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2017 届福建省南平市高三5月质量检查理科数学试题及答案

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'福建省南平市2014年普通高中毕业班质量检查数学(理)试题本试卷分第1卷(选择题)和第Ⅱ卷(非选择题),第Ⅱ卷第21题为选考题,其他题为必考题.本试卷共6页.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0.5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.做选考题时,考生按照题目要求作答,并用213铅笔在答题卡上把所选题目对应的题号涂黑.5.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: 第I卷(选择题共50分)一、选择题:本大题共10小题,每小题5分,共50分.在每小题给出的四个选项中,只有一项是符合题目要求的.1.函数在区间(0,)内的零点个数是A.0B.1C.2D.32.在△ABC中,若角A、B、C的对边分别是a、b、c,则“a2+b2=b2+ac”,是“A、B、C依次成等差数列”的A.既不充分也不必要条件B.充分不必要条件C.必要不充分条件D.充要条件3.已知等比数列{}中,各项都是正数,且成等差数列,则等于 7.设则二项式展开式中的x3项的系数为A.一20B.20C.一160D.160 A.13B.12C.11D.10第Ⅱ卷(非选择题共100分)二、填空题:本大题共5小题,每小题4分,共20分,把答案填在答题卡的相应位置.11.已知是虚数单位,复数为纯虚数,则实数a的值为.12.已知一个几何体是由上下两部分构成的组合体,其三视图如右图所示,若图中圆的半径为1,等腰三角形的腰长为,则该几何体的体积是.13.已知函数是区间[0,4]内任意 两个实数,则事件“f(1)<0”发生的概率为.14.倾斜角为锐角的直线,与抛物线y2=2x相交于A、B两点,三、解答题:本大题共6小题,共80分.解答应写出文字说明,证明过程或演算步骤.16.(本题满分13分)为减少“舌尖上的浪费”,某学校对在该校食堂用餐的学生能否做到“光盘”,进行随机调查,从中随机抽取男、女生各15名进行了问卷调查,得到了如下列联表: 17.(本小题满分13分)已知函数(I)当x∈A时,函数f(x)取得最大值或最小值,求集合A;(Ⅱ)将集合A中x∈(0,+)的所有x的值,从小到大排成一数列,记为{an},求数列{an}的通项公式;(Ⅲ)令,求数列{bn}的前n项和Tn.18.(本小题满分13分) 19.(本小题满分13分)已知椭圆.(I)若椭圆T的离心率为,过焦点且垂直于z轴的直线被椭圆截得弦长为.(i)求椭圆方程;(ii)过点P(2,1)的两条直线分别与椭圆F交于点A,C和B,D,若AB//CD,求直线AB的斜率;(II)设P(x0,y0)为椭圆T内一定点(不在坐标轴上),过点P的两条直线分别与椭圆厂交于点A,C和B,D,且彻∥CD,类比(I)(ii)直接写出直线彻的斜率.(不必证明)20.(本题满分14分) 21.本题有(1)、(2)、(3)三个选答题,每小题7分,请考生任选2个小题作答,满分14分.如果多做,则按所做的前两题记分.作答时,先用2B铅笔在答题卡上把所选题目对应的题号涂黑,并将所选题号填入括号中.(1)(本小题满分7分)选修4--2:矩阵与变换 (2)(本小题满分7分)选修4--4:坐标系与参数方程在平面直角坐标系中,以坐标原点为极点,工轴的非负半轴为极轴建立极坐标系.已知曲线C的参数方程为(参数),直线三的极坐标方程为(I)写出曲线C的普通方程与直线三的直角坐标方程。(II)P为曲线C上一点,求尸到直线工距离的最小值.(3)(本小题满分7分)选修4--5:不等式选讲已知函数(I)求的最大值;(Ⅱ)恒成立,求实数f的取值范围. 2014年南平市高中毕业班适应性考试理科数学试题参考答案及评分标准说明:一、本解答指出了每题要考查的主要知识和能力,并给出了一种或几种解法供参考,如果考生的解法与本解答不同,可根据试题的主要考查内容比照评分标准制定相应的评分细则.二、对计算题,当考生的解答在某一步出现错误时,如果后继部分的解答未改变该题的内容和难度,可视影响的程度决定后继部分的给分,但不得超过该部分正确解答应给分数的一半;如果后继部分的解答有较严重的错误,就不再给分.三、解答右端所注分数,表示考生正确做到这一步应得的累加分数.四、只给整数分数.选择题和填空题不给中间分.一、选择题:本大题考查基础知识和基本运算.