苏州市2018届高三上学期期中考试数学试题

苏州市2018届高三第一学期期中调研试卷数 学一、填空题(本大题共14小题，每小题5分，共70分，请把答案直接填写在答卷纸．．．相应的位置) 1．已知集合{1,2,3,4,5},{1,3},{2,3}U A B ===，则()U A B = ▲ ． 2．函数1ln(1)y x =-的定义域为 ▲ ．3．设命题:4p x >；命题2:540q x x -+≥，那么p 是q 的 ▲ 条件（选填“充分不必要”、“必要不充分”、“充要”、“既不充分也不必要”）．4．已知幂函数22*()m m y x m -=∈N 在(0,)+∞是增函数，则实数m 的值是 ▲ ． 5．已知曲线3()ln f x ax x =+在(1,(1))f 处的切线的斜率为2，则实数a 的值是 ▲ ． 6．已知等比数列{}n a 中，32a =，4616a a =，则7935a a a a -=- ▲ ．7．函数sin(2)(0)2y x ϕϕπ=+<<图象的一条对称轴是12x π=，则ϕ的值是 ▲ ． 8．已知奇函数()f x 在(,0)-∞上单调递减，且(2)0f =，则不等式()01f x x >-的解集为 ▲ ．9．已知tan()24απ-=，则cos2α的值是 ▲ ．10．若函数8,2()log 5,2ax x f x x x -+⎧=⎨+>⎩≤(01)a a >≠且的值域为[6,)+∞，则实数a 的取值范围是 ▲ ． 11．已知数列{},{}n n a b 满足1111,1,(*)21n n n n a a b b n a +=+==∈+N ，则122017b b b ⋅⋅= ▲ ．12．设ABC △的内角,,A B C 的对边分别是,,a b c ，D 为AB 的中点，若cos sin b a C c A =+且2CD =则ABC △面积的最大值是 ▲ ．13．已知函数()sin()6f x x π=-，若对任意的实数5[,]62αππ∈--，都存在唯一的实数[0,]m β∈，使()()0f f αβ+=，则实数m 的最小值是 ▲ ．14．已知函数ln ,0()21,0x x f x x x >⎧=⎨+⎩≤，若直线y ax =与()y f x =交于三个不同的点(,()),(,()),A m f m B n f n(,())C t f t （其中m n t <<），则12n m++的取值范围是 ▲ ． 二、解答题(本大题共6个小题，共90分，请在答题卷区域内作答，解答时应写出文字说明、证明过程或演算步骤) 15．(本题满分14分)已知函数21()sin(2)(0,0)242f x ax b a b π=-+++>>的图象与x 轴相切，且图象上相邻两个最高点之间的距离为2π． （1）求,a b 的值；（2）求()f x 在[0,]4π上的最大值和最小值．16．(本题满分14分)在中，角A ,B ,C 所对的边分别是a ,b ,c ，已知sin sin sin ()B C m A m +=∈R ，且240a bc -=．（1）当52,4a m ==时，求,b c 的值； （2）若角A 为锐角，求m 的取值范围．17．(本题满分15分)已知数列{}n a 的前n 项和是n S ，且满足11a =，*131()n n S S n +=+∈N ． （1）求数列{}n a 的通项公式； （2）在数列{}n b 中，13b =，*11()n n n na b b n a ++-=∈N ，若不等式2n n a b n λ+≤对*n ∈N 有解，求实数λ的取值范围．18．(本题满分15分)如图所示的自动通风设施．该设施的下部ABCD 是等腰梯形，其中AB 为2米，梯形的高为1米，CD 为3米，上部CmD 是个半圆，固定点E 为CD 的中点．MN 是由电脑控制可以上下滑动的伸缩横杆（横杆面积可忽略不计），且滑动过程中始终保持和CD 平行．当MN 位于CD 下方和上方时，通风窗的形状均为矩形MNGH （阴影部分均不通风）． （1）设MN 与AB 之间的距离为5(02x x <≤且1)x ≠米，试将通风窗的通风面积S （平方米）表示成关于x 的函数()y S x =；（2）当MN 与AB 之间的距离为多少米时，通风窗的通风面积S 取得最大值？19．(本题满分16分)已知函数2()ln ,()f x x g x x x m ==--． （1）求过点(0,1)P -的()f x 的切线方程；（2）当0=m 时，求函数()()()F x f x g x =-在],0(a 的最大值；（3）证明：当3m ≥-时，不等式2()()(2)e x f x g x x x +<--对任意1[,1]2x ∈均成立（其中e 为自然对数的底数,e 2.718...=）．20．(本题满分16分)已知数列{}n a 各项均为正数，11a =,22a =，且312n n n n a a a a +++=对任意*n ∈N 恒成立，记{}n a 的前n 项和为n S ．（1）若33a =，求5a 的值；（2）证明：对任意正实数p ，221{}n n a pa -+成等比数列；（3）是否存在正实数t ，使得数列{}n S t +为等比数列．若存在，求出此时n a 和n S 的表达式；若不存在，说明理由．2017—2018学年第一学期高三期中调研试卷数学(附加题部分)21．【选做题】本题包括A 、B 、C 、D 四小题，请选定其中两题，并在相应的答题区域内作答．．．．．．．．．．．．．．．．．．．．若多做，则按作答的前两题评分．解答时应写出文字说明、证明过程或演算步骤． A ．(几何证明选讲)（本小题满分10分）如图，AB 为圆O 的直径，C 在圆O 上，CF AB ⊥于F ，点D 为线段CF 上任意一点，延长AD 交圆O 于E ，030AEC ∠=．