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2020级高三9月数学学科模拟考试试题满分150分,考试时间120分钟2022.09.25一、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。1.已知集合301xAxx−=
+,{|ln(3)}Bxyx==−,则如图中阴影部分表示的集合为()A.[1,3]−B.(3,)+C.(,3]−D.[1,3)−2.已知n,m为正整数,且nm,则在下列各式中错误的是()A.
36A120=;B.77712127ACA=;C.111CCCmmmnnn++++=;D.CCmnmnn−=3.已知1.1log0.9a=,1.10.9b=,0.91.1c=,则a,b,c的大小关系为()A.abcB.acbC.bacD.bca4.若等比数
列na的各项均为正数,且12327aaa=,4283aa−=−,则12naaa的最大值为()A.9B.8C.3D.275.函数22()41xxxfx=−的图象大致为()A.B.C.D.6.函数2()1fxxax=−+在区间1,32上有零点,则实数a的取值范围是()A.(2,)
+B.102,3C.52,2D.[2,)+7.下列命题中,错误的命题有()A.五位同学站成一排合影,张三站在最右边,李四、王五相邻,则不同的站法种数为12B.命题“0[0,1]x,2001xx+”的
否定为“[0,1]x,21xx+”C.设函数22,0()2,0xxxfxx+=,则()fx在R上单调递增D.设x,yR,则“xy”是“2()0xyy−”的必要不充分条件8.设函数()fx的定义域为R,(1)fx+为奇函数,(2)fx+为
偶函数,当[1,2]x时,()2xfxab=+.若(0)(3)6ff+=,则()2log96f的值是()A.-12B.-2C.2D.12二、多项选择题:本题共4小题,每小题5分,共20分。在每小题给出的选项中
,有多项符合题目要求的。全部选对的得5分,部分选对的得2分,有选错的得0分.9.若110ab,则下列结论中正确的是()A.22abB.2abbC.||||||abab++D.33ab10.已知函数()fx是定义在R上的奇函数,当0x时,()(1)xfxex=+
,则下列命题正确的是()A.当0x时,()(1)xfxex−=−B.函数()fx有2个零点C.()0fx的解集为(,1)(0,1)−−D.1x,2xR,都有()()122fxfx−11.已知函数()lnln(2)1fxxx=+−
+,则()A.()fx有一个极值点B.()fx没有零点C.直线yx=是曲线()yfx=的切线D.曲线()yfx=关于直线1x=对称12.英国数学家贝叶斯在概率论研究方面成就显著,根据贝叶斯统计理论,随机事件A、B存在如下关系:()(|)(|)()PAPBA
PABPB=.王同学连续两天在某高校的甲、乙两家餐厅就餐,王同学第一天去甲、乙两家餐厅就餐的概率分别为0.4和0.6.如果他第一天去甲餐厅,那么第二天去甲餐厅的概率为0.6;如果第一天去乙餐厅,那么第二天去甲餐厅的概率为0.5,则王同学()A.
第二天去甲餐厅的概率为0.54B.第二天去乙餐厅的概率为0.44C.第二天去了甲餐厅,则第一天去乙餐厅的概率为59D.第二天去了乙餐厅,则第一天去甲餐厅的概率为49三、填空题:本题共4小题,每小题5分,共20分。13.若函数()fx的定义域为[1,1]−,则(
lg)fx的定义域为____________.14.若212nxx−的展开式中二项式系数的和为64,则该展开式中的常数项是____________.15.数据:1,2,2,3,4,5,6,6,7,8,中位数为m,60%百分位数为a,则m=__________,
a=__________.16.已知正实数a,b满足abab=+,则2ab+的最小值为____________.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.(本小题满分10分)已知等差数列na的前n项的和为nS,2320a
S+=,514a=.(1)求na的通项公式;(2)求数列11nnaa+的前n项和nT.并证明16nT.18.(本小题满分12分)某校为了缓解高三学子复习压力,举行“趣味数学”闯关活动,规定每人从10
道题中至少随机抽3道回答,至少答对2题即可闯过第一关,某班有5位同学参加闯关活动,假设每位同学都能答对10道題中的6道题,且每位同学能否闯过第一关相互独立.(1)求B同学闯过第一关的概率;(2)求这5位同学闯过第一关的人数X的分布列和数学期望.19.(本小
题满分12分)已知奇函数21()21xxafx−=+的定义域为[2,]ab−−(1)求实数a,b的值;(2)当[1,2]x时,2()20xmfx++恒成立,求m的取值范围.20.(本小题满分12分)已知数列na的前n项
和为nS,且满足11a=,12nnSna+=,*nN.(1)求数列na的通项公式;(2)设数列nb满足11b=,12nnnbb+=,*nN,按照如下规律构造新数列nc:1a,2b,3a,4b,5a,6b,7a,8b,…,求数列nc的前2n项和.21.(本小题满分12分)为了
检测某种抗病毒疫苗的免疫效果,需要进行动物与人体试验.研究人员将疫苗注射到200只小白鼠体内,一段时间后测量小白鼠的某项指标值,按[0,20),[20,40),[40,60),[60,80)分组,绘制频率分布直方图
如图所示.试验发现小白鼠体内产生抗体的共有160只,其中该项指标值不小于60的有110只.假设小白鼠注射痕苗后是否产生抗体相互独立.(1)填写下面的22列联表,并根据列联表及0.05=的独立性检验,判断能否认为注射
疫苗后小白鼠产生抗体与指标值不小于60有关.单位:只抗体指标值合计小于60不小于60有抗体没有抗体合计(2)为检验疫苗二次接种的免疫抗体性,对第一次注射疫苗后没有产生抗体的40只小白鼠进行第二次注射疫苗,结果又有20只小白鼠产生抗体.(ⅰ)用频率估计概率,求一只小白鼠注射2次疫苗后产生抗体的概
率p;(ⅱ)以(ⅰ)中确定的概率p作为人体注射2次疫苗后产生抗体的概率,进行人体接种试验,记n个人注射2次疫苗后产生抗体的数量为随机变量X.试验后统计数据显示,当99X=时,()PX取最大值,求参加人体接种试验的人数n及()EX.参考公式:22()()()()()nadbcabcd
