2004年湖北省部分重点中学期中联考

2023-2024学年湖北省武汉市部分重点中学高二上学期期中联考英语试题The ebike has become an essential item, and even the King has been spotted riding one. According to the market research company Mintel, sales have almost tripled over the past five years. While ebikes have become increasingly commonplace, they are still undeniably expensive. You can pick up a perfectly adequate push bike for £300, but most ebikes cost nearly £2,000. Now, Cycling Electric, designed to help consumers choose an ebike, offers the following recommendations.Best folding bikeCarbo Model XPrice: £2, 799 | Weight: 13.3kgRange: up to 34 milesThis is one of the most convenient bikes, from a little-known label. The lightweight Carbo X is the perfect example of convenience thanks to its carbon frame and belt drive, instead of a metal bike chain. This makes it lighter and less likely to get messy than an electric Brompton, with its more traditional, oily chain.Most stylishTemple Cycles ClassicPrice: from £2, 995 | Weight: 19kgRange: up to 74 milesThis Bristol-based brand is known for its elegant pushbikes, which are hand-built in the city. This is its electric version. It’s a very sleek and functional commuter option and not too heavy. Temples have fewer proprietary (品牌专卖的) parts and so are serviceable in any bike shop. Stylish as they may be, the VanMoofs and the Cowboys can be a bit more complex to get back-up and service.The money-no-object optionRiese & Müller Nevo4 GT Vario GXPrice: £5, 380 | Weight: 28.4kgRange: up to 93 milesThis is a prime example of the attention to detail delivering a high-quality experience in functionality, comfort and practicality — if you can afford it. It has the capacity to handle 25kg of luggage in a pannier, powerful road lights and a beautifully smooth motor.1. What is the advantage of the Temples over other stylish ebike brands?A．They are more lightweight. B．They have more functions.C．They have easier access to service. D．They are more comfortable.2. Which ebike will a consumer probably choose, who cares most about an excellent experience?A．Carbo Model X. B．The VanMoofs and the Cowboys.C．Temple Cycles Classic. D．Riese & Müller Nevo4 GT Vario GX. 3. Where is the text probably taken from?A．An online post. B．A magazine column.C．A market research report. D．An official document.Since September I’ve been living in a hall, just a stone’s throw from the Thames. Campus is a 25-minute walk from my hall, which is across the river and down Fleet Street. There, in The London School of Economics and Political Science (LSE), I take courses in social policy, philosophy and public policy, public management and behavioural economics.In the first few weeks we covered how to measure well-being, how to distribute health care resources, and the origins of the welfare state, to name only a few. I’ve had great su pport from my academic adviser, who I was told is really somebody and highly respected in the field of social policy.I learn loads of things each day and more importantly, what I learn feels relevant. Looking back, I can