每小题5分,满分50分.1.B;2.D;3.C;4.B;5.A;6.C;7.C;8.A;9.A;10.D.二、填空题:本题考查基础知识和基本运算,每小题4分,满分20分.11.2;12.;13.;14.;15.三、解答题:本大题共6小题,共80分,解答应写出文字说明、证明过程或演算步骤.16.本题满分13分. 男性女性合计做不到“光盘”517能做到“光盘”313合计1515解:(Ⅰ)…………3分由已知数据得,所以,有99%以上的把握认为“在学校食堂用餐的学生能否做到‘光盘’与性别有关”…………6分(Ⅱ)的可能取值为0,1,2…………7分,,…………10分所以的分布列为:012的数学期望为…………13分17.本题满分13分.解:(Ⅰ)…………1分…………3分当函数取得最值时,集合…………4分(Ⅱ)的所有的值从小到大依次是.…………6分即数列的通项公式是…………8分(Ⅲ)由(Ⅱ)得 …………10分…………11分…………13分18.本题满分13分.(Ⅰ)证明:在正方体ABCD-A1B1C1D1中,∵,…………1分∵E,F分别为A1B1,B1C1的中点,…………2分,,,又…………3分…………4分(Ⅱ)∵OP与DM相交,OP与DM确定一个平面,P为正方体底面ABCD上的点…………5分平面,平面…………6分∵在正方体ABCD-A1B1C1D1中,…………7分(Ⅲ)如图以为原点,,,所在直线分别为x轴,ABA1CB1D1C1OEMFDzyxPy轴,z轴建立空间直角坐标系,则 ,…………8分设,由,得=解得,,即…………10分由(Ⅱ)可知:面CPD与面ABCD共面,面DOP与面DOM共面,面ABCD的一个法向量为设面DOM的一个法向量为,,∴由,可得令,则,,即…………12分,故…………13分19.本题满分13分.解:(Ⅰ)(ⅰ)由题意得解得…………2分则椭圆Г的方程为.…………3分(ⅱ)设点则,故,.…………5分因为点在椭圆上,所以,则 整理得=…………6分由点在椭圆上知,故①…………7分又∥,则同理可得②…………8分①-②得由题意可知,则直线的斜率为.…………10分(Ⅱ)直线的斜率为.…………13分20.本题满分14分.解:(Ⅰ)曲线方程为,设切点为由得切线的斜率,则切线方程为…………2分因为切线过点,所以,即故所求切线方程为…………3分(Ⅱ)函数的定义域为,.令并结合定义域得对应一元二次方程的判别式.…………5分①当,即时,,则函数的增区间为; ②当时,函数的增区间为(0,;③当时,函数的增区间为…………7分(Ⅲ),令得由题意知方程有两个不相等的正数根,则解得,解方程得,则.…………9分又由得,所以=,当时,,即函数是上的增函数所以,故的取值范围是.则.…………11分同理可求,=,,即函数是上的减函数所以,故的取值范围是则=或=0…………12分当=时,;当=时,.…………14分 21.本题满分14分.(1)解(Ⅰ)由…………1分得解得,…………2分…………3分(Ⅱ)…………4分设为上任一点,在变换作用下的对应点为,则…………5分得,代入得…………6分即所求的曲线方程为…………7分(2)解:(Ⅰ)由得曲线C的普通方程:…………1分可化为…………2分即直线L的直角坐标方程:…………3分(Ⅱ)解法一:设曲线C上任一点为它到直线的距离为…………5分当时,…………7分解法二:设与直线L平行且与曲线C相切的直线方程为由可化得…………5分 得…………6分当时,曲线C上点到直线L距离取最小值…………7分(3)解:(Ⅰ)≤…………2分当且仅当≤-2时等号成立,…………3分(说明:通过数形结合直接答出也给3分)(Ⅱ)由恒成立得…………4分即,或…………5分解得:或…………6分实数的取值范围是…………7分内部资料仅供参考9JWKffwvG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXp6X4NGpP$vSTT#UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmA&UE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmA&UE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz84!z89Amv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu##KN&MuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%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