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2017 届福建省三明市高三5月质量检查理科数学试题及答案

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'2014年三明市普通高中毕业班质量检查理科数学本试卷分第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)笔或碳素笔书写,字体工整、笔记清楚.4.做选考题时,考生按照题目要求作答,并用2B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: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.若复数满足(其中为虚数单位),则复数为A.B.C.D.2.已知集合,集合,则等于A.B.C.D.3.观察下列关于两个变量和的三个散点图,它们从左到右的对应关系依次为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.设是两条不同直线,是两个不同平面,下列四个命题中正确的是A.若与所成的角相等,则B.若,,,则C.若,,,则D.若,,,则5.在二项式的展开式中恰好第5项的二项式系数最大,则展开式中含项的系数是A.-56B.-35C.35D.566.设且,命题:函数在上是增函数,命题:函数在上是减函数,则是的A.充分不必要条件B.必要不充分条件C.充分必要条件D.既不充分也不必要条件7.已知双曲线与椭圆有相同的焦点,则该双曲线的渐近线方程为 A.B.C.D.8.如图是某个四面体的三视图,若在该四面体的外接球内任取一点,则点落在四面体内的概率为A.B.C.D.9.已知函数则函数的零点个数为A.1B.2C.3D.410.在数列中,,且,,若数列满足,则数列是A.递增数列B.递减数列C.常数列D.摆动数列 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.曲线与直线及轴所围成的图形的面积是.12.执行如图所示的程序框图,若输入的,则输出的结果是____.13.已知变量满足约束条件若取整数,则目标函数的最大值是. 14.已知矩形的周长为36,矩形绕它的一条边旋转形成一个圆柱,则旋转形成的圆柱的侧面积的最大值为.15.对于集合,如果定义了一种运算“”,使得集合中的元素间满足下列4个条件:(ⅰ),都有;(ⅱ),使得对,都有;(ⅲ),,使得;(ⅳ),都有,则称集合对于运算“”构成“对称集”.下面给出三个集合及相应的运算“”:①,运算“”为普通加法;②,运算“”为普通减法;③,运算“”为普通乘法.其中可以构成“对称集”的有 .(把所有正确的序号都填上)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)某食品厂为了检查一条自动包装流水线的生产情况,从该流水线上随机抽取40件产品作为样本,测得它们的重量(单位:克),将重量按如下区间分组:,,,,,得到样本的频率分布直方图(如图所示).若规定重量超过495克但不超过510克的产品为合格产品,且视频率为概率,回答下列问题:(Ⅰ)在上述抽取的40件产品中任取2件,设为合格产品的数量,求的分布列和数学期望;(Ⅱ)若从流水线上任取3件产品,求恰有2件合格产品的概率. 17.(本小题满分13分)如图,在四棱锥中,底面是直角梯形,,,PABCDO17题图平面平面,若,,,,且.(Ⅰ)求证:平面;(Ⅱ)设平面与平面所成二面角的大小为,求的值.18.(本小题满分13分)已知点是抛物线上不同的两点,点在抛物线的准线上,且焦点到直线的距离为. (I)求抛物线的方程;(Ⅱ)现给出以下三个论断:①直线过焦点;②直线过原点;③直线平行轴.请你以其中的两个论断作为条件,余下的一个论断作为结论,写出一个正确的命题,并加以证明.19.(本小题满分13分)若函数,非零向量,我们称为函数的“相伴向量”,为向量的“相伴函数”.(Ⅰ)已知函数的最小正周期为,求函数的“相伴向量”;(Ⅱ)记向量的“相伴函数”为,将图象上所有点的横坐标伸长到原来的2倍(纵坐标不变),再将所得的图象上所有点向左平移个单位长度,得到函数,若,求的值;(Ⅲ)对于函数,是否存在“相伴向量”?若存在,求出“相伴向量”; 若不存在,请说明理由.20.(本小题满分14分)已知函数,,且在点处的切线方程为.(Ⅰ)求的值;(Ⅱ)若函数在区间内有且仅有一个极值点,求的取值范围;(Ⅲ)设为两曲线,的交点,且两曲线在交点处的切线分别为.若取,试判断当直线与轴围成等腰三角形时值的个数并说明理由.21.本题设有(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:矩阵与变换若二阶矩阵满足:.(Ⅰ)求二阶矩阵;(Ⅱ)若曲线在矩阵所对应的变换作用下得到曲线,求曲线的方程.(2)(本小题满分7分)选修4—4:坐标系与参数方程已知在平面直角坐标系中,圆的方程为.以原点为极点,以轴正半轴为极轴,且与直角坐标系取相同的单位长度,建立极坐标系,直线的极坐标方程为.(Ⅰ)求直线的直角坐标方程和圆的参数方程;(Ⅱ)求圆上的点到直线的距离的最小值.(3)(本小题满分7分)选修4—5:不等式选讲设函数.(Ⅰ)求不等式的解集;(Ⅱ)若存在实数,使得 成立,求实数的取值范围.2014年三明市普通高中毕业班质量检查理科数学试题参考解答及评分标准一、选择题1.D2.B3.D4.C5.A6.D7.A8.C9.B10.C二.填空题:11.12.6213.514.15.①、③三、解答题:16.解:(Ⅰ)由样本的频率分布直方图得,合格产品的频率为.