better understand the work I did in the government, and how the approaches I’ve learned work in some situations or fail in other situations. After one term, I can feel my viewpoint change from 50 feet to 50,000 feet.Probably the most difficult part of attending graduate school overseas isn’t the education itself but rather the act of “putting the previous life on hold”. I think this is especially true for students who have already started their professional life or are a couple of years senior.But there are plenty of benefits, and I feel that time at LSE passes faster than I realize. Already, I’ve made friends with students from dozens of other countries, which brings great advantages and fun. Perhaps what unites us is the great British tradition of meeting each other at the end of the week for entertainment at the cafe. In fact, I’m about to make my way there now.4. What do we know about the author?A．He majors in political science.B．He walks to campus every day.C．He has a tight study schedule.D．His adviser enjoys a good reputation.5. What does the underlined part in paragraph 3 mean?A．He stands higher and sees farther.B．He can handle various situations.C．His life attitude has varied with time.D．His mind has been greatly broadened.6. Which might be the hardest for overseas students?A．Dealing with the heavy study load.B．Adapting to the local educational system.C．Leaving one’s former life behind for now.D．Gaining a deep understanding of British traditions.7. What can be inferred about the author from the last paragraph?A．He isn’t aware he’ll graduate soon.B．He will attend a get-together at the cafe.C．He intends to make friends with more students.D．He has a strong preference for British traditions.This fall, students at the University of Massachusetts found a new menu at their dining commons: the “diet for a cooler planet” menu. This meant herb-roasted lamb, raised with a carbon-friendly approach. It included sweet potatoes that had be en picked from a local farm’s field post-harvest. The options were plant-heavy, locally grown, and involved little to no packaging.“We wanted to let students participate in climate action by making choices about their food,” says Kathy Wicks, sustainability director for UMass Dining. The university is not alone in this effort. Increasingly, American consumers and institutions are thinking about how their food choices factor into climate change. For many, small choices at the grocery store, dining hall, and restaurant can feel more accessible than big-ticket options like buying a fuel-efficient car or installing home solar panels.Small changes in dietary habits may make a big difference. Climate activists often target fossil fuels and transportation systems, but studies point to the food system as a significant contributor to global warming. According to Project Drawdown, a research organization that evaluates climate solutions, the way food is grown, transported, and consumed accounts for about a quarter of the world’s greenhouse gas emissions. Beef is a regular target. “If, on