（1）求证：AF FO =；（2）若3CF =AD AE ⋅的值．B ．(矩阵与变换)（本小题满分10分）已知矩阵1221⎡⎤=⎢⎥⎣⎦A ，42α⎡⎤=⎢⎥⎣⎦，求49αA 的值．C ．(极坐标与参数方程)（本小题满分10分）在平面直角坐标系中，直线l 的参数方程为42525x t y t ⎧=-+⎪⎪⎨⎪=⎪⎩(t 为参数)，以原点O 为极点，x 轴正半轴为极轴建立极坐标系，圆C 的极坐标方程为2cos()(0)4a a ρθπ-≠． （1）求直线l 和圆C 的直角坐标方程；（2）若圆C 任意一条直径的两个端点到直线l 5a 的值．D ．(不等式选讲)（本小题满分10分）设,x y 均为正数，且x y >，求证：2212232x y x xy y ++-+≥．D EFOBC【必做题】第22、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)在小明的婚礼上，为了活跃气氛，主持人邀请10位客人做一个游戏．第一轮游戏中，主持人将标有数字1,2,…,10的十张相同的卡片放入一个不透明箱子中，让客人依次去摸，摸到数字6,7,…,10的客人留下，其余的淘汰，第二轮放入1,2,…,5五张卡片，让留下的客人依次去摸，摸到数字3,4,5的客人留下，第三轮放入1,2,3三张卡片，让留下的客人依次去摸，摸到数字2,3的客人留下，同样第四轮淘汰一位，最后留下的客人获得小明准备的礼物．已知客人甲参加了该游戏． （1）求甲拿到礼物的概率；（2）设ξ表示甲参加游戏的轮数．．，求ξ的概率分布和数学期望()E ξ．23．(本小题满分10分)（1）若不等式(1)ln(1)x x ax ++≥对任意[0,)x ∈+∞恒成立，求实数a 的取值范围； （2）设*n ∈N ，试比较111231n ++++与ln(1)n +的大小，并证明你的结论．2017—2018学年第一学期高三期中调研试卷数 学 参 考 答 案一、填空题(本大题共14小题，每小题5分，共70分)1．{1} 2．(1,2)(2,)+∞ 3．充分不必要 4．1 5．136．4 7．3π 8．(2,0)(1,2)- 9．45- 10．(1,2]11．120181221 13．2π 14．1(1,e )e +二、解答题(本大题共6个小题，共90分) 15．(本题满分14分)解：（1）∵()f x 图象上相邻两个最高点之间的距离为2π， ∴()f x 的周期为2π，∴202||2a a ππ=>且，······································································2分 ∴2a =，··················································································································4分此时21())242f x x b π=+++， 又∵()f x 的图象与x 轴相切，∴12||022b b +=>，·······················································6分 ∴212b =-；··········································································································8分 （2）由（1）可得22())4f x x π=+， ∵[0,]4x π∈，∴4[,]444x ππ5π+∈，∴当444x π5π+=，即4x π=时，()f x 有最大值为212；·················································11分当442x ππ+=，即16x π=时，()f x 有最小值为0．························································14分 16．(本题满分14分)解：由题意得b c ma +=，240a bc -=．···············································································2分（1）当时，，解得212b c =⎧⎪⎨=⎪⎩或122b c ⎧=⎪⎨⎪=⎩；································································································6分 （2）2222222222()()22cos 23222a ma abc a b c bc a A m a bc bc--+-+--====-，····························8分 ∵A 为锐角，∴2cos 23(0,1)A m =-∈，∴2322m <<，····················································11分又由b c ma +=可得0m >，·························································································13分62m <·····································································································14分 17．