金陵中学2019高一期中考试试卷(学生版)

江苏省南京金陵中学2019届高三年级第一学期期中考试数学试题一、填空题（本大题共14小题，每小题5分，共70分，请将答案填写在答题卷相应的位置上．）1.设集合A＝，B＝{﹣1，0，1，2，4}，则A B＝_____________．【答案】{1，2}【解析】【分析】先化简集合A，然后求交集即可.【详解】集合A＝，又B＝{﹣1，0，1，2，4}∴A B＝{1，2}【点睛】本题考查交集的求法，考查交集定义等基础知识，考查对数函数的单调性，是基础题．2.已知复数，其中i是虚数单位，则的值是_____________．【答案】【解析】【分析】利用复数的运算法则、模的计算公式即可得出．【详解】复数z=（1+i）（1+3i）=1﹣3+4i=﹣2+4i，∴|z|==．故答案为：．【点睛】本题考查了复数的运算法则、模的计算公式，考查了计算能力，属于基础题．3.已知一组数据2，4，5，6，8，那么这组数据的方差是_____________．【答案】4【解析】【分析】先求出这组数据的平均数，再求这组数据的方差．【详解】一组数据2，4，5，6，8，这组数据的平均数==5，这组数据的方差S2=[（2﹣5）2+（4﹣5）2+（5﹣5）2+（6﹣5）2+（8﹣5）2]=4．故答案为：4．【点睛】本题考查方差的求法，是基础题，解题时要认真审题，注意方差公式的合理运用．4.从2男3女共5名同学中任选2名（每名同学被选中的机会均等）作为代表，则这2名代表都是女同学的概率为_____________．【答案】【解析】【分析】计算从2男3女共5名同学中任选2名学生和选出的2名都是女同学的选法种数，利用古典概型概率公式计算可得答案．【详解】从2男3女共5名同学中任选2名学生有=10种选法；其中选出的2名都是女同学的有=3种选法，∴2名都是女同学的概率为．故答案为：．【点睛】本题考查了古典概型的概率计算，解题的关键是求得符合条件的基本事件个数．5.如图是一个算法的流程图，则输出的a的值是．【答案】9【解析】:试题分析：由题意可得，a是在不断变大的，b是在不断变小，当程序运行两次时，a=9,b=5，a>b,跳出程序，输出a="9;"考点：算法的流程图的计算6.在平面直角坐标系xOy中，若抛物线的焦点与椭圆的右焦点重合，则实数p的值为_____________．【答案】2【解析】【分析】先根据椭圆方程求出椭圆的右焦点坐标，因为抛物线y2=2px的焦点与椭圆的右焦点重合，所以抛物线的焦点坐标可知，再根据抛物线中焦点坐标为（，0），即可求出p值．【详解】∵中a2=4，b2=3，∴c2=1，c=1∴右焦点坐标为（1，0）∵抛物线y2=2px的焦点与椭圆的右焦点重合，根据抛物线中焦点坐标为（，0），∴，则p=2．故答案为：2【点睛】本题主要考查了椭圆焦点与抛物线焦点的求法，属于圆锥曲线的基础题．7.已知，则＝_____________．【答案】﹣【解析】【分析】利用sin2x=cos（2x+）=2sin2（x+）即可得到结果.【详解】∵，∴sin2x=cos（2x+）=2sin2（x+）=﹣1=，故答案为：﹣【点睛】此题考查了二倍角的余弦函数公式，以及诱导公式的作用，熟练掌握公式是解本题的关键．8.设a＞0，若a n＝且数列{a n}是递增数列，则实数a的范围是__________．【答案】2＜a＜3【解析】由{a n}是递增数列，得解得∴2＜a＜39.在平面直角坐标系xOy中，若曲线(a，b为常数)过点P(2，﹣5)，且该曲线在点P处的切线与直线垂直，则2a＋3b的值是_______．【答案】﹣8【解析】【分析】由曲线y=ax2+（a，b为常数）过点P（2，﹣5），且该曲线在点P处的切线与直线2x﹣7y+3=0垂直，可得y|x=2=﹣5，且y′|x=2=﹣，解方程可得答案．【详解】∵直线2x﹣7y+3=0的斜率k=，∴切线的斜率为﹣，曲线y=ax2+（a，b为常数）过点P（2，﹣5），且该曲线在点P处的切线与直线2x﹣7y+3=0垂直，∴y′=2ax﹣，∴，解得：a=﹣1，b=﹣2，故2a＋3b =﹣8，故答案为：﹣8【点睛】本题考查的知识点是利用导数研究曲线上某点切线方程，其中根据已知得到y|x=2=﹣5，且y′|x=2=﹣，是解答的关键．10.若函数在上不单调，则的取值范围是____．【答案】0【解析】此题考查导数的应用；，所以当时，原函数递增，当原函数递减；因为在上不单调，所以在上即有减又有增，所以11.如下图，在中，．若，则__________．【答案】【解析】因为,又因为，所以，也即，所以，又，故，由余弦定理得，则，应填答案。

南京市金陵中学 2018-2019 学年高一下学期期中考试英语试题题号一二三四五六总分得分1．答题前填写好自己的姓名、班级、考号等信息； 2．请将答案正确填写在答题卡上。

