2014年普通高等学校招生全国统一考试（四川卷)数学（文科）第Ⅰ卷（共50分)一、选择题：本大题共10小题,每小题5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）【2014年四川卷，文1，5分】已知集合{|(1)(2)0}A x x x =+-≤，集合B 为整数集,则A B =（ ）（A ）{1,0}- (B ）{0,1} (C){2,1,0,1}-- （D ）{1,0,1,2}- 【答案】D【解析】由已知得{}12A x x =-，又集合B 为整数集，所以{}1,0,1,2A B =-,故选D ．（2)【2014年四川卷,文2，5分】在“世界读书日”前夕，为了了解某地5000名居民某天的阅读时间，从中抽取了200名居民的阅读时间进行统计分析．在这个问题中，5000名居民的阅读时间的全体是（ ） (A ）总体 （B ）个体 （C ）样本的容 （D ）从总体中抽取的一个样本 【答案】A【解析】由题目条件知，5000名居民的阅读时间的全体是总体；其中1名居民的阅读时间是个体;从5000名居民某天的阅读时间中抽取的200名居民的阅读时间是从总体中抽取的一个样本，样本容量是200,故选A ．(3）【2014年四川卷,文3，5分】为了得到函数sin(1)y x =+的图象，只需把函数sin y x =的图象上所有的点（ )（A ）向左平行移动1个单位长度 （B ）向右平行移动1个单位长度（C ）向左平行移动π个单位长度（D ）向右平行移动π个单位长度 【答案】A【解析】根据平移法则“左加右减"可知，将函数sin y x =的图像上所有的点向左平移移动1个单位长度即可得到函数()sin 1y x =+的图像，故选A ．(4)【2014年四川卷，文4，5分】某三棱锥的侧视图、俯视图如图所示，则该三棱锥的体积是( ）（锥体体积公式：13V Sh =，其中S 为底面面积,h 为高)(A ）3 （B ）2 （C ）3 （D)1【答案】D【解析】由俯视图可知,三棱锥底面是边长为2的等边三角形．由侧视图可知，三棱锥的高为3．故该三棱锥的体积11233132V =⨯⨯⨯⨯=，故选D ．（5)【2014年四川卷，文5，5分】若0a b >>，0c d <<,则一定有（ )(A)a b d c > （B ）a b d c < （C ）a b c d > （D ）a bc d<【答案】B【解析】因为0c d <<,所以110c d >>,两边同乘1-,得110d c ->->，又0a b >>，故由不等式的性质可知0a b d c ->->，两边同乘1-，得a bd c<，故选B ．(6)【2014年四川卷,文6，5分】执行如图的程序框图，如果输入的,x y R ∈,那么输出的S 的最大值为( ) (A ）0 （B ）1 （C ）2 （D ）3【答案】C【解析】由程序框图可知，若输入的x ，y 满足约束条件001x y x y ⎧⎪⎨⎪+⎩，则输出目标函数2S x y =+的值，否则，输出1S =．如图，作出满足条件的可行域．当1x =，0y =时，目标 侧视图俯视图112222111y函数2S x y =+取得最大值2,21>,故输出的S 的最大值为2，故选C ．（7）【2014年四川卷,文7，5分】已知0b >，5log b a =，lg b c =,510d =，则下列等式一定成立的是（ )（A)d ac = （B ）a cd = （C ）c ad = (D)d a c =+ 【答案】B【解析】5log b a =，0b >，故由换底公式得lg lg5ba =,所以lg lg5b a =．因为lg b c =，所以lg5a c =，因为510d =， 所以5log 10d =，即1lg5d =,将其代入lg5a c =中得ac d=，即a cd =，故选B ．（8）【2014年四川卷，文8，5分】如图,从气球A 上测得正前方的河流的两岸B ，C 的俯角分别为75，30，此时气球的高是60cm ，则河流的宽度BC 等于( ） （A)240(31)m - （B)180(21)m - （C ）120(31)m - （D ）30(31)m +【答案】C【解析】如图，30ACD ∠=，75ABD ∠=，60AD =m ,在Rt ACD △中,60603tan tan30AD CD=ACD ==∠m ,在Rt ABD △中，()60606023tan tan 7523AD BD =ABD ===-∠+m ，所以()()603602312031BC CD BD =-=--=-m ,故选C ．（9）【2014年四川卷，文9，5分】设m R ∈，过定点A 的动直线0x my +=和过定点B 的动直线30mx y m --+=交于点(,)P x y ，则||||PA PB +的取值范围是（ ) （A ）[5,25] （B)[10,25] （C ）[10,45] （D ）[25,45] 【答案】B 【解析】直线0x my +=过定点()0,0A ，直线30mx y m --+=过定点()1,3B ． ①当0m =时，过定点A 的直线方程为0x =，过定点B 的直线方程为3y =，两条 直线互相垂直，此时()0,3P ,所以4PA PB +=．②当0m ≠时，直线0x my +=的斜率为1m -，直线30mx y m --+=的斜率为m ，因为11m m -⨯=-,所以两条直线互相垂直，即点P 可视为以AB 为直径的圆上 的点．当点P 与点A 或点B 重合时，PA PB +有最小值10．当点P 不与点A ，点B 重合时,PAB△为直角三角形，且22210PA PB AB +==．由不等式性质知222252PA PBPA PB++=，所以10,25PA PB ⎡⎤+∈⎣⎦．综合①②得10,25PA PB ⎡⎤+∈⎣⎦，故选B ．(10)【2014年四川卷，文10，5分】已知F 为抛物线2y x =的焦点，点A ,B 在该抛物线上且位于x 轴的两侧，2OA OB ⋅=(其中O 为坐标原点）,则ABO ∆与AFO ∆面积之和的最小值是( ）(A ）2 (B ）3 （C ）1728(D)10【答案】B【解析】如图所示，设()11,A x y ，()22,B x y ，则12122x x y y +=（*)．不妨设A 点在第一象限,则10y >，20y <．设直线AB ：x my n =+，代入2y x =中，得20y my n --=，则12y y n =-，代入（*)式，有220n n --=，解得2n =或1n =-(舍)，故直线AB 过定点()2,0,所以ABO AFO S S +=△△1211112224y y y ⨯⨯-+⨯⨯1298y y =-()12923382ny y -==≥，故选B ． 第II 卷（共100分）30°75°60mCBA D CBA 75°30°60 mmx-y-m+3=0x+my=0yx 213-1-2-1321PBA二、填空题：本大题共5小题,每小题5分(11）【2014年四川卷，文11,5分】双曲线2214x y -=的离心率等于 ．【解析】由双曲线方程2214x y -=知24a =，21b =，2225c a b =+=，所以c e a == （12）【2014年四川卷，文12，5分】复数22i1i-=+ ．【答案】2i -【解析】()()()()()2222i 1i 22i 1i 12i i 2i 1i 1i 1i ---==-=-+=-++-．（13）【2014年四川卷，文13，5分】设()f x 是定义在R 上的周期为2的函数，当[1,1)x ∈-时,242,10,(),01,x x f x x x ⎧-+-≤<=⎨≤<⎩，则3()2f =________．【答案】1【解析】()f x 是定义域在R 上的圆周期为2的函数，且()2421001x x f x x x ⎧-+-<=⎨<⎩,所以231142121222f f ⎛⎫⎛⎫⎛⎫=-=-⨯-+=-+= ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭．（14)【2014年四川卷,文14,5分】平面向量(1,2)a =，(4,2)b =，c ma b =+（m R ∈），且c 与a 的夹角等于c 与b 的夹角，则m =_______．【答案】2【解析】()1,2=a ,()4,2=b ，则()4,22m m m +=++c =a b，a =，b =58m ⋅=+a c ，820m ⋅=+b c ．因为c 与a 的夹角等于c 与b 的夹角，所以⋅⋅=⋅⋅a c b c a c b c,解得2m =． （15）【2014年四川卷,文15，5分】以A 表示值域为R 的函数组成的集合，B 表示具有如下性质的函数()x ϕ组成的集合:对于函数()x ϕ，存在一个正数M ,使得函数()x ϕ的值域包含于区间[,]M M -．例如，当31()x x ϕ=，2()sin x x ϕ=时，1()x A ϕ∈，2()x B ϕ∈．现有如下命题：①设函数()f x 的定义域为D ，则“()f x A ∈”的充要条件是“b R ∀∈，x R ∃∈，()f a b =”； ②若函数()f x B ∈，则()f x 有最大值和最小值;③若函数()f x ，()g x 的定义域相同，且()f x A ∈，()g x B ∈，则()()f x g x B +∉;④若函数2()ln(2)1xf x a x x =+++（2x >-，a R ∈）有最大值，则()f x B ∈．其中的真命题有____________．（写出所有真命题的序号）． 【答案】①③④【解析】对于①,()f x A ∈⇔()f x 的值域为R ⇔b ∀∈R ，a D ∃∈，()f a b =，故①正确;对于②，当()1f x x=，1x >时，()1f x <,即()()[]1,00,11,1-⊆-，但()f x 无最值,故②不正确; 对于③,因为x D ∀∈，()g x M ≤，所以总存在0x D ∈,使得()()00f x g x +趋近于无穷大，即()()f x g x B +∉，故③正确；对于④，令2()1x g x x =+,则()()()2222222121'11x x x g x x x +--==++()()()22111x x x +-=+,令()'0g x >，解得11x -<<，故()g x 在()1,1-上单调递增,且()112g =，()112g -=-，又()g x 在()1,+∞上单调递减，1x >时,()0g x >，又()g x 为奇函数，故()12g x ≤．