acbd−=++++(其中nabcd=+++为样本容量)参考数据:()20Pk0.500.400.250.150.1000.0500.0250k0.4550.7081.3232.0722.7063.8415.02422.(本小题满分12分)已知函数()(2)ln(2)fxxx=+
+,2g()(3)2(1)xxaxa=+−+−(aR).(1)求函数()fx的极值;(2)若不等式()g()fxx在(2,)x−+上恒成立,求的取值范围;(3)证明不等式:1323111111114444ne++++(*nN).2
020级高三9月数学学科模拟考试试题答案及评分标准1.C2.C3.A4.D5.A6.B7.C8.B9.AC10.ACD11.AD12.AC13.[10,100]14.423−15.4016.1017.解:(1)设na的公差为d,由题意得11
4420414adad+=+=,解得123ad==.所以1(1)31naandn=+−=−.·················································
··········································5分(2)令11nnnbaa+=,则1111(31)(32)33132nbnnnn==−−+−+所以1231111111111132558313232326nnTbbbbnnn
=++++=−+−++−=−−++·······10分18.解:(1)B同学闯过第一关的情况有答对2题和答对3题,故B同学闯过第一关的概率321664310CCC2C3P+==·····················
····················································································2分(2)由题意可知X的所有可能取
值为0,1,2,3,4,5,且X服从二项分布,即2~5,3XB.·····3分511(0)3243PX===,4152110(1)C33243PX===,23252140(2)C33243
PX===,32352180(3)C33243PX===,4452180(4)C33243PX===,5232(5)3243PX===.················
····································9分故X的分布列为X012345P12431024340243802438024332243··························································
·········································································11分所以1104080803210()0123452432432432432432433EX=+++++=或210()533E
X==.·····································································································12分(1)因为函数f(x)-4-2*
-1是奇函数,所以f(-x)=-f(x),19.解:(1)因为函数21()21xxafx−=+是奇函数,所以()()fxfx−=−,即21212121xxxxaa−−−−=−++,即2212121xxxxaa−−+=++,即221xxaa
−=−+,整理得()(1)210xa−+=,所以10a−=,即1a=,·····························································4分则23a−−=−,因为定义域为[2,]ab−−关于原点对称,所
以3b=;·······································6分(2)因为[1,2]x,所以21()021xxfx−=+,又当[1,2]x时,2()20xmfx++恒
成立,所以()()222121xxxm++−−,[1,2]x时恒成立,令21xt=−,则2(3)(2)5665ttttmtttt++++−==++,[1,3]t时恒成立,而66525265tttt+++=+,当且仅当6tt=,即6t=时,等号成立,所以265m−
−,即m的取值范围是(265,)−−+.·······················································12分20..解:(1)当1n=时,由11a=且12nnSna+=得22a=······························
···························1分当2n时,由12(1)nnSna−=−得12(1)nnnanana+=−−,所以11nnaann+=+(2n).·················2分所以212naan==,nan=(2n)
,··················································································3分又当1n=时,11a=,适合上式.··························
·······························································4分所以.nan=.*Nn········································································
··································5分(2)因为12nnnbb+=,1122nnnbb+++=,所以22nnbb+=(*nN),···········································6分又122bb=,
所以22b=.····································································································7分所以数列nb的偶数项构成
以22b=为首项、2为公比的等比数列.·············································8分故数列nc的前2n项的和()()21321242nnnTaaabbb−=+++++++,()122212
