THUSSAT中学生标准学术能力2022-2023学年高三下学期3月月诊断性测试各科试卷及答案.rar

第1页 共7页 中学生标准学术能力诊断性测试中学生标准学术能力诊断性测试 2023 年年 3 月测试月测试 数学参考答案数学参考答案 一、一、单项单项选择题：本题共选择题：本题共 8 小小题，每小题题，每小题 5 分，共分，共 40 分分在每小题给出的四个选项中，只有一在每小题给出的四个选项中，只有一项是符合题目要求的项是符合题目要求的 1 2 3 4 5 6 7 8 A D B D C C A B 二、多项选择题：二、多项选择题：本题共本题共 4 小题，每小题小题，每小题 5 分，共分，共 20 分在每小题给出的四个选项中，有多项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符合题目要求全部选对的得 5 分，部分选对但不全的得分，部分选对但不全的得 2 分，有错选的得分，有错选的得 0 分分 9 10 11 12 AC BCD ABD AD 三三、填空题：本题共填空题：本题共 4 小题，每小题小题，每小题 5 分，共分，共 20 分分 1352 14(),1 15108 161 四四、解答题：本题共、解答题：本题共 6 小题，共小题，共 70 分解答应写出文字说明、证明过程或演算步骤分解答应写出文字说明、证明过程或演算步骤 17（10 分）（1）()()()23+1391=2nnnnana+()()()()12233221nnnanann+=+即()()()()122321321nnnanann+=+2 分 又()()1231211 1a+=+，所以数列()()221nnan+是首项为13，公比为13的等比数列 从而()()21321nnnan+=+，则()()2123nnnan+=+5 分（2）()()2111+123233nnnnnnnnann+=+6 分 2123313+3nnnS+设12+123333nnnT=+，则23131233+331nnnT+=+第2页 共7页 两式相减得：12111111229311+121+3333333131nnnnnnnT+=+=1112111525136362 33nnnnn+=+=9 分 从而52544 3nnnT+=，故52544 3nnnS+10 分 18（12 分）（1）由已知及正弦定理得：2sincossin2sinBAAC=又在ABC中，()sinsinsincoscossinCABABAB=+=+2 分 2sincossin2sincos2cossinBAAABAB=+即2sincos=sinABA 又sin0A，1cos2B=4 分 又0B，2=3B，即角B的大小为23 5 分（2）13sin24ABCSacBac=6 分 BD是ABC的角平分线，而ABCABDBCDSSS=+311sin60sin60422acABBDBD BC=+即33()44acBDac=+，acBDac=+8 分 2BD=，()2acac=+2acac+，4acac，即16ac 10 分 当且仅当4ac=时取等号，则13sin164 324ABCSacB=即ABC的面积的最小值为4 3 12 分 第3页 共7页 19（12 分）（1）在BE上取一点N，使得12BNNE=，连接FN，NM 6BD=，116BNBD=，2NE=，3ED=12AFFE=，12BNAFNEFE=则FNAB 2 分 FN 面ABC，AB面ABC，FN面ABC 15BNCMNDMD=，NMBC 4 分 NM 面ABC，BC 面ABC，NM面ABC FNNMN=，面FNM面ABC FM 面FNM，FM面ABC 5 分（2）AEBD，CEBD 所以二面角ABDC的平面角为23AEC=6 分 又AECEE=，BD面AEC BD面ABD，面ABD面AEC 面ABD面=AEC AE，过点C作CHAE，则CH 面ABD 则3sin32CHCECE=()223 333 2CE=，33 63 222CH=8 分 即C到面ABD的距离为3 62 56MDCD=，M到面ABD的距离为53 65 6624=9 分 计算EM：31cos3 33CDB=在DME中，5 32DM=，3DE=第4页 共7页 2225 332151235 3232EMEM+=11 分 EM与面ABD所成角的正弦值为5 65 34434512=12 分（其他方法酌情给分）20（12 分）（1）列联表如下：感兴趣 不感兴趣 合计 男生 12 4 16 女生 9 5 14 合计 21 9 30 2 分 ()223012 54 90.40822.07216 14 21 9K =所以没有 85%的把握认为学生对“数学建模”选修课的兴趣度与性别有关 5 分（2）由题意可知X的取值可能为 0，1，2，3 则()35395042CP XC=6 分()21453910121C CP XC=7 分()5243912145C CP XC=8 分()34391321CP XC=9 分 故X的分布列为 X 0 1 2 3 P 542 1021 514 121 第5页 共7页()5105140123422114213E X=+=12 分 21（12 分）（1）因为双曲线C以250 xy=为渐近线 设双曲线方程为()()2525xyxy+=，即2245xy=1 分()0,3F，0，即：22154yx=954=，9920=，即20=3 分 所以双曲线C的方程为：22145yx=4 分（2）设直线:3l ykx=+，()11,P x y，()22,Q x y ()22225420534203yxkxxykx=+=+化简得：()225430250kxkx+=6 分 此方程的两根为12,x x，则12212230542554kxxkx xk+=()()()2222222222954900100110 1545454kkkPQkkkkk=+=+()()2222222014410 15454kkkkk+=+=8 分 PQ中点M坐标为221512,54 54kkk 9 分 PQ中垂线方程为：22121155454kyxkkk+=+令0 x=，227=54yk，2270,54Tk 第6页 共7页 则22227151535454kTFkk+=+=11 分()22221515543420154kTFkPQkk+=+12 分 22（12 分）（1）令()()2ee10eexxxxxxfx=1 分 1x，即函数()f x的单调递增区间为(),1，函数()f x的单调递减区间为()1,+2 分 当12x=时，12111222 eef=，切点为11,2 2 e 又12111222 eef=，()f x在12x=处的切线方程为：1111122 e2 e2 e4 eyxyx=+4 分（2）()2eln1exxkx+，ln1ln1ln1eeexxxxxkkx+=()1,x+，ln1ln10exx+，ln1eln1exxxkx+6 分 由（1）可知()=exxf x在()1,+上单调递减，下证：ln1xx+即证：ln1xx在()1,x+恒成立 令()lng xxx=，则()1110 xgxxx=()g x在()1,+上单调递增 又1x，()()11 ln1 1g xg=9 分 ln1 1xx+第7页 共7页()f x在()1,x+上单调递减()()ln1f xfx+，即ln1ln1eexxxx+，ln1e1ln1exxxx+11 分 1k 12 分 第 1 页 共 4 页 第 2 页 共 4 页 中学生标准学术能力诊断性测试中学生标准学术能力诊断性测试 2023 年年 3 月测试月测试 数学试卷数学试卷 本试卷共本试卷共 150 分，考试时间分，考试时间 120 分钟。