已知a 、b 是平面α的两条斜线，则“a 、b 与平面α所成角相等”是“//a b ”的（ ）条件A ．充分非必要B ．必要非充分C ．充分必要D ．既非充分又非必要巩固练习3．（2021·上海奉贤区致远高级中学高二阶段练习）正方体1111ABCD A B C D -的八个顶点中，平面α经过其中的四个顶点，其余四个顶点到平面α的距离都相等，则这样的平面α的个数为（ ）A ．6B ．8C ．12D ．164．（2021·上海奉贤区致远高级中学高二阶段练习）在正方形123SG G G 中，E 、F 分别是12G G 及23G G 的中点，D 是EF 的中点．现在沿SE 、SF 及EF 把这个正方形折成一个空间四边形，使1G 、2G 、3G 三点重合，重合后的点记为G ，那么，在空间四边形S EFG -中必有（ ）A ．SG EFG △⊥所在平面B ．SD EFG ⊥所在平面C ．GF SEF ⊥所在平面D ．GD SEF ⊥所在平面二、填空题 5．（2022·上海市建平中学高二阶段练习）正方体1111ABCD A B C D -的棱长为1，异面直线11A B 与1BC 的距离是_________6．（2021·上海·西外高二期中）空间中两条直线的位置关系有___________.7．（2021·上海市宝山中学高二期中）如图，正三角形ABC 的边长为2，P 是三角形ABC 所在平面外一点，PA ⊥平面ABC ，且1PA =，则P 到BC 的距离为___________.8．（2021·上海·位育中学高二期中）在30°二面角的一个面内有一个点，它到另一个面的距离是10cm ，则这个点到二面角的棱的距离为___________.9．（2021·上海市控江中学高二期中）两个平面最多可以将空间分为___________部分.10．（2021·上海市洋泾中学高二期中）在棱长为a 的正方体中，与AD 成异面直线且距离等于a 的棱共有_____________条.11．（2021·上海市吴淞中学高二阶段练习）一个山坡面与水平面成120°的二面角，坡脚的水平线（即二面角的睨）为AB ，甲沿山坡自P 朝垂直于AB 的方向走30m ，同时乙沿水平面自Q 朝垂直于AB 的方向走30m ，P ､Q 都是AB 上的点，若10m PQ =，这时甲､乙两个人之间的距离为___________.12．（2021·上海奉贤区致远高级中学高二阶段练习）如图，已知PA ⊥平面ABC ，AC BC ⊥，2AB =，2BC =，PB =6，则二面角P BC A --的大小为________三、解答题13．（2022·上海浦东新·高二期末）如图，在正方体1111ABCD A B C D -中.(1)求异面直线1A B 和1CC 所成的角的余弦值；(2)求证：直线1//A B 平面11DCC D .14．（2021·上海南汇中学高二阶段练习）已知正方体1111ABCD A B C D -中，P ､Q 分别为对角线BD ､1CD 上的点，且123CQ BP QD PD ==. (1)作出平面PQC 和平面11AA D D 的交线（保留作图痕迹），并求证：//PQ 平面11A D DA ；(2)若R 是AB 上的点，当AR AB的值为多少时，能使平面//PQR 平面11A D DA 请给出证明.15．（2021·上海奉贤区致远高级中学高二阶段练习）如图，已知1111ABCD A B C D -是底面为正方形的长方体，1160D A A ∠=，14AD =，点P 是1AD 上的动点（1）试判断不论点P 在1AD 上的任何位置，是否都有平面BPA 垂直于平面11AA D D ，并证明你的结论（2）当P 为1AD 的中点时，求异面直线1AA 与1B P 所成角的余弦值; （3）求1PB 与平面11AA D 所成角的正切值的最大值。
上海徐汇国际小学招生条件
上海徐汇国际小学招生条件上海国际学校众多，招生要求也各不相同，对此我们整理汇总了上海2022年主流国际学校最新的招生要求，供家长们参考。
看文字太麻烦，可以直接咨询菠萝老师哦~上中国际上海中学国际部成立于1993，现有来自60多个不同国家和地区的约3000名学生。
1995年，上中国际加入了IB国际文凭组织。
上中国际同时也是由College Board授权的AP、SAT和PSAT的考试中心和由ETS授权的托福考试中心。
2014 年上中国际成为国际AS& A-Level学校，2018年通过美国AdvancED (COGNIATM)认证。
课程体系：美高+AP、A-level招生年级：•浦西校区：1-5年级(小学)；6-8年级( 初中 )；9-12年级(高中)。
•浦东校区：1-5年级（小学）招生范围:在上海工作的外国人或港澳台人士的子女。
考试要求：汉语、英语、数学（申请10年级及以上年级加试物理和化学；所有科目活动均包括笔试和面试）。
学费：1-5年级（小学）: 5万元/学期；6-8年级（初中）：5.3万元/学期；9-12年级（高中）: 5.5万元/学期。
WLSAWLSA上海学校(WLSAShanghaiAcademy)，是世界名中学联盟于2012年9月在上海复旦附中进修学校开设的国际高中课程项目。
WLSA上海学校的WLSA即为世界名中学联盟（World Leading Schools Association）的缩写，其为“英国政治家的摇篮”伊顿公学、北京四中、哈佛西湖中学、“上海四校之一”复旦附中等世界知名的51所中学联合发起成立的组织。
课程体系：AP、A-level招生年级：初三或高一在读学生招生范围：国内在读学生，户籍不限。
考试要求：需通过学校的笔试（英语+数学）、多邻国英语能力测试和英文面试。
学费：美高课程/A-level：17.2万元/年。
WLSA入学考试英语测试卷WLSA入学考试英语测试卷，需要的同学↓↓↓↓世外中学上海市世界外国语中学（以下简称世外中学），创办于1996年，是一所在上海市具有优秀学术声誉和社会声誉的民办中学。
2024-2025学年上海复旦附中高一上学期数学开学考试卷及答案(2024.09)
1复旦附中2024学年第一学期高一年级数学开学考2024.09一、填空题（本大题共有12题,满分54分,第1-6题每题4分,第7-12题每题5分） 1．用列举法表示《沁园春·长沙》前三句的意象所组成的集合________．2．设高一（5）班全体学生的集合为A （A 中有17名男生，23名女生），高一（5）班全体女生的集合为B ，则A ________B ． 3．用区间法表示实数集R =________．4．已知集合{}0,1,2A =，集合{}|23x B x =>，则AB =________．5．已知集合{}1,2A =，{}532|24670B x x x x x =−+++=，则A B =________．6．已知集合{}1,2,3,4A =、{}1,2B =，满足A C BC =的集合C 有________个．7．已知集合()(){}2|10,x x x x a x R −−+=∈中的所有元素之和为1，则实数a 的取值集合为________．8．设集合{}2,0,1M =−，{}|1N x x a =−<，若M N 真子集个数为1，则正实数a 的取值范围为________． 9．关于的方程221x k xx x x−=−−的解集中只含有一个元素，则k 的所有可能值组成的集合 是________．10．设集合{}{}123,,,,1,2,3,,107n A a a a a =⊆，且A 中任意两数之和不能被5整除，则n 的最大值为________．11．已知集合A 满足若n A ∈且n Z ∈，则1A n∈，小张同学迅速得出3个结论：（1）0A ∉（2）集合A 不可能是单元素集（3）当n 取遍可以取的所有数时，集合元素的个数一定是偶数，其中错误结论的序号为________． 12.若规定集合{}0,1,2,,E n =的子集{}123,,,,m a a a a 为E 的第k 个子集，其中23122222m a a a a k =++++，则E 的第211个子集的真子集个数为________．二、选择题（4题共18分，13~14每题4分，15~16每题5分）13．必要的习题训练，也是数学训练的必须。
孩子如何渡过6-9岁的三个转折点
我是⼀位上海妈妈，有⼀个⼋岁的⼥⼉，她正在处于6-9岁的转折点，下⾯我就分享⼀下我⼥⼉6-9岁的三个转折点是如何渡过的。
⼥⼉今年三年级，是⼀个认真、内向、独⽴并且有着丰沛想象⼒的孩⼦。
如果在另⼀个语境中评论的话，她⼜是⼀个胆⼩、散漫、固执且尚未被⼤⼈思路所束缚的孩⼦。
⼥⼉出⽣后，我看了很多育⼉书，也许是70后这⼀代从⼩做惯了“乖⼩囡”从⽽对“⾃由”特别渴望的缘故，我对⼥⼉⼀直本着“理解”的态度。
现在想起来，我的价值观对⼥⼉后来的表现有着⾮常直接的影响。
⼀直到上⼩学前，我对⾃⼰这个育⼉观念还是⾮常⾃信的，⼩学前，⾯对诸多“幼⼩衔接”的培训班，我们⼀个也没去过，我也没有给她教过识字、拼⾳、算术或英语。
她感兴趣的是做实验、玩各种破玩具以及所有的户外活动。
第⼀个转折：幼升⼩“择校” 进⼩学，对于我这样的⽗母来说，只是后⾯漫漫长路的开始。
由于对⼝学校是所谓的“菜场⼩学”，我和⼥⼉开始了懵懵懂懂的幼升⼩征战。
别说⼥⼉是“考试菜鸟”（她连⼀堂正规的课也没有上过），我们⾃⼰也是“菜鸟⽗母”，所以也就随⼤流既去考了上海北区的“XX附⼩”，也去浦东与3000多⼤军参与了10年⼀贯制⼩学的考试。
记得当时看着⼥⼉⼩⼩的圆滚滚的⾝体⼀个⼈挂着牌⼦找考试教室并且还要上机考试（她以前从未使⽤过计算机），隐隐觉得随着进⼩学后， “⾃由”也许是越来越奢侈的事情。
在门⼝等她的时候，听到⾝边的⽗母在交流经验，越听越汗，普通的拼⾳学习、英语学习根本不算什么，思维训练正正经经学了3年的⼤有⼈在。
“不然考什么考，考了也是⽩考。
”有的家长说。
在⾯试环节，学校⽼师甚⾄和家长明说：“如果⽗母之间没有⼀个能‘全职’陪伴孩⼦学习，建议还是不要来读我们学校，吃不消的。
”幼升⼩考了4所学校，最后双向选择了徐汇⼀所以艺术教育和快乐教育著称的私校，⼩学⽣活就这样开始了。
⼀年级⼀开学，⽴马就给颜⾊看了，由于不识字，考卷上的题⽬都看不懂，语⽂考试只考了六⼗⼏分。
上海徐汇区小学盘点
202X上海徐汇区小学盘点202X上海徐汇区小学盘点作为上海的传统教育大区，徐汇也是大家公认教育质量最好的区，拥有全市最好的高中上海中学，拥有全市数一数二的初中华育中学，简直令人羡慕！下面一起来看看202X上海徐汇区热门小学盘点。
徐汇有且仅有这4所民办小学，就算拿到全市，这几所民办小学都是综合实力排名前列的，同时，学费比其他区那些双语学校廉价的多，性价比可想而知。
世界外国语小学——徐汇四大民办之首，在全市绝对是数一数二的名校。
升学成绩、杯赛成绩亮出来都有鲜明的优势，使之称为沪上最热门的民办小学之一。
进世外的孩子都是千挑万选出来的，资质好、家境好，“牛蛙+牛爹”配置。
英语是世外强项。
从近几年招生情况看，对学生的外语能力要求也越来越高，前几年还要求英语跟读，开始要判断多国语言了，对六国语言都要有所了解，语感要强。
世外小学前110左右大概率能进入世外中学，也有近30人去上外120和华育，小升初占绝对优势。
这也是造成它年年幼升小报名火爆，开分校也能迅速招生的重要原因。
盛大花园小学是目前上海最热门的民办小学之一。
一是他本身的办学成绩喜人；二是他是徐汇区里唯一一所全市招生的民办小学，而且区内外招生人数各占一半（区外招生人数全市第4位）；三是他招收第二志愿学生，有众多抱着万一的态度第二志愿填报了盛大花园，但其实二志愿录取凤毛麟角。
盛大学生毕业择校越来越多元，华育、世外、华二、上宝、张江、西南模范、西南位育等都是盛大学子的择校方向。
而且据在读家长称，盛大的可贵之处，在于其中下游的学生大都也能有个好去向，这是其他很多公办学校无法比的。
逸夫小学曾经有“杯赛最强者”之称号，偏重数理思维，其数学成绩有目共睹，在杯赛场上，成绩可观，以被禁的四大杯赛来说，xx年的获奖人数位列全市第一：中环18人、小机灵22人、亚太11人、走美15人！逸夫小学最大的优势就是去华育的概率比拟大，每年都有30~40人考进华育。
考逸夫的牛蛙，为了小升初能进华育或其他好学校，都是目标明确且愿意为此付出时间、精力、金钱，且能够为之努力好几年。
2020届上海高考翻译句型分类整理
2020届上海高考英语翻译句型分类汇编1.before句型：It will be+ 一段时间before sb does/It was +一段时间before sb did例句：1）（2019届奉贤一模)75. 不久之后，地铁5号线奉贤段即将通车，这让翘首以盼的奉贤人民激动不已。
（before）75. It won’t be long before the Underground/Subway/Metro Line 5 in Fengxian is open to traffic, which e xcites the Fengxian people who have been looking/are looking forward to it.2）(2015届青浦一模)2.要不了多久我的同学们定能学会抵制网络游戏的诱惑。
（It）2.It won’t be long before my classmates (fellow students) can learn to resist the temptation of (playing) online 3）(2013届静安杨浦青浦宝山四二模)1.过不了几天他们就会把面试结果寄给你的。
(It)1.It won’t be a few days/ before/ they mail you/ the result about the interview.4）(2013届松江一模 ) 2. 不久以后，一些药品的价格就会再一次下调。
（before)2. It will not be long before the price of some medicine is reduced again.4）(2011届高考二模翻译) 3．你不会等很长时间意味着过不了多久下一班车就来了。
( before)3. That you won't wait for long means that it won't be long before the next bus arrives.5）(2011届闸北二模 ) 2. 两周以后她才从事故的恐惧中恢复过来。
上海往年华育中学九上英语单元测试卷Unit-1 含答案
华育初三U1测试卷Part II Vocabulary and GrammarII.Choose the best answer:30%26.You can never get a straight answer out of him. Which of the following is correct for the underlined word in thesentence?A./stret/B./strait/C./streit/D./straut/27.All the employees except the manager ________ to work online at home.A. encouragesB. encourageC. is encouragedD. are encouraged28.Under a big tree ________,half asleep.A. did sat a fat manB. a fat man satC. did a fat man satD. sat a fat man29.Polar bears live mostly on ________ sea ice, which they use as ________ platform for hunting seals.A. a; aB. a; theC.不填; aD. the; 不填30.Ten years ago the population of our village was ________ that of theirs.A. as twice large asB. twice as large asC. twice as much asD. as twice much as31. A poet and artist ________ coming to speak to us about Chinese literature and painting tomorrow afternoon.A. isB. areC. wasD. were32.He found it increasingly difficult to read, ________ his eyesight was beginning to fail.A. thoughB. forC. butD. so33.Have you known Dr. Jackson for a long time?—Yes, since she ________ the Chinese Society.A. has joinedB. joinsC. had joinedD. joined34.________ his eyes has gone blind, to our great joy.A. Neither ofB. Either ofC. Both ofD. None of35.The suit fitted him well ________ the colour was a little brighter.A. except forB. except thatC. except whenD. besides36.Could we see each other at 3 o’clock this afternoon? —Sorry, let’s make it ________ time.A. other’sB. the otherC. anotherD. other37.There are four bedrooms, ________ with its own bathroom.A. allB. eachC. everyD. either38.________ came that we should soon started off, but none of us would hear of parting with the villagers.A. WordsB. The wordC. WordD.A Word39.Finally a taxi ________ by the boss.A. sent forB. has sent forC. had sent forD. was sent for40.He said nothing when he ________.A. was spoken to B spoke to C. spoke D. was speaking41.The computers on the table ________ Professor Smith.A. belongsB. are belonged toC. belongs toD. belong to42.________ great the year 2010 is! The Expo took place in Shanghai.A. WhatB. HowC. How aD. What a43.________ the day went on, the weather got worse.A. WithB. SinceC. WhileD.As44.Jack said there ________ a heavy rain there.A. happened to beB. used to haveC. is going to beD. would have45.I am afraid I can’t return the book to you before Friday.—________.A. Don’t be afraidB. Be carefulC. Not at allD. Take your time46.This morning I went to work in such a hurry that I ________ the key ________ my office at home.A. forgot; ofB. left; forC. left; toD. missed; to47. —Alice, have you heard that the grocery store down the street was ________, and nearly all the things there werestolen?—God! Really? That’s horrible!A. broken downB. broken upC. broken intoD. broken out48. —He is an excellent cook.—________. The fish he cooked makes me mouth —watering.A. I couldn’t agree moreB. Oh, come onC. You can’t be seriousD. That couldn’t be better49. She didn’t feel like ________,so he suggested ________ the day in the garden.A. to work, to spendB. working, to spendC. to work, spendingD. working, spending50. ________ it is!A. What frozen weatherB. What a freezing weatherC. How freezing weatherD. What freezing weather51. Measurements ________ in 1911 showed that the Pisa Tower would collapse by 2050.A. takingB. tookC. takenD. were taken52. The cakes are delicious. He’d like to have ________ third one because ________ second one is rather too small.A. a, aB. the, theC. a, theD. the, a53. It is not rare in ________ that people in ________ fifties are going to university for further education.A. 90s,theB. the 90s,/C.90s,theirD. the 90s,their54.