average, Americans cut a quarter pound of beef per week from their diet, it’s like taking 10 million cars off the road a year,” says Sujatha Bergen, director of health campaigns for the Natural Resources Defense Council. Food waste ranks third among climate solutions. While much waste occurs before consumers are involved-food left on the field or thrown away because it does not fit appearance standards, Americans also throw out a lot of food they have purchased: about $1, 600 worth a year per family of four.“People are beginning to understand that their food choices make a big impact on climate, says MeganLarmer, director of regional food at the Glynwood Center for Regional Food and Farming in New York. But, she cautions substantial change will need to come from the whole food system.8. Why did the University of Massachusetts adopt the new menu?A．To promote low carbon awareness.B．To reduce the cost of food packaging.C．To popularize the plant-heavy diet.D．To market the local farming products.9. What can be inferred from paragraph 2?A．Food choices matter much to climate change.B．Food is a decisive factor in global warming.C．Universities are working together in climate action.D．Fuel-efficient cars are not affordable for consumers.10. What does Sujatha Bergen say about beef?A．It is popular among millions of car drivers.B．It has a great influence on carbon reduction.C．It plays a significant role in American diet.D．It accounts for a quarter of the greenhouse gas emission.11. Which of the following can be a suitable title for the text?A．Global Warming: A Disastrous ThreatB．Low Carbon Diet: A Craze, or More?C．Carbon Emission: A Killer, or Healer?D．Food System Reform: A Successful TrialNew smart wheelchairs are designed to allow users more freedom of movement. Some even allow them to raise or lower the seats or to go up and down stairs. Now, Scewo, a Swiss company, has designed their futuristic motorized wheelchair Scewo Bro that is a complete solution with multitasking innovation. This smart wheelchair is so advanced and multifunctional that it won the Design Prize in 2021 and has been recognized as a pioneer in the medical technology industry.With clearly defined lines and a reduction to the essentials, the design team developed a clear design language. That way, Scewo Bro is seen as a new generation of wheelchairs at first glance. Scewo Bro users can control the wheelchair either by a touch interface (界面) or a smartphone. They can switch from one to the other and can also choose from three different modes: park mode, drive mode, and track mode. This allows for any terrain (地形) and allows people with mobility disabilities to be independent.Built with safety in mind, the seat always balances itself so the user stays in a sitting position. This feature allows users to deal with different terrains without the risk of tipping over. Besides, the foot rests, the seat, and the arm rests are designed to be comfortable and adjustable.The smart wheelchair can go up to 35 kilometers on one charge and can go 10 