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高二数学参考答案第1页(共8页)2022—2023学年下期第一次联考高二数学参考答案一、选择题(本大题共12小题,每小题5分,共60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.D【解析】因为()cos33fππ=为常数,所以()03fπ′=.故选D
.2.C【解析】2418≠×,A选项错误;2214≠−×,B选项错误.因为642963==,所以9,6,4依次成等比数列,C选项正确.2648≠×,D选项错误.故选C.3.A【解析】直线130kxyk−+−=可化为()13ykx−
=−,表示过点()3,1,斜率为k的直线,所以所有直线都通过定点()3,1.故选A.4.B【解析】若方程22151xymm+=−−表示焦点在y轴上的椭圆,则150mm−>−>,解得:35m<<.所以p成立的充要条件是:35m<<.结合四个选项可知:p成立的充分不必要
条件是45m<<,故选B.5.C【解析】由题意得()()()00000limlim()xxfxxfxfxabxax∆→∆→+∆−′==+∆=∆,故选C.6.D【解析】∵a,b,c成等差数列,∴2b=a+c,又a+1,b,c与a,b,c+2都分别成等比数列,∴b2=(a
+1)·c,b2=a·(c+2).联立222,(1),(2),bacbacbac=+=+=+解得b=12.故选D.7.A【解析】()1112CMAMACABAD=−=+�����������������������-()1ABBCCC++�������������=1122−−
−���abc,111122MCCMabc=−+=+�������������.故选A.8.D高二数学参考答案第2页(共8页)【解析】根据题意:()0,4M−,()23,8A−−,故6412116m−=,解
得4m=,即221164yx−=,当水面宽度为46米时,即26x=−时,47y=−,拱顶M到水面的距离为()474−.故选D.9.A【解析】设等差数列{}na的公差为d,()()()110105661055111202aaSaaa+==
+=+=,所以613a=,则652daa=−=,所以()52521naann=+−=+,所以1111122123nnnbaann+==−⋅++,所以()11111111112355721232323323
nnTnnnn=−+−++−=−=++++⋯,因为17kT=,所以()13237kk=+,解得9k=.故选A.10.C【解析】因为'()fa、'(1)fa+分别是函数()fx在xa=、1xa=+处的切线斜率,由图可知'(1)'()0fafa+<<,又
0(1)()(1)()'()(1)fafafafafxaa+−+−==+−,0(,1)xaa∈+,所以()()()()11fafafafa′′++−<<,故选C.11.C【解析】依题意,()3,0,3S,()3,0,0A
,()0,1,0B,()0,1,0C−,313,,222E;若13SDSC=,则231,,233D−,则333,,222CE=����,1333,,222CE=����∵,故A正确;232,,233CD=
����,()3,1,0AC=−−����,313,,222AE=−����,故D点到直线CE的距离228721CDCEdCDCE⋅=−=����������������,故B正确;设(
),,nxyz=�为平面ACE的法向量,则00ACnAEn⋅=⋅=����������,即303130222xyxyz−−=−++=,令2z=−,则高二数学参考答案第3页(共8页)()3,
3,2n=−−�为平面ACE的一个法向量,故C错误;而232,,233CD=����,故点D到平面ACE的距离11CDndn⋅==������,故D正确.故选C.12.A【解析】如图所示,设点Q所在椭圆的另一焦点为F,则44445QPQOQPQFPFPO=≤++−+=−+=.故选
A.二、填空题(本大题共4小题,每小题5分,共20分)13.514.20xye−+=15.224ayxb=16.1024013.【解析】(2PA=−����,0,1)−,点P到平面α的距离为|||401|5||5nPAn⋅−+−==��������.14.【解析】由题意,(
)()1ln()1lnfxxxxx−+=′−=+−−故()()e12,eekflnef=−=+=−=−′,则曲线()yfx=在ex=−处的切线方程为:()2y+e=x+e,即:20x-y+e=15.【解析】设抛物线方程为()220ypxp=>,依题意,2aAb
,代入()220ypxp=>得222,244aapbpb==,所以抛物线标准方程为224ayxb=.16.【解析】由题意得2142Sa=+,所以12142aaa+=+,解得28a=,又因为221144nnn
nnaSSaa++++=−=−,于是()211222nnnnaaaa+++−=−,高二数学参考答案第4页(共8页)因此数列{}12nnaa+−是以2124aa−=为首项、2为公比的等比数列,故1112422nnnnaa−++−=×=,于是11122nnnnaa++−=,因此数列2nna
是以1为首项、1为公差的等差数列,故1(1)2nnann=+−=,故2nnan=⋅,所以101010241020a==⋅.三、解答题(本大题共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.【解析】(1)由020xyxy−=+−=可得11xy==,·
·························································2分所以1l,2l的交点为()1,1,································································
····3分故过()1,1且斜率为2的直线的方程为()121yx−=−即210xy−−=.················5分(2)根据正弦定理由2sinsinBA=可得222ba=,·········
·························7分设顶点C的坐标为(),xy,则()()2222222122xyxyxy++=++⇒+=,·············································8分
因为构成三角形,故0y≠,··································································9分故轨迹方程为()2202xyy+≠=.·········
····················································10分18.【解析】(1)设等差数列{}na的公差为d,由19a=得到436624Sd=+=,解得2d=−,······································
···············································2分得112nan=−,21()102nnaanSnn+==−+.·······································
·········6分(2)当5n≤时,2121210nnnnTaaaaaaSnn=+++=+++==−+⋯⋯.····························7分当6n≥时,1212567nnnTaaaaaaaaa=+++=+++−−−−⋯⋯⋯2521050nSSnn
=−=−+.··································································9分高二数学参考答案第5页(共8页)故2210,51050,6nnnnnT
nnnn∗∗−+≤∈=−+≥∈NN且且;·····················································10分所以603600600503050T=−+=.(不求Tn,直接求T60
也得分)···················12分19.【解析】(1)如图过点O作AB的平行线OD交劣弧�BC于点D,连接1OO,1OD,因为1OO∥1AA,1AA⊂平面1AAB,1OO⊄平面1AAB,则1OO∥平面1AAB同理可证
