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2018河北高考理科数学试题及答案解析【Word真题试卷】

2018-06-10 11:15:39文/李木子

 

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绝密★启用前

2018年普通高等学校招生全国统一考试

理科数学

注意事项:

              1.答题前,考生先将自己的姓名、准考证号填写清楚,将条形码准确粘贴在条形码区域内。

2.选择题必须使用2B铅笔填涂;非选择题必须使用0.5毫米黑色字迹的签字笔书写,字体工整,笔迹清楚

3.请按照题号顺序在答题卡各题目的答题区域内作答,超出答题区域书写的答案无效;在草稿纸、试卷上答题无效

4.作图可先使用铅笔画出,确定后必须用黑色字迹的签字笔描黑。

5.保持卡面清洁,不要折叠、不要弄破、弄皱,不准使用涂改液、修正带、刮纸刀

 

一、选择题:本题共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。

 

一、选择题

1

www.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.com(   )

A.www.ziyuanku.com                            B.www.ziyuanku.com                            C.www.ziyuanku.com                            D.www.ziyuanku.com

2已知集合www.ziyuanku.com ,则www.ziyuanku.com(   )

A.www.ziyuanku.com
B.www.ziyuanku.com
C.www.ziyuanku.com
D.www.ziyuanku.com

3.某地区经过一年的新农村建设,农村的经济收入增加了一倍,实现翻番。为更好地了解该地区农村的经济收入变化情况,统计了该地区系农村建设前后农村的经济收入构成比例。得到如下饼图:

www.ziyuanku.com

则下面结论中不正确的是(   )

A.新农村建设后,种植收入减少
B.新农村建设后,其他收入增加了一倍以上
C.新农村建设后,养殖收入增加一倍
D.新农村建设后,养殖收入与第三产业收入的总和超过了经济收入的一半

4记www.ziyuanku.com为等差数列www.ziyuanku.com的前www.ziyuanku.com项和,若www.ziyuanku.com,则www.ziyuanku.com(   )

A.-12        B.-10        C.10         D.12

5设函数www.ziyuanku.com,若www.ziyuanku.com为奇函数,则曲线www.ziyuanku.com在点www.ziyuanku.com处的切线方程为(   )

A.www.ziyuanku.com                                          B.www.ziyuanku.com

C.www.ziyuanku.com                                          D.www.ziyuanku.com

6 在www.ziyuanku.com中,www.ziyuanku.comwww.ziyuanku.com边上的中线,www.ziyuanku.com为的中点,则www.ziyuanku.com(   )

A.www.ziyuanku.com                                          B.www.ziyuanku.com
C.www.ziyuanku.com                                          D.www.ziyuanku.com

7某圆柱的高为2,底面周长为16,其三视图如下图。圆柱表面上的点M在正视图上的对应点为A,圆柱表面上的点N在左视图上的对应点为B,则在此圆柱侧面上,从M到N的路径中,最短路径的长度为(   )

www.ziyuanku.com

A.www.ziyuanku.com                                          B.www.ziyuanku.com                                          C.www.ziyuanku.com                                          D.www.ziyuanku.com

8 设抛物线www.ziyuanku.com的焦点为www.ziyuanku.com,过点www.ziyuanku.com且斜率为www.ziyuanku.com的直线与www.ziyuanku.com交于两点,则www.ziyuanku.com(  )

A.5          B.6          C.7          D.8

9 已知函数www.ziyuanku.com,www.ziyuanku.com,在www.ziyuanku.com存在www.ziyuanku.com个零点,则的取值范围是(   )

A.www.ziyuanku.com                                          B.www.ziyuanku.com                                          C.www.ziyuanku.com                                          D.www.ziyuanku.com

10下图来自古希腊数学家希波克拉底所研究的几何图形,此图由三个车圈构成,三个半圆的直径分别为直角三角形www.ziyuanku.com的斜边www.ziyuanku.com,直角边www.ziyuanku.com.www.ziyuanku.com的三边所围成的区域记为Ⅰ,黑色部分记为Ⅱ,其余部分记为Ⅲ,在整个图形中随机取一点,此点取自Ⅰ 、Ⅱ 、Ⅲ的概率分别记为www.ziyuanku.com,则(   )

www.ziyuanku.com

A.www.ziyuanku.com                                          B.www.ziyuanku.com                                          C.www.ziyuanku.com                                          D.www.ziyuanku.com