第I 卷（选择题）评卷人得分1． To understand someone you must listen to them. Surprise! The problem is that most of us don't know how to listen. It's so typical of us to use one of the five poor listening styles. The response of the man sitting on the sofa is an example ________2． "If you find yourself wanting to bury your feelings deep inside you, don't do it. Instead express them in a responsible way." This advice encourages us to ______C ． practice mirroringD ． listen with our ears and heart3． Have you ever seen a flock of geese heading south for the winter flying along in a V formations? Learning to synergize is like learning to form V formations with others instead of trying to fly through life solo. You"ll be amazed at how much faster and farther you' ll go! Therefore, synergy is not .A ． self-centered listeningB ． spacing outC ． pretend listeningD ． selective listeningA ． stand in others ' shoesB ． seek to be understood4． Your soul is a very private area of your life. Naturally, there are many different ways to feed it.Which of the following ideas might not contribute to feeding your soul positively.A ． Getting back nature.B．Keeping a journal. C． Reading inspiring books D．Staying up all night 5．“ What are you doing? ” Bruno asked in _ _____ , although he wasn't happy to comehomeand find someone going through his possessions. A ． as a polite tone as he could muster B ． as polite a tone as he could muster C ． as polite as a tone he could musterD ． a tone polite as he could muster6． Martin had said that his father was a chef, which Bruno knew to be true because he sometimes Martin from school, wearing a white smock and a tartan apron. A ．pickedB ．caughtC ．collectedD ． brought7． Bruno was secretly pleased when he knew that it could be weeks before they went back because he had himself to the idea of spending one more month there. A ． contributedB ．stuckC ．devotedD ． resigned8． As far as she could _____ , there was nothing ______ low huts. A ． turn out; besides B ． make out; besides C ． turn out; butD ． make out; but9． The whole city seemed to spring to life almost overnight around 1050, a phenomenon now as aA ． celebrating differencesC ． open-mindedness B ． working independentlyD ． finding new betterwaysmiracle.A ．having referred to B．to be referring toC ． referred toD ． referring to金陵中学 2018-2019 学年高一下学期期中考试英语试卷 第3页共 14页10．—Many parents do almost everything for their children, but the children still fail them.— That's _____ these parents are mistaken.A ．howB ．whereC ．whatD ．why11．It's an inspirational story, and the well-researched facts really bring the story and characters . A ． alive B ．livingC ．livelyD ． live12．to the gift was a note on which he expressed his gratitude for our reception during hisstay here.A ．Being attachedB ． AttachingC ． AttachedD ． having attached13 ． When confronted with the audience 's suspicions, the spokesman found himself ________for words.14 ． Drivers must strictly ____ the speed limit when driving on the freeways to keep away from any possible accidents. A ．graspB ．representC ．observeD ． reduce15．—I couldn 't tell Jack from his twin brother.—Me too, and there is no obvious _______ between them.A ． contributionB ．distinctionC ．promotionD ． expressionFor many years my parents subscribed to “ Reader 's Digest ”. Usually I enjoyed reading each issue （期） as much as they did. The 16 in the magazine I liked most was“The BestAdvice I Ever Had. ”Even today, one of those best advice stories sticks with me and 17 a great lesson. Writing about his 18 . Harry Fosdick —longtime beloved minister of Riverside Church in New York City — 19 a morning when his father was leaving the house. He 20to Harry 's mother and said“Tell Harry he can cut thaey g, rass2 t1o d he feels like it. ”Then after a few steps the father turned back and 22a meaningful suggestion:“Tell Harry he had better feel like it.” The reason why I treasure Fosdick's father 'A ． loseB ．lostC ．to loseD ． having losts instructmany of us know what we have to do in our 23 tasks — we are told what to do, whilethe major 24is that in many cases we delay action until we feel like it.So think of the25 that Harry ' s father tells him that he “ had better feel like itwhen we must do the 26things. Already, I am confident you could add a dozen similar27tasks and responsibilities to this list — possibly two dozen. If you28 until youfeel like doing them, tasks will become more difficult. Getting back to young Harry29 that a small boy pushing the mower （ 割草机 ） with great effort back and forth, around flowers and bushes on this 30 day, wiping the sweat from his face until he got the job 31 . Let ' s be aware that this was one of those long ago mowers, 32a motor,appropriately called “ push mowers. ” So th3e3 t ask .i sH arry got the demanding job finished, because he knew that he“ had better feel like it. ”34 you are to finish your task or do your work, you ' ll have to feel like it. After you35 feel like it you ' ll surprisingly find it isn' t difficult to finish.16．A ． articleB ． commentC ． pictur eD ． advertisement 17．A ．shows B ． includesC ． impliesD ． remains18．A．boyhood B ． job C ． life D ． joke19．A． spent B ． consideredC ． rememberedD ． learned20．A ． returned B ．moved C ．wroteD ． turned 21．A．becauseB ． ifC ． thoughD ． since22．A． mentioned B ．understoodC ． addedD ．explained23．A．given B ． dailyC ． nextD ． favorite24．A． factB ． conditionC ． concernD ． problem25．A ．resultB ． wayC ． adviceD ． reason 26．A ．unpleasant B ． awesome C ． aggressive D ． supreme 27．A ． strangeB ． annoyingC ． particularD ． easy 28．A． waitB ． changeC ． stickD ． resist 29．A．believeB ． expectC ． imagineD ． suppose 30．A． freezingB ． hotC ．comfortableD ． starry 31．A ． started B ．done C ． paid D offered 32．A．withoutB ． throughC ． withD ．for s ta33．A．funny B ．different C．interesting D ．rough34．A ．After B．ButCWhenDUnless35．A．mainly B ．really C．naturally D ．hardly评卷人得分三、阅读理解The Omron HEM-780 IntelliSenseTM Automatic Blood Pressure Monitor quickly measures your blood pressure and your measurement reading can be clearly displayed on a large digital panel.KNOW YOUR UNITIMPORTANT SAFETY NOTES* Do not use a cell phone near the unit, which may result in operational failure.* Changes or modifications not approved by Omron Healthcare will declare the user warranty invalid. Do not disassemble or attempt to repair the unit or components.SUGGESTIONS BEFORE TAKING1. Avoid eating, smoking, and exercising for 30 minutes before taking a measurement. Rest for at least 15 minutes before taking the measurement.2. Stress raises blood pressure. Avoid taking measurements during stressful times.3. Rest your left arm on a table so that the cuff is at the same level as your heart.4. Wait 2-3 minutes between measurements. The wait time allows the arteries （动脉） to return to the condition prior to taking the blood pressure measurement.36．Your measure reading might be inaccurate, if ______ .a. you have just finished your mealb. you have just cycled home from workc. your child is using the cell phone in the next roomd. your wife has just finished measuring her blood pressuree. you have just been scolded by your boss because of the delayed workA．a b c B．a b e C．b c e D．c d e 37．Which of the following monitors is under warranty?A ．The monitor that was bought six years ago.B ．The monitor that goes wrong due to false use.C．The monitor whose purchase proof is missing.D ．The monitor whose system broke down when first used.A hospital has been forced to ban Pokemon Go players from the site after a monster hub（妖怪枢纽站） was found in the A&E department. Royal Stoke University Hospital discovered that its casualty unit （急诊室） is on the same spot as a Pokemon Go ‘whgeyrme 'pla y—er—scan train their newly caught Nintendo creatures.The University Hospitals of North Midlands Trust agreed last week that patients can play Pokemon Go on wards because walking around is healthy. But the Trust has been forced to post a warning on its website about public access to A&E. It said if Pokemon Go becomes a major annoyance it would ask Nintendo ——which decides on the locations of the virtual gyms according using GPS ——to have it removed from the premises（道馆）.Kevin Parker, associate chief nurse, said, “ Members of thoet pnuebeldic t ow hboe daot nRoyal Stoke should not attempt to enter A&E or any other part of the hospital building to play the game. The A&E department is incredibly busy this summer. We want the public to understand that anybody who visits the hospital solely to play the game will provide an unwanted distraction to the important work of the hospital. I ' m also aware of various reports inthe media of unsafe areas that the game has been played in. ”“ Royal Stoke University Hospital is a safe area where gamers can enjoy Pokemon Go.”Michelle Harris, the Trust ' s manager, said the game could still be played by those already inhospital. “ We recognize that the Pokemon Go game encourages walking and exercise, which issomething that the Trust is equally keen to promote, said. ” sheThere are a number of “ walking routes ” established throughout the Trust that can be usedto combine walking and playing the game. “ Walking just 30 minutes, five times a week, canhelp reduce the risk of preventable illnesses such as obesity, diabetes , and heart disease. ”There have been several warnings about the game since its UK release. Last week a group of teenagers in Wiltshire were left stranded almost 100ft underground after they got carried away searching for Pokemon Go characters. The four boys, aged 16 and 17, ended up getting lost and had to wait to get a phone signal before they could call for help. Eventually, they contacted Dorset and Wiltshire Fire and Rescue team, who took them to safety. Damien Bence, of the fire and rescue team, said: “ Pokemon Go is obviously leading people into dangerous situations. ” 38．It seems that Pokemon Go is a game __________ .A ．encouraging players to walk and exercise instead of staying indoorsB ．helping cure such diseases as obesity, diabetes and heart diseaseC．designed to help patients in hospital recover soonerD ．warning teenagers of the places easy to get lost or attacked 39．The A&E department is incredibly busy because _________________________________________________ .A ．more patients ' arrival increases the workload of the A&E departmentB ．doctors 'lay ping the game makes the A&E department less efficient C．players ' injury increases the workload of the A&E department D ．players ' arrival disturbs the work of the A&E department40．The passage is mainly about _____ .A ．the popularity of Pokemon Go since its UK releaseB ．the applications of Pokemon Go in hospitals since its UK releaseC．the negative effects of Pokemon Go since its UK releaseD ．