kilometers per hour. If a break is needed, just put the chair into the relaxation position. The smart wheelchair comes with two secure storage areas for people on the go to store valuables. It can fit in the trunk of a car and can be charged in just five hours, whose awesomeness is more than you expected.Scewo Bro is available in Switzerland, Austria, and Germany. While the first batch (首批) has sold out, people can pre-order for delivery in the near future. This futuristic wheelchair will allow increased movement and a better quality of life for people with mobility issues.12. What do we know about the wheelchair from paragraph 1?A．It features more functions.B．It targets smartphone users.C．It is designed for household use.D．It is the first motorized wheelchair worldwide.13. What is mainly discussed about the wheelchair in paragraph 3?A．Its fashionable elements. B．Its working principle.C．Its customer experience. D．Its humanity design.14. How does the author introduce the wheelchair in paragraph 4?A．By listing reasons. B．By making comparisons.C．By giving exact figures. D．By offering examples.15. What can be inferred about Scewo Bro from the text?A．It needs further improvement.B．It has to be charged every five hours.C．It will not be purchased without reservations.D．It helps users move around more conveniently.For many of us, there’s nothing better than burying our head in a good book. Whether it’s a striking crime story or a biography of someone’s amazing life, it’s good to read a book to switch off from the distractions of everyday life. 16It may seem obvious that reading is beneficial to our health. It can be educational and broaden the mind. A good novel can make us happy or make us cry as we empathize with characters. 17 This power of reading books was first recognized a few years ago by the British charity Reading Agency,which published a list of books that doctors could offer to patients, dealing with topics from depression, dementia (痴呆) to chronic pain.Professor Philip Davis, the author of a book called Reading for Life, studied people in reading groups and found that it’s reading literature-only written works thought to have artistic value-that has the best effect. “When reading something for pleasure,” he said, “the brain begins to work from different parts. It gets excited. It gets pre-emotional. 18In our smartphone time, we tend to abandon a good book and play with our phones instead. But opening up a book could be a simple way to help maintain our mental health. 19 And it may allow you to see that awkward situation you’ve been anxious about for long from someone else’s view.20 Refer to the Reading Agency’s book list, or you could read book reviews, visit a library or bookshop or ask a friend. Once you’ve found the right one, you’ll be fascinated!