“________ your chemistry teacher ________?”“Tall and a little fat.”A. What does...likeB.How is...likeC. What does...look likeD. How does...look like55.The annual loss in illness and early death in their country is about 28,000 million dollars —about ________ theincome.A. two-sixs of B .two-sixth of C. two-sixths of D. two-six ofIII. C omplete the following passage with the words or phrases in the box. Each word can only be used once.8%You can’t always predict a heavy rain or remember your umbrella. But designer Mikhail Belvacv doesn’t think that forgetting to check the ____56____ forecast before heading out should result in you getting wet. That’s why he created lampbrella, a lamp post with its own rain sensing umbrella.The designer says he come up with the idea after watching people get wet on streets in Russia. “once,I was driving on a central Saint Petersburg street and saw the street lamps lighting up and people ____57____ to hide from the rain. I thought it would be ____58____ to have a canopy(伞蓬) built into a street lamp.” he said.The lampbrella is a standard-looking street lamp fitted ____59____ an umbrella canopy. It has a built-in electric motorwhich can open or close the umbrella on demand. Sensors then ensure that the umbrella offers pedestrians shelter whenever it starts raining.In addition to the rain sensor, there’s also a 360°motion sensor on the biber glass street lamp which detects whether anyone’s using the lampbrella. After three minutes of not being used the canopyis ____60____.According to the designer, the lampbrella would move at a relatively low speed, ____61____ as notto cause harm to the pedestrians. Besides, it would be grounded to protect from possible ____62____ strike. Each lampbrella would offer enough shelter for several people. Being installed at 2 meters off the ground, it would only be a danger for the tallest of pedestrians.While there are no plans to take lampbrella into production, Belyacv says he recently introduced his creation one Moscow Department, and insists this creation could be installed on my street ____63____ a lot of people walk but there are no canopies to provide shelter.plete the sentences with the given words in their proper forms:7%64.Since then, most have believed that the Odyssey is fiction with a single historical reference–the __________War.(Troy)65.Boarding taught me confidence, __________,how to survive and compete.(depend)66. A dream job __________ not only what you want to do, but who you are doing it with. You should truly enjoy workingwith these people.(including)67.To those who seek peace and __________,we support you.(secure)68.We’re __________ to listen to her beautiful voice.(please)69.The doctor gave him a __________(medicine)examination.70.Astronomy is the __________ study of the sun, moon, and stars and other heaven bodies.(science)plete the sentences with the given verbs in their proper forms:7%71.He moved slowly, __________ his sore feet.(drag)72.We __________ (empty) the rubbish bin and __________ (mend) the hole in the carpet yesterday.73.By the time my parents reached home yesterday, I __________ (have) the dinner already.74.The opportunity is there. Why not __________ (seize)it?75.Cissie __________ (obey)her mother without question when she was young.76.He told me he __________ (ride)on a light rail train to the airport the next day.IV.Rewrite the sentences as required:14%77.She starts from home at 7:30 every morning.(句意相同)She __________ __________ from home at 7:30 every morning.77.They drove the patient to the hospital.(改为被动语态)The patient __________ __________ to the hospital.79.The book was so difficult that neither of us could read it.(句意相同)The book was too difficult __________ __________ of us to read.80.The street is five kilometers long from east to west.(划线提问)__________ __________ is the street from east to west?81.This word is no longer in current use.(句意相同)This word __________ in current use __________ __________.82.The girl is a good teacher.(改为感叹句)__________ __________ the girl teaches!83.an expert guide, visitors, the history, can, of, explore, through, a tour, with, the castle________________________________________________________________________________Part III Reading and WritingA.Choose the best answer:10%Homestay provides English language students with the opportunity to speak English outside the classroom and the experience of being part of a British home.What to ExpectThe host will provide accommodation and meals. Rooms will be cleaned and bedcovers changed at least once a week. You will be given the house key and the host is there to offer help and advice as well as to take an interest in your physical and mental health.Accommodation ZonesHomestays are located in London mainly in Zones 2,3 and 4 of the transport system. Most hosts do not live in the town centre as much of central London is commercial and not residential(居住).Zones3 and 4 often offer larger accommodation in a less crowded area. It is very convenient to travel in London by Underground.Meal Plans Available◆Continental Breakfast◆Breakfast and Dinner◆Breakfast, Packed Lunch and DinnerIt’s important to note that few English families still provide a traditional cooked breakfast. Your accommodation includes. Continental Breakfast which normally consists of fruit juice, cereal, bread and tea or coffee. Cheese, fruit and cold meat are not normally part of a Continental Breakfast in England. Dinners usually consist of meat or fish with vegetables followed by dessert, fruit and coffee.FriendsIf you wish to invite a friend over to visit, you must first ask your host’s permission. You have no rightto entertain friends in a family home as some families feel it is an invasion of their privacy.Self Catering Accommodation in Private HomesAccommodation on a room only basis includes shared kitchen and bathroom facilities and often a main living room. This kind of accommodation offers an independent lifestyle and is more suitable for the longstay student. However, it does not provide the same family atmosphere as an ordinary homestay and may not benefit those who need to practise English at home quite as much.84.The passage is probably written for ________.A. hosts willing to receive foreign studentsB. foreigners hoping to build British cultureC. travellers planning to visit families in LondonD. English learners applying to live in English homes85.Which of the following will the host provide?A. Room cleaning.B. Medical care.C. Free transport.D. Physical training.86.What can be inferred from Paragraph3?A. Zone 4 is more crowded than Zone 2.B. The business centre of London is in Zone 1.C. Hosts dislike travelling to the city centre.D. Accommodation in the city centre is not provided.87.According to the passage, what does Continental Breakfast include?A. Dessert and coffee.B. Fruit and vegetables.C.Bread and fruit juice.D. Cereal and cold meat.88.Why do some people choose selfcatering accommodation?A. To experience a warmer family atmosphere.B. To enrich their knowledge of English.C. To entertain friends as they like.D. To enjoy much more freedom.B.Choose the words or expressions and complete the passage:16%Some personal characteristics play a vital role in the development of one’s intelligence. But people fail to realize the importance of cultivating these factors in young people.The so-called “non-intelligence factor” include one’s feelings, will, motivation, interests and habits. After a 30-year follow-up study of 8000 males, American psychologists ____89____ that the main cause of disparities in intelligence is not intelligence ____90____,but non-intelligence factors including the desire to learn, will-power and self-confidence.Though people all know that one should have definite objectives, a strong will and good learning habits, quite a number of teachers and parents don’t pay much attention to ____91____ these factors.Some parents are greatly worried ____92____ their children fail to do well in their studies. They blame either genetic factors, malnutrition, or laziness, but they never take ____93____ consideration these non-intelligence factors. At the same time, some teachers don’t inquire into these, as reason why students do poorly. They simply give them more course and exercises, or even rebuke or ridicule them. After all, these students lose self-confidence. Some of them just feel defeated and ____94____ themselves up as hopeless. Others may go astray because they are sick of learning. An investigation of more than1000 middle students in Shanghai showed that 46.5 percent of them were afraid of learning, because of examination,36.4 percent lacked persistence, initiative and conscientiousness and 10.3 percent were sick of learning.If we don’t start now to strengthen the cultivation of non-intelligence factors, it will not only obstruct the development of the ____95____ of teenagers, but also affect the quality of a whole generation. Some experts have put forward proposals about how to cultivate student’s non-intelligence factors.First, parents and teachers should ____96____ understand teenage psychology. On this basic, they can help them to pursue the objectives of learning, stimulating their interests and toughening their will power.89. A. came out B. found out C. figured out D. worked out90. A. in itself B. by itself C. itself D. on its own91. A. believing B. studying C. cultivating D. developing92. A. about B. when C. how D. whether93. A. for B. in C. into D. over94. A. put B. get C. handle D. give95. A. intelligence B. diligence C. maturity D. performance96. A. fully B. greatly C. very D. highlyC.Read the passage and fill in