数学试题第 1 页（共 5 页）数 学 注意事项：1答卷前，考生务必将自己的姓名、准考证号填写在答题卡上。2回答选择题时，选出每小题答案后，用铅笔把答题卡上对应题目的答案标号涂黑。如需改动，用橡皮擦干净后，再选涂其他答案标号。回答非选择题时，将答案写在答题卡上。写在本试卷上无效。3考试结束后，将本试卷和答题卡一并交回。一、选择题：本题共 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设1iz ，则2iz Ai Bi C1 D1 2设集合22,3,23Aaa，0,3B，2,Ca若BA，2AC，则a A3 B1 C1 D3 3甲、乙、丙、丁四名教师带领学生参加校园植树活动，教师随机分成三组，每组至少一人，则甲、乙在同一组的概率为 A16 B14 C13 D12 4 平面向量a与b相互垂直，已知(6,8)a，|5b，且b与向量(1,0)的夹角是钝角，则b A(3,4)B(4,3)C(4,3)D(4,3)5已知点A，B，C为椭圆D的三个顶点，若ABC是正三角形，则D的离心率是 A12 B23 C63 D32 6三棱锥ABCD中，AC 平面BCD，BDCD若3AB，1BD，则该三棱锥体积的最大值为 A2 B43 C1 D23 7设函数()f x，()g x在R的导函数存在，且()()fxg x，则当(,)xa b时 A()()f xg x B()()f xg x C()()()()f xg ag xf a D()()()()f xg bg xf b 8已知a，b，c满足5log(23)bba，3log(52)bbc，则 A|acbc，|abbc B|acbc，|abbc C|acbc，|abbc D|acbc，|abbc 数学试题第 2 页（共 5 页）二、选择题：本题共 4 小题，每小题 5 分，共 20 分。在每小题给出的选项中，有多项符合题目要求。全部选对的得 5 分，部分选对的得 2 分，有选错的得 0 分。9已知()f x是定义在R上的偶函数，()g x是定义在R上的奇函数，且()f x，()g x在(,0单调递减，则 A(1)(2)f ff f B(1)(2)f gf g C(1)(2)g fg f D(1)(2)g gg g 10已知平面平面l，B，D是l上两点，直线AB且ABlB，直线CD且CDlD下列结论中，错误的有 A若ABl，CDl，且ABCD，则ABCD是平行四边形 B若M是AB中点，N是CD中点，则MN/AC C若，ABl，ACl，则CD在上的射影是BD D直线AB，CD所成角的大小与二面角l 的大小相等 11质点P和Q在以坐标原点O为圆心，半径为1的O上逆时针作匀速圆周运动，同时出发P的角速度大小为2rad/s，起点为O与 x 轴正半轴的交点；Q的角速度大小为5rad/s，起点为射线3(0)yx x 与O的交点则当Q与P重合时，Q的坐标可以为 A22(cos,sin)99 B55(cos,sin)99 C(cos,sin)99 D(cos,sin)99 12下图改编自李约瑟所著的中国科学技术史，用于说明元代数学家郭守敬在编制授时历时所做的天文计算图中的AB，AC，BD，CD都是以O为圆心的圆弧，CMNK是为计算所做的矩形，其中M，N，K分别在线段OD，OB，OA上，MNOB，KNOB 记AOB，AOC，BOD，COD，则 Asinsincos Bcoscos cos Csinsincos Dcoscoscoscos 数学试题第 3 页（共 5 页）三、填空题：本题共 4 小题，每小题 5 分，共 20 分。13某工厂生产的产品的质量指标服从正态分布2(100,)N质量指标介于99至101之间的产品为良品，为使这种产品的良品率达到95.45%，则需调整生产工艺，使得至多为 （若2(,)XN，则|2 0.9545PX）14若 P，Q 分别是抛物线2xy与圆22(3)1xy上的点，则|PQ的最小值为 15数学家祖冲之曾给出圆周率的两个近似值：“约率”227与“密率”355113它们可用“调日法”得到：称小于3.1415926的近似值为弱率，大于3.1415927的近似值为强率由3411，取3为弱率，4为强率，得13471 12a，故1a为强率，与上一次的弱率3计算得23710123a，故2a为强率，继续计算，若某次得到的近似值为强率，与上一次的弱率继续计算得到新的近似值；若某次得到的近似值为弱率，与上一次的强率继续计算得到新的近似值，依此类推已知227ma，则m ；8a 16右图为一个开关阵列，每个开关只有“开”和“关”两种状态，按其中一个开关1次，将导致自身和所有相邻的开关改变状态 例如，按(2,2)将导致(1,2)，(2,1)，(2,2)，(2,3)，(3,2)改变状态如果要求只改变(1,1)的状态，则需按开关的最少次数为 四、解答题：本题共 6 小题，共 70 分。解答应写出文字说明、证明过程或演算步骤。17（10 分）如图，四边形ABCD是圆柱底面的内接四边形，AC是圆柱的底面直径，PC是圆柱的母线，E是AC与BD的交点，ABAD，60BAD（1）记圆柱的体积为1V，四棱锥PABCD的体积为2V，求12VV；（2）设点F在线段AP上，4PAPF，4PCCE，求二面角FCDP的余弦值(1,1)(1,2)(1,3)(2,1)(2,2)(2,3)(3,1)(3,2)(3,3)数学试题第 4 页（共 5 页）18（12 分）已知函数()sin()f xx在区间(,)6 2单调，其中为正整数，|2，且2()()23ff（1）求()yf x图像的一条对称轴；（2）若3()62f，求 19（12 分）记数列na的前n项和为nT，且11a，1nnaT（2n）（1）求数列na的通项公式；（2）设m为整数，且对任意nN，1212nnmaaa，求m的最小值 20（12 分）一个池塘里的鱼的数目记为N，从池塘里捞出200尾鱼，并给鱼作上标识，然后把鱼放回池塘里，过一小段时间后再从池塘里捞出500尾鱼，X表示捞出的500尾鱼中有标识的鱼的数目（1）若5000N，求X的数学期望；（2）已知捞出的500尾鱼中15尾有标识，试给出N的估计值（以使得(15)P X 最大的N的值作为N的估计值）21（12 分）已知双曲线2222:1xyCab(0,0)ab过点(4 2,3)A，且焦距为10（1）求C的方程；（2）已知点(4 2,3)B，(2 2,0)D，E为线段AB上一点，且直线DE交C于G，H两点证明：|GDHDGEHE 数学试题第 5 页（共 5 页）22（12 分）椭圆曲线加密算法运用于区块链 椭圆曲线2332(,),4270Cx yyxaxbab|PC关于x轴的对称点记为PC在点(,)(0)P x yy处的切线是指曲线3yxaxb+在点P处的切线定义“”运算满足：若PC，QC，且直线PQ与C有第三个交点R，则PQR；若PC，QC，且PQ为C的切线，切点为P，则PQP；若PC，规定*0PP，且*00PPP.（1）当324270ab时，讨论函数3()h xxaxb零点的个数；（2）已知“”运算满足交换律、结合律，若PC，QC，且PQ为C的切线，切点为P，证明：PPQ；（3）已知11(,)P x yC，22(,)Q xyC，且直线PQ与C有第三个交点，求PQ的坐标 参考公式：3322()()mnmn mmnn 英语英语 注意事项：1.答卷前，考生务必将自己的姓名、准考证号填写在答题卡上。2.回答选择题时，选出每小题答案后，用铅笔把答题卡上对应题目的答案标号涂黑。如需改动，用橡皮擦干净后，再选涂其他答案标号。回答非选择题时，将答案写在答题卡上，写在本试卷上无效。3.考试结束后，将本试卷和答题卡一并交回。第一部分 听力（共两节，满分 30分）做题时，先将答案标在试卷上。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。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。例：How much is the shirt?A.19.15.B.9.18.C.9.15.答案是 C。1.What will the woman probably do next?A.Pay for the clothes.B.Go to another store.C.Try on some dresses.2.What will the man do about the camera?A.Have it repaired.B.Lend it to Larry.C.Change it for a new one.3.Why does the woman suggest having a party?A.Bill has got a pay raise.B.Bill has found a good job.C.Bill has finished a project.4.Where will the woman go to celebrate her moms birthday?A.Caf Blue.B.Chez Marie.C.Leblanc.5.What are the speakers mainly talking about?A.A park.B.The weather.C.Bobs pets.