分钟。一、一、单项单项选择题：本题共选择题：本题共 8 小题，每小题小题，每小题 5 分，共分，共 40 分在每小题给出的四个选项中，只有一项是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符合题目要求的 1 已知集合2430Ax xx=+，2112xBy y=，则AB=A)2,3B()1,3C)2,+D()3,+2 设z是纯虚数，若31iz+是实数，则z的虚部为 A3 B1C1D3 3 已知函数()()()()3sincos0,f xxx=+，则“函数()f x是偶函数”是“=3”的 A充分不必要条件 B必要不充分条件 C充要条件 D既不充分也不必要条件 4 若圆()()22320 xay+=上有四个点到直线210 xy+=的距离为5，则实数a的取值范围是 A1317,22+B13 17,22 C37,22+D3 7,2 2 5 若111111777nnnnnnnCCC+是 9 的倍数，则自然数n为 A4 的倍数 B3 的倍数 C奇数 D偶数 6 现将 0-9 十个数字填入右方的金字塔中，要求每个数字都使用一次，第一行的数字中最大的数字为a，第二行的数字中最大的数字为b，第三行的数字中最大的数字为c，第四行的数字中最大的数字为d，则满足abcd的填法的概率为 A110B15C215D257 在矩形ABCD中，已知24ABAD=，E是AB的中点，将ADE沿直线DE翻折成1ADE，连接1AC当二面角1ADEC的平面角的大小为60时，则三棱锥1ACDE外接球的表面积为 A563B18 C19 D5338 已知0a 且1a，若集合22logaAxxx=，1lnln2Bx yxx=+，且AB，则实数a的取值范围是 A14e10,1,e4B14e10,e,4+C12e1,11,e4 D12e1,1e,4+二、二、多项多项选择题：本题共选择题：本题共 4 小题，每小题小题，每小题 5 分，共分，共 20 分分在每小题给出的选项中，有多项符合题目在每小题给出的选项中，有多项符合题目要求，全部选对的得要求，全部选对的得 5 分，部分选对的得分，部分选对的得 2 分，有选错的得分，有选错的得 0 分分 9 设0,0ab，满足321ab+=，下列说法正确的是 Aab的最大值为124B21ab+的最小值为8 3 C22ab+的最小值为113D2294ab+的最小值为 110已知等差数列 na的前n项和为nS，满足12321aaa+=，525S=，下列说法正确的是 A23nan=+B210nSnn=+C nS的最大值为5SD 11nna a+的前 10 项和为109911已知ABC的内角,A B C所对边的长分别为,a b c，已知4,6bc=，ABC的面积S满足()()224 38bcSa+=+，点O为ABC的外心，满足AOABAC=+，则下列结论正确的是（第 7 题图）（第 6 题图）第 3 页 共 4 页 第 4 页 共 4 页 A=6S B10CB AO=C2 213AO=D2 323=12已知()()1122,P x yQ x y是椭圆229144xy+=上两个不同点，且满足121292x xy y+=，则下 列说法正确的是 A 1122233233xyxy+的最大值为62 5+B 1122233233xyxy+的最小值为35 C 11223535xyxy+的最大值为2 102 55+D 11223535xyxy+的最小值为102 2三三、填空题：本题共、填空题：本题共 4 小题，每小题小题，每小题 5 分，共分，共 20 分分 13已知点M为抛物线28yx=上的动点，点N为圆()2245xy+=上的动点，则点M到y轴的距离与点M到点N的距离之和最小值为 14已知()f x为R上的偶函数，函数()()2=h xx f x在)0,+上单调递增，则不等 式()()()()2211330 xfxxfx+的解集为 15用0,1,2,3,4,5这六个数字组成无重复数字的六位数，要求任意两个偶数数字之间至少有一个奇数数字，则符合要求的六位数的个数有 个 16若关于x的不等式()e23xkxx+对任意的()0,x+恒成立，则整数k的最大值为 四四、解答题：本题共、解答题：本题共 6 小题，共小题，共 70 分解答应写出文字说明、证明过程或演算步骤分解答应写出文字说明、证明过程或演算步骤 17（10 分）在数列 na中，149a=，()()()2313912nnnnana+=+（1）求 na的通项公式；（2）设 na的前n项和为nS，证明：52544 3nnnS+18（12 分）已知ABC的内角,A B C的对边分别为,a b c，且2 cos2bAac=（1）求角B；（2）设ABC的角平分线BD交AC于点D，若BD=2，求ABC的面积的最小值 19（12 分）如图所示，在三棱锥ABCD中，满足3 3BCCD=，点M在CD上，且5DMMC=，ABD为边长为 6 的等边三角形，E为BD的中点，F为AE的三等分点，且2AFFE=（1）求证：FM面ABC；（2）若二面角ABDC的平面角的大小为23，求直线EM与面ABD所成角的正弦值 20（12 分）为提高学生的数学应用能力和创造力，学校打算开设“数学建模”选修课，为了解学生对“数学建模”的兴趣度是否与性别有关，学校随机抽取该校 30 名高中学生进行问卷调查，其中认为感兴趣的人数占 70%（1）根据所给数据，完成下面的22列联表，并根据列联表判断是否有 85%的把握认为学生对“数学建模”选修课的兴趣度与性别有关？