(本题满分15分)解：（1）∵*131()n n S S n +=+∈N ，∴*131(,2)n n S S n n -=+∈N ≥，∴*13(,2)n n a a n n +=∈N ≥，·························································································2分 又当1n =时，由2131S S =+得23a =符合213a a =，∴*13()n n a a n +=∈N ，······························3分 ∴数列{}n a 是以1为首项，3为公比的等比数列，通项公式为1*3()n n a n -=∈N ；·····················5分 （2）∵*113()n n n na b b n a ++-==∈N ，∴{}n b 是以3为首项，3为公差的等差数列，····················7分 ∴*33(1)3()n b n n n =+-=∈N ，·····················································································9分 ∴2n n a b n λ+≤，即1233n n n λ-⋅+≤，即2133n n nλ--≤对*n ∈N 有解，··································10分 设2*13()()3n n nf n n --=∈N ， ∵2221(1)3(1)32(41)(1)()333n n nn n n n n n f n f n -+-+---++-=-=， ∴当4n ≥时，(1)()f n f n +<，当4n <时，(1)()f n f n +>，∴(1)(2)(3)(4)(5)(6)f f f f f f <<<>>>，∴max 4[()](4)27f n f ==，···························································································14分 ∴427λ≤．·············································································································15分 18．(本题满分15分)解：（1）当01x <≤时，过A 作AK CD ⊥于K （如上图），则1AK =，122CD AB DK -==，1HM x =-， 由2AK MH DK DH ==，得122HM x DH -==， ∴322HG DH x =-=+，∴2()(1)(2)2S x HM HG x x x x =⋅=-+=--+；·······························································4分 当512x <<时，过E 作ET MN ⊥于T ，连结EN （如下图）， 则1ET x =-，22239(1)(1)224MN TN x x ⎛⎫==---- ⎪⎝⎭∴292(1)4MN x =-- ∴29()2(1)(1)4S x MN ET x x =⋅=---，······································································8分 综上：222,01()952(1)(1)142x x x S x x x x ⎧--+<⎪=⎨---<<⎪⎩≤；·································································9分 （2）当01x <≤时，2219()2()24S x x x x =--+=-++在[0,1)上递减， ∴max ()(0)2S x S ==；································································································11分2︒当512x <<时，2229(1)(1)994()2(1)(1)2424x x S x x x -+--=---⋅=≤，当且仅当29(1)(1)4x x -=--3251(1,)42x =+∈时取“=”， ∴max 9()4S x =，此时max 9()24S x =>，∴()S x 的最大值为94，············································14分 答：当MN 与AB 之间的距离为3214+米时，通风窗的通风面积S 取得最大值．····················15分解：（1）设切点坐标为00(,ln )x x ，则切线方程为0001ln ()y x x x x -=-， 将(0,1)P -代入上式，得0ln 0x =，01x =，∴切线方程为1y x =-；·······························································································2分 （2）当0m =时，2()ln ,(0,)F x x x x x =-+∈+∞， ∴(21)(1)(),(0,)x x F x x x+-'=-∈+∞，············································································3分 当01x <<时，()0F x '>，当1x >时，()0F x '<，∴()F x 