而()ln(2)h x a x =+，当2x >-时,若0a ≠，则()h x A ∈由③知，()()h x g x B +∉，即()f x 无最大值， 所以0a =时，()f x 有最大值,此时()2()1xf xg x B x ==∈+，故④正确．综上：真命题的有①③④．三、解答题：本大题共6题，共75分．(16)【2014年四川卷，文16，12分】一个盒子里装有三张卡片，分别标记有数字1,2，3，这三张卡片除标记的数字外完全相同．随机有放回地抽取3次，每次抽取1张,将抽取的卡片上的数字依次记为a ,b ,c ． （1）求“抽取的卡片上的数字满足a b c +=”的概率；（2）求“抽取的卡片上的数字a ，b ，c 不完全相同"的概率． 解:（1）由题意知，(),,a b c 所有可能的结果为()1,1,1，()1,1,2，()1,1,3，()1,2,1，()1,2,2，()1,2,3，()1,3,1，()1,3,2，()1,3,3,()2,1,1，()2,1,2,()2,1,3,()2,2,1,()2,2,2,()2,2,3，()2,3,1，()2,3,2，()2,3,3，()3,1,1，()3,1,2,()3,1,3，()3,2,1，()3,2,2,()3,2,3，()3,3,1，()3,3,2，()3,3,3，共27种．设“抽取的卡片上的数字满足a b c +=”为事件A ，则事件A 包括()1,1,2，()1,2,3，()2,1,3，共3种．所以()31279P A ==．因此，“抽取的卡片上的数字满足a b c +=”的概率为19．（2)设“抽取的卡片上的数字a ，b ，c 不完全相同"为事件B ，则事件B 包括()1,1,1，()2,2,2，()3,3,3,共3种．所以()()3811279P B P B =-=-=．因此“抽取的卡片上的数字a ，b ，c 不完全相同"的概率为89．（17）【2014年四川卷，文17，12分】已知函数()sin(3)4f x x π=+．（1）求()f x 的单调递增区间;(2）若α是第二象限角，4()cos()cos2354f απαα=+，求cos sin αα-的值．解：（1）因为函数sin y x =的单调递增区间为ππ2π,2π22k k ⎡⎤-++⎢⎥⎣⎦,k ∈Z ．由πππ2π32π242k x k -+++，k ∈Z ,得π2ππ2π43123k k x -++,k ∈Z ．所以函数()f x 的单调递增区间为π2ππ2π,43123k k ⎡⎤-++⎢⎥⎣⎦，k ∈Z ． （2）由已知,有()22π4πsin cos cos sin 454αααα⎛⎫⎛⎫+=+- ⎪ ⎪⎝⎭⎝⎭,所以()22ππ4ππsin cos cos sin cos cos sin sin cos sin 44544αααααα⎛⎫+=-- ⎪⎝⎭,即()()2ππ4sin cos cos sin cos sin sin cos 445αααααα+=-+．当sin cos 0αα+=时，由α是第二象限角，知3π2π4k α=+,k ∈Z ．此时cos sin αα-=当sin cos 0αα+≠时，有()25cos sin 4αα-=．由α是第二象限角,知cos sin 0αα-<，此时cos sin αα-=．综上所述，cos sin αα-=（18）【2014年四川卷,文18,12分】在如图所示的多面体中，四边形11ABB A 和11ACC A 都为矩形． (1）若AC BC ⊥，证明：直线BC ⊥平面11ACC A ；(2)设D ，E 分别是线段BC ，1CC 的中点，在线段AB 上是否存在一点M ，使直线//DE平面1A MC ？请证明你的结论．解:（1)因为四边形11ABB A 和11ACC A 都是矩形，所以1AA AB ⊥，1AA AC ⊥．因为AB ，AC 为平面ABC 内两1A 1条相交直线，所以1AA ⊥平面ABC ．因为直线BC ⊂平面ABC ，所以1AA BC ⊥．又AC BC ⊥，1AA ，AC 为平面11ACC A 内两条相交直线,所以BC ⊥平面11ACC A ．（2）取线段AB 的中点M ,连接1A M ，MC ，1A C ，1AC ，设O 为1A C ,1AC 的交点．由已知可知O 为1AC 的中点．连接MD ，OE ,则MD ,OE 分别为ABC ∆,1ACC ∆的中位线，所以=1//2MD AC ,=1//2OE AC ,因此=//MD OE ．连接OM ,从而四边形MDEO 为平行四边形，则//DE MO ．因为直线DE ⊄平面1A MC ，所以直线//DE 平面1A MC ，即线段AB 上存在一点M (线段AB 的中点），使直线//DE 平面1A MC ．(19）【2014年四川卷，文19，12分】设等差数列{}n a 的公差为d ，点(,)n n a b 在函数()2x f x =的图象上（n N *∈）．(1）证明：数列{}n b 为等差数列；（2）若11a =，函数()f x 的图象在点22(,)a b 处的切线在x 轴上的截距为12ln 2-，求数列2{}n na b 的前n 项和 n S ．解：（1)证明：由已知可知，20na nb =>，当1n 时，1122n na a d n nb b +-+==，所以数列{}n b 是首项为12a ，公比为2d等比数列．（2）函数()2x f x =在()22,a b 处的切线方程为()()22222ln 2a a y x a -=-，它在x 轴上的截距为21ln 2a -． 由题意知，2112ln 2ln 2a -=-，解得22a =．所以211d a a =-=，n a n =，2n n b =，24n n n a b n =⋅． 于是，()231142434144n n n T n n -=⨯+⨯+⨯++-⨯+⨯，()23141424144n n n T n n +=⨯+⨯++-⨯+⨯，因此()1121113444444444439n n nn n n n n T T n n ++++-+--=+++-⨯=-⨯=．所以()113449n n n T +-+=． （20）【2014年四川卷，文20,13分】已知椭圆C ：22221x y a b+=（0a b >>）的左焦点为(2,0)F -，离心率为63．（1)求椭圆C 的标准方程；（2）设O 为坐标原点，T 为直线3x =-上一点，过F 作TF 的垂线交椭圆于P ，Q ．当四边形OPTQ 是平行四边形时，求四边形OPTQ 的面积．解:（1）因为(2,0)F -，所以2c =，又63e =，所以6a =，2222b a c =-=,即椭圆C 的方程为22162x y +=．（2）如图所示，由题意可设直线PQ 的方程为2x my =-．当0m =时，2x =-，此时()3,0T -，P ，Q 关于点F 对称，但DF TF ≠，故四边形OPTQ 不是平行四 边 形，与题意不符，故0m ≠．直线TF ：()2y m x =-+,令3x =-，得y m =， 即()3,T m -，连接OT ,设OTPQ E =，则3,22m E ⎛⎫- ⎪⎝⎭，联立方程222162x my x y =-⎧⎪⎨+=⎪⎩，消去x 整理得()22236my y -+=，即()223420m y my +--=，显然()2216830m m ∆=++>， 令()11,P x y ，()22,Q x y ．则12243m y y m +=+，12223y y m -=+，则1222232E y y m my m +===+，解得21m =． 此时()()221212PQ x x y y =-+-()22121214m y y y y =++-2126=+=，112TF =+=．所以四边形OPTQ 的面积1262232S PQ TF =⨯⨯⨯=⨯=．（21）【2014年四川卷，文21，14分】已知函数2()1x f x e ax bx =---，其中,a b R ∈， 2.71828e =⋅⋅⋅ 为自然对数的底数．（1）设()g x 是函数()f x 的导函数,求函数()g x 在区间[0,1]上的最小值；（2）若(1)0f =，函数()f x 在区间(0,1)内有零点，证明:21e a -<<．解：（1）()2e 1x f x ax bx =---,()()e 2x g x f x ax b '==--．()e 2xg x a '=-．当[]0,1x ∈时，()[]12,e 2g x a a '∈--．当12a 时，()0g x ',所以()g x 在[]0,1上单调递增．因此()g x 在[]0,1上的最小值是()01g b =-；当e 2a 时,()0g x '，所以()g x 在[]0,1上单调递减．因此()g x 在[]0,1上的最小值是()1e 2g a b =--;当1e22a <<时，令()0g x '=，得()()ln 20,1x a =∈．所以函数()g x 在区间()0,ln 2a ⎡⎤⎣⎦上单调递减， 在区间()(ln 2,1a ⎤⎦上单调递增．于是，()g x 在[]0,1上的最小值是()()()ln 222ln 2g a a a ab =--．综上所述，当12a时，()g x 在[]0,1上的最小值是()01g b =-;当1e22a <<时，()g x 在[]0,1上的最小 值是()()()ln 222ln 2g a a a ab =--;当e2a 时,()g x 在[]0,1上的最小值是()1e 2g ab =--．（2)设0x 为()f x 在区间()0,1内的一个零点，则由()()000f f x ==可知，()f x 在区间()00,x 上不可能单调递增，也不可能单调递减．则()g x 不可能恒为正,也不可能恒为负．故()g x 在区间()00,x 内存在零点1x ．同理()g x 在()0,1x 区间内存在零点2x ．所以()g x 在区间()0,1内至少有两个零点．由(1）知，当12a时,()g x 在[]0,1上单调递增，故()g x 在()0,1内至多有一个零点． 当e 2a 时，()g x 在[]0,1上单调递减，故()g x 在()0,1内至多有一个零点．所以1e 22a <<．此时()g x 在区间()0,ln 2a ⎡⎤⎣⎦上单调递减，在区间()(ln 2,1a ⎤⎦上单调递增．因此()(10,ln 2x a ∈⎤⎦,()()2ln 2,1x a ∈,必有()010g b =->，()1e 20g a b =-->．由()10f =，有e 12a b +=-<,有()01e 20g b a =-=-+>，()1e 210g a b a =--=->，得e 21a -<<． 所以函数()f x 在区间()0,1内有零点时，e 21a -<<．。