(121)22212nnnnnTn+−+−=+=+−−······································································11分所以数列nc的前2n项和为1222nn++−.·······
···································································12分21.(1)由频率分布直方图,知200只小白鼠按指标值分布为:在[0,20
)内有0.00252020010=(只);在[20,40)内有0.006252020025=(只);在[40,60)内有0.008752020035=(只);在[60,80)内有0.02520200100=(只);在[80,100]内有0.00752020030=(只
).······································································1分由题意,有抗体且指标值小于60的有50只:而指标值小于60的小白鼠共有10253570++=只,所以指
标值小于60且没有抗体的小白鼠有20只,同理,指标值不小于60且没有抗体的小白鼠有20只,故列联表如下:抗体指标值合计小于60不小于60有抗体50110160没有抗体202040合计70130200···············································
······················································································3分零假设为0H:注射疫苗后小白鼠产生抗
体与指标值不小60无关联.根据列联表中数据,得220.05200(502020110)4.9453.8411604070130x−==.···························5分根据0.05=的独立性检验,推断0H不成立,即认为注射疫苗后小白鼠
产生抗体与指标值不小于60有关,此推断犯错误的概率不大于0.05.··························································································6分(2)(ⅰ)令事
件A=“小白鼠第一次注射疫苗产生抗体”,事件B=“小白鼠第二次注射疫苗产生抗体”,事件C=“小白鼠注射2次疫苗后产生抗体”.记事件A,B,C发生的概率分别为()PA,()PB,()PC,则160()0.8200PA==,20()0.540PB==,()1()()10.20.50.9P
CPAPB=−=−=.所以一只小白鼠注射2次疫苗后产生抗体的概率为0.9.·····························································8分(ⅱ)由题意,知随机变量~(,0.9)XBn,()C0.90.1kknknPXk−==(0
k=,1,2,…,n).因为(99)PX=最大,所以999999989898999999100100100C0.90.1C0.90.1C0.90.1C0.90.1nnnnnnnn−−−−,解得11091109n,因n是整数,故109n=或11
0n=,所以接受接种试验的人数为109或110.···············································································10分
①当接种人数为109时,()1090.998.1EXnp===;②当接种人数为110时,()1100.999EXnp===.···························································1
2分22.(1)解:∵()(2)ln(2)fxxx=++(2)x−),()ln(2)1fxx=++,由()0fx可得12,ex−+,此时()fx是增函数,由()0fx可得1,2ex−−,此时()fx是减函数,·················
·······································2分所以当12ex=−时()fx有极小值,极小值为1e−,无极大值····································
················3分(2)解:由不等式()g()fxx在(2,)x−+上恒成立得2(2)ln(2)(3)2(1)xxxaxa+++−+−即(2)ln(2)(2)(1)xxxxa++++−,因为(2,)x−+,所以1ln(2)axx+−+在(2,)x−+上
恒成立··························································································································
···········5分设()1ln(2)hxxx=+−+,(2,)x−+,由1()02xhxx+==+得1x=−,所以()hx在(2,1)−−上递减,在(1,)−+上递增,所以min()(1)0hxh=−=即0a,所以a的取值范围为(,0]−···················
············································································7分(3)证明:由(2)得1ln(2)xx++在
(1,)−+上恒成立,令114nx=−,则有11ln144nn+,·············································································
···8分所以2211111111ln1ln1ln1144444434nnn+++++++++=−即211111ln111144434nn+++−·························
············································10分因为*nN,所以111114343n−即21111ln1114443n+++,所以13231
11111114444ne++++.····································································12分获得更多资源请扫码加入
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