………………………………………………2分所以抽取的40件产品中,合格产品的数量为.……………………………3分则 可能的取值为0,1,2, …………………………………………4分所以,,,因此的分布列为012……………………7分故数学期望.…………………9分(Ⅱ)因为从流水线上任取1件产品合格的概率为,……………10分所以从流水线上任取3件产品,恰有2件合格产品的概率为.……………………………………………13分17.解:(Ⅰ)因为,,所以 ,……………1分在中,由余弦定理,得,……………………………………3分,,………………………………………………4分,…………………………………………………………………5分又平面平面,平面平面,平面,平面.………………………………………………………………6分EABCDOPxzy(Ⅱ)如图,过作交于,则, ,两两垂直,以为坐标原点,分别以,,所在直线为轴,建立空间直角坐标系,…………………………7分则,,………8分,,……………………9分设平面的一个法向量为,由得即取则,所以为平面的一个法向量.……………………………11分平面,为平面的一个法向量.所以,……………………………… 12分.…………………………………………………13分18.解:(I)因为,依题意得,…………………………2分解得,所以抛物线的方程为…………………………………4分(Ⅱ)①命题:若直线过焦点,且直线过原点,则直线平行轴.…………………………………5分设直线的方程为,,………………………6分由得, ,……………………………………………8分直线的方程为,……………………………………………9分所以点的坐标为,,……………………………………………………12分直线平行于轴.………………………………………………………13分②命题:若直线过焦点,且直线平行轴,则直线过原点.…………………………………5分设直线的方程为, ,………………………6分由得,,……………………………………………8分即点的坐标为,……………………………………………9分∵直线平行轴,∴点的坐标为,…………………………10分∴,,由于,∴∥,即三点共线,……………………………………………12分∴直线过原点.………………………………………………………13分 ③命题:若直线过原点,且直线平行轴,则直线过焦点.…………………………………5分设直线的方程为,则点的坐标为,…………6分∵直线平行轴,∴,∴,即点的坐标为,……………………8分由得,∴即点的坐标为,……………………………10分∴,由于,∴∥,即三点共线,………………………………………12分 ∴直线过焦点.………………………………………………………13分19.解:(Ⅰ),………………………………………1分依题意得,故.………………………………………2分∴,即的“相伴向量”为(1,1).   ………3分(Ⅱ)依题意,,……………………………4分将 图象上所有点的横坐标伸长到原来的2倍(纵坐标不变),得到函数,………………………………………………………5分再将所得的图象上所有点向左平移个单位长度,得到,即,……………………………6分∵,∴,∵,∴,∴,……………8分∴.………………………………………………………10分(Ⅲ)若函数存在“相伴向量”, 则存在,使得对任意的都成立,……………11分令,得,因此,即或,显然上式对任意的不都成立,所以函数不存在“相伴向量”.…………………………13分(注:本题若化成,直接说明不存在的,给1分)20.解:(Ⅰ),∴,又,∴.…………………………………3分(Ⅱ);∴由得, ∴或.…………………………………5分∵,当且仅当或时,函数在区间内有且仅有一个极值点.…………………………………6分若,即,当时;当时,函数有极大值点,若,即时,当时;当时,函数有极大值点,综上,的取值范围是.…………………………………8分(Ⅲ)当时,设两切线的倾斜角分别为,则,∵,∴ 均为锐角,…………………………………………9分当,即时,若直线能与轴围成等腰三角形,则;当,即时,若直线能与轴围成等腰三角形,则.由得,,得,即,此方程有唯一解,直线能与轴围成一个等腰三角形.……11分由得,,得,即,设,,当时,,∴在单调递增,则在单调递增,由于,且,所以,则, 即方程在有唯一解,直线能与轴围成一个等腰三角形.因此,当时,有两处符合题意,所以直线能与轴围成等腰三角形时,值的个数有2个.………………………………………14分21.(1)解:(Ⅰ)设,则,,…………2分.   …………………………3分(Ⅱ),即 …………………………………………4分代入可得,即,故曲线的方程为.……………………………………7分21.(2)解:(Ⅰ)由,得,,即,………………………1分设………………………2分所以直线的直角坐标方程为;圆的参数方程为参数.…………………………………3分(Ⅱ)设,则点到直线的距离为 ,………………………5分当即时,.圆上的点到直线的距离的最小值为.………………………7分(21)(3)解:(Ⅰ)当时,由得,所以;当时,由得,所以;当时,由得,所以.…………2分综上不等式的解集.………………3分(Ⅱ), ……………………………………4分由柯西不等式得,,…………………………………………………………5分当且仅当时取“=”,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%&qYp@Eh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tn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