“I am leaving home.” When Timmy, my 10-year-old son, eagerly announced that he couldn’t wait to ______, the feeling of guilt peaked as I tried to figure out the reason. “So, I can get a dog,” he______.But I stayed ______—we didn’t need a dog to be involved in our life-and he gave up. Until the day I overheard Timmy, with his brother, discussing his moving plan. Suddenly, the situation ______: gain a dog or ______ a child. I felt my resolution ______. “We can look into it,” I replied doubtfully, but the kind we wanted cost incredible $1,000! Yes, love had a price. I ______ for thought again.I shared my ______ with my friend whose love for dogs was quite sensible. He connected me to his pet keeper friend who got a pregnant dog and the baby dogs were due soon. A few days later, we took a surprise trip to the countryside when the little dogs were born. I would admit that all my previous ______ reappeared after watching a group of blind things, but it was too late. Among the screams of ______, we selected one of the little dogs as ours.We attempt to ______ our kids in a healthy way —I didn’t realize the added dimension of love and ______ that a dog would teach my kids. On a hot day, Timmy cut down his outing to check that Odie, the dog, had enough water. When Odie was sick, my elder son made fresh soup and ______ fed it to him.My children no longer talk about leaving home because there is the ______ issue of who will take care of the dog. They quietly learn how to give and receive love. If having a dog means linking my children even closer together, it will be a mother’s prayer ______.21.A．set out B．move out C．sum up D．pull out 22.A．claimed B．inquired C．insisted D．commanded 23.A．determined B．committed C．ashamed D．confused 24.A．ceased B．adjusted C．shifted D．worsened 25.A．lose B．intervene C．teach D．punish26.A．strengthening B．occurring C．disappearing D．weakening 27.A．promised B．paused C．refused D．appealed 28.A．preference B．conflict C．experience D．dilemma 29.A．doubt B．embarrassment C．passion D．reaction 30.A．terror B．relief C．delight D．shock31.A．bring up B．straighten up C．take up D．cheer up 32.A．ability B．interaction C．wisdom D．responsibility 33.A．instantly B．regularly C．securely D．patiently 34.A．academic B．general C．troublesome D．abnormal 35.A．adopted B．answered C．denied D．obeyed阅读下面短文，在空白处填人1个适当的单词或括号内单词的正确形式。

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湖北省部分重点中学2024-2025学年高二英语上学期期中试题留意事项：1．本试卷分第Ⅰ卷（选择题）和第Ⅱ卷（非选择题）两部分。

答题前，考生务必将自己的姓名、考号填写在答题卡上。

2．回答第第Ⅰ卷时，选出每小题答案后，用2B铅笔把答题卡上对应题目的答案桔涂黑，如需改动，用橡皮擦干净后，再选涂其它答案标号，写在本试卷上无效。

3．回答第Ⅱ卷时，将答案写在答题卡上。

写在本试卷上无效。

第Ⅰ卷选择题（共60小题，计100分）第一部分听力（共两节, 满分30分）做题时, 现将答案标在试卷上, 录音内容结束后, 你将有两分钟的时间将试卷上的答案转涂到答题卡上。

第一节（共5小题；每小题1.5分, 满分7.5分）听下面5段对话, 每段对话后有一个小题。

从题中所给的A,B,C三个选项中选出最佳选项, 并标在试卷的相应位置。

听完每段对话后, 你都有10秒钟的时间来回答有关小题和阅读下一小题。

每段对话仅读一遍。

1. Why doesn't the girl want to go to school according to the man?A. She is ill.B. She feels nervous.C. She lost her book report.2. Who does the woman have to wait for?A. Her sister.B. The man.C. Her parents.3. What is the man probably trying on?A. A tie.B. A suit.C. A pair of socks.4. What might the man be doing now?A. Doing some cooking.B. Washing dishes.C. Setting the table.5. What does the woman think about the book?A. She likes it very much.B. There are too many pages.C. It will be hard to understand.其次节（共15小题；每小题1.5分, 满分22.5分）听下面5段对话或独白。

