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参考答案第1页共9页2023~2024学年福州市高三年级4月份质量检测参考答案与评分细则一、选择题:本大题考查基础知识和基本运算.每小题5分,满分40分.1.D2.C3.A4.C5.D6.B7.B8.A二、选择题:本大题考
查基础知识和基本运算.每小题6分,满分18分.全部选对的得6分,部分选对的得部分分,有选错的得0分.9.ABC10.BC11.ACD三、填空题:本大题考查基础知识和基本运算.每小题5分,满分15分.12.2−13.814.6,22四、解答题:本大题共5小题,共77分.解答应写出文字说明、证明过程
或演算步骤.15.【考查意图】本小题主要考查递推数列与数列求和等基础知识,考查运算求解能力、推理论证能力等;考查分类与整合、化归与转化等思想方法;考查数学运算、逻辑推理等核心素养;体现基础性和综合性.满
分13分.解:(1)因为12,2nnaann−=+�,所以12nnaan−−=,···································1分当2n�时,112211()()()nnnnnaaaaaaaa−−−=−
+−++−+L,所以22242nann=+−+++L,·························································3分所以(22),22nnnan+=�,所以2,2na
nnn=+�,··································4分又因为12a=,·················································
······························5分所以2*,nannn=+∈N.································································
······6分(2)由(1)可知2*(1),nannnnn=+=+∈N,·············································7分所以()111111nannnn=
=−++,····························································9分所以11111223(1)(1)nSnnnn=++++××−+L1111111122311nnnn=−+−++−+−−+L,··
···································11分所以111nSn=−+,·········································································1
2分又因为1n�,所以1nS<.·································································································13分参考答案第2页共9页16.【考查意图】本小题主要考查正态分布、全
概率公式、条件概率等基础知识,考查数学建模能力、逻辑思维能力和运算求解能力等,考查分类与整合思想、概率与统计思想等,考查数学建模、数据分析、数学运算等核心素养,体现基础性、综合性和应用性.满分15分.解:(1)依题意得,0,0.2µσ==,·······················
····································1分所以零件为合格品的概率为(0.60.6)(33)0.9973PXPXµσµσ−<<=−<<+=,···················································
················································2分零件为优等品的概率为(0.20.2)()0.6827PXPXµσµσ−<<=−<<+=,·····3分所以零件为合格品但非优等品的概率为0.99730.6827
0.3146P=−=,···········5分所以从该生产线上随机抽取100个零件,估计抽到合格品但非优等品的个数为1000.314631×≈.·························································
····················6分(2)设从这批零件中任取2个作检测,2个零件中有2个优等品为事件A,恰有1个优等品,1个为合格品但非优等品为事件B,从这批零件中任取1个检测是优等品为事件C,这批产品通过检测为事件D,··············
··············································8分则DABC=+,且A与BC互斥,·······················································9分所以()()()P
DPAPBC=+·································································10分()()(|)PAPBPCB=+··································
·······················11分221220.68270.68270.31460.6827CC=×+×××21.62920.6827=×,·······································
·····················12分所以这批零件通过检测时,检测了2个零件的概率为()(|)()PADPADPD=·········································
··································13分220.68271.62920.6827=×11.6292=0.61≈.···············································
······························15分答:这批零件通过检测时,检测了2个零件的概率约为0.61.17.【考查意图】本小题主要考查直线与平面平行的判定定理、直线与平面垂直的判定与性质定理、平面与平面的夹角、空间向量
、三角函数的概念等基础知识,考査直观想象能力、逻辑推理能力、运算求解能力等,考查数形结合思想、化归与转化思想等,考査直观想象、逻辑推理、数学运算等核心素养,体现基础性、综合性.满分15分.解法一:(1)在正方形ABEF中,连接AH并延长,交BE的延长线于点K,连
接PK.···································································································2分参考答案第3页共9页因为,GH分别为线段,APEF中点,所以HFHE
=,所以RtAFH△≌RtKEH△,所以AHKH=,·····························4分所以GHPK∥.································5分又因为,GHBCE
PKBCE⊄⊂面面,所以GHBCE∥面.···········································································7分(2)依题意得,
ABBCE⊥面,又因为BPBCE⊂面,所以ABBP⊥.又因为BPAE⊥,ABAEA=I,,ABAEABEF⊂面,所以BPABEF⊥面,································································