the establishment of Pokemon Go virtual gyms since its UK releaseHow sharks navigate the vast and seemingly featureless ocean has long been a mystery. Now there's evidence they may follow their noses. Sharks rely on their sense of smell to help chart a path through the ever-shifting waters of the deep seas, according to a new study. Scientists have supposed that the animals navigate by monitoring smell clues or the Earth's magnetic field, but no one knew for sure.In new experiments near San Diego, scientists ferried wild leopard sharks about 6 miles(10 kilometers)away from their preferred habitats, fitted them with tracking devices, and stuffed some of the animals' noses with cotton balls. Just 30 minutes after being released facing the wrong way, sharks with full use of their smells made a corrective U-turn and then headed straight back to shore, "says study leader Andrew Nosal, a researcher at the Scripps Institution of Oceanography, California. Sharks with stuffed noses, meanwhile, "appeared lost, "wandering aimlessly and swimming more slowly than those that could smell freely.To test sharks' sense of direction, Nosal and colleagues captured several dozen leopard sharks, a small species found along the coast from Washington State to northern Mexico.After blocking some of the animals' noses, the scientists then took the animals on a cruise to deeper waters before slipping them, safe and unharmed, back into the sea. Even the sharks with plugged noses made it partway back to shore before their tracking devices fell off. But those with unplugged noses "took very straight paths" toward home. Nosal supposes the sharks likely sniffed out chemical molecules found in higher and higher doses nearer to land.Other scientists, however, remain unconvinced. Maybe the animals with plugged noses "were confused by the fact that they had something stuffed in their nose," says Kim Holland, a marine biologist at the University of Hawaii at Manoa. It's also unlikely the animals were following a smell that grew stronger closer to land, adds Jayne Gardiner, a sensory biologist at the New College of Florida. Animals that couldn't smell a thing still turned toward the beach, which "suggests something else is really guiding them," she says.Study leader Nosal responds that sharks with cotton-stuffed nose willingly eat, suggestingthat a congested nose doesn't upset them much. He agrees that sharks use a variety of markers to find their way, "but the point is that smell participates in navigation, "he says. "Our study was only the first step in solving this mystery. " 41．This passage is mainly about ______ .A ．why sharks have a good sense of smellB ．why sharks are clever animalsC．how scientists do the experiment D ．how sharks find their way in the vast ocean42．Which is true about the experiments done by the scientists according to the passage? A ．Scientists plugged sharks' noses to test whether they navigate through their sense of smell. B ．Scientists chose sharks' preferred habitats in order to track the animals easily.C．After being taken to deeper waters, sharks with plugged noses couldn't make it to the shore. D ．The sharks in the experiments aren't willing to eat due to the plugged noses.43．We can infer from the passage that _____ .A ．there are various markers for sharks to find their way and eyes play an important role in itB ．some scientists hold different opinions from the findings of the new studyC ．the new study has completely solved the mystery of sharks' easy navigation in the vast oceanD ．the sharks likely sniffed out chemical molecules found in higher doses nearer to land 44．What's the purpose of writing this passage?A ．To call on people to protect the sharks.B ．To present different theories on the mystery of the sharks.C．To let people learn more about the sharks.D ．To inform the readers of the findings of a new study on shark.On the school playground in Los Tomes, Jos , a lone chiléd, plays a ball- and-cup game. The eight- year-old is the school's only pupil. His teacher, Nilda, herself a former pupil, says that enrolment（注册入学） has dropped from 65 when she started teaching 43 years ago. Drought has driven families away, she says, “Only the old remain. ”Los Tomes is an agricultural cooperative, one of 178 in Chile's Coquimbo region. Nineteen communities try to grow wheat and raise sheep and goats on 2,800 hectares. A decade-long drought has made that harder. Hilltop springs where the animals once drank have dried up. As the number of herds（畜群） decrease, farmers' children moved away to take jobs in cities or at copper mines.① Hope for Los Tomes comes in the form of three 60- square-metre nets stretched between poles on a ridge （山脊） above the community. These nets capture （捕捉）droplets （水珠 ） from the fog that rolls in from the sea 4 kilometers away. They flow down to two troughs （ 槽 ）, from which animals drink. The nets can harvest 650 litres of water a day.② Chile has been investigating fog capture since the 1950s. The fog can be harvested with the help of a coastal mountain range and strong winds. Earlier attempts to turn the mist into usable water failed. In 1990 fog nets at a fishing village captured 8,000 litres a day. Villagers argued about how to share responsibility for maintaining the nets.Climate change, which is expected to decrease rainfall in the region, has inspired a new search for sources ofwater. The project at Los Tomes is part of an attempt to capture fog.question is not whether the fog collectors work but who's going to provide and maintain them, says Daniela.At a community north of Los Tomes, three 150- square-metre fog catchers feed a plantation of young olive trees. When the trees mature, they will produce 750 litres of organic olive oil a year. The water source will be a big selling point. A privately owned brewery （ 啤酒 厂 ）in Pena Blanca was quick to spot fog water's marketing appeal.③ The development fund paid 5.6 million pesos each piece to put up the structures in Los Tomes ； when the nets wear out, the villagers will have to replace them at a cost of 100,000 pesos each. Coquimbo has more than 40,000 hectares of land with the right conditions for putting up fog- catchers. If it were fully employed, the region could harvest 1,400 litres a second, enough to supply all its drinking water.④ That might attract back educated young people from the cities. A chance to develop tourism near the FrayJorge national park, a rainforest which has survived thanks to its ownnatural fog-collection mechanism, brought Salvador to his birthplace.“ Roots, the land and the desire to start this brought me back, says Salvador.45． The boy in the first paragraph is used as an example to show46． The ideal place for nets should be .D ． at the foot of the ridge 47． The concern of the fishing village's people is .A ． the seriousness of droughtB ． the poverty of the areaC ． the trend of the moveD ． the lack of teachersA ． in the rough seaB ． on a coastal ridgeC ． over the seaA ．whether the fog- catcher worksB ．whether the fog- catcher can provide enough waterC．how to make the fog- catcher run well continuouslyD ．how to make use of the water48．The sentence “It makes a profit, but most fog-harvesting projects require investment in their early stages. o”uld sbhe put in .A ．①B ．②C．③ D ．④49．According to the passage, which of the following statement is right ？A ．Water collected from fog can be sold as beer on the market.B ．Daniela suggests that olive trees should be planted in the plantation.C．The products made with fog water will probably appeal to the consumers.D ．Part of temperate rainforest's survival is due to the use of man- made fog nets.50．Salvador returning to his birthplace mainly wants to .A ．protect the remaining forest B．build more fog- catchersC．sell handicrafts on the road stands D．develop local tourism第II 卷（非选择题）Open Hearts to lifeWe often close ourselves off when unfortunate events happen in our live. We try to reduce the hurt and pain by pretending it doesn't exist, 51 ．We need to learn to open our hearts to the potentials of life and let the world soften us.Whenever we start to let our fears and seriousness get the best of us, we should take a step back and re-evaluate our behavior. The items listed below are the ways you can open your heart more fully and completely.Breathe into painWhenever a painful situation arises in your life, try to embrace it instead of running away.52．When we run away from sadness, it will get stronger and more real. By making use of our breath, we soften our experiences and allow more newness and greater experiences to blossom.53．I know we ' ve all tf edlecisions or actions that we had to take simply due to our "gut" impulses: when asked, we can ' t explain the reasons behind -d-jouisntg a s doeep knowing thatit had to get done. To start this process, take few deep breaths and then ask, “ Heart, whdecision should I make here? What action feels the most right? ”See what comes up, then engage and evaluate the outcome.Spend time alone.For most of our lives we ' re surrounded by people: our friends, colleagues, peers, familymembers, and loved ones, and strangers. How often do we really spend time alone? When you spend time in solitude, you're free from the influences of other people, and can truly open yourself and explore whatever you ' d like. See where your thoughts5 4ta．ke you.It might be painful or even scary at first, but by opening yourself up to these new feelings,you ' ll add a whole new layer of depth, experience, and understanding into your life.Get outside of yourselfThis may seem a little contradictory to the last tip, but in reality, they actually work hand-in- hand. After you 've explored the depths of yourself, you come away with a newunderstanding. Now, it's time to share that -not through telling others, but through being with others.When you ' re in a group of people, try to give them you rfull energy and attention so you can understand them just as you did yourself.55．Lose yourself in the beauty of others; see what they can teach you about yourself.A ．Embrace the uncomfortable.B ．Ask your heart what it wantsC．In the end, though, we can't hide from ourselvesD ．We should take a step back and re-evaluate our behaviorE．When the sadness strikes, take a deep breath and lean into it.F．The golden ticket here is to not let yourself become distracted; just see what it's like to be alone.G ．Appreciate their uniqueness, as if they are an extension of you.Texting while walking is something that most of us are guilty of. We can 56．(reply) ' t helpto that message we just received. However, while 57．is fun to keep up with the latest news, we may actually be putting ourselves in danger. 58．(spot) a p“erIs on in front of me walking very slowly and weaving, and I thought,‘ Is this person drunk? ' But it turned out that the person was just texting. ”Ma sttahidew Timmis. 59 ．(inspire) by this, Timmis and his team set out to seekthe effects of phone use on 60 ．(passer-by). A group of 21 volunteers were asked to walk around a certain street. The participants traveled the course a total of 12 times each, either writing 61 ．reading a message, making a call, or with no phone at all. It took the volunteers 118 percent 62．(long) to complete the course when using a phone. They also focused 63．the path 51 percent more when they weren ' t using a phone. Althoeu wghe rteh enro accidents, Timmis believes we should still be aware of 64 ．is going on around us. “ You are not going to be able to respond to danger efficiently, 65 ．increases the risk of injury. ” He added.66．请阅读下列文字，并按照要求写一篇120 词左右的文章。