the blanks with proper words:14%Without proper planning, tourism can cause problems. For example, too many tourists can crowdp____97____ places that are also enjoyed by the inhabitant of a country. If tourists create too much traffic, the inhabitants become annoyed andunhappy. They begin to dislike tourists and to treat them impolitely. They forget how much tourism can help the country’s economy. It isi____98____ to think about the people of a destination country and h____99____ tourism affects tourists.Tourism should also advance the well-being (health and happiness)of local inhabitants.Too much tourism can be a problem. If tourism grows too q____100____, people must leave other jobs to work in the tourism industry. This means that other parts of the country’s economy can suffer.On the other hand, if there is not enough tourism, people can lose jobs. Business can also lose money. It costs a great deal of money to build large hotels, airports, air terminals, first-class roads, and other support facilities n____101____ by tourist attractions. For example, a major international-class tourism hotel can cost as much as 50 thousand dollars per room to build. If this room is not used most of the time, the owners of the hotel lose money.Building a hotel is just a b____102____.There must be many facilities as well, including roadstog____103____ to the hotel, electricity, sewers to handle waste, and water. All of these support facilities cost money. If they are not used because there are not enough tourists, jobs and money are lost.D.Answer the questions:14%“It was all his own idea,” says Pat Peters, the 38-year-old wife of Palo Alto, California high school. Bob had just drawn up a “motherhood contract” —a document(文件)stating that for 70 days this summer he would take over the care and feeding of the couple’s four children, plus all household chores(杂务).Although he didn’t even know how to make coffee when he signed, he was quite confident.(He thought the experience would make a nice book.)After 40 of the 70 days, he was ready to give up.“I was beaten down, completely humbled,” admits Peters. Three weeks later he spoke to the local press (also part of the bargain), stating, “Not only is motherhood a difficult task, not only is it never-ending, it is an impossible job for any normal human being.”Bob and Pat were high school sweethearts. After they were married in1960, she worked as a secretary to help put him through university. Since then Bob has been the football and wrestling coach at Palo Alto’s Cubberley High while Pat raised the kids. Then two years ago Patwent back to work as a secretary at Cubberley.“I had been around children so much,” she sighs, “I couldn’t talk to a grown-up.”She continued to run the household, however-until Bob signed the contract, whereupon she decided to relax and enjoy it. Although Peters had consulted with his school’s home economics teachers and the head of the cafeteria, his meals were sometimes a disaster.“I tried to slip the butter I’d forgotten under the eggs after they were frying,” he says. For the last three weeks, the family ate out a lot-sometimes having Macdonald’s hamburgers for lunch and dinner. As for housekeeping, a home economics teacher had told Bob that a room always looks clean if the bed is made.“I found an easier way —I shut the doors,” he says. Soon the kids were wearing the same clothes for a week.“I made them wear their shirts inside out, and when we went to pick up Pat at work they turned them right side out so they would look clean.”Now that Bob has publicly admitted he was wrong, he is routinely sharing the child-raising and household tasks with Pat. The tentative(暂定的)title of his book about the summer is taken from something he shouted at the kids one day. 104.Where does Bob work?________________________________________________________________________________105.What should Bob do during 70days this summer?________________________________________________________________________________106.What does Pat do at Cubberley?________________________________________________________________________________107.What would Bob have to do if he failed to keep to the contract as it was agreed?________________________________________________________________________________108.Did Bob try to cook good meals for his children?________________________________________________________________________________ 109.How did Bob keep the room clean?________________________________________________________________________________ 110.How do you like the contract? Why?________________________________________________________________________________Keys:26-30 CDDCB 31-35 ABDAB 36-40 CBCDA 41-45DBDAD 46-50 CCADD 51-55 CCDCC 56-63 EACB AEBD64-70 Trojan independence includes security pleased medical scientific71. dragging 72. emptied mended 73. had had 74. seize 75.obeyed 76. would ride77.sets off 78. was driven 79.for either 80.How long 81. isn’t any longer 82. How well 83.Visitors can explore the history of the Castle through a tour with an expert guide.84-88 DABCD89-96 BCCBCDAA97-103 public important how quickly needed beginning get104.At Palo Alto,California high school.105.He should take over the care and feeding of the couple’s four children,plus all household chores 106.She works as a secretary107. He would share the child-raising andhousehold tasks with Pat.108.Yes, he did.109.He shut the door.。
2025年上海市数学高考一轮复习重难点 专题1集合与逻辑(考点练+模拟练)含详解
专题01集合与逻辑（考点练+模拟练）一、填空题1．（23-24高三上·上海·期中）已如全集U =R ，集合10,x A x x x ⎧⎫-=≥∈⎨⎬⎩⎭R ，则A =．2．（23-24高三上·上海黄浦·开学考试）“0x ≠或0y ≠”是“220x y +≠”的条件.3．（2023·上海普陀·模拟预测）已知命题p ：任意正数x ，恒有()1e 1xx +>，则命题p 的否定为．4．（23-24高三上·上海·期中）已知集合()2,1A =-，()()4,11,2B =-- ，则A B =．5．（22-23高一上·上海复旦附中分校·阶段练习）已知全集U =R ，集合{|1},{|2}A x x B x x =≤=≥，则A B ＝．6．（23-24高三上·上海奉贤·阶段练习）已知集合{}ln M x y x ==，集合11N y y x ⎧⎫==⎨⎬-⎩⎭，则M N ⋂=．7．（23-24高三上·上海松江·期中）已知2:280,:123p x x q a x a --<-<<-，且p 是q 的充分不必要条件，则实数a 的取值范围是.8．（23-24高三上·上海静安·开学考试）集合{}1,2,A a =，{}21,2B a =-，若集合A B ⋃中有三个元素，则实数=a .9．（23-24高一上·河北邯郸·阶段练习）若集合{}N |12A x x =∈-<≤，{},,B x x ab a b A ==∈，则集合B 的非空真子集的个数为.10．（20-21高三上·上海崇明·阶段练习）已知:31x m α<-或x m >-，:2x β<或4x ≥，若α是β的必要条件，则实数m 的取值范围是.11．（20-21高一上·上海闵行·期中）已知集合M ＝25|0ax x x a -⎧⎫<⎨⎬-⎩⎭，若3,5M M ∈∉，则实数a 的取值范围是．12．（23-24高三上·上海浦东新·期中）M 是正整数集的子集，满足：1,2022,2023M M M ∈∈∉，并有如下性质：若a 、b M ∈，则222a b M ⎤+∈⎥⎥⎦，其中[]x 表示不超过实数x 的最大整数，则M 的非空子集个数为．二、单选题13．（23-24高三上·上海浦东新·阶段练习）已知集合π,2m A x x m ⎧⎫==∈⎨⎬⎩⎭Z ，集合π,4n B x x n ⎧⎫==∈⎨⎬⎩⎭Z ，则A B = （）A ．∅B ．AC ．BD ．{}π,x x k k =∈Z 14．（16-17高一上·上海浦东新·期中）已知集合A ，B ，若A 不是B 的子集，则下列命题中正确的是（）A ．对任意的a A ∈，都有aB ∉B ．对任意的a B ∈，都有a A ∈C ．存在0a ，满足0a A ∈，且0a B∉D ．存在0a ，满足0a A ∈，且0a B∈15．（21-22高三上·上海浦东新·阶段练习）集合,A B 各有8个元素,A B ⋂有6个元素,若集合C 满足:()()A B C A B ⊆⊆ ,则满足条件的集合C 共有（）A ．32个B ．16个C ．8个D ．4个16．（20-21高三上·浙江·开学考试）设集合,S T 中至少两个元素，且,S T 满足：①对任意,x y S ∈，若x y ≠，则x y T +∈，②对任意,x y T ∈，若x y ≠，则x y S -∈，下列说法正确的是（）A ．若S 有2个元素，则S T 有3个元素B ．若S 有2个元素，则S T 有4个元素C ．存在3个元素的集合S ，满足S T 有5个元素D ．存在3个元素的集合S ，满足S T 有4个元素三、解答题17．（23-24高三上·上海静安·阶段练习）设全集()(){}4230,0A x ax x a a =+-+>>，B x y ⎧⎪==⎨⎪⎩.(1)若2a =，求A B ⋂，A B ；(2)若“x B ∈”是“x A ∈”的充分不必要条件，求实数a 的取值范围.18．（22-23高三上·上海青浦·期中）已知集合{}(2)(3)0A x x x =--≤，{}3B x a x a =<<，且0a >．(1)若x A ∈是x B ∈的充分条件，求实数a 的取值范围；(2)若命题“A B ⋂=∅”为假命题，求实数a 的取值范围．19．（22-23高三上·上海崇明·阶段练习）已知R 为全集，集合R 21|1,1x A x x x -⎧⎫=≤∈⎨⎬+⎩⎭，集合{}1,R B x x a x =-≤∈．(1)求集合A ；(2)若B A B ⋂=，求实数a 的取值范围．20．（22-23高三上·上海浦东新·阶段练习）设全集U 为R ，集合{}11A x x =-<，{}2320B x x x =--≥.(1)求A B ；(2)若{}22430C x x ax a A B =-+≥⊇⋃，求a 的取值范围.21．（23-24高一上·上海·期中）集合{}12,,,n A a a a =⋅⋅⋅是由()3n n >个正整数组成的集合，如果任意去掉其中一个元素()1,2,,i a i n =⋅⋅⋅之后，剩余的所有元素组成的集合都能分为两个交集为空的集合，且这两个集合的所有元素之和相等，就称集合A 为“可分集合”．(1)判断集合{}1,2,3,4、{}1,3,5,7,9,11,13是否为“可分集合”（不用说明理由）；(2)求证：五个元素的集合{}12345,,,,A a a a a a =一定不是“可分集合”；(3)若集合{}12,,,n A a a a = 是“可分集合”，证明n 是奇数．一、填空题1．（2022·上海·模拟预测）已知集合{}2=|40,A x x x x N *-<∈，则用列举法表示集合A =2．（2022·上海浦东新·模拟预测）已知集合()0,2A =，()1,3B =，则A B ⋃=.3．（2024·上海·三模）已知集合{}0,1,2A =，{}331B x x x =-≤，则A B =4．（2024·上海·三模）已知集合{}1,3,4A =，{},1B a a =+，若A B B = ，则=a ．5．（2024·上海·三模）已知集合{}11A x x =-<，11B x x ⎧⎫=<⎨⎬⎩⎭，则A B =．6．（2023·上海静安·二模）若集合{}22,log A a =，{},B a b =，且{}0A B ⋂=，则A B ⋃=.7．（2023·上海青浦·二模）已知集合(){}{}|ln 3,|A x y x B x x a ==-=>，若A B ⋂=∅，则实数a 的取值范围为.8．（2024·上海宝山·二模）已知集合{}2,1,3A a a =++，且1A ∈，则实数a 的值为.9．（2017·上海奉贤·一模）已知互异实数0mn ≠，集合{}{}22,,m n m n =，则m n +=．10．（2023·上海金山·一模）若集合()(){}2,20A x y x y x y =+++-≤，()()(){}222,211B x y x a y a a =-+--≤-，且A B ⋂≠∅，则实数a 的取值范围是.11．（2022·上海青浦·二模）已知集合1,[,1]6A s s t t ⎡⎤=++⎢⎥⎣⎦，其中1A ∉且16s t +<，函数()1xf x x =-，且对任意a A ∈，都有()f a A ∈，则t 的值是．12．（2022·上海普陀·一模）设非空集合Q M ⊆，当Q 中所有元素和为偶数时（集合为单元素时和为元素本身），称Q 是M 的偶子集，若集合{}1,2,3,4,5,6,7=M ，则其偶子集Q 的个数为.二、单选题13．（2022·上海·模拟预测）已知集合(){},2A x y x y =+=，(){},24B x y x y =-=-，则A B = （）A ．{}0,2B ．()0,2C ．∅D ．(){}0,214．（2023·上海普陀·二模）设,a b 为实数，则“0a b >>”的一个充分非必要条件是（）A>B ．22a b >C ．11b a>D ．a b b a->-15．（2023·上海普陀·一模）设1A 、2A 、3A 、L 、7A 是均含有2个元素的集合，且17A A ⋂=∅，()11,2,3,,6i i A A i +⋂=∅= ，记1237B A A A A =⋃⋃⋃⋃ ，则B 中元素个数的最小值是（）A ．5B ．6C ．7D ．816．（2021·上海青浦·一模）设函数,()1,x x P f x x Mx-∈⎧⎪=⎨∈⎪⎩，其中,P M 是实数集R 的两个非空子集，又规定()(){},A P y y f x x P ==∈，()(){},A M y y f x x M ==∈，则下列说法：（1）一定有()()A P A M ⋂=∅；（2）若P M R ⋃≠，则()()A P A M R ⋃≠；（3）一定有P M ⋂=∅；（4）若P M R ⋃=，则()()A P A M R ⋃=.其中正确的个数是（）A ．1B ．2C ．3D ．4三、解答题17．（2017·上海浦东新·三模）数列{}n a 的前n 项12,,,n a a a ⋅⋅⋅()*N n ∈组成集合{}12,,,n n A a a a =⋅⋅⋅，从集合n A 中任取(1,2,3,,)k k n =⋅⋅⋅个数，其所有可能的k 个数的乘积的和为k T （若只取一个数，规定乘积为此数本身），例如：对于数列{21}n -，当1n =时，1{1},A =11;T =2n =时，2{1,3},A =113,T =+213T =⋅；(1)若集合{1,3,5,,21}n A n =⋅⋅⋅-，求当3n =时，1,T 2,T 3T 的值；(2)若集合{}1,3,7,,21nn A =⋅⋅⋅-，证明：n k =时集合k A 的m T 与1n k =+时集合1k A +的m T （为了以示区别，用m T '表示）有关系式()1121k m m m T T T +-'=-+，其中*,N ,m k ∈2m k ≤≤；(3)对于（2）中集合n A ．定义12=+++…n n S T T T ，求n S （用n 表示）．专题01集合与逻辑（考点练+模拟练）一、填空题1．（23-24高三上·上海·期中）已如全集U =R ，集合10,x A x x x ⎧⎫-=≥∈⎨⎬⎩⎭R ，则A =．