11已知双曲线www.ziyuanku.comwww.ziyuanku.com,www.ziyuanku.com为坐标原点,www.ziyuanku.comwww.ziyuanku.com的右焦点,过www.ziyuanku.com的直线与www.ziyuanku.com的两条渐近线的交点分别为www.ziyuanku.comwww.ziyuanku.com为直角三角形,则www.ziyuanku.com(   )

A.www.ziyuanku.com                                          B.www.ziyuanku.com                                          C.www.ziyuanku.com                                          D.www.ziyuanku.com

12已知正方体的棱长为1,每条棱所在直线与平面www.ziyuanku.com所成的角都相等,则www.ziyuanku.com截此正方体所得截面面积的最大值为(   )

A.www.ziyuanku.com                                          B.www.ziyuanku.com                                          C.www.ziyuanku.com                                          D.www.ziyuanku.com

二、填空题

13若www.ziyuanku.com满足约束条件www.ziyuanku.comwww.ziyuanku.com的最大值为        。

14记www.ziyuanku.com为数列www.ziyuanku.com的前n项的和,若www.ziyuanku.com,则www.ziyuanku.com        。

15.从2位女生,4位男生中选3人参加科技比赛,且至少有1位女生入选,则不同的选法共有__________种.(用数字填写答案)

16已知函数www.ziyuanku.com,则www.ziyuanku.com的最小值是        。

三、解答题

17

在平面四边形www.ziyuanku.com中,www.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.com

1.求www.ziyuanku.com;

2.若www.ziyuanku.comwww.ziyuanku.com

18如图,四边形www.ziyuanku.com为正方形,www.ziyuanku.com分别为www.ziyuanku.com的中点,以www.ziyuanku.com为折痕把www.ziyuanku.com折起,使点www.ziyuanku.com到达点www.ziyuanku.com的位置,且www.ziyuanku.com.

www.ziyuanku.com

1. 证明:平面www.ziyuanku.com平面www.ziyuanku.com;

2.求www.ziyuanku.com与平面www.ziyuanku.com所成角的正弦值

19 设椭圆www.ziyuanku.com的右焦点为www.ziyuanku.com,过www.ziyuanku.com得直线www.ziyuanku.comwww.ziyuanku.com交于www.ziyuanku.com两点,点www.ziyuanku.com的坐标为www.ziyuanku.com.

1.当www.ziyuanku.com与轴垂直时,求直线www.ziyuanku.com的方程;

2.设www.ziyuanku.com为坐标原点,证明:www.ziyuanku.com

20某工厂的某种产品成箱包装,每箱产品在交付用户前要对产品作检验,如检验出不合格品,则更换为合格品,检验时,先从这箱产品中任取20件作检验,再根据检验结果决定是否对余下的所有产品作检验。设每件产品为不合格的概率为品(www.ziyuanku.com),且各件产品是否为不合格品相互独立

1.记20件产品中恰有2件不合格品的概率为www.ziyuanku.com,求www.ziyuanku.com的最大值点www.ziyuanku.com

2.现对一箱产品检验了20件,结果恰有2件不合格品,以(1)中确定的  作为www.ziyuanku.com的值。已知每件产品的检验费用为2元,若有不合格品进入用户手中,则工厂要对每件不合格品支付25元的赔偿费用

①若不对该箱余下的产品作检验,这一箱的检验费用与赔偿费用的和记为www.ziyuanku.com,求www.ziyuanku.com;

②检验费用与赔偿费用和的期望值为决策依据,是否该对这箱余下的所有产品作检验?    

21已知函数www.ziyuanku.com

1.讨论www.ziyuanku.com的单调性;

2.若www.ziyuanku.com存在两个极值点www.ziyuanku.com,证明:www.ziyuanku.com

22[选修4—4:坐标系与参数方程]

    在直角坐标系www.ziyuanku.com中,曲线www.ziyuanku.com的方程为www.ziyuanku.com,以坐标原点为极点,轴正半轴为极轴建立极坐标系,曲线www.ziyuanku.com的极坐标方程为www.ziyuanku.com    

1.求www.ziyuanku.com的直角坐标方程

2. 若www.ziyuanku.comwww.ziyuanku.com有且仅有三个公共点,求www.ziyuanku.com的方程

23[选修4—5:不等式选讲]

已知www.ziyuanku.com    

1.当www.ziyuanku.com时,求不等式www.ziyuanku.com的解集

2.若www.ziyuanku.com时,不等式www.ziyuanku.com成立,求的取值范围


参考答案

 