成都七中2014级考试数学试卷(文科)命题:方廷刚 审题:巢中俊 一、选择题(共50分,每题5分)1.设22{|10},{|log 0}A x x B x x =->=<,则A B ⋂=A.{|1}x x >B.{|0}x x >C.{|1}x x <-D.Φ2.设i 是虚数单位,若()(1)2(1)a bi i i ++=-,其中,a b R ∈,则a b +的值是A.12-B.2-C.2D.323.有一正方体,六个面上分别写有数字 1、2、3、4、5、6,有3个人从不同的角度 观察,结果如图所示.若记3的对面的数字为 m ,4的对面的数字为n ,则m n +=A.3B.7C.8D.114.设554log 4,log ((23),log 17a b c ==-=,则A.a c b <<B.b c a <<C.a b c <<D.b a c <<5.设,A B 是锐角ABC ∆的两内角,(sin ,1),(1,cos )p A q B =-=u r r ,则p u r 与q r的夹角是A.锐角B.钝角C.直角D.不确定 6.下列判断错误．．的是 A.“22am bm <”是“a b <”的充分不必要条件B.“3210x x --≤对x R ∈恒成立”的否定是“存在0x R ∈使得320010x x -->”C.若“p q Λ”为假命题,则,p q 均为假命题D.若随机变量ξ服从二项分布:ξ～1(4,)4B ,则1E ξ= 7.设0ω>,函数sin()23y x πω=++的图像向右平移43π个单位后与原图像重合,则ω的最小值是A.32 B.43 C.3 D.238.已知双曲线22221(0,0)x y a b a b -=>>的离心率2e =,则2a eb＋的最小值为A.23B.26C.23D.269.在ABC ∆内部随机取一点P ,则事件“PBC ∆的面积不大于ABC ∆面积的13”的概率是A.13 B.49 C.59 D.2310.已知函数2()22ln (,0)f x x ax a x a R a =--∈≠,则下列说法错误．．的是 A.若0a <,则()f x 有零点 B.若()f x 有零点,则12a ≤且0a ≠ C.0a ∃>使得()f x 有唯一零点 D.若()f x 有唯一零点,则12a ≤且0a ≠二、填空题(共25分,每题5分)11.已知函数2()2x x f x =在区间(0,)a 内单调,则a 的最大值为__________.12.若方程3log (3)20xa x -+-=有实根,则实数a 的取值范围是___________.13.已知直线l :330x y --=与抛物线Γ:24y x =交于,A B 两点,与x 轴交于F ,若()OF OA OB λμλμ=+≤u u u r u u r u u u r, 则λμ=_______. 14.正方体1111ABCD A B C D -中,E 是棱1CC 的中点, F 是侧面11BCC B 内的动点,且1//A F 平面1D AE ,若正方 体1111ABCD A B C D -的棱长是2,则F 的轨迹被正方形 11BCC B 截得的线段长是________.15.已知函数()122014122014f x x x x x x x =+++++++-+-++-L L 的定义域为R ,给定两集合4222{((12101)(2))(2)}A a R f a a a f a =∈-++=+及B ={()(),}a R f x f a x R ∈≥∈,则集合A B ⋂的元素个数是_________.三、解答题(共75分) 16.(12分)设()f x p q=⋅u u r u r,而2(24sin ,1),(cos ,3sin 2)()2xp q x x x R ωωω=-=∈u u ru r.(1)若()3f π最大,求ω能取到的最小正数值.(2)对(1)中的ω,若()23sin 1f x x =+且(0,)2x π∈,求tan x .17.(12分)小区统计部门随机抽查了区内60名网友4月1日这天的网购情况,得到如下数据统计表(图(1)).网购金额超过2千元的顾客被定义为“网购红人”,网购金额不超过2千元的顾客被定义为“非网购红人”.已知“非网购红人”与“网购红人”人数比恰为3:2.(1)确定,,,x y p q 的值,并补全频率分布直方图(图(2)).(2)为进一步了解这60名网友的购物体验,从“非网购红人”和“网购红人”中用分层抽样的方法确定10人,若需从这10人中随机选取3人进行问卷调查,设ξ为选取的3人中“网购红人”的人数,求ξ的分布列和数学期望.18.(12分)执行如图所描述的算法程序,记输出的一列a 的值依次为12,,,n a a a L ,其中*n N ∈且n ≤.(1)若输入2λ=,写出全部输出结果.(2)若输入2λ=,记*1()1n n b n N a =∈-,求1n b +与n b 的关系(*n N ∈). 19.(12分)如图,已知平面ABCD ⊥平面BCEF , 且四边形ABCD 为矩形,四边形BCEF 为直角梯形,090CBF ∠=,//BF CE ,BC CE ⊥,4DC CE ==, 2BC BF ==.(1)作出这个几何体的三视图(不要求写作法). (2)设,P DF AG Q =⋂是直线DC 上的动点, 判断并证明直线PQ 与直线EF 的位置关系.(3)求三棱锥F ADE -的体积.20.(13分)椭圆Γ:2221(0)25x y r r +=>的左顶点为A ,直线4x =交椭圆Γ于,B C 两点(C 上B 下),动点P 和定点(4,6)D -都在椭圆Γ上.(1)求椭圆方程及四边形ABCD 的面积. (2)若四边形ABCP 为梯形,求点P 的坐标.(3)若,m n 为实数,BP mBA nBC =+uu r uu r uu u r,求m n +的最大值.21.(14分)已知函数()2sin f x x x =-,()()(2)2g x f x π=--.(1)讨论()g x 在(0,)6π内和在(,)62ππ内的零点情况. (2)设0x 是()g x 在(0,)6π内的一个零点,求()f x 在0[,]2x π上的最值. (3)证明对*n N ∈恒有11()12212n k n n π=<<+∑.成都七中2014级考试数学试卷(文科)参考答案一、DBCD BCAB CB 二、11.2ln 2 12.6a ≥ 13.1314.2 15.7 三、16.(1)12. (2)33.17.解.(1)96x y =⎧⎨=⎩,0.150.10p q =⎧⎨=⎩,补全频率分布直方图如图所示.(2)选出的10人中,“网购达人”有 4人,“非网购达人”有6人,故ξ的可能 取值为0,1,2,3,且易得ξ的分布列为65E ξ=.18.解.(1)输出结果共2个,依次是:20,2.(2)*11()n n b b n N +=-∈. 19.(1)如右图. (2)垂直. (3)83. 20.(1)22125100x y +=; 78ABCD S =. (2)748(,)55-. (3)13510+.21.解.(1)()2cos 1g x x '=-在(0,)2π有唯一零点3x π=,易知()g x 在(0,)3π单增而在(,)32ππ内单减,且()(3)(2)0332g πππ=--->,故()g x 在(0,)3π和[,)32ππ内都至多有一个零点.又(0)0,()(1)(2)106623g g ππππ<=---=->,故()g x 在(0,)6π内有唯一零点;再由()02g π=知()g x 在(,)62ππ内无零点. 4242俯视图侧视图正视图(2)由(1)知()g x 在[0,]2π有最大值())(2)332g πππ=--,故()f x 在0[,]2x π有最大值()33f ππ=;再由(1)的结论知()f x 在0[,]2x π的最小值应为0min{(),()}2f x f π.由0()0g x =知0()2()22f x f ππ=-=,于是()f x 在0[,]2x π的最小值0()()222f x f ππ==-.(3)由(2)知0[,]2x x π∈时,有2()23f x ππ-≤≤,即111sin 2426x x x ππ+-≤≤+- ①取*)2k x k N π=∈,则2k x π<且0126k x x ππ≥->>,将k x 的值代入①中,可得112π≤≤+111)2122n nn k k k n n π===⇒-≤≤-∑②再由1111221)nn n nk k k k =====>==∑,得1)1)12nk n π=<+-∑ ③相仿地,2n ≥时,1221121n n nk k k ====+<+=∑,故1111)22nk n n =>-=∑ ④ 而1n =时④即01cos1cos 602>=,显然也成立.故原不等式成立.。
成都七中高2014届一诊模拟数学试卷（文科）考试时间：120分钟总分：150分 命题人：张世永刘在廷审题人：巢中俊一．选择题（每小题5分，共50分，在每小题给出的四个选项中，只有一项是符合要求．） 1.已知集合{}1,0,A a =-，{}|01B x x =<<，若A B ≠∅ ，则实数a 的取值范围是（） A {}1B (,0)-∞C (1,)+∞D (0,1)2.复数1()1ii i-⋅+的虚部为（ ） A -2 B -1 C 0 D 13.定义行列式运算：12142334,a a a a a a a a =-将函数3cos ()1 sin xf x x =的图象向左平移m个单位(0)m >，若所得图象对应的函数为偶函数，则m 的最小值是（）A 23πB 3πC 8πD 56π 4．阅读下边的程序框图，若输出S 的值为－14，则判断框内可填写（ ） A ．i<6 ? B ．i<8 ? C ．i<5 ? D.i<7 ?5.在平面直角坐标系中,若角α的顶点在坐标原点,始边 在x 轴的非负半轴上,终边经过点(3,4)P a a -(其中0a <) 则sin cos αα+的值为( ) A 15-B 4 5-C 53D15 6.已知命题:(,0),34x x p x ∃∈-∞<；命题:(0,),sin q x x x ∀∈+∞>则下列命题中真命题是（ ） A p q ∧ B ()p q ∨⌝ C ()p q ∧⌝ D ()p q ⌝∧7.已知正项等比数列{}n a 满足7652a a a =+。
