中学生标准学术能力诊断性测试2023-2024学年高三上学期11月测试 数学答案

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中学生标准学术能力诊断性测试2023-2024学年高三上学期11月测试 数学答案
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第1页共8页中学生标准学术能力诊断性测试2023年11月测试数学参考答案一、单项选择题: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.12345678CABBABDD二、多项选择题:本题共4小题,每小题5

分,共20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5分,部分选对但不全的得2分,有错选的得0分.9101112BDACABACD三、填空题:本题共4小题,每小题5分,共20分.13.114.7215.1024+16.35四、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)(1)2sinsincoscBCbB=,所以由正弦定理得2sinsinsinsincosCBCBB=,1cos2B=,得3B=·············

·······························································3分()33tantan4tantan531tantan3134ABCABAB++=−+=−=−=−−−··············

···5分(2)ABC内切圆的面积为,所以内切圆半径1r=,由圆的切线性质得23,2333cabbca+−==+−=+·····························7分由余弦定理得222bcaac=+−,()()2222333caacacac+=+−=+−,将333ac+

=+代入,{#{QQABLYQUggCgQBBAAQhCAwEQCgAQkBACAAoGxAAIMAIBgRFABAA=}#}第2页共8页1843,sin23323ABCacSac=+==+·····················

·························10分(或()1643,2332ABCabcSrabc++=+=++=+······················10分)18.(12分)(1)过D作AB的垂线交AB于H点,设ACABa

==,则()22,2,2,212BCaBDaHDHBBDaAHBHBAa======−=−,22522ADAHDHa=+=−,由题意得,二面角CSAD−−的平面角为CAD,·········································2分()345222

cos17522DHCADDA+===−············································4分(2)分别以AB,AC,AS为x轴,y轴,z轴建立直角坐标系,()()0,0,0,,0,0ABa,()()0,,

0,0,0,CaSh,则()()()(),0,1,0,,1EahFah−−,SBSC=且,SESFSESFSBSC===·····················································6分又,SABSACBSACSA=,

那么SAESAF,则AEAF=···························································································

··8分故SEF与AEF都是等腰三角形,取EF的中点G,则SG与AG均垂直于EF,()()111111,,1,,,,,,1222222GaahSGaahAGaah−=−=−,平面AEF⊥平面

SBC等价于SGAG⊥·················10分()22221044aaSGAGhhh=+−−=,又260,,3SCBah===························12分19.(12分)(1)()1122,22

2nnnnaaaa++=+−=−,即()11222nnaa+−=−,{#{QQABLYQUggCgQBBAAQhCAwEQCgAQkBACAAoGxAAIMAIBgRFABAA=}#}第3页共8页2na−是公比为12的等比数列···············

················································2分111123,2322nnnnaa−−−==+······································

···············4分(2)211211112312612222nnnnSaaann−=+++=+++++=+−·················································

····························································6分111266,266222023nnnnSnSn−−=−−−=,

即126069,nn−取最小值14··································································8分(3)()11212,133nnnnnCCn

Cn−++==,得2n,即1nnCC+,有345CCC,又1234,3CCC===,故nC中最大项为23,CC·············································

··························10分又mb中最小值为()()2minmax7,3mnbC−,即24733−,()()3710−+,又70,3················

·································12分20.(12分)(1)由题意可知:X的所有可能取值为2.3,0.8,0.5,()1342.30.3245PX===····································

··································1分0.8X=包含的可能为“高低高”“低高高”“低低高”,()1141341140.80.5245245245PX==++=·················

·························2分()0.510.30.50.2PX==−−=·······················································

············3分X的分布列为:X2.30.80.5P0.30.50.2{#{QQABLYQUggCgQBBAAQhCAwEQCgAQkBACAAoGxAAIMAIBgRFABAA=}#}第4页共8页·······························

···············································································4分数学期望()1.19EX=································

················································5分(2)设升级后一件产品的利润为Y,Y的所有可能取值为2.3,0.8,0.5aaa−−−···············

······································6分()36142.3245103PYabb=−=+=+·························································7分

()1341141140.82452452450561PYabbbb=−=−+++−=−·············8分()635610.5110105bbPYa+−=−=−−=···········

·············································9分()()()()()635612.30.80.51.190.910105bbEYaaaba+−=−+−

+−=+−·····························································································

