2017年福建省莆田市普通高中毕业班质量检查文科数学试题 及答案

'www.ks5u.com2015年莆田市高中毕业班教学质量检查试卷数学（文科）本试卷分第I卷(选择题)和第II卷(非选择题)两部分，满分150分．考试时间120分钟．注意事项：1．答题前，考生先将自己的姓名、准考证号填写在答题卡上．2．考生作答时，将答案答在答题卡上．请按照题号在各题的答题区域(黑色线框)内作答，超出答题区域书写的答案无效．在草稿纸、试题卷上答题无效．3．选择题答案使用2B铅笔填涂，如需改动，用橡皮擦干净后，再选涂其他答案标号；非选择题答案使用O．5毫米的黑色中性(签字)笔或碳素笔书写，字体工整、笔迹清楚．4．保持答题卡卡面清洁，不折叠、不破损．考试结束后，将本试卷和答题卡一并交回．参考公式：样本数据x1，x2，…，xn的标准差锥体体积公式s=　　　V=Sh其中为样本平均数其中S为底面面积，h为高柱体体积公式-26- 球的表面积、体积公式V=Sh，其中S为底面面积，h为高其中R为球的半径第Ⅰ卷（选择题共60分）一、选择题：本大题共12小题，每小题5分，共60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．把答案填涂在答题卡相应位置．1．已知，则等于（）A．B．C．D．2．已知复数（，i为虚数单位）在复平面上对应的点为M，则“且”是“点M在第四象限”的（）A．充分而不必要条件B．必要而不充分条件C．充分必要条件D．既不充分也不必要条件3．抛物线的准线方程是（）A．x=－2B．x=－1C．y=－2D．y=－1-26- 4．根据如下样本数据6810122356得到的线性回归方程为，则的值为（）A．－2B．－2．2C．－2．3D．－2．65．阅读右图所示的程序框图，运行相应的程序，输出的k的值等于（）A．3Ｂ．4Ｃ．5Ｄ．66．若实数满足不等式组则的最大值是（）A．－1B．0C．1D．27．已知函数f(x)为奇函数，当x≥0时，f(x)=x2－x，则不等式f(x）>0的解集为（）A．（－∞，－1）∪（0，1）B．（－∞，－1）∪（1，+∞）C．（－1，0）∪（0，1）D．（－1，0）∪（1，+∞）8．已知若向量与垂直，则实数的值为（）A．B．C．D．-26- 9．请在“垂直于同一①的两②平行”①和②处中填入“直线”或“平面”，使之组成四个不同的命题，则其中真命题的个数为（）A．1B．2C．3D．410．已知双曲线的一个焦点在圆上，则双曲线的渐近线方程为(　　)A．B．C．D．11．函数的导函数的图象是如图所示的一条直线，该直线与轴的交点坐标为（1，0），则与的大小关系是（）A．B．C．D．无法确定12．如图，所在平面上的点均满足，（其中，是以1为首项的正项数列），则等于（）A．4B．8C．16D．32第Ⅱ卷（非选择题共90分）二、填空题:本大题共4小题，每小题4分，共16分．-26- 把答案填写在答题卡的相应位置．13．若集合则集合=．14．某校对100名参加“妈祖杯”知识竞赛的选手成绩进行统计，得到样本频率分布直方图(如图)，则在这100名学生中，成绩不低于80分的人数为．15．函数的一条切线与直线垂直，则该切线方程为_______．16．定义：表示不超过的最大整数．例如：，．给出下列结论：①函数是周期为的周期函数；②函数是奇函数；③函数的值域是；④函数不存在零点．其中正确的是_____________．（填上所有正确结论的编号）三、解答题：本大题共6小题，共74分．解答应写出文字说明、证明过程或演算步骤．把答案填在答题卡相应位置．17．（本小题满分12分）已知等差数列{an}的首项为1，公差d≠0，且a1，a2，a4成等比数列．（Ⅰ）求数列{an}的通项公式与前n项和Sn；（Ⅱ）设（），求使不等式成立的最小正整数n．-26- 18．（本小题满分12分）已知函数经化简后利用“五点法”画其在某一周期内的图象时，列表并填入的部分数据如下：①010－10（Ⅰ）请直接写出①处应填的值，并求函数的单调递增区间；（Ⅱ）的内角所对的边分别为，已知，求的面积．19．（本小题满分12分）《聪明花开——莆仙话挑战赛》栏目共有五个项目，分别为“和一斗”、“斗麻利”、“文士生”、“讲头知尾”、“正功夫”．《聪明花开》栏目组为了解观众对项目的看法，设计了“你最喜欢的项目是哪一个”的调查问卷（每人只能选一个项目），对现场观众进行随机抽样调查，得到如下数据（单位：人）：合一斗斗麻利文士生讲头知尾正功夫115230115345460（I）在所有参与该问卷调查的人中，用分层抽样的方法抽取n人座谈，其中恰有4人最喜欢“斗麻利”，求n的值及所抽取的人中最喜欢“合一斗”的人数；-26- （II）在（I）中抽取的最喜欢“合一斗”和“斗麻利”的人中，任选2人参加栏目组互动，求恰有1人最喜欢“合一斗”的概率．