2019届江苏省南京金陵中学高三第一学期期中考试数学试题数学注意事项：1．答题前，先将自己的姓名、准考证号填写在试题卷和答题卡上，并将准考证号条形码粘贴在答题卡上的指定位置。

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一、填空题1．设集合A＝{x|log2x<2}，B＝{﹣1，0，1，2，4}，则A∩B＝_____________．2．已知复数z=(1+i)(1+3i)，其中i是虚数单位，则|z|的值是_____________．3．已知一组数据2，4，5，6，8，那么这组数据的方差是_____________．4．从2男3女共5名同学中任选2名（每名同学被选中的机会均等）作为代表，则这2名代表都是女同学的概率为_____________．5．如图是一个算法的流程图，则输出a的值是_____________．6．在平面直角坐标系xOy中，若抛物线y2=2px的焦点与椭圆x24+y23=1的右焦点重合，则实数p的值为_____________．7．已知sin(x +π4)=35，则sin2x ＝_____________．8．设a ＞0，若a n ＝63377n a n n a n ≤⎧⎨⎩－（－）－，，，＞，且数列{a n }是递增数列，则实数a 的范围是__________．9．在平面直角坐标系xOy 中，若曲线y =ax 2+bx (a ，b 为常数)过点P(2，﹣5)，且该曲线在点P 处的切线与直线2x −7y +3=0垂直，则2a ＋3b 的值是_______．10．若函数f(x)=−12x 2+4x −3lnx 在[t,t +1]上不单调，则t 的取值范围是____．11．如下图，在ABC ∆中，1,2,,2AB AC BC AD DC AE EB ====u u u v u u u v u u u v u u u v ．若12BD AC ⋅=-u u u v u u u v ，则CE AB ⋅=u u u v u u u v __________．12．已知函数f(x)={2x +1，x ≤0|lnx |，x >0，则关于x 的方程f[f(x)]=3的解的个数为_____________．13．已知正数a ，b ，c 满足b 2+2(a +c)b −ac =0，则ba+c 的最大值为_____________．14．若存在正数x ，y ，使得(y −2ex)(lny −lnx)s +x =0，其中e 为自然对数的底数，则实数s 的取值范围是_____________．二、解答题15．如图，在四棱锥P —ABCD 中，四边形ABCD 是矩形，平面PCD ⊥平面ABCD ，M 为PC 中点．求证： （1）PA ∥平面MDB ； （2）PD ⊥BC ．16．已知α∈(0，π2)，β∈(π2，π)，cosβ=−13，sin(α+β)=4−√26．（1）求tan2β的值； （2）求α的值．17．如图，某污水处理厂要在一个矩形污水处理池(ABCD)的池底水平铺设污水净化管道(管道构成Rt △FHE ，H 是直角项点)来处理污水．管道越长，污水净化效果越好．设计要求管道的接口H 是AB 的中点，E ，F 分别落在线段BC ，AD 上．已知AB ＝20米，AD ＝10√3米，记∠BHE ＝θ．（1）试将污水净化管道的长度L 表示为θ的函数，并写出定义域； （2）当θ取何值时，污水净化效果最好？并求出此时管道的长度L ．18．在平面直角坐标系xOy 中，圆O ：x 2+y 2=4与坐标轴分别交于A 1，A 2，B 1，B 2(如图)．（1）点Q 是圆O 上除A 1，A 2外的任意点(如图1)，直线A 1Q ，A 2Q 与直线y +3=0交于不同的两点M ，N ，求线段MN 长的最小值；（2）点P 是圆O 上除A 1，A 2，B 1，B 2外的任意点(如图2)，直线B 2P 交x 轴于点F ，直线A 1B 2交A 2P 于点E ．设A 2P 的斜率为k ，EF 的斜率为m ，求证：2m ﹣k 为定值．（图1）（图2）19．设函数f(x)=e xx 3−3k x−klnx ，其中x ＞0，k 为常数，e 为自然对数的底数．（1）当k ≤0时，求f(x)的单调区间；（2）若函数f(x)在区间(1，3)上存在两个极值点，求实数k 的取值范围；（3）证明：对任意给定的实数k ，存在x 0(x 0>0)，使得f(x)在区间(x 0，+∞)上单调递增．20．若数列{a n }同时满足：①对于任意的正整数n ，a n+1≥a n 恒成立；②若对于给定的正整数k ，a n−k +a n+k =2a n 对于任意的正整数n (n ＞k )恒成立，则称数列{a n }是“R(k )数列”．（1）已知a n ={2n −1，n 为奇数2n ，n 为偶数，判断数列{a n }是否为“R(2)数列”，并说明理由；（2）已知数列{b n }是“R(3)数列”，且存在整数p (p ＞1)，使得b 3p−3，b 3p−1，b 3p+1，b 3p+3成等差数列，证明：{b n }是等差数列．21．二阶矩阵M 对应的变换将点(1，﹣1)与(﹣2，1)分别变换成点(﹣1，﹣1)与(0，﹣2)． （1）求矩阵M 的逆矩阵M −1；（2）设直线l 在变换M 作用下得到了直线m ：2x −y =4，求l 的方程．22．在极坐标系中，设圆ρ=3上的点到直线ρ(cosθ+√3sinθ)=2的距离为d ，求d 的最大值．23．如图，已知三棱锥O —ABC 的侧棱OA ，OB ，OC 两两垂直，且OA ＝1，OB ＝OC ＝2，E 是OC 的中点． （1）求异面直线BE 与AC 所成角的余弦值； （2）求二面角A —BE —C 的余弦值．24．已知f n (x)=(1+√x)n ，n ∈N ∗．（1）若g(x)=f 4(x)+2f 5(x)+3f 6(x)，求g(x)中含x 2项的系数；（2）若p n 是f n (x)展开式中所有无理项的系数和，数列{a n }是由各项都大于1的数组成的数列，试用数学归纳法证明：p n (a 1a 2⋯a n +1)≥(1+a 1)(1+a 2)⋯(1+a n )．2019届江苏省南京金陵中学高三第一学期期中考试数学试题数学答案参考答案1．{1，2}【解析】【分析】先化简集合A，然后求交集即可.【详解】集合A＝{x|log2x<2}={x|0＜x<4}，又B＝{﹣1，0，1，2，4}∴A∩B＝{1，2}【点睛】本题考查交集的求法，考查交集定义等基础知识，考查对数函数的单调性，是基础题．2．2√5【解析】【分析】利用复数的运算法则、模的计算公式即可得出．【详解】复数z=（1+i）（1+3i）=1﹣3+4i=﹣2+4i，∴|z|=√(−2)2+42=2√5．故答案为：2√5．【点睛】本题考查了复数的运算法则、模的计算公式，考查了计算能力，属于基础题．3．4【解析】【分析】先求出这组数据的平均数，再求这组数据的方差．【详解】一组数据2，4，5，6，8，(2+4+5+6+8)=5，这组数据的平均数x=15[（2﹣5）2+（4﹣5）2+（5﹣5）2+（6﹣5）2+（8﹣5）2]=4．这组数据的方差S2=15故答案为：4．【点睛】本题考查方差的求法，是基础题，解题时要认真审题，注意方差公式的合理运用．4．310【解析】【分析】计算从2男3女共5名同学中任选2名学生和选出的2名都是女同学的选法种数，利用古典概型概率公式计算可得答案．【详解】从2男3女共5名同学中任选2名学生有C52=10种选法；其中选出的2名都是女同学的有C32=3种选法，．∴2名都是女同学的概率为310故答案为：3．10【点睛】本题考查了古典概型的概率计算，解题的关键是求得符合条件的基本事件个数．5．10【解析】【分析】根据已知的程序框图可得，该程序的功能是利用循环结构计算并输出变量a的值，模拟程序的运行过程，可得答案．【详解】当a=1，b=12时，不满足a＞b，故a=4，b=10，当a=4，b=10时，不满足a＞b，故a=7，b=8，当a=7，b=8时，不满足a＞b，故a=10，b=6，当a=10，b=6时，满足a＞b，故输出的a值为10，故答案为：10【点睛】本题考查的知识点是程序框图，当循环次数不多，或有规律可循时，可采用模拟程序法进行解答．6．2【解析】【分析】先根据椭圆方程求出椭圆的右焦点坐标，因为抛物线y2=2px的焦点与椭圆x24+y23=1的右焦点重合，所以抛物线的焦点坐标可知，再根据抛物线中焦点坐标为（p2，0），即可求出p值．【详解】∵x 24+y23=1中a2=4，b2=3，∴c2=1，c=1∴右焦点坐标为（1，0）∵抛物线y2=2px的焦点与椭圆x24+y23=1的右焦点重合，根据抛物线中焦点坐标为（p2，0），∴p2=1，则p=2．故答案为：2【点睛】本题主要考查了椭圆焦点与抛物线焦点的求法，属于圆锥曲线的基础题．7．﹣725【解析】【分析】利用sin2x=−cos（2x+π2）=2sin2（x+π4）−1即可得到结果.【详解】∵sin(x+π4)=35，∴sin2x=−cos（2x+π2）=2sin2（x+π4）−1=1825﹣1=−725，故答案为：﹣725【点睛】此题考查了二倍角的余弦函数公式，以及诱导公式的作用，熟练掌握公式是解本题的关键．8．2＜a＜3【解析】由{a n }是递增数列，得87301a a a a ⎧⎪⎨⎪⎩－＞，＞，＞，解得1?392a a a ⎧⎨⎩＜＜，＜－或＞，∴2＜a ＜39．﹣8 【解析】 【分析】由曲线y=ax 2+bx （a ，b 为常数）过点P （2，﹣5），且该曲线在点P 处的切线与直线2x ﹣7y+3=0垂直，可得y|x=2=﹣5，且y′|x=2=﹣72，解方程可得答案．【详解】∵直线2x ﹣7y+3=0的斜率k=27，∴切线的斜率为﹣72，曲线y=ax 2+bx （a ，b 为常数）过点P （2，﹣5），且该曲线在点P 处的切线与直线2x ﹣7y+3=0垂直， ∴y′=2ax ﹣bx 2，∴{4a +b2=−54a −b 4=−72，解得：a=﹣1，b=﹣2， 故2a ＋3b =﹣8， 故答案为：﹣8 【点睛】本题考查的知识点是利用导数研究曲线上某点切线方程，其中根据已知得到y|x=2=﹣5，且y′|x=2=﹣72，是解答的关键．10．0<t <1或2<t <3 【解析】此题考查导数的应用；f ′(x)=−x +4−3x =−x 2−4x+3x=−(x−1)(x−3)x，所以当x ∈(0,1),(3,+∞)时，原函数递增，当x ∈(1,3)原函数递减；因为在[t,t +1]上不单调，所以在[t,t +1]上即有减又有增，所以{0<t <11<t +1<3或{1<t <33<t +1∴0<t <1或2<t <3 11．43-【解析】因为()12AD DC BD BA BC ==+u u u v u u u v u u u v u u u v u u u v，所以,又因为AC AB CB =-u u u v u u u v u u u v ，所以()()()()111222BD AC BA BC AB CB BA BC AB CB ⋅=+-+-=-u u u v u u u v u u u v u u u v u u u v u u u v u u u v u u u v u u u v u u u v 。