2024年秋季湖北省部分高中联考协作体期中考试高二数学试卷（答案在最后）考试时间：2024年11月16日8：00-10：00试卷满分：150分★祝考试顺利★注意事项：1.答卷前，考生务必将自己的学校、考号、班级、姓名等填写在答题卡上.2.选择题的作答：每小题选出答案后，用2B 铅笔把答题卡上对应题目选项的答案信息点涂黑，如需改动，用橡皮擦干净后，再选涂其他答案标号，答在试题卷、草稿纸上无效.3.填空题和解答题的作答：用0.5毫米黑色签字笔直接答在答题卡上对应的答题区域内，答在试题卷、草稿纸上无效.4.考生必须保持答题卡的整洁.考试结束后，将试题卷和答题卡一并交回.第Ⅰ卷选择题（共58分）一、单选题（本大题共8小题，每小题5分，共40分.在每小题给出的四个选项中，只有一项是符合题目要求的.）1.已知向量()()()2,3,1,2,0,3,0,0,2a b c =-==，则()a b c⋅+ ＝（）A .6B.7C.9D.13【答案】C 【解析】【分析】根据空间向量加法与数量积的坐标运算即可.【详解】因为()()()2,3,1,2,0,3,0,0,2a b c =-==所以()()()2,3,12,0,54059a b c ⋅+=-⋅=++=.故选：C .2.椭圆221x my +=的焦点在x 轴上，长轴长是短轴长的两倍，则m 的值为（）A.14B.12C.2D.4【答案】D【解析】【分析】先将方程化为标准方程，再求出长轴和短轴，再由已知列方程可求出m 的值【详解】由221x my +=，得2211y x m+=，因为椭圆的焦点在x 轴上，所以2211,a b m==，因为长轴长是短轴长的两倍，所以224a b =，即41m=，得4m =，故选：D3.直线()2120x m y ++-=与直线320mx y +-=平行，那么m 的值是（）A.2 B.3- C.2或3- D.2-或3-【答案】B 【解析】【分析】根据两直线平行的等价条件列方程组，解方程组即可求解.【详解】因为直线()2120x m y ++-=与直线320mx y +-=平行，所以()()2312123m m m ⎧⨯=+⎪⎨-+≠-⨯⎪⎩，解得：3m =-，故选：B.4.在空间直角坐标系中，已知()1,0,1A ，()1,1,1B ，10,0,2C ⎛⎫⎪⎝⎭，则点A 到直线BC 的距离为（）A.3B.63C.3D.5【答案】A 【解析】【分析】先利用空间两点间距离公式，把AB ，AC ，BC 的长度求出来，再利用余弦定理，面积公式等求出点A 到直线BC 的距离【详解】利用空间两点间距离公式，1AB ==，2AC ==，32BC ==所以222951244cos 323212AB BC AC B AB BC +-+-===⋅⨯⨯sin 3B ==所以113sin 122234ABC S AB BC B =⋅⋅=⨯⨯⨯= 设点A到直线BC 的距离为d则2254332ABC S d BC⨯===故选：A5.如图，在平行六面体1111ABCD A B C D -中，M为11A C 与11B D 的交点，若AB a=，AD b = ，1AA c=，则下列向量中与BM相等的向量是（）A .1122-++ a b c B.1122a b c ++C.1122a b c--+ D.1122a b c-+ 【答案】A 【解析】【分析】利用向量运算的三角形法则、平行四边形法则表示出BM即可.【详解】11,BM BB B M =+12c BD =+ 1()2c BA BC =++ =()12c a b +-+ =1122-++a b c故选：A.6.过点(3,1)A 的圆C 与直线0x y -=相切于点(1,1)B ，则圆C 的方程为（）A.22(2)2x y -+=B.22(2)(1)1x y -+-=C.22(3)(4)9x y -+-=D.22()(31)8x y -++=【答案】A 【解析】【分析】求得圆心和半径，由此求得圆的方程.【详解】设圆心为(),a b ，半径为r ，则()()()()22223111111a b a b b a ⎧-+-=-+-⎪⎨-=-⎪-⎩，解得2,0a b ==，所以圆心为()2,0，半径r ===.所以圆C 的方程为22(2)2x y -+=.故选：A7.已知空间A 、B 、C 、D 四点共面，且其中任意三点均不共线，设P 为空间中任意一点，若64BD PA PB PC λ=-+，则λ=（）A.2B.2- C.1D.1-【答案】B 【解析】【分析】根据空间四点共面的充要条件代入即可解决.【详解】64BD PA PB PC λ=-+，即64PD PB PA PB PCλ=-+- 整理得63PD PA PB PCλ=-+由A 、B 、C 、D 四点共面，且其中任意三点均不共线，可得631λ-+=，解之得2λ=-故选：B8.如图，椭圆的中心在坐标原点,O 顶点分别是1212,,,A A B B ，焦点分别为12,F F ，延长12B F 与22A B 交于Р点，若12B PA ∠为钝角，则此椭圆的离心率的取值范围为（）A.510,4⎛⎫+ ⎪ ⎪⎝⎭B.51,14⎛⎫+ ⎪⎪⎝⎭C.510,2⎛⎫- ⎪ ⎪⎝⎭D.51,12⎛⎫- ⎪ ⎪⎝⎭【答案】D 【解析】【分析】由题意，12B PA ∠就是22B A 与21F B 的夹角，所以22B A 与21F B的夹角为钝角，从而有22210B A F B ⋅<，结合222b a c =-即可求椭圆离心率的取值范围．【详解】解：由题意，设椭圆的长半轴、短半轴、半焦距分别为a ，b ，c ，则22(,)B A a b =-，21(,)F B c b =--，因为12B PA ∠就是22B A 与21F B 的夹角，所以22B A 与21F B的夹角为钝角，所以22210B A F B ⋅<，即20ac b -+<，又222b a c =-，所以220a ac c --<，两边同时除以2a ，得210e e --<，即210e e +->，解得15e --<或152e -+>，又01e <<，所以1512e -+<<，所以椭圆离心率的取值范围为51,12⎛⎫- ⎪ ⎪⎝⎭，故选：D ．二、多选题（本大题共3小题，每小题6分，共18分.在每小题给出的四个选项中，有多项符合题目要求.全部选对的得6分，部分选对的得部分分，有选错或未选的得0分.）9.下列说法正确的是（）A.过11(,)x y ，22(,)x y 两点的直线方程为112121y y x x y y x x --=--B.点(0,2)关于直线1y x =+的对称点为(1,1)C.直线20x y --=与两坐标轴围成的三角形的面积是2D.经过点(1,1)且在x 轴和y 轴上截距都相等的直线方程为20x y +-=【答案】BC 【解析】【分析】运用直线的两点式方程判断A 的正误；利用对称知识判断B 的正误；求出直线在两坐标轴上的截距可得到三角形的面积判断C 的正误；利用直线的截距相等可判断D 的正误．