2018年福建省莆田市普通高中毕业班质量检查理科数学试题 及答案

'www.ks5u.com2015年莆田市高中毕业班教学质量检查试卷数学（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O．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．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-20- 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．把答案填涂在答题卡相应位置．1．下列函数中，为奇函数的是（）A．y=x+1B．y=x2C．y=2xD．y=x|x|2．已知，复数（i为虚数单位）在复平面内对应的点为M，则“”是“点M在第四象限”的（）A．充分而不必要条件B．必要而不充分条件C．充分必要条件D．既不充分也不必要条件3．若a＞0，b＞0，a+b=1，则的最小值是()A．2B．3C．4D．5-20- 4．函数图象的一条对称轴方程为（）A．x＝－B．C．x＝D．x＝5．某几何体的三视图如图所示，其中正视图是腰长为2的等腰三角形，俯视图是半径为1的半圆，则其侧视图的面积是（）A．B．C．1D．6．阅读右图所示的程序框图，运行相应的程序，若输出的的值等于126，则判断框中的①可以是（　　）A．i>4？B．i>5？C．i>6？D．i>7？7．若直线y=kx－k交抛物线于A，B两点，且线段AB中点到轴的距离为3，则=（）A．12B．10C．8D．68．学校将5个参加知识竞赛的名额全部分配给高一年段的4个班级，其中甲班级至少分配2个名额，其它班级可以不分配名额或分配多个名额，则不同的分配方案共有（）A．20种B．24种C．26种D．30种9．常用以下方法求函数的导数：先两边同取以e为底的对数（e≈2．71828…，为自然对数的底数）得-20- ，再两边同时求导，得，即．运用此方法可以求函数(x>0)的导函数．据此可以判断下列各函数值中最小的是()A．B．C．D．10．如图，所在平面上的点均满足与的面积比为3；1，（其中，是首项为1的正项数列），则等于（）A．65B．63C．33D．31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．把答案填写在答题卡的相应位置．11．集合，，则________．12．某工厂的某种型号的机器的使用年限和所支出的维修费用(万元)的统计资料如下表：6810122356-20- 根据上表数据可得y与x之间的线性回归方程，据此模型估计，该机器使用年限为14年时的维修费用约为万元．13．向区域内随机投点，则该点与坐标原点连线的斜率大于1的概率为．14．已知圆和双曲线．若对双曲线上任意一点A（点A在圆外），均存在与圆外切且顶点都在双曲线上的菱形ABCD，则___________．15．定义：表示不超过的最大整数．例如，．给出下列结论：①函数是奇函数；②函数是周期为的周期函数；③函数不存在零点；④函数的值域是．其中正确的是_____________．（填上所有正确命题的编号）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．把答案填在答题卡相应位置．16．本小题满分13分-20- 已知数列{an}的首项为1，前n项和Sn满足．（Ⅰ）求Sn与数列{an}的通项公式；（Ⅱ）设（n∈N*），求使不等式成立的最小正整数n．17．本小题满分13分已知函数经化简后利用“五点法”画其在某一个周期内的图象时，列表并填入的部分数据如下表：①010－10（Ⅰ）请直接写出①处应填的值，并求函数f(x)在区间上的值域；（Ⅱ）的内角所对的边分别为，已知，，求的面积．18．本小题满分13分甲、乙两位选手为为备战我市即将举办的“推广妈祖文化·印象莆田”知识竞赛活动，进行针对性训练，近8次的训练成绩如下（单位：分）：甲8381937978848894乙8789897774788898-20- （I）依据上述数据，从平均水平和发挥的稳定程度考虑，你认为应派哪位选手参加？并说明理由；（II）本次竞赛设置A、B两问题，规定：问题A的得分不低于80分时答题成功，否则答题失败，答题成功可获得价值100元的奖品，问题B的得分不低于90分时答题成功，否则答题失败，答题成功可获得价值300元的奖品．答题顺序可自由选择，但答题失败则终止答题．选手答题问题A，B成功与否互不影响，且以训练成绩作为样本，将样本频率视为概率，请问在（I）中被选中的选手应选择何种答题顺序，使获得的奖品价值更高？