江苏省南京市金陵中学2019-2020学年高一下学期期中数学试题学校:___________姓名：___________班级：___________考号：___________一、单选题1．在平面直角坐标系xOy 中，已知过点()2,A m -和(),4B m 的直线与直线210x y +-=平行，则m 的值为（ ） A ．0B ．10C ．2D ．8-2．已知tan 24πα⎛⎫+= ⎪⎝⎭，tan 3β=，则tan αβ（ ）A ．34-B ．43- C ．34D ．433．在平面直角坐标系xOy 中，若点1,0A ，()0,1B ，()2,3C --，则ABC 的面积为（ ）A ．3B ．2C ．1D ．124．若实数,x y 满足22log log 1x y +=，则x y +的最小值是（ ）A ．2B ．C ．3D ．45．如图，在ABC 中，CD 为角C 的平分线，若2B A =，23AD BD =，则cos A 等于（ ）A ．13B ．12C ．34D ．06．已知函数()sin sin3f x x x =-，[]0,x π∈，则()f x 的所有零点之和等于（ ） A ．54π B ．3πC ．πD ．2π7．在平面直角坐标系xOy 中，已知圆()22:29C x y -+=，,E F 是直线:2l y x =+上的两点，若对线段EF 上任意一点P ，圆C 上均存在两点,A B ，使得cos 0APB ∠≤，则线段EF 长度的最大值为（ ）A ．2BC ．D ．4二、多选题8．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 ．9．在平面直角坐标系xOy 中，直线l 与圆()2222x y -+=相切，则直线l 的方程可以是（ ） A ．0x y +=B ．20x y +-=C ．0x y -=D ．40x y +-=10．已知ABC 的内角,,A B C 所对的边分别为,,a b c ，若2222cos 2c a b ab C ≤++，则C 的取值可以是（ ） A ．6πB ．4π C ．3π D ．2π 11．在正方体1111ABCD A BC D -，点,,E F G 分别是棱11111,,A D A A A B 的中点，下列说法正确的是（ ） A ．1EF B C ⊥ B ．1//BC 平面EFGC ．1AC ⊥平面EFGD ．异面直线FG 、1BC 所成角的大小为4π12．集合{}22(,)|4A x y x y =+=，{}222(,)|(3)(4)B x y x y r =-+-=，其中0r >，若A B中有且仅有一个元素，则r 的值是（ ）. A ．3 B ．5C ．7D ．9三、填空题13．若锐角α满足3sin 5α=，则cos 4πα⎛⎫+ ⎪⎝⎭的值为________.14．在平面直角坐标系xOy 中，若圆224x y +=和圆224440x y x y ++-+=关于直线l 对称，则直线l 的方程为________.15．已知ABC 的面积为2AC AB -=，1cos 4A =-，则BC 的长为________.16．在ABC 中，若cos B =()2tan 3sin 2A C -的最小值为________.四、解答题17．在平面直角坐标系xOy 中，已知ABC 的三个顶点(),A m n ，()2,1B ，()2,3C -. （1）求BC 边所在直线的方程；（2）BC 边上中线AD 的方程为2360x y -+=，且ABC 的面积等于7，求点A 的坐标. 18．在△ABC 中，角A ，B ，C 所对的边分别为a ，b ，c ，且a cos B ＝b cos A ． （1）求ba的值； （2）若sin A ＝13，求sin(C －π4) 的值．19．如图，已知四棱锥P ABCD -的底面ABCD 是菱形，PA ⊥平面ABCD ，点E 为PC 的中点.（1）求证：//PA 平面BDE ； （2）求证：PC BD ⊥.20．围建一个面积为2360m 的矩形场地，要求矩形场地的一面利用旧墙（利用旧墙需维修，假设旧墙足够长），其它三面围墙要新建，在旧墙的对面的新墙上要留一个宽度为2m 的进出口.如图所示，已知旧墙的维修费用为45元/m ，新墙的造价为180元/m ，设利用的旧墙的长度为x （单位：m ），围建场地的总费用为y （单位：元）. （1）将y 表示为x 的函数；（2）试确定x ，使修建此矩形场地围建的总费用最小，并求出最小总费用.21．如图，在ABC 中，120BAC ∠=︒，3AC =，ABC ，D 为边长BC上一点，158AD =.（1）求BC 的长； （2）求sin ACD ∠的值； （3）求cos CAD ∠的值.22．已知圆()22:11M x y -+=，15,22A ⎛⎫ ⎪⎝⎭，()0,B t ，()()0,404C t t -<<，直线,PB PC都是圆M 的切线，且点P 在y 轴右侧.（1）过点A 的直线l 被圆M l 的方程； （2）当1t =时，求点P 的横坐标； （3）求PBC 面积的最小值.参考答案1．D 【分析】利用过点()2,A m -和(),4B m 的直线与直线210x y +-=的斜率相等可得答案. 【详解】∵直线210x y +-=的斜率等于2-，∴过点()2,A m -和(),4B m 的直线的斜率也是2-， 422mm -∴=-+，解得8m =-， 故选：D. 2．B 【分析】先由tan 24πα⎛⎫+= ⎪⎝⎭求出tan α的值，再利用两角差的正切公式计算()tan αβ-【详解】解：因为tan 24πα⎛⎫+= ⎪⎝⎭，所以tan tan421tan tan 4παπα+=-，即tan 121tan αα+=-， 解得1tan 3α=， 因为tan 3β=，所以13tan tan 43tan 11tan tan 3133αβαβαβ， 故选：B 3．A 【分析】将三角形ABC 拆分为三角形ABD 和三角形ACD ，由两点式计算BC 所在直线，计算直线与x 轴的交点，分别求出三角形的底和高，公式求面积即可.【详解】解：因为1,0A ，()0,1B ，()2,3C --所以BC 所在直线为()()()131002y x ---=---,即210x y -+=，设直线BC 与x 轴的交点为D ，过点C 做x 轴的垂线，垂足为E ，则1,02D ⎛⎫- ⎪⎝⎭，()2,0E -则11322ABC ABD ACD S S S AD BO AD CE =+=⨯⨯+⨯⨯=故选：A 4．B 【分析】由题得2xy =，再利用基本不等式求解. 【详解】因为22log log 1x y +=， 所以2log 1,0,0,2xy x y xy =>>=.所以x y +≥（当且仅当x y =所以x y +的最小值是故选：B 5．C 【分析】由CD 为角C 的平分线，23AD BD =，可得32AC BC =，设3AC x =，2BC x =，然后在ABC 中利用正弦定理可得322sin cos sin x xA A A=，化简计算可得答案【详解】解析：因为CD 为角C 的平分线 所以AD ACBD BC= 因为23AD BD = 所以32AC BC = 所以不妨设3AC x =，2BC x = 因为在ABC 中，sin sin AC BCB A=，2B A = 所以32sin 2sin 2sin cos sin AC BC x xA A A A A=⇒= 因为在ABC 中，sin 0A ≠，0x ≠ 所以32322sin cos sin 2cos x x A A A A=⇒=所以3cos 4A =. 故选：C 6．D 【分析】将()f x 变形为()()sin 2sin 2x x x x --+，利用两角和、差的正弦公式化简求零点，进而求出零点之和. 