第二节（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。每段对话或独白后有几个小题，从题中所给的 A、B、C 三个选项中选出最佳选项。听每段对话或独白前，你将有时间阅读各个小题，每小题 5 秒钟；听完后，各小题将给出 5秒钟的作答时间。每段对话或独白读两遍。听第 6 段材料，回答第 6、7 题。6.What is the probable relationship between the speakers?A.Doctor and patient.B.Hu4sband and wife.C.Salesman and customer.7.What does the man suggest the woman do?A.Do more exercise.B.Drink plenty of water.C.Stop eating chocolate.听第 7 段材料，回答第 8 至 10题。8.Why does Joshua make the call?A.To ask for advice.B.To make an apology.C.To check the time.9.Where is Joshua now?A.In Cider Street.B.On Elmer Road.C.In Poplar Street.10.What is Joshuas destination?A.Marias house.B.The National Bank.C.Westin Hospital.听第 8 段材料，回答第 11 至 13题。11.How much rent would the man like to pay every month?A.400.B.800.C.950.12.What is an advantage of the flat?A.Its convenient for shopping.B.Its in a quiet neighborhood.C.Its next to the mans college.13.What is the man going to do tomorrow?A.Find another flat.B.Move into the flat.C.Check on the flat.听第 9 段材料，回答第 14 至 17题。14.Where are the speakers?A.On a bus.B.At a store.C.In a church.15.What was the mans first challenge this week?A.To do some volunteer work.B.To encourage others to smile.C.To treat a stranger to a dinner.16.What day is it today?A.Tuesday.B.Wednesday.C.Thursday.17.What will the woman do next?A.Make an application.B.Call her husband.C.Pay for her groceries.听第 10 段材料，回答第 18 至 20题。18.What should a public speaker avoid doing according to the woman?A.Talking too loudly.B.Speaking too fast.C.Saying too much.19.Why should the closing of a speech be carefully planned?A.Its usually the most remembered part.B.Its the last chance to make your point.C.Its as important as the opening words.20.What does the woman suggest a public speaker do during the speech?A.Show a good sense of humor.B.Motivate the audience to act.C.Meet challenges with courage.第二部分 阅读（共两节，满分 50分）第一节（共 15小题；每小题 2.5 分，满分 37.5 分）阅读下列短文，从每题所给的 A、B、C、D 四个选项中选出最佳选项。A Family-Friendly Events in January ZooLights:Glow Wild Jan.1-19 The Phoenix Zoos yearly holiday light show is on until Jan.19,allowing families one or more opportunities to enjoy the citys zoo,with millions of lights giving an added dimension to the festivities.Glow Wild,455 N.Galvin Pkwy.,Phoenix,phoenixzoo.org,$11.95 members,$13.95 general admission.Downtown Mesa Festival of the Arts Jan.4-18 The Downtown Mesa Festival of the Arts features the work of established and emerging artists,including those who create woodwork,metal crafts,food items,art,photography and gifts.On Macdonald,off of Main Street in Downtown Mesa,,free admission.Family Fun Winterfest Jan.4 OdySea Aquarium in the Desert is hosting the third annual Family Fun Winterfest in its Desert Courtyard,featuring real snow for the kids to play in.This free event features everything from bounce houses to rides,games,snowflake crafts and face painting to go with various stands set up by local sellers,with food and other offerings for sale at the event.9500 E.Via de Ventura,Scottsdale,,free.Youth Fine Arts Course Jan.18-Mar.7 Mesa Arts Center is hosting an eight-week youth arts course on Saturdays to teach artistic skills and knowledge through fun and challenging art classes in a wide variety of art materials,including painting,drawing,mixed media and sculpture,ensuring mentally stimulating sessions for all.Mesa Art Center,1 E.Main St.,Mesa,,$93.21.How can you get a discounted ticket to the ZooLights show?A.Bring a friend.B.Get a membership.C.Join a tour group.D.Book a ticket online.22.What can you do at Family