在(0,1)递增，在(1,)+∞递减，·············································································5分 ∴当01a <≤时，()F x 的最大值为2()ln F a a a a =-+；当1a >时，()F x 的最大值为(1)0F =；········································································7分 （3）2()()(2)e x f x g x x x +<--可化为(2)e ln x m x x x >-+-，设1()(2)e ln ,[,1]2x h x x x x x =-+-∈，要证3m ≥-时()m h x >对任意1[,1]2x ∈均成立， 只要证max ()3h x <-，下证此结论成立．∵1()(1)(e )x h x x x'=--，∴当112x <<时，10x -<，·······················································8分 设1()e x u x x =-，则21()e 0x u x x '=+>，∴()u x 在1(,1)2递增，又∵()u x 在区间1[,1]2上的图象是一条不间断的曲线，且1()e 202u <，(1)e 10u =->，∴01(,1)2x ∃∈使得0()0u x =，即001e x x =，00ln x x =-，····················································11分 当01(,)2x x ∈时，()0u x <，()0h x '>；当0(,1)x x ∈时，()0u x >，()0h x '<； ∴函数()h x 在01[,]2x 递增，在0[,1]x 递减， ∴0max 00000000012()()(2)e ln (2)212x h x h x x x x x x x x x ==-+-=-⋅-=--，····························14分 ∵212y x x =--在1(,1)2x ∈递增，∴0002()121223h x x x =--<--=-，即max ()3h x <-，∴当3m ≥-时，不等式2()()(2)e x f x g x x x +<--对任意1[,1]2x ∈均成立．··························16分解：（1）∵1423a a a a =，∴46a =，又∵2534a a a a =，∴54392a a ==；·······································2分（2）由3121423n n n n n n n n a a a a a a a a +++++++=⎧⎨=⎩，两式相乘得2134123n n n n n n n a a a a a a a ++++++=，∵0n a >，∴2*42()n n n a a a n ++=∈N ，从而{}n a 的奇数项和偶数项均构成等比数列，···································································4分 设公比分别为12,q q ，则1122222n n n a a q q --==，1121111n n n a a q q ---==，······································5分 又∵312=n n n n a a a a +++，∴42231122a a qa a q ===，即12q q =，···························································6分 设12q q q ==，则2212223()n n n n a pa q a pa ---+=+，且2210n n a pa -+>恒成立，数列221{}n n a pa -+是首项为2p +，公比为q 的等比数列，问题得证；····································8分 （3）法一：在（2）中令1p =，则数列221{}n n a a -+是首项为3，公比为q 的等比数列，∴22212223213 ,1()()()3(1),11k k k k k k k q S a a a a a a q q q---=⎧⎪=++++++=-⎨≠⎪-⎩， 12122132 ,13(1)2,11k k k k k k k q q S S a q q q q ---⎧-=⎪=-=⎨--≠⎪-⎩，·····································································10分 且12341,3,3,33S S S q S q ===+=+，∵数列{}n S t +为等比数列，∴22132324()()(),()()(),S t S t S t S t S t S t ⎧+=++⎪⎨+=++⎪⎩ 即22(3)(1)(3),(3)(3)(33),t t q t q t t q t ⎧+=+++⎪⎨++=+++⎪⎩，即26(1),3,t q t t q +=+⎧⎨=-⎩ 解得14t q =⎧⎨=⎩（3t =-舍去）， (13)分∴224121k k k S =-=-，212121k k S --=-，从而对任意*n ∈N 有21n n S =-，此时2n n S t +=，12n n S t S t-+=+为常数，满足{}n S t +成等比数列， 当2n ≥时，111222n n n n n n a S S ---=-=-=，又11a =，∴1*2()n n a n -=∈N ，综上，存在1t =使数列{}n S t +为等比数列，此时1*2,21()n n n n a S n -==-∈N ．