【详解】()()()sin sin3sin 2sin 22sin cos2f x x x x x x x x x =-=--+=-， []0,x π∈，由()2sin cos20f x x x =-=，得10x =，24x π=，334x π=，4x π=，则零点之和为2π. 故选：D 7．C 【分析】设圆的切线为PM 、PN ，由cos 0APB ∠≤得90APB ∠≥，即90MPN ∠≥， 再求得PC 的取值范围，求得点P 的坐标，即可求得EF 的最大值． 【详解】由题意，圆心到直线:2l y x =+的距离为3d ==（半径）故直线l 和圆相交；当点P 在圆外时，从直线上的点向圆上的点连线成角， 当且仅当两条线均为切线时，APB ∠才是最大的角，不妨设切线为PM ，PN ，则由cos 0APB ∠≤， 得90APB ∠≥， 90MPN ∴∠≥；当90MPN ∠=时，32sin sin 452MPC PC ∠===，PC ∴=设()00,2P x x +，PC =解得：0x =设())2,2E F，如图，EF 之间的任何一个点P ，圆C 上均存在两点,A B ，使得90APB ∠≥，线段EF 长度的最大值为EF ==故选：C 8．BC【分析】根据线面平行的判定定理判断． 【详解】A 中如下图，由中位线定理//MQ CD ，而//CD AB ，从而//MQ AB ，AB ⊄平面MNQ ，有线面平行；B 中，如下图，BC MN O =，在平面ABC 上，OQ 与AB 显然相交，因此AB 与平面MNQ相交，不平行．C 中，如下图，C 是所在棱中点，则//CQ MN ，即CQ ⊂平面MNQ ，而在底面ABQ 上，直线CQ 与直线AB 相交，AB 与平面MNQ 相交，不平行．D 中，如下图，由中位线定理//MN CD ，而//CD AB ，从而//MN AB ，AB ⊄平面MNQ ，有线面平行；故选：BC ． 【点睛】关键点点睛：本题考查线面平行的判断，关键是找到线线平行．正方体中平行线很多，易于寻找，在说明线面相交时，注意线线相交一定要在一个平面内的两条直线才能确定是否相交． 9．ACD 【分析】根据圆心到直线的距离等于半径，结合选项逐个验证. 【详解】因为圆()2222x y -+=的圆心为()2,0对于A ，圆心到直线的距离d =对于B ，圆心到直线的距离0d =，不正确；对于C ，圆心到直线的距离d对于D ，圆心到直线的距离d 故选：ACD. 10．ABC 【分析】利用222=cos 2a b c C ab+-代入条件可得22cos cos 10C C +-≥,进而求解即可【详解】因为2222cos 2c a b ab C ≤++所以()2222cos 2cos cos 2cos 2cos 12a b c C C C C C ab+-≥-⇒≥-⇒≥-- 所以22cos cos 10C C +-≥解得1cos 2C ≥或cos 1C ≤- 因为()0,C π∈ 所以1cos 2C ≥所以0,3C π⎛⎤∈ ⎥⎝⎦.故选：ABC. 11．ABC 【分析】对于A ：连接1BD ，由中位线定理得1//EF AD ，又由11BC B C ⊥，可判断； 对于B ：根据线面平行的判定定理可判断； 对于C ：根据线面垂直的判定定理可判断；对于D ：连接1AB AC ，，可得1AB C ∠（或其补角）就是异面直线FG 、1BC 所成的角，由此可判断. 【详解】对于A ：连接1BD ，因为,E F 分别是棱111,A D A A 的中点，所以1//EF AD ，又11//BC AD ，所以1//BC EF ，又11BC B C ⊥，所以1EF B C ⊥，故A 正确；对于B ：由A 选项的解析得1//BC EF ，又EF ⊂平面EFG ，1BC ⊄平面EFG ，所以1//BC 平面EFG ，故B 正确；对于C ：连接11B D ，则1111B D AC ⊥，又因为,E G 分别是棱1111,A D A B 的中点，所以11//EG B D ，所以11EG AC ⊥，又1AA ⊥面1111D C B A ，EG ⊂面1111D C B A ，所以1AA EG ⊥，又11A A AC A =，11A A AC ⊂，面11A AC ，所以EG ⊥面11A AC ，所以1EG AC ⊥，同理可证1FG AC ⊥，又EG FG G =，EG FG ⊂，面EFG ，所以1AC ⊥平面EFG ，故C 正确；对于D ：连接1AB AC ，，因为,F G 分别是棱111,A A A B 的中点，所以1//FG AB ，所以1AB C ∠（或其补角）就是异面直线FG 、1BC 所成的角，又1ABC 是正三角形，所以13AB C π∠=，所以异面直线FG 、1BC 所成角的大小为3π，故D 不正确， 故选：ABC.12．AC 【分析】题意说明两个圆只有一个公共点，两个圆相切（外切和内切）时，只有一个公共点． 【详解】圆224x y +=的圆心是(0,0)O ，半径为2R =， 圆222(3)(4)x y r -+-=圆心是(3,4)C ，半径为r ，5OC =，当25r +=，3r =时，两圆外切，当25r -=，7r =时，两圆内切，它们都只有一个公共点． 故选：AC ． 【点睛】本题考查集合与集合的关系，解题关键是确定集合中的元素，本题实质是考查圆与圆的位置关系．13【分析】由同角三角函数关系求出cos α，由两角和的余弦公式即可求出cos 4πα⎛⎫+ ⎪⎝⎭.【详解】 解：3sin 5α=，且α为锐角，所以4cos 5α=，cos cos cos sin sin 444πππααα⎛⎫+=-=⎪⎝⎭.14．20x y -+= 【分析】直线l 为两个圆心的中垂线，分别求圆心，利用点斜式求解即可. 【详解】若圆224x y +=和圆224440x y x y ++-+=关于直线l 对称， 则直线l 为两个圆心的中垂线， 224x y +=的圆心为1(0,0)O ，224440x y x y ++-+=的圆心为2(2,2)O -.121O O k =-，中点为(1,1)-可得直线l 为11y x -=+ ，整理得：20x y -+=. 故答案为：20x y -+=. 15．8 【分析】求得sin A 的值，利用三角形的面积公式求得24bc =，结合余弦定理可求得a 的值，即为所求. 【详解】在ABC 中，1cos 4A =-，所以sin A =由1sin 2ABC S bc A ==△24bc ∴=，由余弦定理得()22222221552cos 22464222a b c bc A b c bc b c bc =+-=++=-+=+⨯=，因此，8BC a ==. 故答案为：8. 【点睛】本题考查利用余弦定理解三角形，同时也考查了三角形面积公式以及同角三角函数基本关系的应用，考查计算能力，属于中等题.16．6 【分析】由题知4B π=，34C A π=-，所以()224cos 22cos 2tan 3sin 21cos 2A A A C A+-=+，进而令1cos 2A t +=，[)0,2t ∈，再根据基本不等式求解即可得答案.【详解】解：在ABC 中，因为()cos 0,B B π=∈，所以4B π=，34C A π=- 所以()()()222223sin 3cos tan 3sin 2tan 3sin 2cos 22cos A A A C A A A A π⎛⎫-⎛⎫-=--=- ⎪ ⎪⎝⎭⎝⎭()24cos 2cos 21cos 2A A A --⎛⎫=- ⎪+⎝⎭24cos 22cos 21cos 2A AA +=+ 令1cos 2A t +=，[)0,2t ∈，原式24622= 4666t t t t t-+=+-≥=，当且仅当[)0,2t =是等号成立.故答案为：617．（1）240x y +-=；（2）()3,4A 或()3,0-. 