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苏州大学2020届高考考前指导卷参考答案一、填空题:本大题共14小题,每小题5分,共计70分.1.{|12}xx≤2.23.2804.1(0]2,5.26.527.568.π2−9.1310.12−11.5306612.413.24[1]33−,14.1655解答与提示:1.{
|12}ABxx=≤.2.2i(2i)(1i)22i1i222aaaaz+++−+===+−.因为z为纯虚数,所以2020aa−=+,,解得2a=.3.由图可知,时速在区间[8090)[110120),,,的频率
为(0.010.02)100.3+=,所以时速在区间[90110),的频率为10.3−,所以时速在区间[90,110)的车辆约为4000.7280=辆.4.由1200xx−≥,,解得102x≤,即函数()fx的定义域为1(0]2,.5.离心率1
31cea+===,所以2=.6.执行第一次循环105Si==,;执行第二次循环207Si==,;执行第三次循环349Si==,;执行第四次循环5211Si==,,终止循环.所以52S=.7.记方案一与方案二坐到“3号”车的概率分别为P1,
P2,三辆车的出车顺序可能为:123,132,213,231,312,321.方案一坐“3号”车可能:132,213,231,所以136P=;方案二坐“3号”车可能:312,321,所以226P=.则该嘉宾坐到“3号”车的概率1256PPP=+=.8.()cossinfxxx
x=−,所以在π2x=处的切线的斜率为ππ()22kf==−.9.2312135616[1()]111(1)131aqaaaqaqSqq−++−===−+−.10.因为π2sincos()4=+,解得1tan3=,所以11
π13tan()14213−−==−+.11.如图,10AB=(寸),则5AD=(寸),1CD=(寸),设圆O的半径为x(寸),则(1)ODx=−(寸).在RtADO△,由勾股定理可得2225(1)xx+−=
,解得13x=(寸),则该木材的体积约为221001316900x==(立方寸).12.函数()fx的图象如右图所示,由题意,30()2fx,即319x,因为123()(
)()fxfxfx==,所以3133()(3)xfxxx=−,令3(1,3)tx=,构造函数32()3gttt=−+,2()36gttt=−+,所以当2t=时,max()(2)4gtg==,所以31()xfx的最大值为4.13.设正方形ABCD的边长为a,
以A为原点,ABAD,所在直线为分别为xy,轴建立平面直角坐标系,则(00)(0)()(0)ABaCaaDa,,,,,,,.设()Pxy,,因为0CPDP=,所以()()0xayaxya−−−=,,,即222()()24aaxya−+−=
,设cos22sin2aaxaya=+=+,.又因为()()22aaEaFa,,,,APAEAF=+,所以()()()22aaxyaa=+,,,,即22axaaya=+=+,,所以2232()[(sincos
)]1sin()332234aaxyaa+=+=++=++,由P为正方形ABCD内部一点(包含边界),可得[2],,所以[]444+,,所以2241sin()[1]3433
+=++−,.14.法一:设ADt=,则3CDt=,4ACt=,在ABD△中,222(2)cos22tcADBt+−=,在BDC△中,222(3)(2)cos223taBDCt+−=,又coscosADBBDC=−,所以222222(2)(3)(2)22223tcta
tt+−+−=−,解得2221238tca=+−,①DCBA在ABC△中,2222(4)2cosACtacacB==+−,即2221162tacac=+−,②由①②可得2239322acac++=.所
以2222333532(3)(3)(3)()(3)2228acacacacac+=+−+−=+≥,即2832(3)5ac+≤,所以16535ac+≤,当且仅当3ac=,即8585,515ac==时等号成立,所以3ABBC+的最大值为
1655.法二:因为3CDAD=,所以3CDDA=,即3()BDBCBABD−=−,整理得到3144BDBABC=+,两边平方后有22291316168BDBABCBABC=++,所以22913216168BABCBABC=
++即2291312||||161684BABCBABC=++,整理得到223329||||||||2BABCBABC=++,设||||cBAaBC==,,所以22239329(3)22caaccaac=++=+−,因为293333(
)2222acacca+=≤,所以222293532(3)(3)(3)(3)288caaccacaca=+−+−+=+≥,832165355ca+=≤,当且仅当8585515ac==,时等号成立,所以3ABBC
+的最大值为1655.二、解答题:本大题共6小题,共计90分.15.(本小题满分14分)解:(1)因为1a=且3cossinCcA=,所以3cossinaCcA=,·····················2分在ABC△中,由正弦定理sinsinacAC
=,所以sinsinaCcA=,所以3sincossinsinACCA=.··························································4分因为(0)A,,所以sin0A,所以3cossinCC=,因为(0)C,,所以si
n0C,所以cos0C,所以tan3C=,·············6分因为(0)C,,所以3C=.······························································8分(2)