20．（本小题满分12分）已知四边形ABCD为平行四边形，，BD=AD，AB=2，四边形ABEF为正方形，且平面平面ABCD．(Ⅰ)求证：平面ADF；(Ⅱ)若M为CD中点，证明在线段EF上存在点N，使得MN∥平面ADF，且MN//平面BDF，并求出此时三棱锥N—ADF的体积．21．（本小题满分12分）如图，O为坐标原点，椭圆：（）的左、右焦点分别是F1、F2，上顶点为P，离心率e=．直线PF2交椭圆于另一点Q，△PQF1的周长为8．（I）求椭圆的方程；（II）若点R满足，求△PQR的面积；（III）若M、N为椭圆E上异于点P的两动点，试探究：是否存在点M、N，使得△PMN为正三角形？若存在，求出M、N两点的坐标；若不存在，请说明理由．-26- 22．（本小题满分14分）已知函数，．（Ⅰ）若，求函数的单调区间；（Ⅱ）求证：对任意给定的正数，总存在实数，使函数在区间上不单调；（Ⅲ）试探究：是否存在实数，使当时，函数的值域为？若存在，试确定实数的取值范围；若不存在，说明理由．2015年莆田市高中毕业班教学质量检查试卷数学（文科）试题参考答案及评分标准-26- 一、选择题：本题考查基础知识和基本运算，本大题共12小题，每小题5分，满分60分．1．A2．A3．B4．C5．C6．D7．D8．C9．B10．A11．B12．C二、填空题：本题考查基础知识和基本运算，本大题共4小题，每小题4分，满分16分．13．{1，2}14．2515．4x+y+3=016．①③④三、本大题共6小题，共74分，解答应写出文字说明、证明过程或演算步骤．17．本小题主要考查等差、等比数列、数列求和等基础知识，考查运算求解能力，考查化归与转化思想．满分12分．解：（Ⅰ）因为a1，a2，a4成等比数列，所以a1a4=a22．…………………………………1分即a1(a1+3d)=(a1+d)2，解得d=1或d=0（舍去）．…………………………………………2分所以an=1+(n－1)1=n，………………………………………………………………………4分．………………………………………………………………………………6分（Ⅱ）由（Ⅰ）知，…………………………………………7分所以-26- ．…………9分解，解得n>9，…………………………………………………………………………11分所以使不等式成立的最小正整数为10．……………………………………………………12分18．本小题主要考查三角函数的图象与性质、两角和与差的三角函数、解三角形等基础知识，考查运算求解能力，考查化归与转化思想．满分12分．解：（Ⅰ）①处应填入．…………………1分……………3分．………………4分因为T=，所以，，．…………5分令，，得，所以函数的单调递增区间为．……………7分（Ⅱ）因为，…………8分解法一：由余弦定理得，得，．…………10分-26- 所以的面积．………12分解法二：由正弦定理得，所以，，而，………8分所以，………10分即，因为，，所以．因此为等边三角形，其面积．……12分19．本小题主要考查分层抽样、古典概型等基础知识，考查数据处理能力、运算求解能力、应用意识，考查必然与或然思想．满分12分．解（I）由已知得，解得n=22．…………3分抽取的人中最喜欢“合一斗”有(人)．……………5分（II）从（I）中抽取的最喜欢“合一斗”和“斗麻利”的人中，最喜欢“合一斗”的有2人，记为A1、A2，最喜欢“斗麻利”的有4人，记为B1、B2、B3、B4．…………………6分从中随机抽取2人，所有的可能结果共有15种，它们是:(A1，A2)、(A1，B1)、(A1，B2)、(A1，B3)、(A1，B4)、(A2，B1)、(A2，B2)、(A2，B3)、(A2，B4)、(B1，B2)、(B1，B3)、(B1，B4)、(B2，B3)、(B2，B4)、(B3-26- ，B4)．…………9分其中，恰有1人最喜欢“合一斗”的可能结果共有8种，它们是：(A1，B1)、(A1，B2)、(A1，B3)、(A1，B4)、(A2，B1)、(A2，B2)、(A2，B3)、(A2，B4)．故所求的概率P=．…………12分20．本小题主要考查空间直线与直线、直线与平面、平面与平面的位置关系、几何体的体积等基础知识，考查空间想象能力、抽象概括能力、推理论证能力、运算求解能力，考查数形结合思想、化归与转化思想．满分12分．(Ⅰ)证：正方形ABEF中，AF⊥AB，∵平面ABEF⊥平面ABCD，又AF平面ABEF，平面ABEF平面ABCD=AB，∴AF⊥平面ABCD．又∵BD平面ABCD，∴AF⊥BD．……………3分又，AFAD=A，AF、AD平面ADF，∴平面ADF．……………5分(Ⅱ)解：当N为线段EF中点时，MN∥平面ADF，且MN//平面BDF．……………6分证明如下：正方形ABEF中，NFBA，平行四边形形ABCD中，MDBA，NFMD，四边形NFDM为平行四边形，MN//DF．