【详解】对于A ：当12x x ≠，12y y ≠时，过11(,)x y ，22(,)x y 两点的直线方程为112121y y x x y y x x --=--，故A 不正确；对于B ：点(0,2)与(1,1)的中点坐标1322⎛⎫ ⎪⎝⎭，，满足直线方程 1y x =+,并且两点的斜率为：−1,所以点(0,2)关于直线y =x +1的对称点为(1,1)，所以B 正确；对于C ：直线20x y --=在两坐标轴上的截距分别为：2,−2,直线20x y --=与坐标轴围成的三角形的面积是12222⨯⨯=，所以C 正确；对于D ：经过点(1,1)且在x 轴和y 轴上截距都相等的直线方程为x +y −2=0或y =x ，所以D 不正确；故选：BC.【点睛】本题考查直线的方程，直线与坐标轴的截距，点关于直线的对称点，注意在考虑截距相等的时候，不漏掉截距为0的情况，属于基础题．10.已知椭圆C ：22148x y +=内一点()1,2M ，直线l 与椭圆C 交于A ，B 两点，且M 为线段AB 的中点，则下列结论不正确的是（）A.C 的焦点坐标为2,0，()2,0-B.C的长轴长为C.直线l 的方程为30x y +-=D.3AB =【答案】AB 【解析】【分析】由椭圆标准方程确定,,a b c ，即可得到选项A ，B 错误；利用点差法可求直线l 方程，得到选项C 正确；联立直线和椭圆方程，利用弦长公式可得选项D 正确.【详解】由22:148x y C +=，得椭圆焦点在y 轴上，且28a =，24b =，则a =2b =，2c ==，所以椭圆的焦点坐标为0,2，()0,2-，长轴长为2a =，故选项A、B 错误；设1,1，2,2，则2211148x y +=，2222148x y +=，两式作差得()()()()1212121248x x x x y y y y -+-+=-，因为()1,2M 为线段AB 的中点，所以122x x +=，124y y +=，所以()1212121222214x x y y x x y y +-⨯=-=-=--+，所以直线l 的方程为()21y x -=--，即30x y +-=，所以选项C 正确；由2214830x y x y ⎧+=⎪⎨⎪+-=⎩，得23610x x -+=，则122x x +=，1213x x =，所以3AB ===，所以选项D 正确．故选：AB ．11.如图，在平行六面体1111ABCD A B C D -中，以顶点A 为端点的三条棱长均为6，且它们彼此的夹角都是60°，下列说法中不正确的是（）A.1126AC =B.BD ⊥平面1ACC C.向量1B C 与1AA的夹角是60°D.直线1BD 与AC 所成角的余弦值为66【答案】AC 【解析】【分析】根据题意，利用空间向量的线性运算和数量积运算，对选项中的命题分析，判断正误即可．【详解】解：对于111:A AC AB BC CC AB AD AA =++=++，∴22221111222AC AB AD AA AB AD AD AA AD AA =+++⋅+⋅+⋅363636266cos 60266cos 60266cos 60216=+++⨯⨯⨯︒+⨯⨯⨯︒+⨯⨯⨯︒=，所以1||2166AC ==，选项A 错误；对于:B 11()()AC BD AB AD AA AD AB ⋅=++⋅-22110AB AD AB AD AB AD AA AD AA AB =⋅-+⋅+⋅--⋅=，所以10AC DB ⋅= ，即1AC DB ⊥，2222()()0AC BD AB AD AD AB AD AB AD AB⋅=+⋅-==--= ，所以0AC BD ⋅=，即ACBD ⊥，因为1AC AC A ⋂=，1,AC AC ⊂平面1ACC ，所以BD ⊥平面1ACC ，选项B 正确；对于C ：向量1B C 与1BB 的夹角是18060120︒-︒=︒，所以向量1B C 与1AA的夹角也是120︒，选项C 错误；对于11:D BD AD AA AB =+- ，AC AB AD =+所以()2222211111222BD AD AA ABAD AA AB AD AA AD AB AA AB =+-=+++⋅-⋅-⋅ ，1||BD ∴==同理，可得||AC =11()()18183636181836AC BD AD AA AB AB AD ⋅=+-⋅+=+-++-=，所以111cos ||||AC BD BD AC AC BD ⋅<⋅>==⋅ ，所以选项D 正确．故选：AC ．第Ⅱ卷非选择题（共92分）三、填空题（本大题共3小题，每小题5分，共15分.）12.已知两平面的法向量分别为()0,1,0m = ，()0,1,1n =，则两平面所成的二面角为____________.【答案】45°或135°【解析】【分析】根据二面角夹角与法向量的关系，结合夹角公式求解即可.【详解】因为两平面的法向量分别为()0,1,0m = ，()0,1,1n =，则两平面所成的二面角与,m n 相等或互补，因为cos ,2m nm n m n ⋅===⋅ ，且0,180m n ︒≤≤︒ ，故,45m n =︒ .故两平面所成的二面角为45°或135°.故答案为：45°或135°13.已知圆1C ：()229x y a +-=与圆2C ：()221x a y -+=有四条公共切线，则实数a 的取值可能是___________.（填序号）①3-；②2-；③；④【答案】①④##④①【解析】【分析】两圆有四条公共切线可知两圆相离，然后两圆心的距离与半径之和作比较计算即可.【详解】圆1C 的圆心()10,C a ，半径13r =，圆2C 的圆心()2,0C a ，半径21r =，因为两圆有四条公切线，所以两圆外离，又两圆圆心距d =31>+，解得a <-或a >故答案为：①④.14.已知12,F F 为椭圆C ：221164x y+=的两个焦点，P ，Q 为C 上关于坐标原点对称的两点，且12PQ F F =，则四边形12PFQF 的面积为________．【答案】8【解析】【分析】根据已知可得12PF PF ⊥，设12||,||PF m PF n ==，利用勾股定理结合8m n +=，求出mn ，四边形12PFQF 面积等于mn ，即可求解.【详解】因为,P Q 为C 上关于坐标原点对称的两点，且12||||PQ F F =，所以四边形12PFQF 为矩形，设12||,||PF m PF n ==，则228,48m n m n +=+=，所以22264()2482m n m mn n mn =+=++=+，8mn =，即四边形12PFQF 面积等于8.故答案为：8.四、解答题（本大题共5小题，共77分.解答应写出文字说明、证明过程或演算步骤.）15.在平面直角坐标系中，ABC V 的顶点坐标分别为A （2，1），B （-2，3），C （-3，0）．（1）求BC 边所在直线的方程；（2）求BC 边上的高AD 所在直线的方程．【答案】（1）3x -y +9=0（2）x +3y -5=0【解析】【分析】（1）由题意可设直线BC 的直线方程为y =kx +b ，将B ，C 的坐标代入即可求解；（2）由题意可知113AD BC k k =-=-，设直线AD 的方程为13y x m =-+，将点A （2，1）代入，即可求解设直线BC 的直线方程为y =kx +b ，将点B （-2，3），C （-3，0）代入，可得3203k b k b =-+⎧⎨=-+⎩，解得39k b =⎧⎨=⎩，∴直线BC 方程为y =3x +9，即3x -y +9=0．