由(1)知,3ACB=,因为1a=,3b=,所以ABC△的面积1333sinsin2234ABCSabACB===△,························10分因为D是AB上的点,CD平分AC
B,所以1sin12613sin26BCDACDaCDSaSbbCD===△△,····················································12分因为ABCACDBCDSSS=
+△△△,所以33339344416ACDABCSS===△△.·············14分16.(本小题满分14分)证:(1)因为四边形ABCD是矩形,所以ABCD∥.··························································
·······2分又AB平面PDC,CD平面PDC,所以AB∥平面PDC,···································5分又因为AB平面ABE,平面ABE∩平面PDCEF=,所以ABEF∥.········
····································7分(2)因为四边形ABCD是矩形,所以AB⊥AD.因为AF⊥EF,(1)中已证ABEF∥,所以AB⊥AF,·················································
·································9分因为AB⊥AD,由点E在棱PC上(异于点C),所以F点异于点D,所以AFADA=,又AFAD,平面PAD,所以AB⊥平面PAD,·······································12分又AB平面A
BCD,所以平面PAD⊥平面ABCD.··································14分17.(本小题满分14分)解:(1)由题意AOCCOD==,设四边形OCDB的面积为()S,因为四边形O
CDB可以分为OCD△和OBD△两部分,所以11()sinsin(2)22OCDOBDSSSOCODOBOD=+=+−△△,···············3分因为1OBOCOD===,所以1()(sinsin2)2S=+.因为020−,,所以02
.所以四边形OCDB的面积1()(sinsin2)(0)22S=+,,.······················6分(2)由(1)1()(sinsin2)(0)22S=+
,,,EFABCDP所以2211()(sin)(sincos)coscossin22S=+=+−21(4coscos2)2=+−,令()0S=,即24coscos20+−=,解得133cos8−+=或133cos8−−=,因为02
,所以存在唯一的0,使得0133cos8−+=.·····················10分当00时,()0S,()S在0(0),单调递增;当02时,()0S,()S在0()2,单调递减,所以0=时,max0()()SS=,·······
···················································12分此时22202cos(2)BDOBODOBOD=+−−22000112cos222(2cos1)4cos=++=+−=,从而0
1332cos4BD−+==(千米).答:当四边形OCDB的面积最大时,BD的长为1334−+千米.··················14分18.(本小题满分16分)解:(1)因为椭圆22221(0)xyabab+=的离心率为22,短轴长为2,所以22222
22babcca==+=,,,解得21ab==,,所以该椭圆的标准方程为2212xy+=.····················································4分(2
)因为点(2)(0)(20)MmmA−,,,,所以直线AM的方程为(2)22myx=+,即2(2)4myx=+.由22122(2)4xymyx+==+,,消去y得2222(4)22280mxmxm+++−=.·············
·7分设00()Cxy,,则2022824mxm−−=+,所以2022424mxm−=−+,所以0244mym=+.连接OM,取OM的中点R,则2()22mR,,·····························
············10分连接CR,因为OCCM=,所以CROM⊥.又3020422242322OMCRmymmmkkmx−−===−−,,所以324124232mmmm−=−−,即42280mm+−=,因为0m,所以2m=,·····················
············································13分所以四边形OBMC的面积2114242222223(2)4ABMAOCSSS=−=−=+△△.··········
···························································································16分19.(本小
题满分16分)解:(1)因为2()2lnfxxaxx=−+,所以222()(0)xaxfxxx−+=.···············2分令2()22pxxax=−+,216a=−,当0≤即44a−≤≤时,()