……………7分又DF平面ADF，MN平面ADF，∴MN//平面ADF，-26- 同理可证MN//平面BDF．……………9分过D作DHAB于H，∵平面ABEF⊥平面ABCD，又DH平面ABCD，平面ABEF平面ABCD=AB，∴DH⊥平面ABEF．在Rt∆ABD中，AB=2，BD=AD，∴DH=1，……………10分所以．……………12分21．本小题主要考查平面向量、点到直线的距离、椭圆的定义与性质、直线与椭圆的位置关系等基础知识，考查运算求解能力、推理论证能力，考查化归与转化思想、数形结合思想、特殊与一般思想、分类与整合思想．满分12分．解法一：（Ⅰ）由已知可得，，4a=8，所以a=2，c=1．·……………2分又由，解得，所以椭圆的方程为．……………3分（Ⅱ）因为，所以，所以R，O，Q三点共线，且R在椭圆E上．……………4分直线PF2的方程为y=(x－1)，由得5x2－8x=0，解得x=或x=0，……………5分所以P（0，），Q（，），R（，）．·……………6分-26- 所以S△PQR=S△POR+S△POQ=|PO|·|xQ－xR|=．……………7分（Ⅲ）存在点M，N，当其坐标为（－，），（，）时，△PMN为等边三角形．…8分证明如下：当MN⊥x轴时，易得△PMN不可能为等边三角形．当MN⊥y轴时，因为为等边三角形，结合椭圆的对称性，以及（Ⅱ）可得M，N的坐标为（－，），（，），符合题意．……………9分当MN不与坐标轴垂直时，设M（x1，y1），N（x2，y2），MN的中点为D（x0，y0），由得，即，所以kMN=．……………10分因为△PMN为等边三角形，所以kMN·kPD=—1，即，解得y0=，与y0∈矛盾，此时不存在M，N使△PMN是等边三角形．·……………11分-26- 综上，存在M，N，且其坐标为（－，），（，）时，△PMN是等边三角形．……12分解法二：（Ⅰ）同解法一；（Ⅱ）同解法一，可得|QR|=2|QO|=．………6分因为直线QR的方程为y=x，即x+8y=0，所以点P（0，）到直线QR的距离d=．所以S△PMN=|QR|·d=．……………7分（Ⅲ）存在点M，N，当其坐标为（－，），（，）时，△PMN为等边三角形．…8分证明如下：当MN⊥x轴时，易得△PMN不可能为等边三角形．（1）当MN垂直于坐标轴时，同解法一．……………9分（2）当MN不与坐标轴垂直时，设直线MN的方程为y=kx+m（k≠0），M（x1，y1），N（x2，y2）．由得，所以-26- ，MN的中点坐标为．因为△PMN为等边三角形，所以kMN·kPD=—1，即，化简得，（*）又因为，即，这与（*）式矛盾，满足条件的M，N不存在．……………11分综上，存在M，N，当其坐标为（－，），（，）时，△PMN是等边三角形．……12分22．本小题主要考查函数导数的几何意义、导数的运算及导数的应用，考查运算求解能力、抽象概括能力、推理论证能力，考查函数与方程思想、数形结合思想、化归与转化思想、分类与整合思想．满分14分．解：（Ⅰ）由，（x>0）．…………1分令，得，f(x)，的变化情况如下表：(0，1)1（1，+∞）+0－单调递增极大值单调递减-26- 所以函数f(x)的增区间为（0，1），减区间为（1，+∞）．………………………………3分（Ⅱ）．（1）当时，恒成立，此时函数在区间（0，+∞）上单调递增，不合题意，舍去．…………………4分（2）当时，令，得，，的变化情况如下表：+0－单调递增极大值单调递减所以函数的增区间为，减区间为．………………6分要使函数在区间上不单调，须且只须，即．所以对任意给定的正数，只须取实数，就能使得函数在区间上不单调．……………………7分（Ⅲ）假设存在实数，使当时，函数f(x)的值域为．由得．………………………8分令．（1）当时，均在区间（0，+∞）上单调递增，由已知得为方程的两个不等正根．（*）-26- 令，即．要使（*）成立，须且只须存在两个零点．………………………9分因为．①当，即时，在区间（0，+∞）上单调递增，（*）不成立．②当，即时，令，得，此时取到最大值．要使（*）成立，须且只须，得．所以当时，要使（*）成立，须且只须．……………………10分（2）当时，由（Ⅱ）知，在处取到最大值．此时要使命题成立，须且只须有两个零点，结合图形可得：①若，由均在区间上单调递增知，存在符合题意；②若，则取符合的解为即可．由①，②，结合（1）得．……………………13分注意到，所以，且．综上，当时，存在符合题意；-26- 当时，存在符合是题意；当时，满足条件的实数不存在．……14分-26- -26- -26- -26- -26- 内部资料-26- 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