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2018年福建漳州市普通高中毕业班质量检查理科数学试题及答案

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'2014年福建省漳州市普通高中毕业班质量检查数学(理科)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.答第Ⅰ卷前,考生务必需将自己的姓名、准考证号、考试科目涂写在答题卡上.2.每小题选出答案后,用2B铅笔把答题卡上对应题目的答案标号涂黑,如需改动,用橡皮擦干净后,再选涂其它答案,不能答在试题卷上.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.将正确答案填写在答题卷相应位置.1.已知i是虚数单位,则等于A.-1+iB.-1-iC.1+iD.1-i2.展开式中的常数项为A.6B.8C.10D.123.已知正三棱柱(侧棱与底面垂直,底面是正三角形)的高与底面边长均为1,其直观图和正(主)视图如图所示,则它的左(侧)视图的面积是A.B.1C.D.4.已知均为单位向量,它们的夹角为,则等于A.1B.C.D.25.执行如图所示的程序框图,如果输入,则输出的的值为A.7B.9C.11D.136.数列的前项和为,若,则等于A.B.C.D.7.已知函数的定义域为[a,b],值域为[-2,1],则的值不可能是-10- A.B.C.D.8.已知正三角形ABC的顶点A(1,1),B(1,3),顶点C在第一象限,若点(x,y)在△ABC内部,则z=-x+y的取值范围是A.(1-,2)B.(0,2)C.(-1,2)D.(0,1+)9.已知为上的可导函数,且,均有,则以下判断正确的是A.B.C.D.大小无法确定10.已知F1,F2分别是双曲线的左、右焦点,过F1的直线与双曲线的左、右两支分别交于A、B两点.若ΔABF2是等边三角形,则该双曲线的离心率为A.2B.C.D.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.=_________.12.等差数列中,,数列是等比数列,且,则的值为.13.在区间[﹣2,4]上随机地取一个数x,则满足|x|≤3的概率为.14.过圆x2+y2=1上一点作圆的切线与x轴、y轴的正半轴交于A、B两点,则|AB|的最小值为.15.定义全集U的非空子集P的特征函数表示集合P在全集U的补集.已知均为全集U的非空子集,给出下列命题:①若,则对于任意;②对于任意;③对于任意;④对于任意.则正确命题的序号为.三、解答题(本大题共6小题,共80-10- 分,解答应写在答题卷相应位置,要写出文字说明、证明过程或演算步骤.)16.(本小题满分13分)已知向量,函数.(I)求函数的最小正周期及单调递增区间;(II)已知中,角的对边分别为,若,,求的面积.17.(本小题满分13分)某电视台组织部分记者,用“10分制”随机调查某社区居民的幸福指数.现从调查人群中随机抽取16名,如图所示的茎叶图记录了他们的幸福指数的得分(以小数点前的一位数字为茎,小数点后的一位数字为叶):(Ⅰ)指出这组数据的众数和中位数;(Ⅱ)若幸福指数不低于9.5分,则称该人的幸福指数为“极幸福”.求从这16人中随机选取3人,至多有1人是“极幸福”的概率;(Ⅲ)以这16人的样本数据来估计整个社区的总体数据,若从该社区(人数很多)任选3人,记表示抽到“极幸福”的人数,求的分布列及数学期望.18.(本小题满分13分)在四棱锥P-ABCD中,侧面PCD底面ABCD,PDCD,底面ABCD是直角梯形,AB∥DC,,,.(I)求证:BC平面PBD:(II)求直线AP与平面PDB所成角的正弦值;(Ⅲ)设E为侧棱PC上异于端点的一点,,试确定的值,使得二面角E-BD-P的余弦值为.19.(本小题满分13分)-10- 已知抛物线C:y2=2px(p>0)的焦点F和椭圆的右焦点重合,直线过点F交抛物线于A、B两点.(I)求抛物线C的方程;(II)若直线交y轴于点M,且,m、n是实数,对于直线,m+n是否为定值?若是,求出m+n的值;否则,说明理由.20.(本小题满分14分)巳知函数,,其中.(Ⅰ)若是函数的极值点,求的值;(II)若在区间上单调递增,求的取值范围;(Ⅲ)记,求证:.21.本题(1)、(2)、(3)三个选答题,每小题7分,请考生任选2题作答,满分14分,如果多做,则按所做的前两题计分。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:矩阵与变换已知矩阵.(Ⅰ)求的逆矩阵;(Ⅱ)求矩阵的特征值、和对应的特征向量、.(2)(本小题满分7分)选修4—4:坐标系与参数方程在直角坐标系中,直线的方程为,曲线的参数方程为.