若存在两项,m n a a 使得14m n a a a =，则19m n+的最小值为( ) A83 B 114 C 145 D 1768.平面四边形ABCD 中，AD=AB=2,CD=CB=5,且AD AB ⊥，现将ABD ∆沿着对角线BD 翻折成/A BD ∆，则在/A BD ∆折起至转到平面BCD 内的过程中，直线/A C 与平面BCD 所成的最大D 1C 1A 1B 1CD EF 角的正切值为( ) A 1 B12 C 33D 39.已知)(x f 、)(x g 都是定义在R 上的函数，()0g x ≠，//()()()()0f x g x f x g x -<，()()x f x a g x =，25)1()1()1()1(=--+g f g f ，则关于x 的方程2520((0,1))2abx x b ++=∈有两个不同实根的概率为（）A51B52 C53 D54 10．已知()f x 是定义在[1,1]-上的奇函数，当12x x ≤时，12()()f x f x ≤。
成都七中高2014届第4次高考模拟考试第Ⅰ卷(选择题, 共90分)Ⅰ. 单项填空(共10小题; 每小题1分, 满分10分)1. Alfonso Cuarón, director and co-writer of Gravity, makes it big success at the 86th.Oscar Academy Awards.A. a; theB. the; theC. the; aD. a; a2. It‟s good for a university student to take up a job if he can get.A. itB. oneC. someD. that3.With the word “PM2.5” constantly in media reports, you might be wondering what on earth it means.A. appearedB. appearingC. to appearD. being appeared4.Naturally a smile the eyes participate is extremely communicative.A. in whichB. to whichC. at whichD. with which5.With the rise of modern communication tools, handwritten letters by emails.A. will replaceB. have replacedC. are replacedD. are being replaced6.H7N9 is a deadly virus, it is still safe to eat thoroughly-cooked chicken according to some experts.A. UntilB. WhileC. IfD. Since7.I wondered they had raised so much money on their own to help youngsters in need.A. whatB. thatC. howD. why8. It is reported that Alain Perrin to serve as the new coach of National Football Team of China after Asian Cup Qualifiers.A. schedulesB. is scheduledC. had scheduledD. had been scheduled9.the picture of the popular singer, I would show it to you right now.A. Had I takenB. Did I takeC. I had takenD. I took10. —Do you think their table tennis team will win the first place at the coming Asian Games?—. Ours is much stronger than theirs.A. Of courseB. It dependsC. Don‟t mention itD. By no meansⅡ. 完形填空(共20小题; 每小题1. 5分, 满分30分)“Meow, meow, meow,” is what I heard as I walked through the alley. I approached the noise and noticed a tail sticking out from under a piece of wood. Under the wood was a 11 black and white cat. I picked him up and 12 he must be freezing to death. I 13 home with the cat held in my jacket.My new best friend, who soon became known as Little Bit, received his name because he was almost 14 when I held him in my hands. He stood about five inches tall. Little Bit‟s small size had a great advantage — he 15 perfectly in the pocket of my jacket, which made 16 him everywhere very easy. Any time I was home, he wouldn‟t leave my 17 . He was always eager toplay with me. When I fell asleep at night, he would always 18 up around my head to ensure that I was warm.Unfortunately, I grew up. My teenage life 19 my relationship with Little Bit. I lived at such a fast pace that I stopped 20 time for him. My free time was spent with my friends instead. I would come in the house on my phone and not 21 him at all. His meows became an annoyance to me, but it wasn‟t his 22 that he wanted his best friend back.Time had caused a 23 to Little Bit. His body began 24 down and by the time I realized something was wrong with him, he had already lost his balance. He lay there and looked at me, and 25 this day I still remember the 26 look in his bright green eyes. I took him to the vet (兽医), but there was nothing he could do. The last time I 27 him he wasn‟t the same tiny cat I had found ten years before. Little Bit filled my arms and he was put to sleep that day.Little Bit‟s 28 made me realize how much he meant to me. He was always there for me when I needed him. I 29 our last years together and I feel sorry for not always being there for him. I will always 30 the special memories we made.11. A. lovely B. tiny C. pretty D. friendly12. A. agreed B. insisted C. realized D. proved13. A. left B. stayed C. drove D. hurried14. A. weightless B. useless C. breathless D. hopeless15. A. grew B. fitted C. played D. existed16. A. showing B. keeping C. taking D. guiding17. A. mind B. heart C. body D. side18. A. rise B. stand C. wake D. roll19. A. weakened B. fastened C. deepened D. sharpened20. A. making B. losing C. wasting D. gaining21. A. overlook B. interrupt C. acknowledge D. recognize22. A. desire B. purpose C. fault D. greed23. A. fear B. loss C. delay D. concern24. A. dropping B. tearing C. calming D. shutting25. A. on B. to C. in D. for26. A. shameful B. hateful C. harmful D. sorrowful27. A. helped B. protected C. held D. cured28. A. illness B. death C. sadness D. pressure29. A. regret B. bother C. confuse D. dream30. A. treasure B. admire C. evaluate D. exploreⅢ. 