一、选择题

答案: C

解析: www.ziyuanku.com,www.ziyuanku.com,故选C

答案: B

解析: 由题得www.ziyuanku.com=www.ziyuanku.comwww.ziyuanku.com,故www.ziyuanku.com,故选B

3.答案:A

解析:设建设前总经济收入为www.ziyuanku.com则建设后总经济收入为www.ziyuanku.com

对于www.ziyuanku.com,建设前种植收入为www.ziyuanku.com,建设后种植收入为www.ziyuanku.comwww.ziyuanku.com借误:

对于www.ziyuanku.com,建设前其他收入为www.ziyuanku.com,建设后其他收入为www.ziyuanku.com,故www.ziyuanku.com正确

对于www.ziyuanku.com,建设前养殖收入为www.ziyuanku.com,建设后养殖收入为www.ziyuanku.com,故www.ziyuanku.com正确:

对于www.ziyuanku.com,建设后,养殖收入占www.ziyuanku.com,第三产业收入占www.ziyuanku.com,www.ziyuanku.comwww.ziyuanku.com正确:

答案: B

解析: 由www.ziyuanku.com为等差数列,且www.ziyuanku.com,故有www.ziyuanku.com,即www.ziyuanku.com又由www.ziyuanku.com,故可得www.ziyuanku.com,故www.ziyuanku.com,故选B

答案: D

解析: 因为www.ziyuanku.com是奇函数,所以www.ziyuanku.com,即www.ziyuanku.com解得www.ziyuanku.com,所以www.ziyuanku.com,故切线方程为:www.ziyuanku.com,故选D

答案: A

解析: 由www.ziyuanku.comwww.ziyuanku.com边上的中线,www.ziyuanku.comwww.ziyuanku.com的中点,故www.ziyuanku.com,故选A

答案: B

解析: www.ziyuanku.com

如图,最小路径www.ziyuanku.com,故选B

答案: D

解析: 由直线过点www.ziyuanku.com且斜率为www.ziyuanku.com故可得直线www.ziyuanku.comwww.ziyuanku.com,联立直线www.ziyuanku.com与抛物线www.ziyuanku.com,解得www.ziyuanku.comwww.ziyuanku.com,故可设www.ziyuanku.com,则www.ziyuanku.com.又由抛物线焦点www.ziyuanku.com,故www.ziyuanku.com,www.ziyuanku.com,所以www.ziyuanku.com,故选D

答案: C

解析: www.ziyuanku.com有两个零点等价于www.ziyuanku.comwww.ziyuanku.com有两个交点,由图可知,当www.ziyuanku.com,即www.ziyuanku.com时,www.ziyuanku.comwww.ziyuanku.com有两个交点,故选C

www.ziyuanku.com

答案: A

解析: 假设www.ziyuanku.com,由三角形www.ziyuanku.com是直角三角形,故有www.ziyuanku.com,即www.ziyuanku.com,即有www.ziyuanku.com,故区域Ⅰ的面积为www.ziyuanku.com,区域Ⅱ的面积为www.ziyuanku.com,区域Ⅲ的面积为www.ziyuanku.com又由于总区域固定,故www.ziyuanku.com·即选A

答案: B

解析: www.ziyuanku.com

www.ziyuanku.com

www.ziyuanku.com中,www.ziyuanku.com

www.ziyuanku.com中,www.ziyuanku.com

答案: A

解析: 如图所示平面www.ziyuanku.com与平面的所有棱缩成角都相等

www.ziyuanku.com

www.ziyuanku.com平面www.ziyuanku.com,构造平面www.ziyuanku.com平面www.ziyuanku.com

www.ziyuanku.com,www.ziyuanku.comwww.ziyuanku.com,www.ziyuanku.com

www.ziyuanku.com=www.ziyuanku.com

www.ziyuanku.comwww.ziyuanku.com

二、填空题

答案: www.ziyuanku.com

解析: 作出约束区域如图所示,

www.ziyuanku.com

目标函数化为www.ziyuanku.com

www.ziyuanku.com直线经过www.ziyuanku.com时有最大截距,且此时www.ziyuanku.com取得最大值。

故当www.ziyuanku.comwww.ziyuanku.com取得最大值www.ziyuanku.com

答案: www.ziyuanku.com

解析: 由题意,当www.ziyuanku.com时,www.ziyuanku.com,解得www.ziyuanku.com

www.ziyuanku.comwww.ziyuanku.com

化简得www.ziyuanku.com

www.ziyuanku.com是以www.ziyuanku.com为首项,www.ziyuanku.com为公比的等比数列,因此www.ziyuanku.com