【分析】（1）利用点斜式求得BC 边所在直线方程；（2）利用点到直线的距离公式求得A 到直线BC 的距离，根据面积7ABC S =△以及点A 在直线2360x y -+=上列方程组，解方程组求得A 点的坐标. 【详解】 （1）∵311222AB k -==---，采用点斜式设直线方程：11(2)2y x -=-- ∴240x y +-=（2）∵A 点在中线AD 上，把A 点坐标代入，2360-+=m n 点A 到直线:240BC x y +-=的距离d =∵11||722ABC S d BC =⋅⋅==△ 即23603 2474m n m m n n -+=⎧=⎧⇒⎨⎨+-==⎩⎩或30m n =-⎧⎨=⎩所以，点A 的坐标为()3,4A 或()30A -,18．（1）1（2 【详解】分析：（1）已知等式利用正弦定理化简，利用两角和与差的正弦函数公式化简，整理得到结果，（2）由（1）可得：C=π-2A ，利用sinA=13，A 为锐角，可得：cosA ，sin2A ，cos2A 的值，利用诱导公式及两角和与差的正弦函数公式即可求值． （1）由a cos B ＝b cos A ，得sin A cos B ＝sin B cos A ， 即sin(A －B )＝0．因为A ，B ∈(0，π)，所以A －B ∈(－π，π)，所以A －B ＝0， 所以a ＝b ，即＝1．（2）因为sin A ＝，且A 为锐角，所以cos A ＝． 所以sin C ＝sin(π－2A )＝sin2A ＝2sin A cos A ＝，cos C ＝cos(π－2A )＝－cos2A ＝－1＋2sin 2A ＝－． 所以sin(C －)＝sin C cos －cos C sin ＝．点睛：此题考查了正弦定理，两角和与差的正弦函数公式的应用，熟练掌握正弦定理是解本题的关键，考查了计算能力和转化思想，属于中档题． 19．（1）证明见解析；（2）证明见解析. 【分析】（1）连接AC 交BD 于O 点，连接EO ，可证//PA EO ，线面平行的判定定理可证//PA 平面BDE ；（2）由菱形对角线垂直和线面垂直可证BD ⊥平面PAC ，从而得出BD PC ⊥.【详解】（1）证：连接AC 交BD 于O 点，连接EO ∵底面ABCD 是菱形 ∴O 为AC 的中点 ∵点E 为PC 的中点 ∴//PA EO∵EO ⊂平面BDE ，且PA ⊄平面BDE ∴//PA 平面BDE（2）证：∵底面ABCD 是菱形 ∴AC BD ⊥ ∵PA ⊥平面ABCD ∴PA BD ⊥∵AC PA A ⋂=，∴BD ⊥平面PACPC ⊂平面PAC ，∴BD PC ⊥20．（1）()23602253602y x x x=+->；（2）当24x =时，min 10440y =元.【分析】(1)设矩形的宽为a 米，则根据围建的矩形场地的面积为360m 2，易得 a ＝360x，此时再根据旧墙的维修费用为45元/m ，新墙的造价为180元/m ，我们即可得到修建围墙的总费用y 表示成x 的函数的解析式； (2)根据(1)中所得函数的解析式，利用基本不等式，我们易求出修建此矩形场地围墙的总费用最小值，及相应的x 值． 【详解】 （1）设宽为a 米 ∵360ax =，∴360a x=∴36045180(2)1802y x x x=+-+⨯⨯2360225360(2)x x x=+->（2）∵2x >利用基本不等式：y 2360225360(2)x x x=+->∴36010440y ≥=当且仅当2360225x x=时，即24x =时取等号，此时min 10440y =元答：当24x =米时，修建费用最小，最小费用为10440元. 21．（1）7；（2；（3）12或7198.【分析】（1）由三角形的面积可求出5c =，再利用余弦定理可求出BC 的长； （2）在ABC 中，利用正弦定理直接求解即可；（3）在ACD △中利用正弦定理求出sin ADC ∠，再利用同角三角函数的关系求出cos ADC ∠的值，而()()cos cos cos CAD ACD ADC ACD ADC π∠=-∠+∠=-∠+∠⎡⎤⎣⎦，然后利用两角和的余弦公化简计算即可 【详解】解：（1）设角,,A B C 所对边为,,a b c，由题知132S c =⨯=，解得5c =， 在ABC 中，由余弦定理可得219252352a ⎛⎫=+-⨯⨯⨯- ⎪⎝⎭，解得7a =，即7BC =；（2）在ABC 中，由余弦定理可得9492511cos 23714ACB +-∠==⨯⨯，又因为()0,ACD π∠∈，则sin sin ACD ACB ∠=∠= （3）在ACD △153sin ADC =∠，解得sin ADC ∠=1cos 7ADC ∠=±，①当1cos 7ADC ∠=时，()()cos cos cos CAD ACD ADC ACD ADC π∠=-∠+∠=-∠+∠⎡⎤⎣⎦ cos cos sin sin ACD ADC ACD ADC =-∠∠+∠∠11111472=-⨯= ②当1cos 7ADC ∠=-时，()()cos cos cos CAD ACD ADC ACD ADC π∠=-∠+∠=-∠+∠⎡⎤⎣⎦ cos cos sin sin ACD ADC ACD ADC =-∠∠+∠∠1117114798⎛⎫=-⨯-= ⎪⎝⎭ 22．（1）12x =或2410370x y +-=；（2）3；（3）163. 【分析】（1）分类讨论：斜率不存在时成立；斜率存在时，先求弦心距，再利用弦长可求斜率，从而可求方程；（2）根据题意易得:1PB y =，设:3PC y nx =-，利用M 到直线PC 距离为1，求得n ，得出直线PC 的方程，从而可得点P 的横坐标；（3）由于BC 长度一定，故求PBC 面积的最小值，即求P 的横坐标的最小值，利用PB ，PC 是圆的切线，可求P 的坐标，根据已知，可求其最小值． 【详解】解：（1）由题意知，圆心M 到直线l的距离12d ==， ①当直线l 的斜率不存在时，1:2l x =，满足圆心M 到直线l 的距离等于12； ②当直线l 的斜率存在时，设斜率为k ，则51:22l y k x ⎛⎫-=- ⎪⎝⎭，即502k kx y --+=，圆心M到直线l的距离12d ==，解得125k =-，则:2410370l x y +-=；综上得，直线l 的方程为12x =或2410370x y +-=（2）当1t =时，此时()0,1B ，()0,3C -，又因为直线,PB PC 都是圆M 的切线，且点P 在y 轴右侧.易得:1PB y =；且PC 斜率存在，设为n ，则:3PC y nx =-，即30nx y --=， M 到直线PC1=，解得43n =， 则4:33PC y x =-，把1y =代入433y x =-，解得3x =， 则P 的横坐标为3；（3）因为4BC =为定值，且B 、C 在y 轴上，所以要求PBC 面积的最小值，即求P 的横坐标的绝对值的最小值， 设点P 的横坐标为p x ，由题知，直线,PB PC 的斜率皆存在，设PB 的斜率为m ，则:PB y mx t =+，即0mx y t -+=．因为PB 与圆M1=，得212t m t -=． 所以21:2t PB y x t t-=+．同理可得21(4):42(4)t PA y x t t --=+--． 由22121(4) 4.2(4)t y x t t t y x t t ⎧-=+⎪⎪⎨--⎪=+-⎪-⎩解得222841P t tx t t -=-+． 则2222411144pt t x t t t t -+==+--．因为04t <<，所以2044t t >-≥-，所以234px ≤，83P x ≥．当2t =时，83P x =，此时18164233ABCS=⨯⨯=． 所以PBC 面积的最小值为163．。