【答案】{}01x x ≤<【分析】解出集合A ，利用补集的定义可求得集合A .【解析】由10x x -≥可得()100x x x ⎧-≥⎨≠⎩，解得0x <或1x ≥，则{0A x x =<或}1x ≥，又因为全集U =R ，则{}01A x x =≤<.故答案为：{}01x x ≤<.2．（23-24高三上·上海黄浦·开学考试）“0x ≠或0y ≠”是“220x y +≠”的条件.【答案】充要【分析】利用充分条件、必要条件的定义判断作答.【解析】命题“若0x ≠或0y ≠，则220x y +≠”是真命题，命题“若220x y +≠，则0x ≠或0y ≠”是真命题，所以“0x ≠或0y ≠”是“220x y +≠”的充要条件.故答案为：充要3．（2023·上海普陀·模拟预测）已知命题p ：任意正数x ，恒有()1e 1xx +>，则命题p 的否定为．【答案】存在正数0x ，使()001e 1xx +≤【分析】含有全称量词的否定，改成特称量词即可.【解析】由全称命题的否定为特称命题知：存在正数0x ，使()001e 1xx +≤.故答案为：存在正数0x ，使()001e 1xx +≤4．（23-24高三上·上海·期中）已知集合()2,1A =-，()()4,11,2B =-- ，则A B = ．【答案】()2,1--【分析】直接由交集的概念、区间的表示即可得解.【解析】因为()2,1A =-，()()4,11,2B =-- ，所以()2,1A B ⋂=--.故答案为：()2,1--.5．（22-23高一上·上海复旦附中分校·阶段练习）已知全集U =R ，集合{|1},{|2}A x x B x x =≤=≥，则A B ＝．6．（23-24高三上·上海奉贤·阶段练习）已知集合{}ln M x y x ==，集合11N y y x ⎧⎫==⎨⎬-⎩⎭，则M N ⋂=．【答案】()0,∞+【分析】根据函数的定义域及值域结合交集的运算求值即可.【解析】由题意可知()()()0,,,00,M N ∞∞∞=+=-⋃+，所以()0,M N ∞⋂=+.故答案为：()0,∞+7．（23-24高三上·上海松江·期中）已知2:280,:123p x x q a x a --<-<<-，且p 是q 的充分不必要条件，则实数a 的取值范围是.8．（23-24高三上·上海静安·开学考试）集合{}1,2,A a =，{}21,2B a =-，若集合A B ⋃中有三个元素，则实数=a .【答案】2-或1-【分析】集合A B ⋃中有三个元素，则222a -=或22a a -=，解方程并检验即可.【解析】集合{}1,2,A a =，{}21,2B a =-，若集合A B ⋃中有三个元素，则222a -=或22a a -=，若222a -=，解得2a =±，其中2a =与元素互异性矛盾舍去，2a =-满足题意；若22a a -=，解得2a =或1a =-，2a =舍去，1a =-满足题意，所以2a =-或1a =-.故答案为：2-或1-9．（23-24高一上·河北邯郸·阶段练习）若集合{}N |12A x x =∈-<≤，{},,B x x ab a b A ==∈，则集合B 的非空真子集的个数为.10．（20-21高三上·上海崇明·阶段练习）已知:31x m α<-或x m >-，:2x β<或4x ≥，若α是β的必要条件，则实数m 的取值范围是.11．（20-21高一上·上海闵行·期中）已知集合M ＝2|0x x a -⎧⎫<⎨⎬-⎩⎭，若3,5M M ∈∉，则实数a 的取值范围是．12．（23-24高三上·上海浦东新·期中）M 是正整数集的子集，满足：1,2022,2023M M M ∈∈∉，并有如下性质：若a 、b M ∈，则M ∈，其中[]x 表示不超过实数x 的最大整数，则M 的非空子集个数为．二、单选题13．（23-24高三上·上海浦东新·阶段练习）已知集合π,2m A x x m ⎧⎫==∈⎨⎬⎩⎭Z ，集合π,4n B x x n ⎧⎫==∈⎨⎬⎩⎭Z ，则A B = （）A ．∅B ．AC ．BD ．{}π,x x k k =∈Z14．（16-17高一上·上海浦东新·期中）已知集合A ，B ，若A 不是B 的子集，则下列命题中正确的是（）A ．对任意的a A ∈，都有aB ∉B ．对任意的a B ∈，都有a A ∈C ．存在0a ，满足0a A ∈，且0a B∉D ．存在0a ，满足0a A ∈，且0a B∈【答案】C【分析】根据子集关系结合元素与集合的关系逐项分析判断.【解析】对于选项A 、B ：例如{}{}1,2,2,3A B ==，满足A 不是B 的子集，但2,2A B ∈∈，故A 错误；3,3A B ∉∈，故B 错误；对于选项C ：对任意的a A ∈，都有a B ∈，则A B ⊆，若A 不是B 的子集，则存在0a ，满足0a A ∈，且0a B ∉，故C 正确；对于选项D ：例如{}{}1,2A B ==，满足A 不是B 的子集，但不存在0a ，满足0a A ∈，且0a B ∈，故D 错误；故选：C.15．（21-22高三上·上海浦东新·阶段练习）集合,A B 各有8个元素,A B ⋂有6个元素,若集合C 满足:()()A B C A B ⊆⊆ ,则满足条件的集合C 共有（）A ．32个B ．16个C ．8个D ．4个【答案】B【分析】根据题意设出集合,A B ,根据()()A B C A B ⊆⊆ 判断集合C 中元素的构成情况,根据子集和集合中元素的个数关系即可得出结果.【解析】解:由题知,A B 各有8个元素,且A B ⋂有6个元素,设{}123456,,,,,c c c c A c c B = ,且{}12123456,,,,,,,,a a c c c c c c A ={}12123456,,,,,,,b bc c c c c B c =,则画Venn 图如下:因为()()A B C A B ⊆⊆ ,所以{}{}1234561212123456,,,,,,,,,,,,,,,c c c c c c C a a b b c c c c c c ⊆⊆所以集合C 中至少有123456,,,,,c c c c c c ,6个元素,最多有1212123456,,,,,,,,,a a b b c c c c c c ,10个元素,只需求出{}1212,,,a a b b 的子集,在每个子集中加入123456,,,,,c c c c c c 6个元素,即可得集合C ,所以集合C 的个数,即是{}1212,,,a a b b 的子集的个数4216=个.故选:B16．（20-21高三上·浙江·开学考试）设集合,S T 中至少两个元素，且,S T 满足：①对任意,x y S ∈，若x y ≠，则x y T +∈，②对任意,x y T ∈，若x y ≠，则x y S -∈，下列说法正确的是（）A ．若S 有2个元素，则S T 有3个元素B ．若S 有2个元素，则S T 有4个元素C ．存在3个元素的集合S ，满足S T 有5个元素D ．存在3个元素的集合S ，满足S T 有4个元素【答案】A【解析】不妨设{,}S a b =，由②知集合S 中的两个元素必为相反数，设{,}S a a =-，由①得0T ∈，由于集合T 中至少两个元素，得到至少还有另外一个元素m T ∈，分集合T 有2个元素和多于2个元素分类讨论，即可求解.【解析】若S 有2个元素，不妨设{,}S a b =，以为T 中至少有两个元素，不妨设{},x y T ⊆，由②知,x y S y x S -∈-∈，因此集合S 中的两个元素必为相反数，故可设{,}S a a =-，由①得0T ∈，由于集合T 中至少两个元素，故至少还有另外一个元素m T ∈，当集合T 有2个元素时，由②得：m S -∈，则,{0,}m a T a =±=-或{0,}T a =.当集合T 有多于2个元素时，不妨设{0,,}T m n =，其中,,,,,m n m n m n n m S ----∈，由于,0,0m n m n ≠≠≠，所以,m m n n ≠-≠-，若m n =-，则n m =-，但此时2,2m n m m m n n n -=≠-=-≠，即集合S 中至少有,,m n m n -这三个元素，若m n ≠-，则集合S 中至少有,,m n m n -这三个元素，这都与集合S 中只有2个运算矛盾，综上，{0,,}S T a a =- ，故A 正确；当集合S 有3个元素，不妨设{,,}S a b c =，其中a b c <<，则{,,}a b b c c a T +++⊆，所以,,,,,c a c b b a a c b c a b S ------∈，集合S 中至少两个不同正数，两个不同负数，即集合S 中至少4个元素，与{,,}S a b c =矛盾，排除C ，D.故选：A.【点睛】解题技巧：解决以集合为背景的新定义问题要抓住两点：1、紧扣新定义，首先分析新定义的特点，把心定义所叙述的问题的本质弄清楚，应用到具体的解题过程中；2、用好集合的性质，解题时要善于从试卷中发现可以使用的集合的性质的一些因素.三、解答题17．（23-24高三上·上海静安·阶段练习）设全集()(){}4230,0A x ax x a a =+-+>>，B x y ⎧⎪==⎨⎪⎩.(1)若2a =，求A B ⋂，A B ；(2)若“x B ∈”是“x A ∈”的充分不必要条件，求实数a 的取值范围.18．（22-23高三上·上海青浦·期中）已知集合{}(2)(3)0A x x x =--≤，{}3B x a x a =<<，且0a >．(1)若x A ∈是x B ∈的充分条件，求实数a 的取值范围；(2)若命题“A B ⋂=∅”为假命题，求实数a 的取值范围．19．（22-23高三上·上海崇明·阶段练习）已知R 为全集，集合R |1,1A x x x -⎧⎫=≤∈⎨⎬+⎩⎭，集合{}1,R B x x a x =-≤∈．(1)求集合A ；(2)若B A B ⋂=，求实数a 的取值范围．20．（22-23高三上·上海浦东新·阶段练习）设全集U 为R ，集合11A x x =-<，{}2320B x x x =--≥.(1)求A B ；(2)若{}22430C x x ax a A B =-+≥⊇⋃，求a 的取值范围.21．（23-24高一上·上海·期中）集合{}12,,,n A a a a =⋅⋅⋅是由()3n n >个正整数组成的集合，如果任意去掉其中一个元素()1,2,,i a i n =⋅⋅⋅之后，剩余的所有元素组成的集合都能分为两个交集为空的集合，且这两个集合的所有元素之和相等，就称集合A 为“可分集合”．(1)判断集合{}1,2,3,4、{}1,3,5,7,9,11,13是否为“可分集合”（不用说明理由）；(2)求证：五个元素的集合{}12345,,,,A a a a a a =一定不是“可分集合”；(3)若集合{}12,,,n A a a a = 是“可分集合”，证明n 是奇数．【答案】(1){}1,2,3,4不是“可分集合”，{}1,3,5,7,9,11,13为“可分集合”(2)证明见解析(3)证明见解析【分析】（1）由“可分集合”的定义判断；（2）不妨设12345a a a a a <<<<，讨论当在集合{}12345,,,,a a a a a 中去掉元素1a 、2a 后，将剩余元素构成的集合，结合“可分集合”的定义进行分拆，得出等式，推出矛盾，即可证得结论成立；（3）根据集合中元素总和与单个元素的奇偶性讨论后证明.【解析】（1）解：对于{}1,2,3,4，去掉3后，{}1,2,4不满足题中条件，故{}1,2,3,4不是“可分集合”，对于{}1,3,5,7,9,11,13，集合{}1,3,5,7,9,11,13所有元素之和为49.当去掉元素1时，剩下的元素之和为48，剩下元素可以组合{}3,5,7,9、{}11,13这两个集合，显然符合题意；当去掉元素3时，剩下的元素之和为46，剩下元素可以组合{}1,9,13、{}5,7,11这两个集合，显然符合题意；当去掉元素5时，剩下的元素之和为44，剩下元素可以组合{}1,3,7,11、{}9,13这两个集合，显然符合题意；当去掉元素7时，剩下的元素之和为42，剩下元素可以组合{}1,9,11、{}3,5,13这两个集合，显然符合题意；当去掉元素9时，剩下的元素之和为40，剩下元素可以组合{}1,3,5,11、{}7,13这两个集合，显然符合题意；当去掉元素11时，剩下的元素之和为38，剩下元素可以组合{}3,7,9、{}1,5,13这两个集合，显然符合题意；当去掉元素13时，剩下的元素之和为36，剩下元素可以组合{}1,3,5,9、{}7,11这两个集合，显然符合题意.综上所述，集合{}1,3,5,7,9,11,13是“可分集合”.（2）证明：不妨设123450a a a a a <<<<<，一、填空题1．（2022·上海·模拟预测）已知集合{}2=|40,A x x x x N *-<∈，则用列举法表示集合A =【答案】{}1,2,3【分析】根据不等式的解法，求得04x <<，进而利用列举法，即可求解.【解析】由不等式240x x -<，可得()40x x -<，解得04x <<，即集合{|04A x x =<<且}{1,2,3}x N *∈=.故答案为：{}1,2,3.2．（2022·上海浦东新·模拟预测）已知集合()0,2A =，()1,3B =，则A B ⋃=.【答案】()0,3【分析】直接根据并集定义求解即可.【解析】因为()0,2A =，()1,3B =，所以()0,3A B ⋃=，故答案为：()0,33．（2024·上海·三模）已知集合{}0,1,2A =，{}331B x x x =-≤，则A B =【答案】{}0,1【分析】把集合中的元素代入不等式331x x -≤检验可求得{0,1}A B = .【解析】当0x =时，303001-⨯=≤，所以0B ∈，当1x =时，313121-⨯=-≤，所以1B ∈，当2x =时，323221-⨯=>，所以2∉B ，所以{0,1}A B = .故答案为：{0,1}.4．（2024·上海·三模）已知集合{}1,3,4A =，{},1B a a =+，若A B B = ，则=a ．【答案】3【分析】根据给定条件，利用交集的结果直接列式计算即得.【解析】集合{}1,3,4A =，{},1B a a =+，由A B B = ，得B A ⊆，又11a a +-=，因此143a a +=⎧⎨=⎩，所以3a =.故答案为：35．（2024·上海·三模）已知集合{}11A x x =-<，11B x x ⎧⎫=<⎨⎬⎩⎭，则A B =．6．（2023·上海静安·二模）若集合{}22,log A a =，{},B a b =，且{}0A B ⋂=，则A B ⋃=.【答案】{}0,1,2【分析】依题意可得0A ∈且0B ∈，即可求出a 、b 的值，从而求出集合A 、B ，再根据并集的定义计算可得.【解析】因为{}22,log A a =，{},B a b =，且{}0A B ⋂=，所以0A ∈且0B ∈，显然0a >，所以2log 0a =且0b =，所以1a =，所以{}2,0A =，{}1,0B =，所以{}0,1,2A B = .故答案为：{}0,1,27．（2023·上海青浦·二模）已知集合(){}{}|ln 3,|A x y x B x x a ==-=>，若A B ⋂=∅，则实数a 的取值范围为.【答案】[)3,+∞【分析】求函数的定义域求得集合A ，根据A B ⋂=∅求得a 的取值范围.【解析】由30x ->解得3x <，所以(),3A =-∞，由于A B ⋂=∅，所以3a ≥，所以a 的取值范围是[)3,+∞.故答案为：[)3,+∞8．（2024·上海宝山·二模）已知集合{}2,1,3A a a =++，且1A ∈，则实数a 的值为.9．（2017·上海奉贤·一模）已知互异实数0mn ≠，集合{}{}22,,m n m n =，则m n +=．【答案】-1【分析】分情况讨论2m m =,2n n =,或2n m =,2m n =再计算即可.【解析】互异实数m n ≠,集合{}{}22,,m n m n =,∴2m m =,2n n =,或2n m =,2m n =,0mn ≠,m n ≠．由2m m =,2n n =,0mn ≠,m n ≠,无解．由2n m =,2m n =,0mn ≠,m n ≠．可得22n m m n -=-,解得1m n +=-．故答案为：1-．【点睛】本题主要考查了根据集合的互异性与集合相等求参数的问题,属于基础题型.10．（2023·上海金山·一模）若集合()(){}2,20A x y x y x y =+++-≤，()()(){}222,211B x y x a y a a =-+--≤-，且A B ⋂≠∅，则实数a 的取值范围是.B 其中()()2221x a y a -+--当1a =±时，B 表示点(1,3)当1a ≠±时，B 表示以(M 其圆心在直线21y x =+上，依题意A B ⋂≠∅，即表示圆当1a =-时，显然满足题意，当当1a <-时，因为A B ⋂≠所以d r ≤，即222a a +++所以()()17110a a ++≤，所以1117a -≤<-；当1a >时，因为A B ⋂≠∅11．（2022·上海青浦·二模）已知集合,[,1]6A s s t t ⎡⎤=++⎢⎥⎣⎦ ，其中1A ∉且6s t +<，函数()1xf x x =-，且对任意a A ∈，都有()f a A ∈，则t 的值是．12．（2022·上海普陀·一模）设非空集合Q M ⊆，当Q 中所有元素和为偶数时（集合为单元素时和为元素本身），称Q 是M 的偶子集，若集合{}1,2,3,4,5,6,7=M ，则其偶子集Q 的个数为.【答案】63【分析】对集合Q 中奇数和偶数的个数进行分类讨论，确定每种情况下集合Q 的个数，综合可得结果.【解析】集合Q 中只有2个奇数时，则集合Q 的可能情况为：{}1,3、{}1,5、{}1,7、{}3,5、{}3,7、{}5,7，共6种，若集合Q 中只有4个奇数时，则集合{}1,3,5,7Q =，只有一种情况，若集合Q 中只含1个偶数，共3种情况；若集合Q 中只含2个偶数，则集合Q 可能的情况为{}2,4、{}2,6、{}4,6，共3种情况；若集合Q 中只含3个偶数，则集合{}2,4,6Q =，只有1种情况.因为Q 是M 的偶子集，分以下几种情况讨论：若集合Q 中的元素全为偶数，则满足条件的集合Q 的个数为7；若集合Q 中的元素全为奇数，则奇数的个数为偶数，共7种；若集合Q 中的元素是2个奇数1个偶数，共6318⨯=种；若集合Q 中的元素为2个奇数2个偶数，共6318⨯=种；若集合Q 中的元素为2个奇数3个偶数，共616⨯=种；若集合Q 中的元素为4个奇数1个偶数，共133⨯=种；若集合Q 中的元素为4个奇数2个偶数，共133⨯=种；若集合Q 中的元素为4个奇数3个偶数，共1种.综上所述，满足条件的集合Q 的个数为771818633163+++++++=.故答案为：63.二、单选题13．（2022·上海·模拟预测）已知集合(){},2A x y x y =+=，(){},24B x y x y =-=-，则A B = （）A ．{}0,2B ．()0,2C ．∅D ．(){}0,214．（2023·上海普陀·二模）设,a b 为实数，则“0a b >>”的一个充分非必要条件是（）A >B ．22a b >C ．11b a >D ．a b b a->-15．（2023·上海普陀·一模）设1A 、2A 、3A 、L 、7A 是均含有2个元素的集合，且17A A ⋂=∅，()11,2,3,,6i i A A i +⋂=∅= ，记1237B A A A A =⋃⋃⋃⋃ ，则B 中元素个数的最小值是（）A ．5B ．6C ．7D ．8【答案】A 【分析】设1x 、2x 、L 、()4n x n ≥是集合B 互不相同的元素，分析可知4n ≥，然后对n 的取值由小到大进行分析，验证题中的条件是否满足，即可得解.【解析】解：设1x 、2x 、L 、()4n x n ≥是集合B 互不相同的元素，若3n =，则12A A ⋂≠∅，不合乎题意.①假设集合B 中含有4个元素，可设{}112,A x x =，则{}24634,A A A x x ===，{}35712,A A A x x ===，这与17A A ⋂=∅矛盾；②假设集合B 中含有5个元素，可设{}1612,A A x x ==，{}2734,A A x x ==，{}351,A x x =，{}423,A x x =，{}545,A x x =，满足题意.综上所述，集合B 中元素个数最少为5.故选：A.【点睛】关键点点睛：本题考查集合元素个数的最值的求解，解题的关键在于对集合元素的个数由小到大进行分类，对集合中的元素进行分析，验证题中条件是否成立即可.16．（2021·上海青浦·一模）设函数,()1,x x P f x x M x -∈⎧⎪=⎨∈⎪⎩，其中,P M 是实数集R 的两个非空子集，又规定()(){},A P y y f x x P ==∈，()(){},A M y y f x x M ==∈，则下列说法：（1）一定有()()A P A M ⋂=∅；（2）若P M R ⋃≠，则()()A P A M R ⋃≠；（3）一定有P M ⋂=∅；（4）若P M R ⋃=，则()()A P A M R ⋃=.其中正确的个数是（）A ．1B ．2C ．3D ．4【答案】B【解析】根据分段函数的定义、一次函数和反比例函数的性质，结合集合交集、并集的运算定义进行判断即可.【解析】函数()f x 是分段函数，故P M ⋂=∅一定成立，因此说法（3）正确；对于（1）：当{}{}1,1P M =-=时，根据已知的规定，有{}{}()1,()1A P A M ==，显然()(){}1A P A M ⋂=≠∅，因此说法（1）不正确；对于（4）：当(,1),[1,)P M =-∞=+∞时，显然满足P M R ⋃=成立，根据已知的规定，有()(1,),()(0,1]A P A M =-+∞=，显然()()(1,)(0,1]A P A M R ⋃=-+∞⋃≠，因此说法（4）不正确；对于（2）来说，当P M R ⋃=时，()()A P A M R ⋃=不一定成立，故当P M R ⋃≠时，显然()()A P A M R ⋃≠一定成立，因此说法（2）正确，所以只有（2）（3）说法正确.故选：B三、解答题17．（2017·上海浦东新·三模）数列{}n a 的前n 项12,,,n a a a ⋅⋅⋅()*N n ∈组成集合{}12,,,n n A a a a =⋅⋅⋅，从集合n A 中任取(1,2,3,,)k k n =⋅⋅⋅个数，其所有可能的k 个数的乘积的和为k T （若只取一个数，规定乘积为此数本身），例如：对于数列{21}n -，当1n =时，1{1},A =11;T =2n =时，2{1,3},A =113,T =+213T =⋅；(1)若集合{1,3,5,,21}n A n =⋅⋅⋅-，求当3n =时，1,T 2,T 3T 的值；(2)若集合{}1,3,7,,21n n A =⋅⋅⋅-，证明：n k =时集合k A 的m T 与1n k =+时集合1k A +的m T （为了以示区别，用m T '表示）有关系式()1121k m m m T T T +-'=-+，其中*,N ,m k ∈2m k ≤≤；(3)对于（2）中集合n A ．定义12=+++…n n S T T T ，求n S （用n 表示）．。