0px≥,即()0fx≥,所以函数()fx单调递增区间为(0)+,.当0即4a−或4a时,2212161644aaaaxx−−+−==,.若4a−,则120xx,所以()0px,即()0fx,所以函数()fx的单调递增区间
为(0)+,.若4a,则210xx,由()0fx即()0px,得10xx或2xx;由()0fx,即()0px得12xxx.所以函数()fx的单调递增区间为12(0)()xx+,,,;单调
递减区间为12()xx,.综上,当4a≤时,函数()fx的单调递增区间为(0)+,,无减区间;当4a时,函数()fx的单调递增区间为12(0)()xx+,,,,单调递减区间为12()xx,.·····6分(2)由(1)得22
2()(0)xaxfxxx−+=,若()fx有两个极值点12xx,,则12xx,是方程2220xax−+=的两个不等正实根,由(1)知4a.则1212212axxxx+==,,故1201xx,···············
·····8分要使12()fxmx恒成立,只需12()fxmx恒成立.因为222311111111111221()2ln222ln22ln1fxxaxxxxxxxxxxxx−+−−+===−−+,········10分令3()22ln(01)h
tttttt=−−+,则2()32lnhttt=−+,··························12分当01t时,()0ht,()ht为减函数,所以()(1)3hth=−.·················
·14分由题意,要使12()fxmx恒成立,只需满足3m−≤.所以实数m的取值范围(3]−−,.·······················································16分20.(本小题满分16分)解:(1)由32nnS=+,可
知1123nnnnaSS++=−=,故1320nnnaS+−=−对一切正整数n都成立,故{}na是P数列.················3分(2)由题意知,该数列的前n项和为(1)2nnnSnd−=−+,11na
nd+=−+,由数列12310aaaa,,,,是P数列,可知211aSa=,故公差0d.213(1)1022nndSandn+−=−++对满足19n≤≤中的每一个正整数n都成立,即23(1)1022dndn−++
对于19n≤≤都成立.·······································6分由2231(1)1022399(1)1022dddd−++−++,,可得8027d,故d的取值范围是8(0)27,.·····8分(3)若{}na是P数列
,则12aSaaq==,若0a,则1q,又由1nnaS+对一切正整数n都成立,可知11nnqaqaq−−,即12()nqq−对一切正整数n都成立,由1()0nq,1()(01)nq,,故20q−≤,可得2q≥.若0a,则1q,
又由1nnaS+对一切正整数n都成立,可知11nnqaqaq−−,即(2)1nqq−对一切正整数n都成立,又当(1]q−−,时,(2)1nqq−当2n=时不成立,故有(01)(2)1qqq−,,,或2(10)(2)1qqq−−,,,解得15(0)(01)2q−
,,.所以{}na是P数列时,a与q所满足的条件为02aq,≥,或015(01)(0)2aq−,,,.············································
·························································12分下面用反证法证明命题“若0a且12TT=,则{}na不是P数列”.假设{}na是P数列,由0a,可知2q
≥且{}na中每一项均为正数,若{}nb中的每一项都在{}nc中,则由这两数列是不同数列,可知12TT,若{}nc中的每一项都在{}nb中,同理可得12TT.若{}nb中至少有一项不在{}nc中且{}nc中至少有一项不在{}
nb中,设{}{}nnbc,是将{}{}nnbc,中的公共项去掉之后剩余项依次构成的数列,它们的所有项和分别为12TT,,不妨设{},{}nnbc中的最大项在{}nb中,设为ma,则2m≥,则21211mmTaaaaT−+++≤≤,故21TT,所以21TT,
故总有12TT,与12TT=矛盾.故{}na不是P数列.·································16分数学Ⅱ(附加题)21.【选做题】本题包括A、B、C三小题,请选定其中两题......,若多做,则按作答的前两题评
分.A.选修4−2:矩阵与变换(本小题满分10分)解:依题意12345x=2yy−,即102320xyxy+=−+=,,解得48xy=−=,,····················3分由逆矩阵公式知,矩阵M1234=
的逆矩阵1213122−−=−M,···················7分所以1xy−M213122−=−48−1610=−.·······
········································10分B.选修4−4:坐标系与参数方程(本小题满分10分)解:直线22(sincos)222l−=:,所以直线l的直角坐标方程为20xy−+=.············
·································3分曲线C的普通方程为22(2)1(32)xyx++=−−≤≤,·································6分2220(2)1
(32)xyxyx−+=++=−,≤≤-,消去y整理得22870xx++=,则222x=−−,所以交点坐标为22(2)22−−−,.·································10分C.