(Ⅰ)已知在极坐标系(与直角坐标系取相同的长度单位,且以原点为极点,以轴正半轴为极轴)中,点的极坐标为,判断点与直线的位置关系;(Ⅱ)设点是曲线上的一个动点,求它到直线的距离的最小值.(3)(本小题满分7分)选修4-5:不等式选讲已知且,若恒成立,(Ⅰ)求的最小值;(Ⅱ)若对任意的恒成立,求实数的取值范围.-10- 2014漳州市高中毕业班质量检查理科数学试卷参考答案一、选择题:1-5D,A,D,C,B,6-10D,C,A,B,B.二、填空题:11.9;12.16;13.;14.2;15..①②③.三、解答题:16.解:(I)依题意,得              ………………………………………………3分   ∴的最小正周期为,………………………………………………4分由得:即的递增区间是.………………………………6分(II)由得,,∴,∵,∴,∴,∴,………………………………………………8分∵,∴根据余弦定理得,,∴,………………………………………………11分∴………………………………………13分17.解:(Ⅰ)众数:8.6;中位数:8.75;………………………………………2分(Ⅱ)设表示所取3人中有个人是“极幸福”,至多有1人是“极幸福”记为事件,则;………………………………6分(Ⅲ)的可能取值为0,1,2,3.………………………………………7分;;…………………9分;.…………………11分的分布列为:…………………12分所以.……………………13分另解:的可能取值为0,1,2,3,-10- 则,因此.…………………9分有;;;.…………………11分的分布列为:…………………12分所以=.……………………13分18.解:(Ⅰ)证明:因为侧面⊥底面,⊥,所以⊥底面,所以⊥.又因为=,即⊥,以为原点建立如图所示的空间直角坐标系,则,,,,所以所以,所以。由⊥底面,可得,又因为,所以⊥平面.……………………4分(Ⅱ)由(Ⅰ)知平面的一个法向量为,所以设直线AP与平面PDB所成角为,则……………………7分(Ⅲ)因为,又,设则所以,.………………………8分设平面的法向量为,因为,由,,得,……………9分令,则可得平面的一个法向量为……………10分所以,……………………………11分解得或,………………………12分又由题意知,故.………………………13分19.解:(Ⅰ)∵椭圆的右焦点,∴,得,-10- ∴抛物线C的方程为.……………………………3分(II)由已知得直线的斜率一定存在,所以设:,与y轴交于,设直线交抛物线于,………………………………………4分由………………………………………5分∴,………………7分又由即m=,同理,………………………………………10分∴………………………………12分所以,对任意的直线,m+n为定值-1………………………………13分20.解:(Ⅰ)由,得,……………………………1分∵是函数的极值点,∴,解得,………………………………2分经检验为函数的极值点,所以.………………………………3分(II)∵在区间上单调递增,∴在区间上恒成立,………………………4分∴对区间恒成立,………………………………5分令,则当时,,有,………………7分∴的取值范围为.………………………………8分(Ⅲ)解法1:,……………………………9分令,则…………………………11分令,则,显然在上单调递减,在上单调递增,-10- 则,则,………………………………13分故.………………………………14分解法2:………………………………9分则表示上一点与直线上一点距离的平方.由得,让,解得,∴直线与的图象相切于点,………………………12分(另解:令,则,可得在上单调递减,在上单调递增,故,则,直线与的图象相切于点),点(1,0)到直线的距离为,则.……………………………14分21(1)解:(Ⅰ),……………………………1分∴.……………………………2分(Ⅱ)矩阵的特征多项式为,………………3分令,得,……………………………5分当时,得,当时,得.…………………………7分21(2)解:(I)把极坐标系下的点化为直角坐标得,满足方程,点在直线上.……………2分(II)解法一、因为点是曲线上的点,故可设点的坐标为,所以点到直线的距离……………5分所以当时,取得最小值……………7分解法二、曲线的普通方程为:,……………1分平移直线到使之与曲线相切,设,-10- 由得:,即:…2分由,解得:,……………5分曲线上的点到距离的最小值.…………7分21(3)解:(Ⅰ),,(当且仅当,即时取等号)又∵恒成立,∴.故的最小值为3.…………………………4分(II)要使恒成立,须且只须.∴或或∴或.…………………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%&qYp@Eh5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