2017 届福建省三明市高三5月质量检查文科数学试题及答案

'2014年三明市普通高中毕业班质量检查文科数学本试卷分第I卷（选择题）和第Ⅱ卷（非选择题）．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．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，…，的标准差锥体体积公式其中为样本平均数其中为底面面积，为高柱体体积公式球的表面积、体积公式 其中为底面面积，为高其中为球的半径第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．1．设是虚数单位，那么复数等于A．B．C．D．2．已知集合，，则为A．B．C．D．3．观察下列关于变量和 的三个散点图，它们从左到右的对应关系依次是A．正相关、负相关、不相关B．负相关、不相关、正相关C．负相关、正相关、不相关D．正相关、不相关、负相关4．命题：“，都有”的否定是A．，都有B．，都有C．，使得D．，使得开始输入是否输出结束5．函数的单调递增区间是A．B．C．D．6.某程序框图如图所示，若输入，则该程序运行后输出的值分别是A．B.C.D.7．直线与圆相交所得线段的长度为111正视图俯视图侧视图A．B．C．D． 8．某几何体的三视图如右图所示，则该几何体的表面积是A．B．C．D．9．若均为区间的随机数，则的概率为A．B．C．D．10.对于函数在定义域内的任意实数及，都有及成立，则称函数为“函数”.现给出下列四个函数：；.其中是“函数”的是A．　　　　　　B．　　　　　C．　　　　　　　D．11．在边长为2的等边中，是的中点，为线段上一动点，则的取值范围是A．B．C．D． 12．设函数的导函数为，那么下列说法正确的是A.若，则是函数的极值点B.若是函数的极值点，则C.若是函数的极值点，则可能不存在D.若无实根，则函数必无极值点第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．把答案填在答题卡相应位置．13．在等差数列中，若,则．14.已知椭圆的焦点是双曲线的顶点，双曲线的焦点是椭圆的长轴顶点，若两曲线的离心率分别为则______.15．已知若直线与直线互相垂直，则的最小值是．16．定义表示所有满足的集合 组成的有序集合对的个数．试探究，并归纳推得=_________.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.频率/组距0.0120.0160.018分80605070901000.0300.02417．（本小题满分12分）某校为了解高一期末数学考试的情况，从高一的所有学生数学试卷中随机抽取份试卷进行成绩分析，得到数学成绩频率分布直方图（如图所示），其中成绩在的学生人数为6.（Ⅰ）估计所抽取的数学成绩的众数；（Ⅱ）用分层抽样的方法在成绩为和这两组中共抽取5个学生，并从这5个学生中任取2人进行点评，求分数在恰有1人的概率.18．（本小题满分12分） ……将数列按如图所示的规律排成一个三角形数表，并同时满足以下两个条件：①各行的第一个数构成公差为的等差数列；②从第二行起，每行各数按从左到右的顺序都构成公比为的等比数列.若，，.（Ⅰ）求的值；（Ⅱ）求第行各数的和.19．（本小题满分12分）如图，在三棱锥中，平面平面，于点，且，，（Ⅰ）求证：（Ⅱ）（Ⅲ）若，， 求三棱锥的体积．20．（本小题满分12分）已知抛物线（）的准线与轴交于点．（Ⅰ）求抛物线的方程，并写出焦点坐标；（Ⅱ）是否存在过焦点的直线（直线与抛物线交于点，），使得三角形的面积？若存在，请求出直线的方程；若不存在，请说明理由．21．（本小题满分12分）设向量，定义一种向量积．已知向量，，点为的图象上的动点，点为的图象上的动点，且满足（其中为坐标原点）．（Ⅰ）请用表示；（Ⅱ）求的表达式并求它的周期； （Ⅲ）把函数图象上各点的横坐标缩小为原来的倍（纵坐标不变），得到函数的图象.设函数，试讨论函数在区间内的零点个数.22．（本小题满分14分）已知函数为自然对数的底数）．（Ⅰ）求曲线在处的切线方程；（Ⅱ）若是的一个极值点，且点，满足条件：.（ⅰ）求的值；（ⅱ）求证：点，，是三个不同的点，且构成直角三角形． 2014年三明市普通高中毕业班质量检查文科数学试题参考解答及评分标准一、选择题：1．B2．B3．D4．C5．C6．A7．D8．A9．D10．A11．A12．B二、填空题：13．21；14．1；15．2；16．．三、解答题：17．解：（Ⅰ）由频率分布直方图可知：样本的众数为75．……………………………3分（Ⅱ）由频率分布直方图可得：第三组的频率：，所以，………………………………………………………………4分第四组的频数：；第五组的频数：；用分层抽样的方法抽取5份得： 第四组抽取：；第五组抽取：．…………7分记抽到第四组的三位同学为，抽到第五组的两位同学为则从5个同学中任取2人的基本事件有：，，共10种．其中分数在恰有1人有：，共6种．所求概率：．………………………………………………………12分18．解：（Ⅰ）依题意得，，所以．……………………………………………2分又，，所以的值分别为 ．…………………………………6分（Ⅱ）记第行第1个数为，由（1）可知：，………………7分又根据此数表的排列规律可知：每行的总个数构成一个以1为首项，2为公差的等差数列，所以第行共有个数，………………………………9分第行各数为以为首项，为公比的等比数列，因此其总数的和．…………………………12分19．解：（Ⅰ），……2分………………3分（Ⅱ）因为平面平面， 且平面平面，平面，，所以平面，……………6分又平面，所以平面平面．…………7分（Ⅲ）由（Ⅱ）可知平面．法一：中，，由正弦定理，得，因为，所以，则，因此，…………8分△的面积．…………………………10分所以三棱锥的体积．…………………………12分法二：中，，，由余弦定理得：，所以，所以．…………………… ……………8分△的面积．……………10分所以三棱锥的体积．……………………12分20．解法一：（Ⅰ）由已知得：，从而抛物线方程为，焦点坐标为．……………………4分（Ⅱ）由题意，设，并与联立，得到方程：，…………………………………………………6分设，，则，．…………………7分∵，∴，……9分 又，∴……………………………………10分解得，………………………………………………………………11分故直线的方程为：．即或．…………………12分解法二：（Ⅰ）（同解法一）（Ⅱ）当轴时，，，不符合题意．……………………………………………………………5分故设（），并与联立，得到方程：，……………………………6分设，，则， ．…………………7分，点到直线的距离为，………………9分∴，…………10分解得，…………………………………………………………11分故直线的方程为：．即或．………12分21．解：（Ⅰ），……………2分（Ⅱ），所以 ，……………………4分因此即………………………………6分所以，它的周期为．………………………………8分（Ⅲ）在上单调递增，在上单调递减，又，……………………………10分函数在区间内只有一个零点；函数在区间内有两个零点；当或时，函数在区间内没有零点．…………………………12分22．解：（Ⅰ） ，……………………………………2分，又，…………………………………………4分所以曲线在处的切线方程为，即．…………………………5分（Ⅱ）（ⅰ）对于，定义域为．当时，，，∴；当时，；当时，，，∴，………………8分所以存在唯一的极值点，∴，则点为．…………………9分（ⅱ）若，则，，与条件不符，从而得． 同理可得．………………………………………………10分若，由，此方程无实数解，从而得．………………………………………………………11分由上可得点，，两两不重合．又从而，点，，可构成直角三角形．………………………14分内部资料仅供参考 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