········8分又BEABEF⊂面,所以BPBE⊥,·····················································9分所以,,BPBEBA两两垂直.以B为原点,,,BPBEBA所在直线分别为,,xyz轴建立空间直角坐标系,如图所示.····
····························································10分不妨设1AB=,则31(1,0,0),(,,1)22PD−,()31=1,0,0,,,122BPBD��=−������uuuruuur,···········
··············································11分设平面BPD的法向量为(),,xyz=m,则0,0,BPBD�⋅=��⋅=��uuuruuurmm即0,310,22xxyz=���−+=��取2y=
,得0,1xz==,所以平面BPD的一个法向量是()0,2,1=m,·········································13分又平面BPA的一个法向量为()0,1,0=n.···········
·····································14分设平面BPD与平面BPA的夹角为θ,则225coscos,551θ⋅=<>===×mnmnmn.KFEPCDHAGBzyxBGAHDCPEF参考
答案第4页共9页所以平面DBP与平面BPA夹角的余弦值为255.····································15分解法二:(1)证明:取BP的中点Q,连接,GQEQ.·····1分因为,GH分别为线段
,APEF的中点,所以GQAB∥,12GQAB=,····························2分又因为,ABEFABEF=∥,所以,GQHEGQHE=∥,·································3分所以四边形GQEH是平行四边形,····
··················································4分所以GHQE∥,·············································································
·5分又因为,GHBCEQEBCE⊄⊂面面,所以GHBCE∥面.············································································7分(2)同解法一.··
··················································································15分解法三:(1)证明:取AB的中点I,连接,GIHI.········1分因为,GH分别为线段,APEF的中点,所以,GI
BPHIEB∥∥,又因为,GIBCEBPBCE⊄⊂面面,所以GIBCE∥面.············································3分因为,HIBCEBEBCE⊄⊂面面,所以HIBCE∥面.·······························
·············································5分又因为,,GIHIIGIGIHHIGIH=⊂⊂I面面,所以GIHBCE面∥面,·······································
······························6分又因为GHGIH⊂面,所以GHBCE∥面.············································································7分(2)同解法一.······
··············································································15分18.【考查意图】本小题主要考查圆、椭圆的标准方程及简单几何性质,直线与椭
圆的位置关系等基础知识,考査直观想象能力、逻辑推理能力、运算求解能力等,考查数形结合思想、化归与转化思想、分类与整合思想等,考査直观想象、逻辑推理、数学运算等核心素养,体现基础性、综合性与创新性.满分17分.解法一:(1)依题意,222bca+=.·························
·································1分IFEPCDHAGBQFEPCDHAGB参考答案第5页共9页COyxPF2F1IG设00(,)Pxy,则2200221xyab+=,0axa−<<,所以2
22222222020000022||()()(1)(1)2xbPFxcyxcbxcxbcaa=−+=−+−=−−++,所以22220002||2||ccPFxcxaxaaa=−+=−,······························
··············3分又ac>,所以0caxa>,20axc>,所以20||cPFaxa=−,20adxc=−所以0220||caxPFcaadaxc−==−,即2||PFd为定值,且这个定值为ca.··············
····4分(2)(ⅰ)依题意,00(,)33xyG,设直线IG与x轴交于点C,因为IG⊥x轴,所以0(,0)3xC,·······················5分所以001202||||()()333xxFCFCccx−=+−−=,·
·········································6分因为△PF1F2的内切圆与x轴切于点C,所以121202||||||||3PFPFFCFCx−=−=,······················
···························7分又因为12||||2PFPFa+=,解得02||3xPFa=−,··········································8分由(1)得20||cPFaxa=−,所以0
03xcaxaa−=−,所以椭圆E的离心率13cea==.·························10分(ⅱ)由26a=,得3a=,又13ca=,所以1c=,2228bac=−=,所以椭圆E的方程为22198xy+=.···············
·······································11分根据椭圆对称性,不妨设点P在第一象限或y轴正半轴上,即003x<�,0022y<�,又1(1,0)F−,2(1,0)F,所以直线PF1的方程为00(1)1yyxx=+
+,设直线IG与PF1交于点D,因为03Dxx=,所以000(3)3(1)Dyxyx+=+,△F1CD的面积1S与△PF1F2的面积S之比为000200100(3)1(1)233(1)(3)118(1)22xyxxxSSxy++×++==+××,···
·············································13分令2(3)()18(1)xfxx+=+(03x<�),则2(3)(1)(1))(18xxxfx−′++=,参考答案第6页共9页当[0,1)x∈,()0fx′<,当(1,3)x∈,()0fx′
>,所以函数()fx在[0,1)单调递减,在(1,3)单调递增.又因为1(0)2f=,4(1)9f=,1(3)2f=,所以()fx的值域是41[,]92,所以14192SS��,···································
·······································15分所以11415SSS−��,················································