感兴趣 不感兴趣 合计 男生 12 女生 5 合计 30（2）若感兴趣的女生中恰有 4 名是高三学生，现从感兴趣的女生中随机选出 3名进行二次访谈，记选出高三女生的人数为X，求X的分布列与数学期望 附：()()()()()22n adbcKabcdacbd=+，其中nabcd=+()20P Kk0.15 0.10 0.05 0.025 0.010 0.005 0.001 0k2.072 2.706 3.841 5.024 6.635 7.879 10.828 21（12 分）已知双曲线C以250 xy=为渐近线，其上焦点F坐标为()0,3（1）求双曲线C的方程；（2）不平行于坐标轴的直线l过F与双曲线C交于,P Q两点，PQ的中垂线交y轴于点T，问TFPQ是否为定值，若是，请求出定值，若不是，请说明理由 22（12 分）设()()exxf xx=R（1）求()f x的单调性，并求()f x在12x=处的切线方程；（2）若()()()eln1xf xkx+在()1,x+上恒成立，求k的取值范围（第 19 题图） 第 1 页 共 1 页 中学生标准学术能力诊断性测试中学生标准学术能力诊断性测试 2023 年年 3 月月测试测试 英语英语参考答案参考答案 1-5 BDCCB 6-10 BAABA 11-15 CDBBB 16-20 FECAG 21-25 ACBDA 26-30 CDBAB 31-35 BACCD 36a 37unity 38to work 39started 40injured 41including 42that/which 43gradually 44to 45taking 书面表达书面表达 The FIFA World Cup 2022 has kicked off in Qatar,fueling passion of the football fans all over the world.Though Team China is missing out on the competition,Chinese elements shine at the World Cup.From a stadium to souvenirs,from transportation to accommodation,“Made in China”appears both on and off the field at the FIFA World Cup Qatar 2022.The Lusail Stadium,the biggest one in Qatar and the venue for the final with 92,000 seats was designed and built by China Railway International Group with advanced energy-saving technologies and sustainable material in 2021.Besides,more than 1,500 buses from Chinas leading bus maker Yuyong are zooming down the streets of Qatar.Additionally,goods from Yiwu in Zhejiang Province also have an enormous presence during the World Cup,accounting for 70%of the souvenir market share.短文续写短文续写 When the big day finally came,I was full of mixed emotions.On one hand,as I saw the performers dressed in costumes performing on the stage,I couldnt help but feel a bit down as I wasnt given the opportunity to do the same.But on the other hand,when the performance finished and the audience clapped and cheered,I felt a great sense of pride.I knew that our script was a success and that our hard work had paid off.What I learned was that you dont have to be on the stage to be a star.As it is said in Records of the Three Kingdoms,“a glimmer of light can make solar and lunar brighter.”It means that even the smallest elements can make a difference through determination,hard work or unity.As long as you try your best and contribute to the overall goal,you can shine every day. 第 1 页 共 8 页 第 2 页 共 8 页 中学生标准学术能力诊断性测试中学生标准学术能力诊断性测试 2023 年年 3 月测试月测试 英语英语试卷试卷 本试卷共本试卷共 150 分，考试时间分，考试时间 100 分钟。分钟。第一部分第一部分 阅读理解（共两节，满分阅读理解（共两节，满分 60 分）分）第一节（共第一节（共 15 小题；每小题小题；每小题 3 分，满分分，满分 45 分）分）阅读下列短文，从每题所给的 A、B、C 和 D 四个选项中，选出最佳选项，并在答题卡上将该项涂黑。A What limits you from hiking more often?Many complain that they dont have a suitable partner to go with.Then why not take a look at the great hiking clubs across Canada?Yukon Outdoors Club Website:yukonoutdoorsclub.ca Membership cost:$10 for a single membership;$15 for a family membership Description:The club coordinates day hikes,backpacking trips,canoe trips,mountain biking,cross-country skiing trips,snowshoeing trips and various workshops for members to gain new skills and valuable information.The trips are open to everyone and range from easy to moderate to difficult.UBC Varsity Outdoor Club Website:ubc- Membership cost:UBC students$40;non-UBC students$60 Description:The club is a social group that hikes,mountaineers,kayaks（皮划艇）,rock climbs and ice climbs.Members lead trips,run skills workshops,and host presentations.Members can borrow equipment from the clubrooms