阅读理解(共两节, 满分50分)第一节(共20小题; 每小题2分, 满分40分)AMonday: Here I am, in the middle of nowhere. This camping trip idea is not getting off to a very good start. It‟s raining and the tent leaks. The hiking seemed t o take forever, and I still can‟t understand how it could all have been up hill! How did I ever let my brother talk me into this? Whenwe get home—if we ever get home—he‟s going to have to do something great to get back on my good side. Maybe he should sponsor a shopping spree (狂购) at the mall!Tuesday: Things are looking up. The sun came out today, so we were able to leave the tents and dry out. We‟re camped at the edge of a small lake that I couldn't see before because of the rain and fog. The mountains are all around us, and the forest is absolutely beautiful. We spent most of the day dragging out everything out of our backpacks or tents and putting it where the sun could dry it out. Later in the afternoon we tried to catch the fish for dinner, but the fish were smarter than we were. At night we built a fire and sang songs happily.Wednesday: We hiked to the far side of the lake and climbed to the top of a small peak. From there we could see how high the other mountains were and how far the forest spread around us. On the way up we passed through a snowfield!Thursday: I caught my first fish! We followed the stream that fed the lake. After about two miles, we came to a section that Carol said “looked …fishy‟”. She had a pack rod (竿), which can be carried in a backpack. I asked to cast it, and I caught a fish on my first try. Carol caught a few more. But they were just too pretty to eat for lunch, so we put them back in the stream.Friday: I can‟t believe we are going home already. It will be nice to get a hot shower, sleep in a real bed, and eat junk food, but the trip has been wonderful. We‟re already talking about another camping adventure next year where we canoe down a river. It‟s hard to believe, but I think this city girl has a little country blood in her veins.31. The writer went on this camping trip because .A. she enjoyed campingB. she wanted to go fishingC. she was influenced by her brotherD. she was tired of staying home32. The whole morning of Tuesday, the writer .A. hiked along the lakeB. dried out her belongingsC. climbed the mountainD. caught the fish for dinner33. It can be inferred that Carol had a pack rod with her because .A. she could not afford to buy a regular fishing poleB. she needed it to get their main source of foodC. she thought the writer of the journal might need itD. she expected to go fishing while they were hiking34. It is likely that the writer will .A. go on another camping tripB. invite Carol to go fishing togetherC. make her brother buy her somethingD. convince her brother to go campingBAn idea came to me, and I turned off the lights in the studio. In the darkness, I took off my shirt and took up the cello (大提琴); it was the first time in my life I‟d felt the instrument against my barechest. I'd never thought about that; music scholars always talk about the resonating properties (共振) of various instr uments, but surely the performer‟s own body must have some effect on the sound. As I dug into the notes I imagined that my own chest and lungs were extensions of the sound box; I seemed to be able to change the sound by the way I sat, and by varying the muscular tightness in my upper body.After improvising for a while, I started playing, still in the darkness. I heard the music through my skin. For the first time I didn't think about how it would sound to anyone else, and slowly, joyfully, gratefully, I started to hear again. The notes sang out, first like a trickle: then like a fountain of cool water bubbling up from a hole in the middle of a desert. After an hour or so I looked up, and in the darkness saw the outline of the cat sitting on the floor in front of me, cleaning her paws and purring loudly, I