·······················16分根据对称性,△PF1F2被直线IG分成两个部分的图形面积之比的取值范围是45[,]54.········17分解法二:(1)同解法一·································
··········································4分(2)(ⅰ)依题意,00(,)33xyG,设直线IG与x轴交于点C,因为IG⊥x轴,所以0(,0)3xC,·······
················5分所以001202||||()()333xxFCFCccx−=+−−=,··········································6分因为△PF1F2的内切圆与x轴切于点C,所以12120
2||||||||3PFPFFCFCx−=−=,·················································7分又因为12||||2PFPFa+=,得0102||,3||.3xPFaxPFa�=+����=−��··
·········································8分所以2200022000(),3(),3xxcyaxxcya�++=+����−+=−��两式平方后取差,得00443cxax=对任意0x成立,所以椭圆E的离心率13cea==
.························································10分(ⅱ)同解法一···········································································
········17分解法三:(1)同解法一···········································································4分(2)(ⅰ)依题意,00(,)33xy
G,因为IG⊥x轴,设点I坐标为0(,)3xt.··········5分可求直线1PF方程为00()yyxcxc=++,则点I到直线1PF的距离0002200()()3||()xyctxctyxc+−+=++,······················
···········6分即()222200000()()()3xyctxctyxc��+−+=++����,化简得22000002()()()033xxyttcxcyc+++−+=,①同理,由点I到直
线2PF的距离等于||t,可得参考答案第7页共9页22000002()()()033xxyttcxcyc+−−−−=,②············································7分将式①−式②,得
00084233tcxycx⋅=⋅,则04yt=.·····································8分将04yt=代入式①,得2200001()()()016233yxxcxcc+++−+=,化简得220022198xycc+=,得229ca=,所以椭
圆E的离心率13cea==.························································10分(ⅱ)同解法一···························································
························17分19.【考查意图】本小题主要考查集合、导数、不等式等基础知识,考查逻辑推理能力、直观想象能力、运算求解能力和创新能力等,考查函数与方程思想、化归与转化思想、分类与整合思想、数形结合思想等,考查数学抽象、逻辑推理、直观想象、数学运算等核心素养,体
现基础性、综合性与创新性.满分17分.解:(1)依题意,因为0(),()fxxlxL∈∈R,所以2,1xxxkx∀∈−++R�,且0xR,20001xxkx−+=+,····················1分令2()(1)1xxkxφ=−+−−,()214k∆=−−,则()0xφ�,且0
()0xφ=,所以0,0,∆��∆���所以0∆=,···································································3分即()2140k−−=,解得3k=或1−.········
······································································4分(2)(ⅰ)先证必要性.若直线()ylx=是曲线()ygx=的切线,设切点为()00,e1xx+,因为()exgx′=
,所以切线方程为()000e1e()xxyxx−+=−,即()000e(1)e1xxlxxx=+−+(*).························································5分一方面,()()00g
xlx=,所以000,()()xgxlx∃∈=R,································6分参考答案第8页共9页另一方面,令()()()000ee(1)exxxGxgxlxxx=−=−−−,则()00Gx=
,因为()0eexxGx′=−,所以当0xx<时,()0Gx′<,()Gx在()0,x−∞单调递减,当0xx>时,()0Gx′>,()Gx在()0,x+∞单调递增,所以()()00GxGx=�,所以()()gxlx�.····························
··················7分即,()()xgxlx∀∈R�,所以()(),gxxlxT∈∈R,即()lx是函数()gx在R上的“最佳下界线”.···············8分再证充分性.若()lx是函数()gx在R上的“最佳下界线”,不妨设()lxkxb=+,由“最佳下界线”的定义,
()(),xgxlx∀∈R�,且()()000,xgxlx∃∈=R,····9分令()()()e1xHxgxlxkxb=−=+−−,则()0Hx�且()00Hx=,所以()min0Hx=.··················
·······················10分因为()exHxk′=−,①若0k�,则()0Hx′�,所以()Hx在R上单调递增,所以10xx∃<,使得()()100HxHx<=,故0k�不符合题意.·····················11分②若0k>,令()
0Hx′=,得lnxk=,当(),lnxk∈−∞时,()0Hx′<,得()Hx在(),lnk−∞单调递减,当()ln,xk∈+∞时,()0Hx>,得()Hx在()ln,k+∞单调递增,所以,当且仅当lnxk=时,()Hx取得最小值()lnHk.················
············12分又由()Hx在0x处取得最小值,()min0Hx=,所以0ln,(ln)0,xkHk=��=�·······················································
····················13分即000e,e10,xxkkxb�=��+−−=��解得0exk=,()001e1xbx=−+,····························14分所以()000e(1)e1xxlxxx=+−+,参考答案第9页共9
页由(*)式知直线()ylx=是曲线()ygx=在点()00,e1xx+处的切线.············15分综上所述,直线()ylx=是曲线()ygx=的一条切线的充要条件是直线()ylx=是函数()gx在R上的“最佳下界线”.(ⅱ)集合(),
(1,)(),hxxgxxLTRI∈+∞∈元素个数为2个.············································17分