2022上海高二数学考试满分攻略(沪教版2020第一册)第3章 空间向量及其应用(单元提升卷)解析
第３章 空间向量及其应用（单元提升卷）（考试时间：120分钟 试卷满分：150分）注意事项：1． 本试卷共4页，21道试题，满分150分，考试时间120分钟．2． 本试卷分设试卷和答题纸．试卷包括试题与答题要求．作答必须涂（选择题）或写（非选择题）在答题纸上，在试卷上作答一律不得分．3． 答卷前，务必用钢笔或圆珠笔在答题纸正面清楚地填写姓名、准考证号码等相关信息． 一、填空题（本大题共有12题，满分54分，第1-6题每题4分，第7-12题每题5分）考生应在答题纸的相应位置直接填写结果．1．（2021·上海市张堰中学高三阶段练习）已知空间向量(),a x y =-，(),b y x =-，则a 与b 的夹角为______. 【答案】2π【解析】 【分析】计算出0a b ⋅=，由此可得出a 与b 的夹角. 【详解】由已知条件可得20a b xy xy xy ⋅=--+=，a b ∴⊥，因此，a 与b 的夹角为2π. 故答案为：2π. 【点睛】本题考查利用空间向量的数量积求空间向量的夹角，考查计算能力，属于基础题. 2．（2022·上海·高三专题练习）设1e ，2e 是平面内不共线的向量，已知122AB e ke =+，123CB e e =+，122CD e e =-，若A ，B ，D 三点共线，则k =____.【答案】8- 【解析】 【分析】求出BD ，利用三点共线，得到AB BD λ=，求出λ和k. 【详解】由题意，()121212234BD CD CB e e e e e e =-=--+=-，又122AB e ke =+，且A 、B 、D 三点共线，由共线向量定理得，存在实数λ使得AB BD λ=成立， 即()121224e ke e e λ+=-，则24k λλ=⎧⎨=-⎩，解得8k =-.故答案为：8-.3．（2022·上海·高三专题练习）若向量()()1,0,1,0,1,1a b ==-，则向量,a b 的夹角为_____． 【答案】23π【解析】 【分析】直接利用空间向量的夹角公式求解. 【详解】根据题意，设向量,a b 的夹角为θ， 向量()()1,0,1,0,1,1a b ==- 则向量2,2,1a b a b ==⋅=- 则1cos2θ==- 又由0θπ≤≤，则23πθ= 故答案为：23π． 4．（2021·上海·高一课时练习）已知向量a 和b 夹角为120︒，且||2||5a b ==，则(2)a b a -⋅=_______． 【答案】2254【解析】 【分析】 由题设可得254a b ⋅=-，应用向量数量积的运算律有2(2)2a b a a b a -⋅=-⋅，即可求值. 【详解】由题意，25125||||cos ,()224a b a b a b ⋅=<>=⨯-=-， ∴225225(2)250()44a b a a b a -⋅=-⋅=--=. 故答案为：22545．（2022·上海·高三专题练习）已知正四棱柱1111ABCD A B C D -中，1AA =2AB ，E 为1AA 中点，则异面直线BE 与1CD 所形成角的余弦值为 ． 【答案】【解析】 【详解】 如图：连结，则，所以即为异面直线与所成角，设，则，，，，，由余弦定理得6．（2021·上海市金山中学高二期末）四棱锥P ABCD -中，底面ABCD 是一个平行四边形，()2,1,4AB =--，()4,2,0AD =，()1,2,1AP =--，则四棱锥P ABCD -的体积为_______ 【答案】16 【解析】 【分析】计算0AP AB ⋅=，0AP AD ⋅=，得到AP ⊥底面ABCD ，计算cos BAD ∠=sin ABCD S AB AD BAD =⋅∠=.【详解】由2240AP AB ⋅=--+=，4400AP AD ⋅=-++=，所以AP ⊥底面ABCD ，8cos 21AB AD BAD AB AD⋅-∠===⋅故sin ABCD S AB AD BAD =⋅∠==，体积为111633ABCD V S AP =⋅=⨯=. 故答案为：16.7．（2022·上海·高三专题练习）设正四面体1234PP P P 在空间直角坐标系中点i P 的坐标为(),,(1,2,3,4)i i i x y z i =，集合A ={y |存在{}1,2,3,4i ∈，使得i y y =}，则集合A 的元素个数可能为__________种.（写出所有可能的值） 【答案】2或3或4 【解析】 【分析】分析正四面体的某个面或棱与坐标面xOz 的关系即可确定四个点的纵坐标的不同取值而得解. 【详解】在空间直角坐标系O -xyz 中，正四面体1234PP P P 的四个顶点不共面，即A 中至少有两个元素，正四面体1234PP P P 的某个面(不妨令平面234P P P )所在平面与平面xOz 平行或重合时，2341y y y y ==≠，即A 中有两个元素；正四面体1234PP P P 的任意一个面所在的平面与平面xOz 都不平行且不重合，而某条棱所在的直线与平面xOz 不相交(不妨令棱34P P )时，y 3=y 4，且y 1，y 2，y 3互不相等，即A 中有三个元素；正四面体1234PP P P 的所有棱所在直线与平面xOz 都相交时，y 1，y 2，y 3，y 4中任意两个都不相等，即A 中有四个元素. 故答案为：2或3或4 【点睛】结论点睛：两个平面平行，一个平面内任意一点到另一个平面的距离都相等；一条直线与一个平面平行，这条直线上任意一点到这个平面的距离都相等.8．（2022·上海·华师大二附中高二期末）如图所示，在平行六面体1111ABCD A B C D -中，1111AC B D F =，若1AF xAB yAD zAA =++，则x y z ++=___________.【答案】2 【解析】 【分析】题中 几何体为平行六面体，就要充分利用几何体的特征进行转化，1111112AF AB BB B F AB BB B D =++=++,再将11A D 转化为AD ，以及将11A B 转化为AB ,11BB AA =,总之等式右边为AB ，AD ，1AA ,从而得出12x y ==，1z =. 【详解】解：因为1111112AF AB BB B F AB BB B D =++=++()1111112AB BB A D A B =++- 11122AB BB AD AB =++- 11122AB AD AA =++， 又1AF xAB AD z AA =++， 所以12x y ==，1z =， 则2x y z ++=. 故答案为：2. 【点睛】要充分利用几何体的几何特征，以及将1AF xAB AD z AA =++作为转化的目标，从而得解. 9．（2021·上海市实验学校高二期中）在三棱锥D ABC -中，已知2AB AD ==，1BC =，3AC BD ⋅=-，则CD =___________【解析】 【分析】用,AB AD 表示BD ，根据条件列出方程建立,,AC BAC DAC ∠∠的关系，利用等量代换计算22||CD AD AC =-即得.【详解】设,BAC DAC αβ∠=∠=，显然||||1AC AB BC -==，则222||||cos 1AC AB AC AB α+-⋅=，即24||cos 3AC AC α-=-， 而3AC BD ⋅=-，即()3AC AD AB AC AD AC AB ⋅-=⋅-⋅=-，于是得2||cos 2||cos 3AC AC βα-=-，2||cos 32||cos AC AC βα=-+， 22222||244||cos CD AD AC AD AC AD AC AC AC β=-=+-⋅=+- 2242(32||cos )104||cos 7AC AC AC AC αα=+--+=+-=，则有||7CD =CD =10．（2020·上海·高三专题练习）正三棱锥的一个侧面与底面的面积之比为2:3，则这个三棱锥的侧面和底面所成二面角的大小为________． 【答案】60︒ 【解析】 【分析】由题意作出正三棱锥S ABC -，设O 为底面ABC 的中心，过S 作SE AB ⊥交AB 于点E ,连接EO ，可得SEO ∠为侧面和底面所成二面角的平面角，由条件23SAB ABCSS =，得出2SE OE=，从而得出答案. 【详解】如图在正三棱锥S ABC -中，设O 为底面ABC 的中心，连接SO ,则SO ⊥平面ABC . 过S 作SE AB ⊥交AB 于点E ,连接EO则SO AB ⊥,又SE AB ⊥，且SE SO S ⋂=,所以AB ⊥平面SEO 则OE AB ⊥,所以SEO ∠为侧面和底面所成二面角的平面角. 在正三角形ABC 中，O 为中心，3++32ABCOBCOABOACOABSSSSSAB OE ===由条件有122332SAB ABCAB SE S SAB OE ⋅==⋅，可得2SE OE = 在直角三角形SOE 中，1cos 2EO SEO ES∠== 所以60SEO ∠=° 故答案为：60︒【点睛】本题考查三棱锥的线面关系，正三棱锥的侧面面积与底面积的关系，考查二面角，属于中档题.11．（2020·上海·复旦附中高二期中）已知正三棱锥P ABC -的侧棱长为2020，过其底面中心O 作动平面α交线段PC 于点S ，交,PA PB 的延长线于,M N 两点，则111PS PM PN++的取值范围为__________【答案】32020⎧⎫⎨⎬⎩⎭【解析】设,,PM x PN y PS z ===,则111333zPA PB PCPO PM PN PS x y =⨯⋅+⨯⋅+⨯⋅,根据空间四点共面的条件，又,,,S M N O 四点共面，则202020202020+1333zx y +=，即得出答案. 【详解】设,,PM x PN y PS z ===. 则PA PA PM x=⋅,PB PB PN y=⋅,PC PC PS z=⋅.由O 为底面ABC 中心, ()2132PO PA AO PA AB AC =+=+⨯+()()133PA PB PCPA PB PA PC PA ++⎡⎤=+-+-=⎣⎦ 111333z PA PB PCPM PN PS x y =⨯⋅+⨯⋅+⨯⋅ 333zPA PB PC PM PN PS x y=⋅+⋅+⋅又因为,,,S M N O 四点共面,所以+1333zPA PB PC xy+=且2020PA PB PC ===.所以202020202020+1333z x y +=，即1113+z 2020x y +=即11132020PS PM PN ++=. 故答案为：32020⎧⎫⎨⎬⎩⎭.【点睛】本题考查空间四点共面的条件的应用，属于中档题.12．（2022·上海·高三专题练习）在空间直角坐标系中，点(,,)P x y z 满足：22216x y z ++=，平面α过点(1,2,3)M ，且平面α的一个法向量(1,1,1)n =，则点P 在平面α上所围成的封闭图形的面积等于__________. 【答案】4π 【解析】 【分析】由题意，点P 在球面上，所以点P 在平面α上所围成的封闭图形即为平面α截球面所得的截面圆，根据球的截面性质求出截面圆的半径r 即可求解. 【详解】解：由题意，点P 在以()0,0,0为球心，半径为4的球面上，所以点P 在平面α上所围成的封闭图形即为平面α截球面所得的截面圆， 因为平面α的方程为()()()1112130x y z ⨯-+⨯-+⨯-=，即60x y z ++-=，所以球心()0,0,0到平面α的距离为d ==，所以截面圆的半径2r =，截面圆的面积为24S r ππ==，所以点P 在平面α上所围成的封闭图形的面积等于4π. 故答案为：4π.二、选择题（本大题共有4题，满分20分，每题5分）每题有且只有一个正确选项，考生应在答题纸的相应位置，将代表正确选项的小方格涂黑．13．（2021·上海市金山中学高二期中）在正方体11ABCD ABC D -中，下列结论错误的是（ ） A ．()2211111113A A A D A B A B ++=B ．()11110AC A B A A ⋅-=C ．向量1AD 与1A B 的夹角是120°D ．正方体111ABCD AB C D -的体积为1AB AA AD ⋅⋅ 【答案】D 【解析】 【分析】根据空间向量的知识对每个选项逐一分析即可. 【详解】正方体1111ABCD A B C D - 如图所示，对于A 选项，22111111()A A A D A B AC ++=，2211111||(3||)3AC A B A B ==，故 A 正确； 对于B 选项，111111()AC A B A A AC AB ⋅-=⋅ ，1A C 在平面11ABB A 内的投影为1A B ，又因为11A B AB ⊥1AC AB ∴⊥，即110A C AB ⋅=，故B 正确； 对于C 选项，11A BC 为等边三角形1160A BC =∴∠︒，11//AD BC ，∴向量1AD 与1A B 的夹角是18060120︒-︒=︒，故 C 正确；对于D 选项，1AB AA ⊥，10AB AA ∴⋅=，故D 显然错误. 故选：D14．（2020·上海市七宝中学模拟预测）已知MN 是正方体内切球的一条直径，点Р在正方体表面上运动，正方体的棱长是2，则PM PN ⋅的取值范围为（ ） A ．[]0,4 B ．[]0,2 C ．[]1,4D ．[]1,2【答案】B 【解析】 【分析】利用向量的线性运算和数量积运算律可得21PM PN PO ⋅=-，根据正方体的特点确定PO 最大值和最小值，即可求解 【详解】设正方体内切球的球心为O ，则1OM ON ==,()()()2PM PN PO OM PO ON PO PO OM ON OM ON ⋅=+⋅+=+⋅++⋅,因为MN 是正方体内切球的一条直径， 所以0OM ON +=，1OM ON ⋅=-， 所以21PM PN PO ⋅=-， 又点Р在正方体表面上运动，所以当P 为正方体顶点时，PO 最大，且最大值为当P 为内切球与正方体的切点时，PO 最小 ，且最小为1； 所以2012PO ≤-≤，所以PM PN ⋅的取值范围为[]0,2， 故选：B15．（2022·上海·高三专题练习）如图一副直角三角板，现将两三角板拼成直二面角，得到四面体ABCD ，则下列叙述正确的是（ ） ①平面BCD 的法向量与平面ACD 的法向量垂直； ②异面直线BC 与AD 所成的角为60︒； ③四面体ABCD 有外接球；④直线DC 与平面ABC 所成的角为30.A ．②④B ．③C ．③④D ．①②③④【答案】C 【解析】 【分析】①由题设四面体相关侧面的关系即可判断正误；②过D 、C 作BC 、BD 的平行线且交于F ，连接AF ，则ADF ∠就是异面直线BC 与AD 的夹角，设1BD =求相关边的长度，再应用余弦定理求cos ADF ∠；③由四面体的性质即可知正误；④由面面垂直确定DC 与平面ABC 所成的角是DCB ∠，即知线面角的大小. 【详解】①平面BCD 的法向量与平面ABC 的法向量垂直，而与平面ACD 的法向量不垂直，故错误； ②过D 作BC 的平行线，过C 作BD 的平行线，两平行线交于点F ，联结AF ，则ADF ∠就是异面直线BC 与AD 的夹角，过A 作AE BC ⊥，联结ED 、EF ，若1BD =，则2,BD BC AB AC ====由BD BC ⊥，面BDC ⋂面ABC BC =，面BDC ⊥面ABC ，BD ⊂面BDC ，∴BD ⊥面ABC ，AB 面ABC ，则BD ⊥AB ，同理可证FC ⊥AC ，∴AD AF ==DF =cos ADF ∠=③由于所有的四面体都有外接球，故正确；④因为BD ⊥平面ABC ，所以DC 与平面ABC 所成的角是30DCB ∠=︒，正确. 故选：C16．（2022·上海·高三专题练习）如图，点A ､B ､C 分别在空间直角坐标系O xyz -的三条坐标轴上，()0,0,2OC =，()1,0,0OA =，()0,2,0OB =，设二面角C AB O --的大小为θ，则cos θ=（ ）A B C D 【答案】B 【解析】 【分析】先求出平面ABC 和平面ABO 的法向量，再利用二面角公式求解即可. 【详解】因为()0,0,2OC =，()1,0,0OA =，()0,2,0OB =， 所以()1,2,0AB =-，()1,0,2AC =-，设平面ABC 的法向量为(),,n x y z =，则00AC n AB n ⎧⋅=⎨⋅=⎩即2020x z x y -+=⎧⎨-+=⎩取()2,1,1n =又因为平面ABO 的法向量为()0,0,2OC =，所以2cos 2OC n OC nθ⋅===⨯⋅故选：B三、解答题（本大题共有5题，满分76分）解答下列各题必须在答题纸的相应位置写出必要的步骤．17．（2022·上海市实验学校高三开学考试）如图，在正三棱住111ABC A B C -中，14AA =，异面直线1BC 与1AA 所成角的大小为3π．(1)求正三棱柱111ABC A B C -的体积；(2)求直线1BC 与平面11AAC C 所成角的大小．（结果用反三角函数值表示）【答案】(1)(2)3arcsin 4【解析】 【分析】（1）由已知可得113C BB π∠=，又114B B A A ==，求得正三棱柱111ABC A B C -的底面边长为111ABC A B C -的体积；（2）在底面三角形ABC 中，过B 作BO AC ⊥，垂足为O ，则1BC O ∠为直线1BC 与平面11AAC C 所成角，求解三角形得答案． (1)解： 异面直线1BC 与1AA 所成角的大小为3π，且11//A A B B ， ∴113C BB π∠=，又114B B A A ==，∴111t n 3a B C BB π==111ABC A B C -的底面边长为∴162ABCS=⨯=则1114ABC A B C V -== (2)解：在底面三角形ABC 中，过B 作BO AC ⊥，垂足为O ，则O 为AC 中点，又平面ABC ⊥平面11ACC A ，平面ABC 平面11ACC A AC =，所以BO ⊥平面11AAC C ，连接1C O ，则1BC O ∠为直线1BC 与平面11AAC C 所成角，因为cos 6BO BC OBC =∠=，18BC ，163sin 84BC O ∴∠==， 13arcsin4BC O ∴∠=． 即直线1BC 与平面11AAC C 所成角的大小为3arcsin4．18．（2022·上海长宁·高二期末）在矩形ABCD 中，E 是BC 的中点，H 是AD 上，AE BH G ⋂=，且AE BH ⊥，如图，将AEB △沿AE 折起至AEF ：(1)指出二面角F AE D --的平面角，并说明理由； (2)若FH GH ⊥，求证：平面FAD ⊥平面AECD ； (3)若L 是线段DF 的中点，求证：直线//LC 平面AEF ； 【答案】(1)FGH ∠为二面角F AE D --的平面角，理由见解析 (2)证明见解析 (3)证明见解析 【解析】 【分析】（1）根据FG AE ⊥，GH AE ⊥结合二面角定义得到答案.（2）证明AE ⊥平面FGH 得到AE HF ⊥，得到FH ⊥平面AECD ，得到证明. （3）延长AE ，DC 交于点M ，连接FM ，证明LC FM ∥即可. (1)连接FG ，则FG AE ⊥，GH AE ⊥，故FGH ∠为二面角F AE D --的平面角.(2)FG AE ⊥，GH AE ⊥，FG HG G ⋂=，故AE ⊥平面FGH ，HF ⊂平面FGH ，故AE HF ⊥，又FH GH ⊥，AE GH G =，故FH ⊥平面AECD ，FH ⊂平面FAD ，故平面FAD ⊥平面AECD .(3)延长AE ，DC 交于点M ，连接FM ，易知ABE EMC ≅△△，故AB CM CD == 故C 是MD 的中点，L 是线段DF 的中点，故LC FM ∥， FM ⊂平面AEF ，且LC ⊄平面AEF ，故直线LC ∥平面AEF .19．（2022·上海·高三专题练习）如图，在四棱锥P ABCD -中，底面ABCD 为直角梯形，//BC AD ，AB BC ⊥，45ADC ∠=︒，PA ⊥平面ABCD ，1AB AP ==，3AD =．（1）求点D 到平面PBC 的距离；（2）求二面角B PC D --的平面角的余弦值．【答案】（1 （2【解析】 【分析】（1）建立空间直角坐标系，计算平面PBC 的法向量，由点面距离的向量公式即得解； （2）计算平面PCD 的法向量，结合（1）中平面PBC 的法向量，利用二面角的向量公式即得解 【详解】（1）由题意，PA ⊥平面ABCD ，//BC AD ，AB BC ⊥， AB AD ∴⊥以A 为坐标原点，,,AB AD AP 所在直线为,,x y z 轴建立如图所示的空间直角坐标系 则P （0，0，1），B （1，0，0），C （1，2，0），D （0，3，0）， 设平面PBC 的一个法向量为n =（x ，y ，z ），PB =(1，0，﹣1)，BC =(0，2，0)，CD =(﹣1，1，0)，则020n PB x z n BC y ⎧⋅=-=⎨⋅==⎩，取x =1，得n =(1，0，1)，∴点D 到平面PBC 的距离22n CD d n⋅==． （2）由（1）可得平面PBC 的一个法向量为n =(1，0，1)， 设平面PCD 的一个法向量为(,,)m a b c =，(1,2,1)PC =-， CD =(﹣1，1，0)，则200m PC a b c m CD a b ⎧⋅=+-=⎨⋅=-+=⎩，取1a =，得(1,1,3)m =，设二面角B PC D --的平面角为α，由图得二面角为钝角故4cos ||||||m n m n α⋅=-=-20．（2021·上海·高二阶段练习）如图，正方形ABCD 所在平面与平面四边形ABEF 所在的平面互相垂直，ABE △是等腰直角三角形，,,45AB AE FA FE AEF ==∠=︒.（1）求证：EF ⊥平面BCE ；（2）设线段CD AE 、的中点分别为P M 、，求证：//PM 平面BCE ； （3）求二面角F BD A --的余弦值.【答案】（1）证明见解析；（2）证明见解析；（3. 