had an audience again, humble as it was.So that's what I do now with my cello. At least once a clay I find time to tune it, close my eyes and listen. It's probably not going to lead to the kind of comeback I'd be thirsty for—years of playing badly have left scars on my technique—but I might eventually try giving a concert if I feel up to it.Occasionally I feel a stab of longing, and I wish I could give just one more concert on a great stage before my lights blink off, but that longing passes more quickly now. I take comfort in the fact that, unlike the way I felt before, I can enjoy playing for myself now. I feel relaxed and expansive when I play, as if I could stretch out my arms and reach from one end of the apartment to the other. A feeling of completeness and dignity surrounds me and lifts me up.35. The writer put the cello against his bare chest to .A. test music scholars' ideas about the resonating propertiesB. experience the effect of his body on the musical soundC. reduce his muscular tightness in his upper bodyD. check the function of the sound box36. In Paragraph 2, the writer intends to .A. explain his feelings of playing before a catB. identify specific pieces of music he playedC. express his feelings of playing against his bodyD. describe the sound when he played against his body37. From the last paragraph we can infer that the writer was .A. optimisticB. discouragedC. nervousD. enthusiastic38. The passage is mainly about .A. a musician playing the cello for an audienceB. a musician's feelings when playing the celloC. a musician finding joy in playing music in a new wayD. a musician's desire to return to his former professionCWhen a group of children politely stop a conversation with you, saying, “We have to go to worknow,” you‟re left feeling surprised and certainly uneasy. After all, this is the 1990s and the idea of children working is just unthinkable. That is, until you are told that they are all pupils of stage schools, and tha t the “work” they go off to is to go on the stage in a theatre.Stage schools often act as agencies (代理机构) to supply children for stage and television work. More worthy of the name “stage school”are those few places where children attend full time, with a training for the theatre and a general education.A visit to such schools will leave you in no doubt that the children enjoy themselves. After all, what lively children wouldn't settle for spending only half the day doing ordinary school work, and acting, singing or dancing their way through the other half of the day?Then of course there are times for the children to make a name and make a little money in some big shows. Some stage schools give their children too much professional work at such a young age. But the law is very tight on the amount they can do. Those under 13 are limited to 40 days in the year; those over 13 do 80 days. The schools themselves admit that not all children will be successful in the profession for which they are being trained. So what happens to those who don't make it? While all the leading schools say they place great importance on children getting good study results, the facts seem to suggest this is not always the case.39. People would stop feeling uneasy when realizing that the children they're talking to _______.A. attend a stage schoolB. are going to the theatreC. have got some work to doD. love singing and dancing40. In the writer's opinion, a good stage school should _______.A. produce star performersB. help pupils improve their study skillsC. train pupils in language and performing artsD. provide a general education and stage training41. “Profession work” as used in the text means _______.A. ordinary school workB. money-making performancesC. stage training at schoolD. acting, singing or dancing after class42. Which of the following best describes how the writer feels about stage schools?A. He thinks highly of what they have to offer.B. He favors an early start in the training of performing arts.C. He feels uncomfortable about children putting on night shows.D. He doubts the standard of ordinary education they have reached.DOnline shoppers would rather receive an offer for a product or service than make their own offer, according to a study led by a Michigan State University scholar that has implications for the fast-growing e-commerce industry. The findings may come as a