OD∥平面1AAB,1OOODO=∩,且1OO⊂平面1OOD,OD⊂平面1OOD所以平面1AAB∥平面1OOD,又因为1OD⊂平面1OOD,所以1OD∥平面1AAB故存在点D满足题意.因为ABC△为底面O⊙
的内接正三角形,所以3BACπ∠=,即6ABOBODπ∠=∠=,又因为3AB=,所以O⊙的半径为332sin3π=,所以劣弧�BD的长度为362326ππππ××=.·································
··············6分(2)如图取BC的中点为M,连接MA,以MB为x轴,MA为y轴,过M作1OO平行线为z轴,建立空间直角坐标系,又因为13AAAB==,设AB中点为N.故()0,0,0M,3,0,02B,330,,02
A,3,0,02C−,30,,02O,130,,32O,333,,044N,易知平面1AAB的法向量33,,044ON=����设平面1CBO的法向量为(),,nxyz=�
,又因为130,,32MO=�����,3,0,02MB=����故1·0·0nMOnMB==�����������即3302302yzx+==,令23y=得()0,23,1n=−�高二数学参考答案第6页(共8页)易知平面1CB
O和平面1BAA夹角为锐角,所以平面1CBO和平面1BAA夹角的余弦值为33921323134nONnON⋅==⋅×����������··········12分20.【解析】(1)依题意动点M到定点()1,0F的距离等于动点M到直线1x
=−的距离,··············2分由抛物线的定义可知,动点M的轨迹是以点F为焦点,直线1x=−为准线的抛物线,所以曲线C的方程为24yx=.·····························
··········································5分(2)联立方程2214yxyx=−=,整理得24810xx−+=.············································6分设()()1122,,,AxyBxy,则
有122xx+=,1214xx=.于是212115ABkxx=+−=.··································································8分设(),0,Pa
P到直线AB的距离为d,因为:210ABlxy−−=,由点到直线的距离公式得215ad−=.··························································9分又12ABPSABd=⋅△,所以2ABPSdAB=△,于是932212515a×−
=.·························10分解得5a=或4a=−,············································································11分故点P的坐标为()5,0或(
)4,0−.·····························································12分21.【解析】(1)2nnaSn+=∵①1122,2nnaSnn−−∴+=−≥②①-②得12nnnaaa−+=−,即122nnaa−=+,··········
··································2分变形可得11222nnaa−=−−,······································································
·3分又112aS+=,得11a=故数列{}2na−是以-1为首项,12为公比的等比数列,·····························4分由等比数列的通项公式可得1122nna−−=−,············································5分高
二数学参考答案第7页(共8页)*1122nnanN−∴=−∈,.·······································································6分(2)令()()()232nfnna=−−,则()
1232nnfn−−=·············································7分()()12123521222nnnnnnfnfn−−−−∴+−=−=·······
··········································8分当1n=或2n=时,()()10fnfn+−>,当3,nnN≥∈时,()()10fnfn+−<又()334f=,()max34fn∴
=,····························································10分因为不等式()()22232nnaλλ−>−−对任意的正整数n恒成立,2324λλ∴−>,························
······················································11分解得λ13<<22.·················································
··························12分22.【解析】(1)设直线:lykxm=+,显然0k≠,令0y=,得mxk=−,则(,0)mNk−,因为M是线段PN的中点,所以Pmxk=,2Pym=,所
以(,2)mPmk,(,2)mQmk−,又200QQymmmkmxk−−−′==−−3k=−,························································4分所以3kk′=−为定值.············
····························································5分(2)联立22142xyykxm+==+,消去y并整理得222()124240kxmkx
m++−=+,········6分则2222164(12)(24)0mkkm∆=−+−>,则2242mk<+,根据韦达定理可得222412APmxxk−⋅=+,所以22(24)(12)Akmxmk−=
+,高二数学参考答案第8页(共8页)所以222(24)(12)AAkmykxmmmk−=+=++2222244(12)kmkmmk−+=+,所以2222222(24)44,(12)(12)kmkmkmAmkmk−−+++
,··············································7分由(1)知,3kk′=−,所以直线QM的方程为3ykxm=−+,联立221423xyykxm+==−+,消去y并整理得2
22(118)12240kxkmxm+−+−=,则22221444(118)(24)0mkkm∆=−+−>,则22362mk<+,根据韦达定理可得2224118BQmxxk−=+,所以22(24)(118)Bkmxmk−=+,·····················8分所以2223
(24)3(118)BBkmykxmmmk−=−+=−++222221212(118)kmkmmk++=+,所以2222222(24)1212,(118)(118)kmkmkmBmkmk−++++,···········································
9分所以ABABAByykxx−=−22222222222222441212(12)(118)(24)(24)(12)(118)kmkmkmkmmkmkkmkmmkmk−+++−++=−−−++2222261224(2)kmmkkm+−−=−222
26(2)24(2)kmmkm−+−=−2614kk+=1342kk=+,··········································································
·····11分因为0k>,所以13136242422kkkk+≥×=,当且仅当66k=时取等号,所以62ABk≥;即tanθ的最小值为62.····································
····························12分获得更多资源请扫码加入享学资源网微信公众号www.xiangxue100.com