【解析】 【分析】（1）根据题意，由面面垂直的性质得出BC ⊥平面ABEF ，进而得出BC EF ⊥，由等腰直角三角形的性质得出FE EB ⊥，最后根据线面垂直的判定定理，即可证明EF ⊥平面BCE ； （2）由题意可知AE AB ⊥，通过线面垂直的判定定理可证出AE ⊥平面ABCD ，从而以,,AD AB AE 方向分别为x ，y ，z 轴正方向，建立空间坐标系，令正方形ABCD 的边长为2，求出各点的坐标，得出(2,1,1)PM →=--以及平面BCE 的一个法向量(0,1,1)EF →=--，利用空间向量的数量积得出0PM EF →→⋅=，通过空间向量法即可证明//PM 平面BCE ；（3）设平面FBD 的一个法向量(,,)n x y z =，通过空间向量求法向量的方法求得(1,1,3)n =，(0,0,2)AE →=为平面ABCD 的一个法向量，最后根据空间向量求二面角的方法，即可求出二面角F BD A --的余弦值. 【详解】解：（1）∵平面ABEF ⊥平面ABCD ，且平面ABEF 平面ABCD AB =，由于正方形ABCD ，所以BC AB ⊥， ∴BC ⊥平面ABEF ， 又由EF ⊂平面ABEF ， ∴BC EF ⊥，又∵ABE △是等腰直角三角形，AB AE =，则45BEA ∠=︒， 在AEF 中，,45FA FE AEF =∠=︒， ∴90FEB ∠=︒，即FE EB ⊥，又∵EB BC B =，且EB BC ⊂，平面BCE ， ∴EF ⊥平面BCE ；（2）因为ABE △是等腰直角三角形，AB AE =，所以AE AB ⊥， 又平面ABEF ⊥平面ABCD ，且平面ABEF平面ABCD AB =，所以AE ⊥平面ABCD ，因此AE AD ⊥，即AD ，AB ，AE 两两垂直，以A 为坐标原点，,,AD AB AE 方向分别为x ，y ，z 轴正方向，建立空间坐标系， 令正方形ABCD 的边长为2，则：()()()()()()()()0,0,0,0,2,0,2,2,0,2,0,0,0,0,2,0,1,1,2,1,0,0,0,1A B C D E F P M -，由（1）得EF ⊥平面BCE ，则(0,1,1)EF →=--为平面BCE 的一个法向量， 又(2,1,1)PM →=--，则()()()()0211110PM EF →→⋅=⨯-+-⨯-+-⨯=， PM EF →→∴⊥，∴//PM 平面BCE ；（3）设平面FBD 的一个法向量(,,)n x y z =，()()2,2,0,0,3,1BD BF →→=-=-，则00n BD n BF ⎧⋅=⎪⎨⋅=⎪⎩，即22030x y y z -=⎧⎨-+=⎩，令1x =，则平面FBD 法向量(1,1,3)n =，又∵AE ⊥平面ABCD ，则(0,0,2)AE →=为平面ABCD 的一个法向量， 令二面角F BD A --的平面角为θ，则6cos 11211AE nAE n θ→→⋅===⨯⋅所以二面角F BD A --.21．（2022·上海市控江中学高二期末）如图，在长方体1111ABCD A B C D -中，11,2DD DA AB ===，点E 在棱AB 上运动．(1)证明：11B C D E ⊥；(2)当E 为棱AB 的中点时，求直线AB 与平面1DEC 所成角的正弦值；(3)AE 等于何值时，二面角1A DE C --的大小为150︒？【答案】(1)证明见解析；(3)AE =【解析】【分析】 （1）连接1AD 、1BC ，长方体、线面垂直的性质有11BC CB ⊥、1AB CB ⊥，再根据线面垂直的判定、性质即可证结论.（2）连接EC ，由已知条件及勾股定理可得DE EC ⊥、1EC DE ⊥，即可求1DEC S △、DEC S ，等体积法11C DEC C DEC V V --=求C 到面1DEC 的距离，又直线AB 与面1DEC 所成角即为DC 与面1DEC 所成角，即可求线面角的正弦值.（3）由题设易知二面角1C DE C --为30，过C 作CF DE ⊥于F ，连接1FC ，可得二面角1C DE C --平面角为130CFC ∠=︒，令AE x =，由长方体的性质及勾股定理构造方程求x 即可.(1)由题设，连接1AD 、1BC ，又长方体1111ABCD A B C D -中11DD DA ==，∴11BCC B 为正方形，即11BC CB ⊥， 又AB ⊥面11BCC B ，1CB ⊂面11BCC B ，即1AB CB ⊥， ∵1BC AB B ，1,BC AB ⊂面11ABC D ， ∴1CB ⊥面11ABC D ，而1D E ⊂面11ABC D ，即11B C D E ⊥.(2)连接EC ，由E 为棱AB 的中点，则1AE EB AD BC ====，∴DE EC ==2DC AB ==，故222DE EC DC +=， ∴DE EC ⊥，又1EC 1DC =22211EC DE DC +=，则1EC DE ⊥，由11C DEC C DEC V V --=，若C 到面1DEC 的距离为d ，又1112DEC S DE EC =⋅=112DEC S DE EC =⋅=，∴111133DEC DEC CC S d S ⋅=⋅，可得d =，又//AB DC ，∴直线AB 与面1DEC 所成角即为DC 与面1DEC 所成角为θ，故sin d DC θ==. (3)二面角1A DE C --的大小为150︒，即二面角1C DE C --为30， 由长方体性质知：1CC ⊥面ABCD ，DE ⊂面ABCD ，则1CC ⊥DE ， 过C 作CF DE ⊥于F ，连接1FC ，又1CC FC C ⋂=， ∴DE ⊥面1CFC ，则二面角1C DE C --平面角为130CFC ∠=︒，∴12FC =，令AE x =，则2EB x =-，故2222211(2)2EC EB BC CC x =++=-+，而22211EC EF FC =+，1EF DE DF =-=，∴22211)46EC x =+=+-∴26x +-2(2)2x =-+，整理得231x =，解得x =∴AE =1A DE C --的大小为150︒.。
上海市尚德实验学校
上海市民办尚德实验学校是由尚德教育发展投资有限公司举办，并于2003年9月正式开学。这是一所幼儿园，小学、初中到高中的十五年寄宿制学校。校园占地 216亩，建筑面积14. 7万平方米。目前，有118个教学班，其中，小学53 个班，初中50个班，高中15个班。在校学生4300多人。(将在2013年下半学期开设幼儿园)现有教职员工410人，专任教师343人，其中中高级以上专业技术职务150人，平均年龄33岁。2010年4月成为IB(DP)正式授权学校。
学校荣誉
总述
尚德连续三年中考成绩名列前茅。2007年全体初三同学的中考平均分达518分（高出区平均分68.6分）；2007年考入上海中学6人、交大附中4人、华师大二附中11人、复旦附中2人、上师大附中4人计27人；市重点达线人数为251人，达线率51.8%（以市重点南汇中学的最低录取线计算），中考成绩综合评价位列南汇区第一。数千名学生获奖。吴雨帷同学荣获“白兰氏杯”上海市第十八届中学生作文竞赛初三年级组一等奖；陈一民同学获首届全国中小学生迎奥运作文大赛初中组一等奖；黄艺禧同学荣获第十八届《双龙杯》全国少年儿童书画大赛金杯奖；刘睿获2006年全国初中化学竞赛（天原杯）一等奖；刘睿获2006年上海初中化学竞赛（天原杯）一等奖（第二名）；刘睿荣获2005年上海市青少年“白猫杯”应用化学与技能竞赛（初中组）一等奖；任瑜冰、强荔、翟聪获19届上海市中学生作文大赛一等奖；龚译凡在《丰田杯》上海少年将棋大赛中通过顽强角逐荣获中学年部门竞赛一等奖；杨晓婕在2005年在中国郑州——新西兰奥克兰举办的和平杯书画艺术国际联展青少年儿童书画作品评奖活动中荣获金奖；田中夏叶、李海东在05上海市“文广杯”国际舞、社交舞公开赛中，获L少儿B组一等奖；体育学科中心指导的校健美操队获得2004年“晶兰杯”上海市健美操锦标赛丙组健身健美操金奖；上海市青少年“庄臣绿色生活行动”——环保儿童剧一等奖……
2016-2017学年上海市复旦大学附中高一上学期开学考英语试卷含详解
上海市复旦大学附中2016-2017学年高一上学期开学考试英语试卷I. Choice1．Rather than read the science fiction alone in the classroom he preferred______the performance put on by the American student’s choir.A．watching B．watch C．having watched D．to watch2．Eggs in the shopping center are sold ______.A．by dozen B．by the dozen C．in dozens D．in the dozens3．The Chinese government has decided to boycott the imported beef from UK for the purpose of ______ the local citizen from mad cow disease.A．protecting B．preserving C．reserving D．releasing4．Paper produced every year is ______ the world’s production of vehicles.A．as three times heavy as B．the three time weight ofC．three times the weight of D．three times heavier as5．Don’t worry about it. My dog will ______ the gate of my house when I am away.A．defend B．protect C．save D．guard6．He set aside much money for his tuition each month and only a ______ amount of money went to his living expense. A．little B．small C．tiny D．few7．This clever boy absorbed almost all ______ his teacher could give him.A．which B．/C．what D．as8．______ in thought, he failed to realize his mother’s approach.A．Losing B．Being lost C．Lost D．To be lost9．The whereabouts of Bin Laden has become a focus of media’s _____.A．observation B．notice C．intention D．attention10．Walk ______ of the street, and you can see a post office just around the corner.A．at the end B．in the end C．to the end D．to an end11．You really don’t know what to do ______ your mother’s kindness.A．in place of B．instead of C．ahead of D．in return for12．The panda ______ to China is threatened with extinction. Something must be done to save her from danger. A．natural B．local C．native D．rural13．Many searchlights were used to ______ the missing passengers on the spot the air crash occurred.A．situate….which B．search ….. whereC．search for….which D．locate ….where14．V enus Fly-Trap has two special leaves and ______ leaf has several hairs in the middle.A．any B．either C．both D．none15．It was ______ late to catch a bus after the party, ______, we called a taxi.A．too much…..moreover B．too much ….. thereforeC．much too …. therefore D．much too…. moreover16．It is believed that ______ type of tree grows ______ than the Giant Redwood.A．not any ….. more highly B．no other …. higherC．no other ….more highly D．not any….. higher17．Too much rainfall failed to ______ the growth of these plants.A．encourage B．discourage C．disappoint D．appoint18．Our English monitor can sing ______, if not better than, Mary.A．as good as B．as well as C．so well as D．so good as19．We know for sure that the Venus Fly-Trap ______ insects for food.A．catches B．holds C．masters D．kidnaps 20．Admission will be free to ______ wears the school uniform and the school badge.A．anyone B．who C．whoever D．whom21．The first prize ______ to Mr. Brown for his outstanding design of his banknote counting machine. A．rewarded B．awarded C．presented D．went22．Nurseries are the areas where plants are grown to be sold or ______ in other places.A．being planted B．to plant C．plant D．planted 23．Ecology is the study of the relationship between plants, animal, people and their ______. A．condition B．situation C．environment D．atmosphere24．We parted from each other at the crossroads and returned to our ______ houses. A．respected B．respectful C．respectable D．respective 25．Mimosa, 15 cm in height, is a sensitive small plant, which will close when ______. A．touching B．touched C．is touched D．to be touched26．As a matter of fact ______, we raise at home are collected from the wild forest.A．most the plants B．most of plants C．most of the plants D．most of plant27．I thought that they were unhappy at the party when they got nothing to eat, ______?A．didn’t I B．do I C．were they D．weren’t they28．______ I referred to in class is the weird world of plants.A．Knowing to be the best chapter B．Known as the best chapterC．The best-known chapter D．To know it as the best chapter29．—Will professor Brown attend her wedding ceremony?—It will not be ______ so.A．particularly B．necessarily C．commonly D．regularly30．—How are we going to make our language teaching more efficient?—______ the teaching methodology.A．To improve B．Improving C．Having improved D．By improvingⅡ.Close Test(A)31 to us all, Martin Luther King was the black leader in America, who devoted his life to the racial equality, and died for his beliefs. The following is a(n) 32 from his famous speech delivered in the large audience of American people. The speech is called “I have a dream”.I say to you today, my friend, even though we face the difficulties of today and tomorrow, I will have a dream. It is a dream deeply 33 in the American dream. I have a dream that one day this 34 will rise up and live out the true meaning of its belief that “all men are 35 equal”.I have a dream that the sons of 36 slaves and the sons of former slave owners will be able to sit down together at the table of 37 . I have a dream that one day even the state of Mississippi, a state sweltering with the heat of injustice, sweltering with the heat of oppression, will be transformed into an oasis(绿洲) of freedom and justice. I have a dream that my four little children will one day live in a nation 38 they will not be judged by the color of their skin39 by the content of their character.I have a dream that one day little black boys and black girls will be able to join hands with little white boys and white girls as sisters and brothers. This is our hope and faith. 40 this faith, we will work together, pray together, struggle together, go to jail together, stand up for freedom together, knowing that we will be free one day.31．A．It is known B．As it is known C．As is known D．We all know32．A．story B．fiction C．speech D．excerpt33．A．rooted B．mixed C．rested D．grown34．A．state B．nation C．government D．community35．A．discovered B．invented C．created D．born36．A．latter B．former C．late D．previous37．A．manhood B．brotherhood C．boyhood D．sisterhood38．A．which B．where C．that D．what39．A．and B．either C．but D．though40．A．As B．With C．For D．By(B)A 25-year-old architectural designer spends at least five hours a day 41 on the Internet, but not just for work42 sending e-mails to friends. To him, the most important thing is 43 he can have on-line talks with his girlfriends, a primary school teacher in south China.A shy young man is active in the virtual world. 44 his girlfriend, he has many on- line good friends .