surprise given that shopping online is an anonymous (匿名的) process that seemingly can give consumers more confidence to drive a hard bargain, said Don Conlon, Professor of Management in MSU‟s Broad College of Business.But the study found that participants who made their own offers were less successful in sealing the deal and, when they were successful, worried they overpaid. Many shoppers found the process of researching an offer to be a hassle. “Americans are very busy, and it‟s less time-consuming to be the one receiving the offer rather than the one proposing the offer” Conlon said, “People tend to be happier when they‟re in the receiver role.”Online spending in the United States is expected to jump 45 percent in the next four years, from $ 226 billion this year to $ 327 billion in 2017, according to Forrester Research Inc. Nevertheless, researches into this prosperous market remain largely focused on the strategies of companies, rather than those of shoppers. Conlon‟s study is, obviously, a rare exception.Conlon got the idea for the study after considering the difference between two popular sites for hotels and airline flights, , which takes bids, and , which provides offers. Using these two models, Conlon and his fellow researchers conducted a series of experiments with more than 850 people who were charged with booking a fictional hotel room and acquiring a fictional antique car. Not only did participants prefer to receive bids, Conlon said, but they also secured more deals in that receiver role. Further, when they had to make the bids, they were left more mentally taxed and regretful.From an industry perspective, putting customers in the receiver role may help fill more hotel rooms and airplane seats. “If you‟re a business with a lot of products” Conlon said, “you may want to be the one making the offers.” However, when selling single items, such as an antique car, accepting bids may be a better option since that typically drives up the price, he said.43. What can we learn from Para. 1?A. The result of the study gives customers more confidence.B. Scholars aren‟t surprised at the findings of the study.C. Online shoppers don‟t bargain as much as expected.D. E-commerce industry drives more hard bargains.44. It can be concluded from Conlon‟s experiments that______.A. online shopping is time-consumingB. given prices help promote online salesC. online businesses provide a lot of productsD. receiving offers makes online shoppers regretful45. The underlined word “taxed” in Para. 4 probably means______.A. burdenedB. numbedC. relievedD. challenged46. The passage is mainly about______.A. the big advantages of online shoppingB. the rapid development of online shoppingC. online shoppers‟ preference for taking offersD. online companies‟ strategies to improve serviceEHomeownership has let us down. For generations, Americans believed that owning a home wasundoubtedly good. Our political leaders hammered home the point. Franklin Roosevelt held that a country of homeowners was “unconquerable.” Homeownership could even save babies, save children, save families and save America. A house with a lawn and a fence wasn‟t just a nice place to live in or a risk-free investment; it was a way to transform a nation. No wonder leaders of all political types wanted to spend more than $100 billion a year on subsidies and tax breaks to encourage people to buy.But the dark side of homeownership is now all too apparent: Indeed, easy lending stimulated by the cult of homeownership may have triggered the financial crisis. Housing remains a drag on the economy. Existing-home sales in April dropped 27% from the prior month, worsening fears of a double-dip. And all that is just the obvious tale of a housing bubble and what happened when it popped. The real story is deeper and darker still.For the better part of a century, politics, industry and culture lined up to create a fetish of the idea of buying a house. Homeownership has done plenty of good over the decades; it has provided stability to tens of millions of families. Yet by idealizing the act of buying a home, we have ignored the downsides. In the bubble years, lending standards slipped dramatically, allowing many Americans to put far too much of their income into paying for their housing. And we ignored longer-term phenomena too. Homeownership contributed to the hollowing out of cities