并说明理由．19．本小题满分13分如图，边长为2的正方形ABCD绕AB边所在直线旋转一定的角度（小于）到ABEF的位置．(Ⅰ)求证:CE//平面ADF；(Ⅱ)若K为线段BE上异于B,E的点，CE=．设直线AK与平面BDF所成角为,当时,求BK的取值范围．20．本小题满分13分如图，椭圆C：的离心率e=，且椭圆C的首项为的短轴长为2．（Ⅰ）求椭圆C的方程；-20- （Ⅱ）设P，M，N椭圆C上的三个动点．（i）若直线MN过点D（0，），且P点是椭圆C的上顶点，求△PMN面积的最大值；（ii）试探究：是否存在△PMN是以O为中心的等边三角形，若存在，请给出证明；若不存在，请说明理由．21．本小题满分14分已知函数f(x)=lnx+ax2+b（a，b∈R）．（Ⅰ）若曲线y=f(x)在x=1处的切线为y=－1，求函数f(x)的单调区间；（Ⅱ）求证：对任意给定的正数m，总存在实数a，使函数f(x)在区间(m，+∞)上不单调；（Ⅲ）若点A（x1，y1），B（x2，y2）（x2>x1>0）是曲线f(x)上的两点，试探究：当a<0时，是否存在实数x0∈（x1，x2），使直线AB的斜率等于？若存在，给予证明；若不存在，说明理由．-20- 2015年莆田市高中毕业班教学质量检查试卷理科数学试题参考解答及评分标准一、选择题（本大题共10小题，每小题5分，共50分）1．D2．A3．B4．A5．B6．C7．C8．A9．B10．D二、填空题（本大题共5小题，每小题4分，共20分）11．12．7.513．14．15．②③④三、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．本小题主要考查数列、不等式等基础知识，考查运算求解能力，考查化归与转化思想．满分13分．解：（Ⅰ）因为，所以是首项为1，公差为1的等差数列，………1分则=1+(n－1)1=n，……………2分从而Sn=n2．…………………3分当n=1时，a1=S1=1，当n>1时，an=Sn－Sn－1=n2－(n－1)2=2n－1．因为也符合上式，-20- 所以an=2n－1．…………………6分（Ⅱ）由（Ⅰ）知，……………8分所以，……………10分由，解得n>12．………………12分所以使不等式成立的最小正整数为13．……………13分17．本小题主要考查三角函数的图象与性质、两角和与差的三角函数、解三角形等基础知识，考查运算求解能力，考查化归与转化思想．满分13分．解：（Ⅰ）①处应填入．………1分………3分．………4分因为T=，所以，，即．………5分因为，所以，所以，从而得到的值域为．………7分（Ⅱ）因为，又所以，得，．………9分由余弦定理得，即，所以．………11分-20- 所以的面积．………13分18．本小题主要考查平均数、方差、古典概型、相互独立事件的概率、离散型随机变量分布列、数学期望等基础知识，考查数据处理能力、运算求解能力、应用意识，考查必然与或然思想、分类与整合思想．满分13分．解：（I）记甲、乙两位选手近8次的训练的平均成绩分别为、，方差分别为、．，．………………2分，．………………4分因为，，所以甲、乙两位选手的平均水平相当，但甲的发挥更稳定，故应派甲参加．………………5分（II）记事件C表示为“甲回答问题A成功”，事件D表示为“甲回答问题B成功”，则P(C)=,P(D)=,且事件C与事件D相互独立．………………6分记甲按AB顺序获得奖品价值为，则的可能取值为-20- 0,100,400．P(=0)=P()=,P(=100)=P()=,P(=400)=P()=．即的分布列为：0100400P所以甲按AB顺序获得奖品价值的数学期望．………………9分记甲按BA顺序获得奖品价值为，则的可能取值为0,300,400．P(=0)=P()=,P(=300)=P()=,P(=400)=P()=，即的分布列为：0300400P所以甲按BA顺序获得奖品价值的数学期望．………………12分因为，所以甲应选择AB的答题顺序，获得的奖品价值更高．………………13分19．本小题主要考查空间直线与直线、直线与平面、平面与平面的位置关系等基础知识，考查空间想象能力、抽象概括能力、推理论证能力、运算求解能力，-20- 考查数形结合思想、化归与转化思想．满分13分．(Ⅰ)证明：正方形ABCD中，CDBA，正方形ABEF中，EFBA．…………2分EFCD，四边形EFDC为平行四边形，CE//DF．…………3分又DF平面ADF，CE平面ADF，CE//平面ADF．…………5分(Ⅱ)解：BE=BC=2，CE=，，∆BCE为直角三角形，BEBC，……………6分又BEBA，BCBA=B，BC、BA平面ABCD，BE平面ABCD．