金陵中学2019学年度高三第一学期期中考试第一部分：听力（共两节，满分20分）做题时，先将答案划在试卷上。

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分，满分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．What’s the topic of the talk?A．On pollution. B．On children. C．On death.2．How can the man get to the Hotel?A．At first taking a left, then taking a right, and then taking a right again.B．At first taking a right, then taking a left, the hotel is on his left.C．At first taking a left, then taking a right, the hotel is on his right.3．How old is the woman’s bicycle possibly?A．Three years old. B．Four years old. C．Five years old.4．Where are the two most possibly?A．At a bookstore. B．At an art gallery. C．At a shop.5．What is the most probable relationship between Jim and Bob?A．Friends. B．Classmates. C．Twins.第二节（共15小题；每题1分，满分15分）听下面5段对话或独白。

2020-2019学年度江苏省金陵中学第一学期高三期中考试化学试卷相对原子质量：H—1 C—12 O—16 Na—23 Al—27 S—32 Ba—137本试卷分第Ⅰ卷（选择题）和第Ⅱ卷（非选择题）两部分。

第Ⅰ卷（选择题）部分请在答题卡上填涂，第Ⅱ卷（非选择题）部分请在答卷纸上作答。

本卷满分120分，考试用时100分钟。

第Ⅰ卷（选择题共48分）一、选择题（本题包括8小题，每小题3分，共24分。

每小题只有一个．．．．选项符合题意）1．2020年的诺贝尔化学奖被三位科学家（见下图）共同获得，以表彰他们在“发现和研究绿色荧光蛋白（green fluorescent protein 简称GFP）方面做出的贡献”。

GFP需要在氧化状态下产生荧光，强化还原剂能使GFP转变为非荧光形式，但一旦重新暴露在空气或氧气中，GFP荧光便立即得到恢复。

下列说法不正确的是（）A．GFP是有氨基酸构成的能发出荧光的一类特殊蛋白质B．可用GFP标记需要研究的蛋白质，研究特定的反应C．GFP的发光是将化学能转化为光能D．GFP产生荧光不受外界环境影响2．下列药品保存方法正确的是（）A．白磷保存在煤油中B．在氯化亚铁试剂中加盐酸和铁粉C．金属钾保存在CCl4中D．过氧化钠保存配有玻璃塞的广口试剂瓶中3．阿伏伽德罗常数约为6.02×1023mol-1，下列叙述正确的是（）A．2.24LCl2中含有的原子数一定为0.2×6.02×1023B．0.1L2mol·L-1的Na2CO3溶液中含有CO32-数目为0.2×6.02×1023C．0.88gC3H8中含有的共价键总数为0.2×6.02×1023D．7.8g过氧化钠粉末与水反应转移的电子数为0.2×6.02×10234．下述实验能达到预期目的的是（）编号实验内容实验目的A 金属钠投入硫酸铜溶液中置换出铜B 向硫酸铝溶液中滴加过量氨水制取氢氧化铝C 将氯化铁溶液加热蒸发制氯化铁晶体D 把铜和银用导线连接后插入稀硫酸中组成原电池验证铜比银活泼5．下列离子方程式书写正确的是（）A．AgNO3溶液中加入过量氨水：Ag++NH3H2O===AgOH↓+NH4+B．把0.4mol铁粉和0.8mol稀硝酸充分反应：Fe+4H++NO3—===Fe3++NO↑+2H2OCO===CaCO3↓C．石灰水与Na2CO3溶液混合：Ca2++-23D．NH4HSO3溶液与足量NaOH溶液也混合：HSO3—+OH—====SO32-+ H2ONO的溶液中，能大量共存的离子组是（）6．在含有大量Ba2+和-3A．H+K+ClO—Cl—B．H+NH4+Fe2+Br—SO D．K+Na+Cl—OH—C．NH4+K+Cl—-247．X、Y、Z、W有如下图所示的转化关系，则X、W可能是（）①C、O2②AlCl3、NaOH ③Fe、HNO3④S、O2A．①②③B．①②C．③④D．①②③④8．早在1807年化学家戴维用电解熔融氢氧化钠制得钠，4NaOH（熔）电解4Na+O2+2H2O；后来盖·吕萨克用铁与熔融氢氧化钠作用也制得钠，反应原理为3Fe+4NaOH 1000Fe3O4+2H2↑+4Na↑。

金陵中学高一年级语文期中测试卷一、选择题（共24分）1．下列加点字的注音有错误的一项是（2分）A．游说（shuì）鞭挞（tà）云气氤氲（y ūn）B．虫豸（zhì）梦魇（yǎn）叱咤风云（z hà）C．凫水（fú）股肱（gōng）洗衣濯足（z huó）D．开凿（záo）端睨（nì）凭舷眺望（x uán）2．下列词语书写全部正确的一项是（2分）A．虚无飘渺毛骨悚然辍学B．金榜题名甘败下风赏鉴C．临风陨泪惹事生非弥漫D．临危不苟突巫而立奚落3．下列加点词语解释有错误的一项是（2分）A．无足称道（值得）积毁销骨（解除）山麓（山脚）B．摧山坼地（裂开）筚路蓝缕（柴车）虔诚（恭敬）C．临虹款步（慢慢走）衣锦还乡（穿）质铺（抵押）D．泽被后世（恩惠）随俗为变（风俗）治学（研究）4．填入下列文字空缺处的词语，全都恰当的一项是（2分）我懂得青年____会变猴子，变虫豸，这是后来的事情。

现在不再给人去补靴子了，不过我____要多做些事情。

_____我努力，他们变猴子和虫豸的机会总可以少一些，____应该少一些。

A．却总是于是而且是B．也总是于是总是C．也还是只要而且是D．却还是但是总是5．下列加点的成语运用不当的一句是（2分）A．对学习中遇到的关键问题不求甚解是要吃苦头的。

B．对于一个偶然的机缘，我却不期而遇地晤见了鲁迅先生。

C．这篇文章对问题的分析非常全面，真可谓入木三分。

D．你别看他平时沉默寡言，可是一旦做起报告来，就口若悬河，滔滔不绝。

6．下列句子表达清晰、没有语病的一句是（2分）A．学校领导要经常深入班级，征求同学们的问题和意见。

B．不论党委书记还是党内其他领导同志，都应当发扬民主，走群众路线。

C．难道我们能否认鲁迅先生不是一个伟大的革命家吗？D．这个现代化大型集团企业的前身是三个乡镇企业合并起来的。

7．下列句子中标点符号使用有错误的一项的（2分）A．我爱热闹，也爱冷静；爱群居，也爱独处。