and kept renters out of the best neighborhoods. It fed America‟s overuse of energy and oil. It made it more difficult for those who had lost a job to find another. Perhaps worst of all, it helped us become casually self-deceiving: By telling ourselves that homeownership was a pathway to wealth and stable communities and better test scores, we avoided dealing with these frightening issues head-on.Now, as the U.S. recovers from the biggest housing bust (破产) since the Great Depression, it is time to rethink how realistic our expectations of homeownership are—and how much money we want to spend chasing them. Many argue that homeownership should not be a goal pursued at all costs.47. Political leaders wanted to spend money encouraging people to buy houses because .A. owning a home was undoubtedly goodB. homeownership was unconquerableC. houses could save families and AmericaD. homeownership could shape a country48. The underlined sentence in Para. 2 means .A. homeownership has quite a lot of bad effectsB. there might be another housing breakdown in the U.S.C. the existing-home sales will keep decreasing in the U.S.D. the result of homeownership is much worse than it appears49. It can be inferred from Para. 3 that .A. it is hard for Americans to get a home loanB. it is the way to wealth to have one‟s own houseC. many Americans choose to live out of urban areasD. homeownership has made many people out of work50. What is the author‟s attitude towards homeownership?A. Favorable.B. Ambiguous.C. Cautious.D. Optimistic.第二节补全对话(共5小题; 每小题2分, 满分10分)Jerry: Hi, Mike. Look like you‟ve got some sun.Mike: I guess so. I spent the weekend on the beach.Jerry: Really? That sounds exciting. __1__Mike: At my friend‟s house. He invited me to stay there for as long as I wanted.Jerry: __2__Mike: Oh, I have a paper to work on.Jerry: __3__ I mean besides lying out in the sun.Mike: I played some volleyball. I never realized how hard it is to run on sand.Jerry: __4__ Did you go swimming?Mike: I intended to. __5__ So I just went fishing.Jerry: All sounds so relaxing.A. What a pity!B. It must be cool.C. Where did you stay?D. But how did you get there?E. So what else did you do out there?F. But the water wasn‟t warm enough.G. Then why not stay there for a longer time?Ⅳ. 阅读表达(共5小题; 每小题2分, 共10分)A 14-year-old girl was attacked by a grey bear while competing in a bike race. She managed to whisper the “bear” to her rescuer. The girl had serious head, neck and leg wounds. Now she is in a dangerous condition in hospital.The girl was taking part in a 24-hour race through Bicentennial Park in Alaska when she was attacked by the bear. Bicentennial Park is next to Chugach State Park where some wild animals live.The animal attacked the girl in a heavily wooded area at about 1:30 am. The girl called emergency services but she was unable to say any word. Another rider Peter Basinger found her lying unconscious on the ground. He told Anchorage Daily News that she managed to say the word “bear”when he stopped to help. Her helmet（头盔）had been knocked off in the attack and rolled into the woods. Mr. Basinger waited with the girl for about 20 minutes until doctors arrived. Armed police travelled 3 km with doctors to pick up the girl.Animal expert Rick Sinnot said something must be done as soon as possible to prevent the bearfrom attacking people again. “It was extremely terrifying,” he said.Mr. Sinnot told Anchorage Daily News that the bear might be a mother. It hurt two runners on a nearby path two weeks ago. He also said that the girl was lucky to have been wearing a bike helmet because the bear had bitten her head several times.56. Who attacked the girl? (Within 3 words)____________________________________________________________________________57. What was the girl doing when she was attacked? (Within 7 words)____________________________________________________________________________58. What did the girl say to emergency services after the attack? (Within 1 word)____________________________________________________________________________59. How did Mr. Basinger help the girl? (Within 7 words)____________________________________________________________________________60. Why did Mr. Sinnot say that the girl was lucky? (Within 10 words)____________________________________________________________________________Ⅴ. 短文改错(共10小题; 每小题1. 5分, 满分15分)下面短文中有10处语言错误。