······················16分 法二：由（2）知，则122n n a q -=，121n n a q --=，且12341,3,3,33S S S q S q ===+=+，∵数列{}n S t +为等比数列，∴22132324()()(),()()(),S t S t S t S t S t S t ⎧+=++⎪⎨+=++⎪⎩ 即22(3)(1)(3),(3)(3)(33),t t q t q t t q t ⎧+=+++⎪⎨++=+++⎪⎩，即26(1),3,t q t t q +=+⎧⎨=-⎩ 解得14t q =⎧⎨=⎩（3t =-舍去），·······················································································11分∴121222n n n a q --==，22212n n a --=，从而对任意*n ∈N 有12n n a -=，····································13分 ∴01211222222112n n n n S --=++++==--， 此时2n n S t +=，12n n S t S t-+=+为常数，满足{}n S t +成等比数列， 综上，存在1t =使数列{}n S t +为等比数列，此时1*2,21()n n n n a S n -==-∈N ．······················16分21．【选做题】本题包括A 、B 、C 、D 四小题，请选定其中两题，并在相应的答题区域内作答．．．．．．．．．．．．．．．．．．．．若多做，则按作答的前两题评分．解答时应写出文字说明、证明过程或演算步骤．A ．(几何证明选讲，本小题满分10分)解：（1）证明 ：连接,OC AC ，∵030AEC ∠=，∴0260AOC AEC ∠=∠=，又OA OC =，∴AOC ∆为等边三角形，∵CF AB ⊥，∴CF 为AOC ∆中AO 边上的中线，∴AF FO =；······································································5分（2）解：连接BE ，∵3CF =AOC ∆是等边三角形，∴可求得1AF =，4AB =， D F E O BC∵AB 为圆O 的直径，∴90AEB ∠=，∴AEB AFD ∠=∠，又∵BAE DFA ∠=∠，∴AEB ∆∽AFD ∆，∴AD AF AB AE=， 即414AD AE AB AF ⋅=⋅=⨯=．··················································································10分B ．(矩阵与变换，本小题满分10分)解：矩阵A 的特征多项式为212()2321f λλλλλ--==----， 令()0f λ=，解得矩阵A 的特征值121,3λλ=-=，····························································2分当11λ=-时特征向量为111α⎡⎤=⎢⎥-⎣⎦，当23λ=时特征向量为211α⎡⎤=⎢⎥⎣⎦，·····································6分 又∵12432ααα⎡⎤==+⎢⎥⎣⎦，······························································································8分 ∴5049494911225031331αλαλα⎡⎤-=+=⎢⎥+⎣⎦A ．···········································································10分C ．(极坐标与参数方程，本小题满分10分)解：（1）直线l 的普通方程为220x y +-=；··········································································3分圆C 的直角坐标方程为222()()222a a a x y -+-=；·······························································6分 （2）∵圆C 任意一条直径的两个端点到直线l 5∴圆心C 到直线l 5|2|525a a +-，·······················································8分 解得3a =或13a =-．·······························································································10分D ．(不等式选讲，本小题满分10分)证：∵0,0,0x y x y >>->，∴22211222()2()x y x y x xy y x y +-=-+-+- 232211()()3()3()()x y x y x y x y x y =-+-+-=--≥， ∴2212232x y x xy y ++-+≥．····················································································10分。