gear cache（器材存放点）.The UBC VOC has also constructed a few huts in the Coast Mountains.Pender Harbor Hiking Group Website: Membership cost:Free Description:Now in its fourth year and boasting about 100 members,the Pender Harbour Hiking Group offers a way for members to meet like-minded individuals and get fit.Hikes are scheduled two months in advance,so get on the mailing list or check the website regularly to find a hike thats right for you!Hikes are usually one and a half to two hours long every Monday and Wednesday morning,but some full-day hikes are scheduled,depending on member interest.Vernon Outdoors Club Website:vernonoutdoorsclub.org Membership cost:a single membership$25;students$10 Description:The Vernon Outdoors Club is an active group that enjoys hiking and cycling.Since the membership is comprised of outdoor lovers in general,they are encouraged to share their activity ideas and interests.The group organizes a Tuesday Rambles event each week and also hosts multi-day trips.1Which club lends hiking equipment to its members?AYukon Outdoors Club BUBC Varsity Outdoor Club CPender Harbor Hiking Group DVernon Outdoors Club 2What is special about Pender Harbor Hiking Group?AHikes are mostly held on weekends.BIt has built huts for its members.CIt organizes workshops for its members.DIt is possible for members to hike for a whole day.3How much does it cost when a couple with a student goes to Vernon Outdoors Club?A$40.B$50.C$60.D$70.B Before Mya Gordon led the Lakeridge High School Districts Diversity Committee,she was a black girl who felt alone in the face of racial discrimination.Gordon,18,moved from New Jersey to Oregon when she was in seventh grade.She had a rough time in middle school,experiencing incidents ranging from microaggressions（微歧视）to obvious racist interactions.As a freshman at Lakeridge,Gordon decided to approach the principal at the time,Jennifer Schiele,to discuss the lack of support for diversity.Schiele encouraged forming an equity（平等）council at Lakeridge,which Gordon ran with.The goal of the group is to deal with racism and other types of prejudice with empathy（共情）.With Schieles support,Gordon recruited volunteers and discussed mental health and privilege.But some students“just joked around”,Gordon said.“It is disheartening to hear that people dont care about the work that you are doing.”Still,Gordon remained resilient in improving diversity,equity and inclusion in the school district.They have been meeting monthly to discuss how to improve district experiences for people of color,those with disabilities and other minority groups.Members looked at data from peer districts,auditing（评估）district policies through an equity lens and looking for ways to include equity into the curriculum.Her equity work in the community didnt stop there.Just before graduation,Gordon completed a documentary on the racist history of Lake Oswego called Lake No Negro.The video currently has over 11,000 views on YouTube.The documentary started as a desire to tell a story about the racially exclusive（排外的）history of Lake Oswego.Gordon learned everything she could about black exclusion laws in Oregon and in Lake Oswego that gave the city the nickname“Lake No Negro”.