【小问2详解】∵AD 为直线BC 的高，∴AD ⊥BC ，∴113AD BC k k =-=-，设直线AD 的方程为13y x m =-+，将点A （2，1）代入，解得53m =，∴直线AD 的方程为1533y x =-+，即x +3y -5=0．16.已知空间三点()()()2,0,2,1,1,2,3,0,4A B C ---，设,a AB b AC == ．（1）若3c = ，//c BC ，求c ；（2）求a 与b 的夹角的余弦值；（3）若ka b + 与2ka b - 互相垂直，求k ．【答案】（1）()2,1,2c =-- 或()2,1,2c =-（2）10（3）52k =-或2k =【解析】【分析】（1）根据向量共线设出向量c 的坐标，由模长公式列出方程，求解即可；（2）利用向量的坐标公式和向量的夹角公式即可得出；（3）根据向量垂直时数量积为0，结合向量的平方即为模的平方，计算即可得到k ．因为()()1,1,2,3,0,4B C --，所以()2,1,2BC =-- ，又因为//c BC ，所以()2,,2c λλλ=-- ，又因为3c = ，31λ=⇒=±，因此()2,1,2c =-- 或()2,1,2c =- ；【小问2详解】因为()()1,1,0,1,0,2a AB b AC ====- 所以a 与b的夹角的余弦值为1010a b a b ⋅==-⋅ ；【小问3详解】因为ka b + 与2ka b - 互相垂直，所以()()2222·20202100ka b ka b k a ka b b k k +-=⇒⋅-⋅-=⇒+-= 52k ⇒=-或2k =.17.已知圆22:1O x y +=和点()1,4M --.（1）过点M 向圆O 引切线，求切线的方程；（2）求以点M 为圆心，且被直线212y x =-截得的弦长为8的圆M 的方程；【答案】（1）1x =-或158170x y --=（2）()()221436x y +++=【解析】【分析】(1)分斜率不存在和斜率存在两种情况求解；（2）根据垂径定理和弦长公式求解即可.【小问1详解】（1）当切线的斜率不存在，直线方程为1x =-，为圆O 的切线；当切线的斜率存在时，设直线方程为()41y k x +=+，即40kx y k -+-=，∴圆心O1=，解得158k =，∴直线方程为158170x y --=综上切线的方程为1x =-或158170x y --=.【小问2详解】点()1,4M --到直线2120x y --=的距离为d ==，∵圆被直线212y x =-截得的弦长为8，∴6r ==，∴圆M 的方程为()()221436x y +++=.18.如图，在三棱锥A BCD -中，AB AD =，O为BD 的中点，OA CD ⊥.（1）证明：平面ABD ⊥平面BCD ;（2）若OCD 是边长为1的等边三角形，点E 在棱AD 上，2DE EA =，三棱锥B ACD -的体积为33，求平面BCD 与平面BCE 的夹角的余弦值.【答案】（1）证明见解析（2）5【解析】【分析】（1）根据线面垂直的判定定理先证明OA ⊥平面BCD ，又OA ⊂平面ABD ，从而由面面垂直的判定定理即可得证；（2）取OD 的中点F ，因为OCD 为正三角形，所以CF OD ⊥，过O 作//OM CF 与BC 交于点M ，则OM OD ⊥，又由（1）知OA ⊥平面BCD ，所以OM ，OD ，OA 两两垂直，以点O 为坐标原点，分别以OM ，OD ，OA 为x 轴，y 轴，z 轴建立空间直角坐标系，然后求出所需点的坐标，进而求出平面的法向量，最后根据向量法即可求解.【小问1详解】证明：因为AB AD =，O 为BD 的中点，所以OA BD ⊥，又OA CD ⊥且BD CD D ⋂=，所以OA ⊥平面BCD ，又OA ⊂平面ABD ，所以平面ABD ⊥平面BCD ；【小问2详解】解：由题意，111224OCD S =⨯⨯⨯=，所以2BCD S = ，由（1）知OA ⊥平面BCD ，所以1133B ACD A BCD BCD V V S OA --=⋅⋅=⨯== ，所以OA =2，取OD 的中点F ，因为OCD 为正三角形，所以CF OD ⊥，过O 作//OM CF 与BC 交于点M ，则OM OD ⊥，所以OM ，OD ，OA 两两垂直，以点O 为坐标原点，分别以OM ，OD ，OA 为x 轴，y 轴，z轴建立空间直角坐标系如图所示，则(0B ，1-，0)，1(,,0)22C ，(0D ，1，0)，A （0，0，2），14(0,,33E ，因为OA ⊥平面BCD ，所以平面BCD 的一个法向量为(0,0,1)m =，设平面BCE 的法向量为(,,)n x y z =，又344,0),(0,)233BC BE == ，所以由00n BC n BE ⎧⋅=⎨⋅=⎩，得302244033x y y z +=⎪⎪⎨⎪+=⎪⎩，令x =1y =-，1z =，所以1,1)n =- ，所以|||cos ,|5||||m n m n m n ⋅<>== ，所以平面BCD 与平面BCE 的夹角的余弦值为5.19.已知椭圆2222:1(0)x y C a b a b+=>>过点M ，且右焦点为(1,0)F .（1）求椭圆C 的标准方程；（2）过点F 的直线l 与椭圆C 交于A ，B 两点，交y 轴于点P .若PA mAF =uu r uuu r ，PB nBF =uu r uu u r，求m n +的值.【答案】（1）22143x y +=（2）83-【解析】【分析】（1）由题干所给条件及椭圆简单的几何性质可求；（2）设直线l 的方程为1x ty =+，联立椭圆方程，消x 可得关于y 的二次方程，再由韦达定理可得两交点纵坐标关系，再根据题意和两交点纵坐标关系求出m n +.【小问1详解】由题意可得1b c ==，2224a b c =+=，故椭圆的方程为22143x y +=【小问2详解】1、若直线l 垂直y 轴，根据椭圆简单的几何性质可知A ，B 两点的坐标，如(2,0)A -，则()2,0B ，(0,0)P ，(2,0)PA - ，(3,0)AF ，(2,0)PB ，(1,0)BF - 又∵PA mAF =uu r uuu r ，求得23m =-PB nBF =uu r uu u r ，2n =-∴83m n +=-2、若直线l 不垂直y 轴，则设直线l 的方程为1x ty =+，联立椭圆方程，消x 可得22(34)690t y ty ++-=，设1122()A x y B x y ,,(,)，则12122269,3434t y y y y t t +=-=-++由PA mAF =uu r uuu r 可得111()(0)y m y t --=-，∴111m ty =--，由PB nBF =uu r uu u r 同理可得211n ty =--∴2121212261111128342()22293334t y y t m n t y y t y y t t -+++=--+=--=--⋅=--=--+。