“It’s not a45 thing for net chatters to have on-line friends of the opposite sex or even 46 a girlfriend or a boyfriend, ”said a graduate student of Beijing university. “Many of 47 classmates and friends have had such an experience.”He and his girlfriend surnamed Wang first met on the Internet, and on-line discussions became the starting 48 of their love story. Nowadays “Net-love” is no longer a strange 49 to many young Chinese, and on-line recreational activities have sprung up.Evidences show that the number of net chatter in China 50 79.5 million by the end of last year.and 32.2 percent of the them said that the purpose of surfing on the Internet was to have fun and make friends.41．A．to surf B．having surfed C．surf D．surfing42．A．or B．and C．either D．neither43．A．which B．what C．that D．whom44．A．Except that B．except C．In addition D．Besides45．A．fair B．bare C．usual D．rare46．A．to chat B．to search C．to hunt D．to locate47．A．his B．her C．my D．their48．A．top B．point C．tip D．spot49．A．word B．vocabulary C．phrase D．letter50．A．had reached B．had arrived C．would rise D．would raise(A)I arrived in the United States on February 6, 1966, but I remember my first day here very clearly. My friend was waiting for me when my plane landed at Kennedy Airport at three o'clock in the afternoon. The weather was very cold and it was snowing, but I was too excited to mind. From the airport, my friend and I took a taxi to my hotel. On the way, I saw the skyline of Manhattan for the first time and I stared in astonishment at the famous skyscrapers and their man-made beauty. My friend helped me unpack at the hotel and then left me because he had to go back to work. He promised to return the next day.Shortly after my friend had left, I went to a restaurant near the hotel to get something to eat. Because I couldn’t speak aword of English, I couldn’t tell the waiter what I wanted. I was very upset and started to make some gestures, but the waiter didn’t understand me. Finally, I ordered the same thing the man at the next table was eating. After dinner, I started to walk along Broadway until I came to Times Square with its movie theatres, neon lights, and huge crowds of people. I did not feel tired, so I continued to walk around the city. I wanted to see everything on my first day. I knew it was impossible, but I wanted to try.When I returned to the hotel, I was exhausted, but I couldn’t sleep because I kept hearing the fire and police sirens during the night. I lay awake and thought about New York. It was a very big and interesting city with many tall buildings and big cars, and full of noise and busy people. I also decided right then that I had to learn to speak English.51．He did not have what he really wanted, because _______.A．he only made some gesturesB．he did not order at allC．he could not make himself understoodD．the waiter was unwilling to serve52．The waiter _______.A．knew what he orderedB．finally understood what he saidC．took the order through his gesturesD．served the same thing the man at the next table was having53．After dinner, he _______.A．walked back to the hotel right awayB．had a walking tour about the cityC．went to the moviesD．did some shopping on Broadway54．That night he could not sleep because _______.A．he did not know what to do the next dayB．he was not tired at allC．he kept hearing the fire and police sirensD．he was thinking about the great city（B）Dear Doctor,My husband and I got married in 1965 and for the first ten years of our marriage I was very happy to stay home and raise our three children. Then four years ago, our youngest child went to school and I thought I might go back to work.My husband was very supportive and helped me to make my decision. He emphasized all of the things I can do around the house, and said he thought I could be a great success in business.After several weeks of job-hunting, I found my present job, which is working for a small public relations firm. At first, my husband was very proud of me and would tell his friends, “My clever little wife can run that company she’s working for.”But as his joking remark approached reality, my husband stopped talking to me about my job. I have received several promotions and pay increases, and I am now making more money than he is. I can buy my own clothes and a new car. Because of our combined incomes, my husband and I can do many things that we had always dreamed of, but we don’t do these things because he is very unhappy.We fight about little things and my husband is very critical of me in front of our friends. For the first time in our marriage, I think there is a possibility that our marriage may come to an end.I love my husband very much, and I don’t want him to feel inferior, but I also love my job. I think I can be a good wife and a working woman, but I don’t know how. Can you give me some advice? Will I have to choose one or the other or can I keep both my husband and my new career?Please help.55．What do you think shows her husband was supportive?A．He took up all the work she used to do.B．He made all the decisions for her.C．He gave her encouragement.D．All of the above.56．As her income increased _______.A．she found a gap emerged between her and her husbandB．she bought more clothes and a new houseC．she did many things she had dreamed forD．she felt very proud of herself57．The dilemma of the working woman is a choice between _______.A．husband and childrenB．children and workC．career and moneyD．job and marriage（C）If we are observant(善于观察的), we can find that most of the flowers in nature are red, orange and yellow. If we haveseen a black flower, it is a chance in a million. People have made census (统计) to colors of more than four thousand kinds of flowers and discovered that only eight of them are black. Why are black flowers so rare?As we know, sunlight is formed by seven different kinds of colored light. The wave length of each light changes, so the quantity of heat in each light changes, too. Flowers, especially their petals (花瓣) , are very weak and easy to the harm caused by high temperature. Black flowers can take in all the light waves, which cause the flowers to dry up in a high temperature. So the black flowers can hardly continue their lives. While red flowers, orange flowers and yellow ones can protect themselves from sunlight by reflecting（反射）the red light, orange light and yellow light, each of which has a large quantity of heat.That is why red, orange and yellow flowers are very common in nature while black flowers are so unusual.58．It is a chance in a million” means______.A．something commonB．something luckyC．something extremely rareD．something impossible59．Sunlight is formed by _____.A．many different kinds of colored lightB．three different kinds of colored light—red, orange and yellowC．seven different kinds of colored lightD．four thousand kinds of colored light60．Black flowers are so rare because ______.A．they are so weak that it’s difficult for them to grow upB．there are actually no black flowers in the worldC．the petals of black flowers are very delicate（易碎的）D．they can take in the light of all wave length which make the flowers dry up because of high temperature.61．Which of the following statements is NOT true?A．Red, orange and yellow flowers can absorb the light of all wave lengths.B．People have found that only a few kinds of flowers are black.C．Most of the flowers are resistant（有抵抗力的）to high temperature owing to their bright colors.D．The black flowers cannot protect themselves from sunlight.62．幸运的是，今天的天气很适合野营。
2024-2025学年安徽省潜山市第四中学数学九年级第一学期开学复习检测试题【含答案】
2024-2025学年安徽省潜山市第四中学数学九年级第一学期开学复习检测试题题号一二三四五总分得分A 卷（100分）一、选择题（本大题共8个小题，每小题4分，共32分，每小题均有四个选项，其中只有一项符合题目要求）1、（4分）小军自制的匀速直线运动遥控车模型甲、乙两车同时分别从A 、B 出发，沿直线轨道同时到达C 处，已知乙的速度是甲的速度的1.5倍，甲、乙两遥控车与A 处的距离1d 、2d （米）与时间t （分钟）的函数关系如图所示，则下列结论中：①AC 的距离为120米；②乙的速度为60米/分；③a 的值为65；④若甲、乙两遥控车的距离不少于10米时，两车信号不会产生互相干扰，则两车信号不会产生互相干扰的t 的取值范围是502t ≤≤，其中正确的有（）个A ．1B ．2C ．3D ．42、（4分）下列四组线段中，不能构成直角三角形的是（）A ．4，5，6B ．6，8，10C ．7，24，25D ．5，3，43、（4分）下列各曲线中能表示y 是x 的函数的是（）A ．B ．C ．D ．4、（4分）《中国诗词大会》是央视科教频道自主研发的一档大型文化益智节目，节目带动全民感受诗词之趣，分享诗词之美，从古人的智慧和情怀中汲取营养，涵养心灵．比赛中除了来自复旦附中的才女武亦姝表现出色外，其他选手的实力也不容小觑．下表是随机抽取的10名挑战者答对的题目数量的统计表，则这10名挑战者答对的题目数量的中位数为答对题数（）答对题数4578人数3421A ．4B ．5C ．6D ．75、（4分）下列根式中属于最简二次根式的是（）A ．B C D ．6、（4分）如图，在ABC ∆中，分别以点,A B 为圆心，大于12AB 长为半径作弧，两弧交于点,M N ，作直线MN 分别交AB ，AC 于点,D E ，连接BE ，下列结论错误的是（）A ．AD BD =B ．2BEC DBE ∠=∠C ．AE BE =D ．BE 平分CBD∠7、（4分）关于2、6、1、10、6的这组数据，下列说法正确的是（）A ．这组数据的众数是6B ．这组数据的中位数是1C ．这组数据的平均数是6D ．这组数据的方差是108、（4分）估计的值应在（）A ．4和5之间B ．5和6之间C ．6和7之间D ．7和8之间二、填空题（本大题共5个小题，每小题4分，共20分）9、（4分）如图，在ABCD 中，点E 是BC 边上的动点，已知4AB =，6BC =，60B ∠=︒，现将ABE ∆沿AE 折叠，点'B 是点B 的对应点，设CE 长为x .（1）如图1，当点'B 恰好落在AD 边上时，x =______；（2）如图2，若点'B 落在ADE ∆内（包括边界），则x 的取值范围是______.10、（4分）当2x =-______.11、（4分）如图，菱形ABCD 的对角线相交于点O ，若5,4AB OA ==，则菱形ABCD 的面积＝____.12、（4分）x 的取值范围是_______13、（4分）若分式3||3x x -+的值为零，则x 的值为_____三、解答题（本大题共5个小题，共48分）14、（12分）如图，在平面直角坐标系中，△ABC 的顶点A 、B 分别落在x 轴、y 轴的正半轴上，顶点C 在第一象限，BC 与x 轴平行．已知BC =2，△ABC 的面积为1．（1）求点C 的坐标．（2）将△ABC 绕点C 顺时针旋转90°，△ABC 旋转到△A 1B 1C 的位置，求经过点B 1的反比例函数关系式．15、（8分）已知关于x 的方程（a ﹣1）x 2+2x +a ﹣1＝1．（1）若该方程有一根为2，求a 的值及方程的另一根；（2）当a 为何值时，方程的根仅有唯一的值？求出此时a 的值及方程的根．16、（8分）某汽车制造商对新投入市场的两款汽车进行了调查，这两款汽车的各项得分如下表所示：汽车型号安全性能省油效能外观吸引力内部配备A 3123B 3222（得分说明：3分﹣﹣极佳，2分﹣﹣良好，1分﹣﹣尚可接受）（1）技术员认为安全性能、省油效能、外观吸引力、内部配备这四项的占比分别为30%，30%，20%，20%，并由此计算得到A 型汽车的综合得分为2.2，B 型汽车的综合得分为_____；（2）请你写出一种各项的占比方式，使得A 型汽车的综合得分高于B 型汽车的综合得分．（说明：每一项的占比大于0，各项占比的和为100%）答：安全性能：_____，省油效能：_____，外观吸引力：_____，内部配备：_____．17、（10分）如图，正方形ABCD 的边长为4，E 是线段AB 延长线上一动点，连结CE ．（1）如图1，过点C 作CF ⊥CE 交线段DA 于点F ．①求证：CF =CE ；②若BE =m （0＜m ＜4），用含m 的代数式表示线段EF 的长；（2）在（1）的条件下，设线段EF 的中点为M ，探索线段BM 与AF 的数量关系，并用等式表示．（3）如图2，在线段CE 上取点P 使CP =2，连结AP ，取线段AP 的中点Q ，连结BQ ，求线段BQ 的最小值．18、（10分）若a ＝312-，b ＝12，请计算a 2+b 2+2ab 的值．B 卷（50分）一、填空题（本大题共5个小题，每小题4分，共20分）19、（4分）如图，直线l 过正方形ABCD 的顶点B，点A、C 到直线l 的距离AE、CF 分别是1cm、2cm，则线段EF 的长为______cm．