Fun Winterfest?A.Have free food.B.Enjoy real snow.C.Take art classes.D.Meet local artists.23.Which event lasts the longest?A.ZooLights:Glow Wild.B.Downtown Mesa Festival of the Arts.C.Family Fun Winterfest.D.Youth Fine Arts Courses.B As a young girl growing up in France,Sarah Toumi dreamed of becoming a leader who could make the world a better place.Her passion to help others was awakened when,from the age of nine,she accompanied her Tunisian father to his birthplace in the east of the country during holidays.There she organized homework clubs and activities for children.Toumi witnessed first-hand the destructive effect of desertification.“Within 10 years rich farmers became worse off,and in 10 years from now they will be poor.I wanted to stop the Sahara Desert in its tracks.A decrease in average rainfall and an increase in the severity of droughts（干旱）have led to an estimated 75 per cent of Tunisias agricultural lands being threatened by desertification.Toumi recognized that farming practices needed to change.She is confident that small land areas can bring large returns if farmers are able to adapt by planting sustainable crops,using new technolo-gies for water treatment and focusing on natural products and fertilizers（肥料）rather than chemi-cals.In 2012,Toumi consolidated her dream to fight the desert.She moved to Tunisia,and set up a programme named Acacias for All to put her sustainable farming philosophy into action.“I want to show young people in rural areas that they can create opportunities where they are.Nobody is better able to understand the impact of desertification and climate change than somebody who is living with no access to water.”By September 2016,more than 130,000 acacia trees had been planted on 20 pilot farms,with farmers recording a 60 per cent survival rate.Toumi estimates that some 3 million acacia trees are needed to protect Tunisias farmland.She expects to plant 1 million trees by 2018.In the next couple of years,Toumi hopes to extend the programme to Algeria and Morocco.24.How did Toumis holiday trips to Tunisia influence her?A.They made her decide to leave the country.B.They helped her better understand her father.C.They fired her enthusiasm for helping others.D.They destroyed her dream of being a teacher.25.What is the main cause of the desertification of Tunisias farmland?A.Low rainfall.B.Soil pollution.C.Cold weather.D.Forest damage.26.Why did Toumi set up Acacias for All in Tunisia?A.To create job opportunities for young people.B.To help the children obtain a basic education.C.To persuade the farmers not to use fertilizers.D.To facilitate the protection of their farmland.27.Which of the following can be the best title for the text?A.Saving Water in Tunisia B.Planting Trees of Native Species C.Holding back the Sahara D.Fighting Poverty in North Africa C For years,David James,who studies insects at Washington State University,had wanted to ex-amine the migration（迁徙）patterns of West Coast monarch butterflies（黑脉金斑蝶）.The route the butterflies travel has been hardly known because the populations are too small to follow.For every 200 monarchs tagged（打标签）by a researcher，only one is usually recovered at the end of its trip,James says,and finding even 200 in the wild to tag is unlikely.Knowing the route is vital to conser-vation efforts,but James had no way to figure it out-until he got a phone call from Washington State Penitentiary in Walla Walla.The prison was looking for new activities to improve the mental health of those serving long-term sentences.So James began working with prisoners to raise monarchs through the whole process of their transformation.The adult insects were then tagged and released from the prison.Over five years,nearly 10,000 monarchs