第 3 页 共 8 页 第 4 页 共 8 页 Although a lot of that information didnt make it into the final product,it was helpful for her to learn about the black exclusion laws that created an ideology that exists in Lake Oswego today.Gordons activism wont end when she leaves for college in the fall.She said shes already signed up to be a part of a college program about civic engagement.“I just really hope that other students speak up for what they believe all the time,even when its difficult.and stand up for others as well,”she said.4What did Gordon do to handle racism at Lakeridge?AShe ignored the racial discrimination.BShe improved the black exclusion laws.CShe turned to the school leader for help.DShe included equity into the curriculum.5What does the underlined word in Paragraph 4 mean?AAmazing BDiscouraging CExciting DShocking 6What can we know about Gordons documentary according to the passage?AGordon finished it after graduation.BIt centers on the racist history of Lake Oswego.CIt includes all the information concerning racism.DIt has received more than 11,000 likes on social media.7What could be the best title of the passage?AWork toward change BRacial discrimination CThe history of racism DAn outstanding college student C Crowded into a tiny plane,1,000 feet above the water off Georgia,the surveyors look down in search of a North Atlantic right whale,among the rarest in the world.There are thought to be fewer than 400 of the right whales alive today,and no more than 100 females that can breed.Now this high-stakes（高风险的）game of“I Spy,”including three teams affiliated（隶属）with the Clearwater Marine Aquarium Research Institute,is one part of an urgent mission to prevent these whales from extinction.Survivors swim to the Southeast,from the Carolinas to Florida,for calving（生幼崽）season.The surveyors note each mother and precious calf they see,sending reports from sky to land,which within minutes,are shared with shipping companies to broadcast out to sea,so captains can be cautious.From December to March,the surveyors fold themselves into planes,any day the weather is fair.They can cover 300 or 400 nautical miles（海里）in about six hours,practicing moderate dehydration（中度脱水）to avoid using the toilet,which is just a tube.They crane（伸长）their necks to look out bubble windows and relax their aching limbs when theyre back on solid ground.Every North Atlantic right whale has a unique pattern of raised skin on its head.This marking allows the surveyors to identify each whale by number and sometimes a name.They get to know the animals they see regularly:like Nauset,who never stays at the surface,taking one breath before diving for 10 minutes at a time.Over the months they track not just whales but moments that remind them of the works purpose.Marcy Lee,48,enjoys every first sighting of a mother and