选修4−5:不等式选讲(本小题满分10分)解:由00xy,,2211274xyxy+++=,得2215316127444xyxyxy−=+++−22338811273273312688444xyxxyy+−=+−=≥.···············
··················6分当且仅当22818xxyy==,,即122xy==,时等号成立.故1534xy−的最小值为6.·····················································
··············10分【必做题】第22题、第23题,每小题10分,共计20分.请在答题卡指定区域.......内作答,解答时应写出文字说明、证明过程或演算步骤.22.(本小题满分10分)解:设O是AD中点,PAD△为正三角形,则POAD⊥.因为平面PAD⊥平面ABCD,平面PAD平面A
BCDAD=,PO平面PAD,所以POABCD⊥面.又因为2ADAE==,60DAB=,所以ADE△为正三角形,所以OEAD⊥.建立如图所示空间直角坐标系Oxyz−,则(003)(030)(230)(100)PE
CD−−,,,,,,,,,,,,于是(233)(033)(103)PCPEDP=−−=−=,,,,,,,,.···················2分(1)设平面PEC的法向量为1()xyz=,,n,由110,0PCPE==nn,得一个法向量为1
(011)=,,n,平面EDC的一个法向量为2(001)=,,n,所以1212cos22==,nn,又由图可得二面角PECD−−为锐角,所以二面角PECD−−的余弦值为22.····································
············4分(2)设(01)PMPC=≤≤,则(233)PM=−−,,,(12333)DMDPPM=+=−−,,,(033)PE=−,,,················6分所以2|63|6|cos|||8||||6
10104DMPEDMPEDMPE−===−+,,·················8分解得13=或23,所以存在点M为线段PC的三等分点.·······················
····10分23.(本小题满分10分)ADCBPEOxyz解:(1)当2n=时,{0}{1}{2}{02}{012}M=,,,,,,,具有性质P,对应的k分别为01211,,,,,故(2)5f=.··························
····················3分(2)设当nt=时,具有性质P的集合M的个数为()ft,则当1nt=+时,(1)()(1)ftftgt+=++,其中(1)gt+表示1tM+时也具有性质P的集合M的个数,下面计算(1)gt+关于t的表达式,此时应有21kt+
≥,即12tk+≥,故对nt=分奇偶讨论.①当t为偶数时,1t+为奇数,故应该有22tk+≥,则对每一个k,1t+和21kt−−必然属于集合M,且t和2kt−,,k和k共有1tk+−组数,每一组数中的两个数必然同时属于或不属于集合M,故对每一个k,对应具有性质P的集合M的个数为01111112t
ktktktktkCCC+−+−+−+−+−+++=,所以21222(1)2221221tttgt−+=++++=−.·········································5分
②当t为奇数时,1t+为偶数,故应该有12tk+≥,同理111222(1)22212221tttgt+−+=++++=−,····································7分综上,可得22()
221(1)()2221ttfttftftt+−+=+−,为偶数,,为奇数,又(2)5f=,由累加法解得212625()425ttttfttt+−−=−−,为偶数,,为奇数,即212625()425nnnnfnnn+
−−=−−,为偶数,,为奇数.·······················································10分