2016届福建省福州八中高三上学期第一次质量检查理科数学试题及答案

'福州八中2015届高三上学期第一次质量检查数学（理）试题考试时间：120分钟试卷满分：150分2014.8.29本试卷分第I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．本试卷满分150分．考试时间120分钟．参考公式：样本数据x1，x2，…，xn的标准差s=其中为样本平均数锥体体积公式V=Sh其中S为底面面积，h为高柱体体积公式V=Sh其中S为底面面积，h为高球的表面积、体积公式，其中R为球的半径第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，在每个小题给出的四个选项中，只有一项是符合题目要求的.把正确选项涂在答题卡的相应位置上．）第1题1．如图，在复平面内，若复数对应的向量分别是，则复数所对应的点位于A．第一象限 B．第二象限C．第三象限D．第四象限第2题2.一个简单几何体的主视图、侧视图如图所示，则其俯视图不可能为①长、宽不相等的矩形；②正方形；③圆；④三角形.其中正确的是A.①②B.②③C.③④D.①④3.命题“对任意，均有”的否定为A.对任意，均有B.对任意，均有C.存在，使得D.存在，使得4.对具有线性相关关系的变量，，有一组观测数据(，)(i=1，2，…，8)，其回归直线方程是：，且，则实数的值是A．B．C．D．5.已知为两条不同的直线，为一个平面。若，则 “”是“”的A.充分不必要条件　B.必要不充分条件C.充要条件　　D.既不充分又不必要条件6.已知在各项均不为零的等差数列中，，数列是等比数列，且，则等于A．2B.4C．8D.167.已知正方体的棱长为2，在正方体的外接球内任取一点，则该点落在正方体内的概率为第8题A．B.C．D.8.执行如图所示的程序框图，若输出的结果是9，则判断框内m的取值范围是A.(42，56]B.(56，72]C.(72，90]D.(42，90) 9.已知向量若则的值为A.B.C.D.10.已知集合，若对于任意，存在，使得成立，则称集合是“集合”.给出下列4个集合：①②③④其中所有“集合”的序号是A.①②④B.②③C.③④D.①③④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．把答案填在答题卡相应位置．11.已知抛物线的焦点是双曲线的一个焦点，则此双曲线的实轴长为.12.设变量、满足约束条件，则 的最大值为.13.若二项式的展开式中的常数项为，则=.14.已知，.若偶函数满足（其中,为常数），且最小值为1，则．15.对于个互异的实数，可以排成行列的矩形数阵，右图所示的行列的矩形数阵就是其中之一．将个互异的实数排成行列的矩形数阵后，把每行中最大的数选出，记为，并设其中最小的数为；把每列中最小的数选出，记为，并设其中最大的数为.两位同学通过各自的探究，分别得出两个结论如下：①和必相等；②和可能相等；③可能大于；④可能大于．以上四个结论中，正确结论的序号是_______（请写出所有正确结论的序号）．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.(本小题满分13分) 为增强市民的节能环保意识,某市面向全市征召义务宣传志愿者.从符合条件的500名志愿者中随机抽取100名志愿者,其年龄频率分布直方图如图所示,其中年龄分组区间是:.（Ⅰ）求图中的值并根据频率分布直方图估计这500名志愿者中年龄在岁的人数;（Ⅱ）在抽出的100名志愿者中按年龄采用分层抽样的方法抽取20名参加中心广场的宣传活动,再从这20名中采用简单随机抽样方法选取3名志愿者担任主要负责人.记这3名志愿者中“年龄低于35岁”的人数为,求的分布列及数学期望.17．（本小题满分13分）某同学用“五点法”画函数在某一个周期内的图象时，列表并填入的部分数据如下表：0000（Ⅰ）请写出上表的、、，并直接写出函数的解析式； （Ⅱ）将的图象沿轴向右平移个单位得到函数的图象，、分别为函数图象的最高点和最低点（如图），求的大小.18.(本小题满分13分)如图直三棱柱中，，是上一点，且平面.（I）求证：平面；（Ⅱ）在棱是否存在一点，使平面与平面的夹角等于，若存在，试确定点的位置，若不存在，说明理由. 19.（本小题满分13分）已知椭圆的两个焦点分别为，，离心率为，过的直线与椭圆交于，两点，且的周长为．（Ⅰ）求椭圆的方程；（Ⅱ）过原点的两条互相垂直的射线与椭圆分别交于，两点，证明：点到直线的距离为定值，并求出这个定值．20.(本小题满分14分)已知函数.（I）若函数在定义域内为增函数，求实数的取值范围；（Ⅱ）在（1）的条件下，若，，，求的极小值；(Ⅲ)设，若函数存在两个零点，且满足，问：函数在 处的切线能否平行于轴？若能，求出该切线方程，若不能，请说明理由.21.本题有（1）、（2）、（3）三个选答题，每小题7分，请考生任选2小题作答，满分14分．如果多做，则按所做的前两题记分．（1）（本小题满分7分）选修4－2：矩阵与变换把曲线先进行横坐标缩为原来的一半，纵坐标保持不变的伸缩变换，再做关于轴的反射变换变为曲线，求曲线的方程．（2）（本小题满分7分）选修4一4：坐标系与参数方程在直角坐标系中，直线的参数方程为：（为参数），以为原点，轴为极轴，单位长度不变，建立极坐标系，曲线的极坐标方程为：①写出直线和曲线的普通方程。②若直线和曲线相切，求实数的值。（3）（本小题满分7分）选修4一5：不等式选讲设函数f(x)=|x－4|+,（Ⅰ）求f(x)的最小值m（Ⅱ）当(a,b,c∈R)时,求的最小值. 福州八中2014—2015学年高三毕业班第一次质量检查数学(理)试卷参考答案及评分标准一、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．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．17.解：（Ⅰ），，……………………3分 ……………………6分（Ⅱ）将的图像沿轴向右平移个单位得到函数…………7分因为、分别为该图像的最高点和最低点，所以…………………………………………9分所以…………………………………………10分………………12分所以…………………………………………13分法2：法3：利用数量积公式， 19.解：（I）由题意知，，所以．因为所以，所以．所以椭圆的方程为．------4分（II）由题意,当直线的斜率不存在，此时可设，.又，两点在椭圆上，所以，．所以点到直线的距离．--------6分当直线的斜率存在时，设直线的方程为． 所以．-----------11分整理得，满足．所以点到直线的距离为定值．--------13分20.解：（Ⅰ）由题意，知恒成立，即.……2分又，当且仅当时等号成立.故，所以.……4分 设，⑤式变为设，所以函数在上单调递增，因此，，即也就是，，此式与⑤矛盾.所以在处的切线不能平行于轴.……14分21.（1）解：先伸缩变换M=后反射变换N=,得A=NM==……………4分 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