……………7分以B为原点，、、的方向分别为x轴、y轴、z轴的正方向，建立空间直角坐标系，则B（0，0，0），F（0，2，2），A（0，2，0），，．设K（0，0，m），平面BDF的一个法向量为．由，，得可取，……………9分又，于是sin=，-20- ，，即…………11分结合，解得，即BK的取值范围为（0，]．……………13分20．本小题考查点到直线的距离公式、椭圆的性质、直线与椭圆的位置关系等基础知识，考查运算求解能力、推理论证能力、分析解决问题能力，考查函数与方程思想、数形结合思想、特殊与一般思想、化归与转化思想．满分14分．解：（Ⅰ）由题意得解得a=2，b=1，…………………………………3分所以椭圆方程为．………………………………………………………………3分（Ⅱ）（i）解法一：由已知，直线MN的斜率存在，设直线MN方程为y=kx－，M（x1，y1），N（x2，y2）．由得(1+4k2)x2－4kx－3=0，所以，又．……5分-20- 所以S△PMN=|PD|·|x1－x2|=……………………………………………6分．…………………………………7分令t=，则t≥，所以S△PMN=，………………………………………………8分令h(t)=，t∈[，+∞),则>0，所以h(t)在[)单调递增，则t=，即k=0时，h(t)的最小值，为h(）=，所以△PMN面积的最大值为．……………………9分解法二：由已知，直线MN的斜率存在，设直线MN方程为y=kx－，M（x1，y1），N（x2，y2）．由得(1+4k2)x2－4kx－3=0，所以．…………………5分所以|MN|．-20- 点P（0，1）到直线MN的距离d=．………6分所以S△PMN=|MN|·d=．…………………………………7分以下同解法一．（ii）假设存在△PMN是以O为中心的等边三角形．（1）当P在y轴上时，P的坐标为（0，1），则M，N关于y轴对称，MN的中点Q在y轴上．又O为△PMN的中心，所以，可知．从而|MN|=，|PM|=，|MN|≠|PM|，与△PMN为等边三角形矛盾．（2）当P在x轴上时，同理可知，|MN|≠|PM|，与△PMN为等边三角形矛盾．……………10分（3）当P不在坐标轴时，设P（x0，y0），MN的中点为Q，则kOP=，又O为的中心，则，可知．设M（x1，y1），N（x2，y2），则，，又x12+4y12=4，x22+4y22=4，两式相减得kMN=-20- ，……11分从而kMN=．……12分所以kOP·kMN=·（）=≠－1，所以OP与MN不垂直，与等边△PMN矛盾．……13分综上所述，不存在△PMN是以O为中心的等边三角形．………………………14分21．本小题主要考查函数导数的几何意义、导数的运算及导数的应用，考查运算求解能力、抽象概括能力、推理论证能力，考查函数与方程思想、化归与转化思想、分类与整合思想．满分14分．解：（Ⅰ）由已知得解得……………2分此时，（x>0）．令，得，f(x)，的变化情况如下表：x(0，1)1（1，+∞）+0－f(x)单调递增极大值单调递减所以函数f(x)的增区间为（0，1），减区间为（1，+∞）．………………4分-20- （Ⅱ）（x>0）．（1）当a≥0时，恒成立，此时，函数f(x)在区间（0，+∞）上单调递增，不合题意，舍去．………5分（2）当a<0时，令，得，f(x)，的变化情况如下表：x(0，)（，+∞）+0－f(x)单调递增极大值单调递减所以函数f(x)的增区间为（0，），减区间为（，+∞）．………………7分要使函数f(x)在区间(m，+∞)上不单调，须且只须>m，即．所以对任意给定的正数m，只须取满足的实数a，就能使得函数f(x)在区间(m，+∞)上不单调．……8分（Ⅲ）存在实数x0∈（x1，x2），使直线AB的斜率等于．…………9分证明如下：令g(x)=lnx－x+1（x>0），则，易得g(x)在x=1处取到最大值，且最大值g(1)=0，即g(x)≤0，从而得lnx≤x－1．（*）………10分-20- 由，得．………………11分令，，则p(x)，q(x)在区间[x1，x2]上单调递增．且，，结合（*）式可得，，．令h(x)=p(x)+q(x)，由以上证明可得，h(x)在区间[x1，x2]上单调递增，且h(x1)<0，h(x2)>0，……13分所以函数h(x)在区间（x1，x2）上存在唯一的零点x0，即成立，从而命题成立．……………14分（注：在（Ⅰ）中，未计算b的值不扣分．）-20- 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