···············11分()()0.90EYEXba−,即:()1100,0.429aba(备注:12不写出不扣分)·······························12分21.(12分)(1

)1142ABAFBF++=,即221142AFBFAFBF+++=,又12122,442,2AFAFBFBFaaa+=+===·······························2分又2e=,1,12ccba===,故椭圆E的方程是2212xy+=···············

·······················································4分(2)依题意知直线BC的斜率存在,设直线:BCykxm=+,代入2212xy+=,得()2212xkxm++=,即22212102kxkmxm

+++−=①,{#{QQABLYQUggCgQBBAAQhCAwEQCgAQkBACAAoGxAAIMAIBgRFABAA=}#}第5页共8页()()22222124122402kmkmmk=−+−=−++,即22210km−+②,设()(

)1122,,,BxyCxy,则()22,Axy−,122212kmxxk−+=+③,2122112mxxk−=+④·····················································6分2,,AFB三点共

线,()21,0F,直线AB不与坐标轴垂直,()()()()12211212,1111yyxkxmxkxmxx−=−+=−−+−−,()()121212220kxxmxxkxxm++−+−=,()22222221222

0111222kmkmkmmkkk−−+−=+++,2222222220kmkkmkmmkm−−+−−=,2,mk=−直线():2BCykx=−,由②得:22212410,2kkk−+,2121BCkx

x=+−,点()11,0F−到直线:20BCkxyk−−=的距离231kdk=+,1121322FBCSBCdkxx==−·································································8分(

)()222212121222441441122kkxxxxxxkk−−=+−=−++{#{QQABLYQUggCgQBBAAQhCAwEQCgAQkBACAAoGxAAIMAIBgRFABAA=}#}第6页共8页()422222221164412421

122kkkkkk−−+−==++,()()()12222243012FBCkkSkk−=+,设2211kt+=,则212tk−=,()()12222123211333231FBCttttS

tttt−−−+−===−+−21313248t=−−+···································································10分所以当13

4t=时,即22441,21,336tkk=+==(符合题意),1FBCS的最大值为324,所以当66k=时,1FBC的面积取最大值为324·······································································

·····································12分22.(12分)(1)定义域为()(),11,−−+,由题意知()()2ln1xxxax−−=,则()()1ln1xxa−−=有三个不同的实数根,当1x时,

令()()()1ln1gxxx=−−,()()()ln11,gxxgx=−+在()1,x+上单调递增,又110eg+=,()gx在11,1ex+上单调递减,在11,ex++上单调递增,()111eegxg+=−··

············································································2分{#{QQABLYQUggCgQBBAAQhCAwEQCgAQkBACAA

oGxAAIMAIBgRFABAA=}#}第7页共8页当1x−时,令()()()1ln1hxxx=−−−,()()()()()221123ln1,1111xxhxxhxxxxx−+=−−+=+=++++,又()30h−=,()hx在(),3−

−上单调递减,在()3,1−−上单调递增,()()32ln20hxh−=+,()hx在(),1−−上单调递增·····································································5

分又()20h−=,当1x−→−时,()hx→+,当x→−时,()hx→−,当1x+→时,()0gx−→,当x→+时,()gx→+,10ea−···········································

················································6分(2)由已知可知()()1ln1axxx−−=有3个零点123,,xxx,不妨设123xxx,显然121x−−,由(

1)中函数()gx性质且11e11e11eea−−++,231112exx+,只需证2322exx++,即3222exx+−····················································8分又

()()()222222122211ln11ln10eeegxgxxxxx+−−=+−+−−−−,上面不等式证明如下:令()()()2211ln11ln1,1,1eeexxxxxx

=+−+−−−−+,{#{QQABLYQUggCgQBBAAQhCAwEQCgAQkBACAAoGxAAIMAIBgRFABAA=}#}第8页共8页()()21ln1ln12,1,1

eexxxx=−+−−−−+()2ln1120exx=+−−−,()x在11,1ex+上单调递增,()110ex+=····································

·······································10分又221211,21eeexx−−−+−+,()22121egxgx+−,又显然有()()23gxgx,23121exx+−

,2322exx++,1232exxx++··················································································12分{#{QQABLYQUggCgQBBAAQhCAwEQCgAQkBACAAo

GxAAIMAIBgRFABAA=}#}获得更多资源请扫码加入享学资源网微信公众号www.xiangxue100.com

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