flew from the facility.Elsewhere in Washington,Oregon and Idaho,research-ers released another few thousand.The tags included email addresses,and soon after the first butterflies took off,James started re-ceiving messages from people who had spotted them.The butterflies,the reports confirmed,wintered in coastal California.Twelve of them landed at Lighthouse Field State Beach in Santa Cruz.Several more headed to Bolinas and Morro Bay.The work helps researchers identify ideal places to plant milkweed and other vegetation that are important to the life cycle of West Coast monarch butterflies.It also brought out the gentler side of some of the prisoners.“They were very worried that they were going to harm the butterflies,”James says.Watching the monarch change their form also touched the men.“This butterfly changed,James recalls prisoners telling him,“and maybe we can too.”28.What was hard for David to do in his study?A.Gain financial support.B.Hire qualified workers.C.Build a new laboratory.D.Find enough monarchs.29.Why are the butterflies tagged before being released?A.To guarantee their safety.B.To enable them to fly longer distances.C.To track their travel routes.D.To distinguish them from other species.30.What makes the prisoners feel that they can change?A.The patience the butterflies showed.B.The hardship the butterflies underwent.C.The transformation of the butterflies.D.The devotion of James to the butterflies.31.What is the last paragraph mainly about?A.The impact of the research.B.The findings of James study.C.The release of the prisoners.D.The life cycle of the butterflies.D We all know that unpleasant feeling when were talking about something interesting and halfway through our sentence were interrupted.But was that really an interruption?The answer depends on whom you ask,according to new research led by Katherine Hilton from Stanford University.Using a set of controlled audio clips（录音片段），Hilton surveyed 5,000 American English speakers to better understand what affects peoples perceptions of interruptions.She had participants listen to audio clips and then answer questions about whether the speakers seemed to be friendly and engaged,listening to one another,or trying to interrupt.Hilton found that American English speakers have different conversational styles.She identified two distinct groups:high and low intensity speakers.High intensity speakers are generally uncomfort-able with moments of silence in conversation and consider talking at the same time a sign of engage-ment.Low intensity speakers find it rude to talk at the same time and prefer people speak one after another in conversation.The differences in conversational styles became evident when participants listened to audio clips in which two people spoke at the same time but were agreeing with each other and stayed on topic,Hilton said.The high intensity group reported that conversations where people spoke at the same time when expressing agreement were not interruptive but engaged and friendlier than the conversations with moments of silence in between speaking turns.In contrast，the low intensity group perceived any amount of simultaneous（同时）chat as a rude interruption,regardless of what the speakers were saying.“People care about being interrupted,and those small interruptions can have a massive effect on the overall communication,Hilton said.“Breaking apart what an interruption means is essential if we want to understand how humans interact with each other.”32.What does Hiltons research focus on?A.What interruptions mean to people.B.Whether interruption is good or not.C.How to avoid getting interrupted.D.Why speakers interrupt each other.33.What do participants of the study need to do?A.Record an audio