20、（4分）如果不等式组841x x x m +<-⎧⎨>⎩的解集是3x >，那么m 的取值范围是______.21、（4分）如图，直线AB ，IL ，JK ，DC ，相互平行，直线AD ，IJ 、LK 、BC 互相平行，四边形ABCD 面积为18，四边形EFGH 面积为11，则四边形IJKL 面积为____.22、（4分）如图，在平面直角坐标系中，O 为原点，四边形OABC 是矩形，A（-10，0），C （0，3），点D 是OA 的中点，点P 在BC 边上运动，当△ODP 是腰长为5的等腰三角形时，点P 的坐标是______．23、（4分）如图，在矩形ABCD 中，按以下步骤作图：①分别以点A 和点C 为圆心，以大于12AC 的长为半径作弧，两弧相交于点M 和N ；②作直线MN 交CD 于点E ，若AB=8，AD=6，则EC=_____________．二、解答题（本大题共3个小题，共30分）24、（8分）阅读下列题目的解题过程：已知a、b、c 为△ABC 的三边，且满足a 2c 2﹣b 2c 2=a 4﹣b 4，试判断△ABC 的形状．解：∵a 2c 2﹣b 2c 2=a 4﹣b 4（A ）∴c 2（a 2﹣b 2）=（a 2+b 2）（a 2﹣b 2）（B ）∴c 2=a 2+b 2（C ）∴△ABC 是直角三角形问：（1）上述解题过程，从哪一步开始出现错误？请写出该步的代号：；（2）错误的原因为：；（3）本题正确的结论为：．25、（10分）先化简()222a 2a 1a 1a 1a 2a 1+-÷++--+，然后a 在﹣1、1、2三个数中任选一个合适的数代入求值．26、（12分）已知m n ==22m mn n ++的值.参考答案与详细解析一、选择题（本大题共8个小题，每小题4分，共32分，每小题均有四个选项，其中只有一项符合题目要求）1、C【解析】根据题意和函数图象中的数据可以判断各个小题中的结论是否成立，从而可以解答本题．【详解】由图可得，AC的距离为120米，故①正确；乙的速度为：（60+120）÷3=60米/分，故②正确；a的值为：60÷60=1，故③错误；令[60+（120÷3）t]-60t≥10，得t≤5 2，即若甲、乙两遥控车的距离不少于10米时，两车信号不会产生相互干扰，则两车信号不会产生相互干扰的t的取值范围是0≤t≤52，故④正确；故选C．本题考查一次函数的应用，解答本题的关键是明确题意，利用一次函数的性质和数形结合的思想解答．2、A【解析】由勾股定理的逆定理，只要验证两小边的平方和是否等于最长边的平方，即可解答．【详解】解：A、42+52≠62，故不是直角三角形，符合题意；B、62+82=102，能构成直角三角形，不符合题意；C、72+242=252，能构成直角三角形，不符合题意；D、32+42=52，能构成直角三角形，不符合题意．故选：A．本题考查勾股定理的逆定理的应用．判断三角形是否为直角三角形，已知三角形三边的长，只要利用勾股定理的逆定理加以判断即可．3、B 【解析】因为对于函数中自变量x 的取值,y 有唯一一个值与之对应,故选B.4、B 【解析】将这组数据从小到大的顺序排列后，根据中位数的定义就可以求解．【详解】解：将这组数据从小到大的顺序排列后，处于中间位置第1和第6个数是1、1，那么由中位数的定义可知，这组数据的中位数是1．故选：B ．本题为统计题，考查中位数的意义，中位数是将一组数据从小到大（或从大到小）重新排列后，最中间的那个数（或最中间两个数的平均数），叫做这组数据的中位数，如果中位数的概念掌握得不好，不把数据按要求重新排列，就会出错．5、A 【解析】根据最简二次根式的定义选择即可.【详解】A 是最简二次根式，故本选项正确；B =C 3=不是最简二次根式，故本选项错误；D2=不是最简二次根式，故本选项错误.故选：A .本题考查了最简二次根式，掌握最简二次根式的定义是解题的关键.6、D【解析】根据题意可知DE 是AB 的垂直平分线，由此即可得出△AEB 是等腰三角形，据此作出判断．【详解】由题可知，DE 是AB 的垂直平分线，∴AD BD =，AE BE =，故A 、C 选项正确；∵BEC ∠是等腰ABE ∆的外角，∴2BEC DBE ∠=∠，故B 选项正确；D 无法证明，故选：D.本题考查了线段的垂直平分线的性质、等腰三角形的性质，解题时注意：线段垂直平分线上任意一点，到线段两端点的距离相等．7、A 【解析】根据方差、算术平均数、中位数、众数的概念进行分析.【详解】数据由小到大排列为1，2，6，6，10，它的平均数为15（1+2+6+6+10）=5，数据的中位数为6，众数为6，数据的方差=15[（1﹣5）2+（2﹣5）2+（6﹣5）2+（6﹣5）2+（10﹣5）2]=10.1．故选A ．考点：方差；算术平均数；中位数；众数．8、D 【解析】先合并后，再根据无理数的估计解答即可．【详解】＝＝＝∵78，∴的值应在7和8之间，故选D．本题考查了估算无理数的大小，解决本题的关键是估算出无理数的大小．二、填空题（本大题共5个小题，每小题4分，共20分）9、2；22x ≤≤【解析】（1）根据折叠的性质可得4AB BE ==，由此即可解决问题；（2）作AH ⊥DE 于H ．解直角三角形求出AH 、HB′、DH ，再证明6DE AD ==，求出EB′即可解决问题；【详解】解：（1）∵折叠，∴'BAE B AE ∠=∠.∵AD BC ∥，∴'B AE AEB ∠=∠，∴BAE AEB ∠=∠，∴4AB BE ==，∴2CE BC BE =-=.（2）当'B 落在DE 上时，过点A 作AH DE ⊥于点H .∵'60AB H B ∠=∠=︒，'4==AB AB ，∴1''22HB AB ==，∴AH =.在Rt ADH ∆中，DH ==，∴''2DB DH HB =-=.∵AD BC ∥，∴DAE AEB AED ∠=∠=∠，∴6DE AD ==.∴()'628EB BE ==-=-，∴(682EC BC BE =-=--=，∴22x ≤≤.本题考查翻折变换、平行四边形的性质、解直角三角形等知识，解题的关键是学会添加常用辅助线，构造直角三角形解决问题，属于中考常考题型．【解析】把x=-2代入根式即可求解.【详解】把x=-2此题主要考查二次根式，解题的关键是熟知二次根式的性质.11、3.【解析】先求出菱形对角线AC 和BD 的长度，利用菱形面积等于对角线乘积的一半求解即可．【详解】因为四边形ABCD 是菱形，所以AC ⊥BD ．在Rt △AOB 中，利用勾股定理求得BO=1．∴BD=6，AC=2．∴菱形ABCD 面积为12×AC×BD=3．故答案为：3．本题主要考查了菱形的性质，解题的关键是熟记菱形面积的求解方法，运用对角线求解面积是解题的最优途径．12、3x 【解析】根据二次根式有意义的条件：被开方数为非负数求解即可．【详解】解：有意义，260x ∴- ，解得：3x ．故答案为：3x ．本题考查了二次根式有意义的条件，解答本题的关键是掌握被开方数为非负数．13、1【解析】分式的值为零：分子等于零，且分母不等于零，由此得到1-|x|=2且x+1≠2，从而得到x 的值．【详解】依题意得：1-|x|=2且x+1≠2，解得x=1．故答案是：1．本题考查了分式的值为零的条件．若分式的值为零，需同时具备两个条件：（1）分子为2；（2）分母不为2．这两个条件缺一不可．三、解答题（本大题共5个小题，共48分）14、（1）C （2，1）；（2）经过点B 1的反比例函数为y =6x ．【解析】（1）过点C 作CD ⊥x 轴于点D ，BC 与x 轴平行可知CD ⊥BC ，ABC 1S BC CD 12∆=⋅=即可求出CD 的长，进而得出C 点坐标；（2）由图形旋转的性质得出CB 1的长，进而可得出B 1的坐标，设经过点B 1（2，3）的反比例函数为k y x =，把B 1的坐标代入即可得出k 的值，从而得出反比例函数的解析式．【详解】解：（1）作CD ⊥x 轴于D ．∵BC 与x 轴平行，∴S △ABC =12BC•CD ，∵BC=2，S △ABC =1，∴CD=1，∴C （2，1）；（2）∵由旋转的性质可知CB 1=CB=2，∴B 1（2，3）．设经过点B 1（2，3）的反比例函数为k y x =，∴3=2k ，解得k=6，∴经过点B 1的反比例函数为y=6x ．本题考查的是反比例函数综合题，涉及到图形旋转的性质及三角形的面积公式、用待定系数法求反比例函数的解析式，涉及面较广，难度适中．15、（3）a=15，方程的另一根为12；（2）答案见解析.【解析】（3）把x=2代入方程，求出a 的值，再把a 代入原方程，进一步解方程即可；（2）分两种情况探讨：①当a=3时，为一元一次方程；②当a≠3时，利用b 2－4ac ＝3求出a 的值，再代入解方程即可．【详解】（3）将x ＝2代入方程2(a 1)x 2x a 10-++-=，得4(a 1)4a 10-++-=，解得：a ＝15．将a ＝15代入原方程得24x 2054x 5-+-=，解得：x 3＝12，x 2＝2．∴a ＝15，方程的另一根为12；（2）①当a ＝3时，方程为2x ＝3，解得：x ＝3.②当a≠3时，由b 2－4ac ＝3得4－4(a －3)2＝3，解得：a ＝2或3．当a ＝2时，原方程为：x 2＋2x ＋3＝3，解得：x 3＝x 2＝－3；综上所述，当a＝3，3，2时，方程仅有一个根，分别为3，3，－3.考点：3.一元二次方程根的判别式；2.解一元二次方程；3.分类思想的应用.16、（1）2.1；（2）10%；10%；10%；50%【解析】（1）根据加权平均数的计算公式列式计算即可；（2）要使得A型汽车的综合得分高于B型汽车的综合得分，根据这两款汽车的各项得分，将A型汽车高于B型汽车得分的项（内部配备）占比较高，同时将A型汽车低于B型汽车得分的项（省油效能）占比较低即可.【详解】(1)B型汽车的综合得分为：1×10%+2×10%+2×20%+2×20%=2.1．故答案为2.1；（2）∵A型汽车的综合得分高于B型汽车的综合得分，∴各项的占比方式可以是：安全性能：10%，省油效能：10%，外观吸引力：10%，内部配备50%．本题考查的是加权平均数的求法，掌握公式是解题的关键.17、（1）①详见解析；②；（2）BM=AF；（3）【解析】（1）①根据正方形的性质以及余角的性质即可证明△DCF≌△BCE，再根据全等三角形对应边相等即可得出结论；②根据全等三角形的性质可得DF=BE=m．在Rt△ECF中，由勾股定理即可得出结论；（2）在直线AB上取一点G，使BG=BE，由三角形中位线定理可得FG=2BM，可以证明AF=AG．在Rt△AFG中由勾股定理即可得出结论．（3）在AB的延长线上取点R，使BR=AB=4，连结PR和CR，由三角形中位线定理可得BQ=PR．在Rt△CBR中，由勾股定理即可得出CR的长，再由三角形三边关系定理即可得出结论．【详解】（1）解：①证明：∵正方形ABCD ，∴BC =CD ，∠DCB =∠CBE =90°．∵CF ⊥CE ，∠FCE =90°，∴∠DCF =∠BCE ，∴△DCF ≌△BCE （ASA ），∴CE =CF ．②∵△DCF ≌△BCE ，∴DF =BE =m ，∴AF =4-m ，AE =4+m ，由四边形ABCD 是正方形得∠A =90°，∴EF ==；（2）解：在直线AB 上取一点G ，使BG =BE ．∵M 为EF 的中点，∴FG =2BM ，由（1）知，DF =BE ，又AD =AB ，∴AF =AG ．∵∠A =90°，∴FG =AF ，∴2BM =AF ，∴BM =AF ．（3）解：在AB 的延长线上取点R ，使BR =AB =4，连结PR 和CR ．∵Q 为AP 的中点，∴BQ =PR ．∵CP =2，CR ==，∴PR ≥CR -CP =，∴BQ 的最小值为．本题考查了正方形的性质以及三角形中位线定理．作出恰当的辅助线是解答本题的关键．18、1．【解析】将a 、b 的值代入原式＝（a +b ）2计算可得．【详解】当a ＝12 ，b ＝2时，原式＝（a +b ）221,122⎛⎫= ⎪ ⎪⎝⎭-+2,=＝1．本题主要考查考查二次根式的运算，解题的关键是掌握完全平方公式和二次根式的混合运算顺序和法则．一、填空题（本大题共5个小题，每小题4分，共20分）19、3【解析】∵四边形ABCD 为正方形，∴AB=BC,∠ABC=90°.∵AE ⊥l ，CF ⊥l ，∴∠E=∠F=90°,∠EAB+∠ABE=90°,∠FBC+∠BCF=90°.∵∠ABE+∠ABC+∠FBC=180°，∴∠ABE+∠FBC=90°，∴∠EAB=∠FBC.在△ABE 和△BCF 中,{E F EAB FBC AB BC ∠=∠∠=∠=，∴△ABE ≌△BCF(AAS)，∴BE=CF=2cm ，BF=AE=1cm ，∴EF=BE+BF=2+1=3cm.故答案为3.20、3m ≤.【解析】先用含有m 的代数式把原不等式组的解集表示出来，然后和已知的解集比对，得到关于m 的不等式，从而解答即可．【详解】在841x x x m +<-⎧⎨>⎩中，由（1）得，3x >，由（2）得，x m >，根据已知条件，不等式组解集是3x >.根据“同大取大”原则3m ≤.故答案为：3m ≤.本题是已知不等式组的解集，求不等式中另一未知数的问题．可以先将另一未知数当作已知数处理，求出解集与已知解集比较，进而求得另一个未知数．21、1【解析】由平行四边形的性质可得EHB EIH S S ∆∆=，AEF EFJ S S ∆∆=，DFG FKG S S ∆∆=，GCH GHL S S ∆∆=，由面积和差关系可求四边形IJKL 面积．【详解】解：//AB IL ，//IJ BC ，∴四边形EIHB 是平行四边形，EHB EIH S S ∆∆∴=，同理可得：AEF EFJ S S ∆∆=，DFG FKG S S ∆∆=，GCH GHL S S ∆∆=，∴四边形IJKL 面积=四边形EFGH 面积-（四边形ABCD 面积-四边形EFGH 面积）11(1811)4=--=，故答案为：1．本题考查了平行四边形的判定与性质，由平行四边形的性质得出EHB EIH S S ∆∆=是解题的关键．22、（-4，3），或（-1，3），或（-9，3）【解析】∵A （-10，0），C （0，3）,10OA CD ==,3OC AB ==.∵点D 是OA 的中点，152OD AD OA ∴===.当1OP OD =时，4PC ===,()4,3P ∴-.当2OD P D =时，4DE ===,549OE OD DE =+=+=,()9,3P ∴-当3OP OD =时，541OF OD DF =-=-=,()1,3P ∴-.当PD PO =时，不合题意.故答案有三种情况.【点睛】本题考查了矩形的性质，等腰三角形的概念，平面直角坐标系中点的坐标及分类的思想.涉及等腰三角形的计算，不管是角的计算还是腰的计算，一般都要进行分类讨论.像本题就要分四种情况进行计算.23、254【解析】连接EA ，如图，利用基本作图得到MN 垂直平分AC ，所以EC=EA ，设CE=x ，则AE=x ，DE=8-x ，根据勾股定理得到62+（8-x ）2=x 2，然后解方程求出x 即可．【详解】解：连接EA ，如图，由作图得到MN垂直平分AC，∴EC=EA，∵四边形ABCD为矩形，∴CD=AB=8，∠D=90°，设CE=x，则AE=x，DE=8-x，在Rt△ADE中，62+（8-x）2=x2，解得x=25 4，即CE的长为25 4．故答案为25 4．本题考查了作图-基本作图：熟练掌握基本作图（作一条线段等于已知线段；作一个角等于已知角；作已知线段的垂直平分线；作已知角的角平分线；过一点作已知直线的垂线）．也考查了线段垂直平分线的性质．二、解答题（本大题共3个小题，共30分）24、（1）C；（2）没有考虑a=b的情况；（3）△ABC是等腰三角形或直角三角形．【解析】【分析】（1）根据题目中的书写步骤可以解答本题；（2）根据题目中B到C可知没有考虑a=b的情况；（3）根据题意可以写出正确的结论．【详解】（1）由题目中的解答步骤可得，错误步骤的代号为：C，故答案为：C；（2）错误的原因为：没有考虑a=b的情况，故答案为：没有考虑a=b的情况；（3）本题正确的结论为：△ABC是等腰三角形或直角三角形，故答案为：△ABC 是等腰三角形或直角三角形．【点睛】本题考查因式分解的应用、勾股定理的逆定理，解答本题的关键是明确题意，写出相应的结论，注意考虑问题要全面．25、5【解析】解：原式=()()()()22a 1a 1a 112a 1a 3a 1a 1a 1a 1a 1a 1++-++⋅+=+=-+----．取a=2，原式23521+==-．先根据分式混合运算的法则把原式进行化简，再选取合适的a 的值（使分式的分母和除式不为0）代入进行计算即可．26、11【解析】先求出m+n 和mn 的值，再根据完全平方公式变形，代入求值即可.【详解】∵m n ==∴，mn=1∴22m mn n ++＝222()111m n mn +-=-=.此题考查了二次根式的混合运算法则，完全平方公式的应用，主要考查了学生的计算能力，题目较好.。
大兴区2019年高一上学期语文期中考试试卷B卷
大兴区2019年高一上学期语文期中考试试卷B卷姓名:________ 班级:________ 成绩:________一、选择题 (共1题；共6分)1. （6分）(2016·成都模拟) 欣赏下面的漫画，按要求答题。
（1）给这幅漫画拟定标题。
（不超过 6 个字，不能以“无题”为题）（2）概括漫画的寓意，不超过 30 个字。
二、现代文阅读 (共3题；共27分)2. （6分） (2017高一上·牡丹江月考) 阅读下面的文字，完成后面题目。
永远的烛之武秦晋围郑，郑危在旦夕，“若使烛之武见秦君，师必退”，佚之狐的力荐可谓慧眼识英雄，烛之武临危受命，夜缒而出，智说秦君，秦穆公心悦诚服，秦军乃还。
晋文公面对风雨突变，亦毅然去之。
郑国终于转危为安。
在国家危难面前，烛之武深明大义、义无反顾；在强秦面前，烛之武不卑不亢、能言善辩、聪慧机智。
他的不计个人得失，处处为郑国安危着想的爱国主义精神，他的义无反顾赴敌营的信心和勇气都值得我们学习。
烛之武的智勇及强烈的爱国主义精神溢于字里行间。
1955年，著名科学家钱学森冲破重重阻挠回国，当时美国的海军处长金波尔说：“我宁可把这家伙枪毙了，也不让他离开美国，无论他在哪里，都抵得上五个师。
”一个人的力量能抵得上五个师吗？《烛之武退秦师》告诉了我们答案。
曹操有句诗：“老骥伏枥，志在千里。
烈士暮年，壮心不已。
”古往今来，有多少志存高远之士，哪怕社会、命运对他如何不公，但到了紧要关头，依然挺身而出，为国家、为苍生献出自己的一份光和热。
烛之武就是这样一个让我们敬佩不已的人。
刘备三顾茅庐见到了诸葛亮，诸葛亮对他畅谈天下大势。
可以说诸葛亮未出山前就做着出山的准备，烛之武也是如此，尽管怀才不遇，但他没有放弃自己的希望。
烛之武不是不想做一番事业，他的心始终没有放弃过“一飞冲天”的渴望，他一刻不停地关注着天下形势，思考着定天下济苍生一逞抱负的路径。
你想，一个小小的“弼马温”，怎么对晋国的历史、对晋国国君的为人那么清楚，对秦伯的心理、对秦晋之间的恩怨又是如此了然？“老骥伏枥，志在千里”的豪情无疑是后人景仰他们的主要缘由。
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I. Choice
复旦附中
高一开学考考卷题型分析
复旦附中高一英语练习单选分析
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1.Rather than read the science fiction alone in the classroom he preferred the
performance put on by the American student’ｓ
choir.
A.watching
B. watch
C. having watched
D.to watch
【考点分析】答案选D,考察prefer to do sth rather than do sth
【核心单词】performance 表演put on 上映表演
2.Eggs in the shopping center are sold .
A.by dozen
B. by the dozen
C. in dozens
D. in the dozens
【考点分析】答案选C，考察固定搭配in dozens 按打计算
【核心单词】in dozens
3.The Chinese government has decided to boycott the imported beef from UK for the purpose of
the local citizen from mad cow disease.
A.protecting
B. preserving
C. reserving
D. releasing
【考点分析】答案选A，根据词意及句意来回答，protect 保护，preserve 保存，
reserve 储备保存，release 释放
【核心单词】import 进口，mad cow disease 进口
4. Paper produced every year is the world’s production of
vehicles.
A. as three times heavy as
B. the three time weight of
C. three times the weight of
D. three times heavier as
【考点分析】答案选C，考察倍数的表达法倍数the n
of…..
倍数比较级than…
倍数as…as…
【核心单词】produce 产生，vehicle 交通工具
5.Don’ｔ
worry about it. My dog will the gate of my house when I am away.
A.defend
B. protect
C. save
D. guard
【考点分析】答案选D，根据词意及句意来选择，defend 防卫，protect 保护，save拯救，guard 守卫【核心单词】worry about 担心…
6.He set aside much money for his tuition each month and only a amount of money
went to his living expense.
A.little
B. small
C. tiny
D. few
【考点分析】答案选B，修饰amount 只能用large/small
【核心单词】set aside 留出
7.This clever boy absorbed almost all his teacher could give him.。