15.答案:16

解析:在www.ziyuanku.com人中任选www.ziyuanku.com人的选法总共有www.ziyuanku.com种;选出的www.ziyuanku.com人劝慰男生的选法共有www.ziyuanku.com

故至少有一位女生入选的选法共有www.ziyuanku.com

答案: www.ziyuanku.com

解析: 显然www.ziyuanku.com,故www.ziyuanku.com是以为www.ziyuanku.com周期的函数

www.ziyuanku.com

故当www.ziyuanku.com,即www.ziyuanku.com时,www.ziyuanku.com单调递增

www.ziyuanku.com,即www.ziyuanku.com时,www.ziyuanku.com单调递减

所以www.ziyuanku.com时,www.ziyuanku.com取得最小值

不妨令www.ziyuanku.com,取www.ziyuanku.com代入www.ziyuanku.comwww.ziyuanku.com

三、解答题

答案: 1.在www.ziyuanku.com中,由正弦定理可知:www.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.com

www.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.com
2.∵www.ziyuanku.com,www.ziyuanku.com

又由余弦定理知:www.ziyuanku.com

解得:www.ziyuanku.comwww.ziyuanku.com

答案: 1.证明:∵www.ziyuanku.com分别为www.ziyuanku.com的中点,四边形www.ziyuanku.com为正方形∴www.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.com,∴www.ziyuanku.com

而:www.ziyuanku.comwww.ziyuanku.com平面www.ziyuanku.com,而www.ziyuanku.com平面www.ziyuanku.com,∴平面www.ziyuanku.com平面www.ziyuanku.com
2.记正方形www.ziyuanku.com边长为www.ziyuanku.com则:www.ziyuanku.com,且由翻折的性质可知:www.ziyuanku.com

www.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.com连接www.ziyuanku.com,由1知:平面www.ziyuanku.com平面www.ziyuanku.com,平面www.ziyuanku.com平面www.ziyuanku.com,∴www.ziyuanku.com平面www.ziyuanku.com,∴www.ziyuanku.com即为www.ziyuanku.com与平面www.ziyuanku.com所成的角.记www.ziyuanku.com,则www.ziyuanku.com,∴www.ziyuanku.com,在www.ziyuanku.com中,由勾股定理得:www.ziyuanku.com,即www.ziyuanku.com,解得www.ziyuanku.comwww.ziyuanku.com

www.ziyuanku.comwww.ziyuanku.com与平面www.ziyuanku.com所成的角的正弦值为www.ziyuanku.com

www.ziyuanku.com

 

答案: 1.依题意,右焦点www.ziyuanku.com,当www.ziyuanku.com与轴垂直时,则点www.ziyuanku.com的坐标为www.ziyuanku.com,所以当www.ziyuanku.com时,直线方程为www.ziyuanku.com

所以当www.ziyuanku.com时,直线www.ziyuanku.com方程为www.ziyuanku.com
2.①当直线www.ziyuanku.com与轴垂直时,www.ziyuanku.com两点分别为www.ziyuanku.comwww.ziyuanku.com根据对称性可知,www.ziyuanku.com所以www.ziyuanku.com

②当直线www.ziyuanku.com不与垂直时,设直线的方程为www.ziyuanku.com联立方程组www.ziyuanku.com

www.ziyuanku.com,则www.ziyuanku.comwww.ziyuanku.com

www.ziyuanku.com

www.ziyuanku.com

www.ziyuanku.com

www.ziyuanku.com

 

答案: 1.www.ziyuanku.com

www.ziyuanku.com

www.ziyuanku.com,www.ziyuanku.com

www.ziyuanku.com时,www.ziyuanku.com单调递增

www.ziyuanku.com时,www.ziyuanku.com,www.ziyuanku.com单调递减

所以,当www.ziyuanku.com时,www.ziyuanku.com有最大

www.ziyuanku.com
2.①有题意可知www.ziyuanku.com

设剩余www.ziyuanku.com件产品恰有www.ziyuanku.com件是不合格品,则www.ziyuanku.com

www.ziyuanku.com

www.ziyuanku.com

②若对余下产品进行检查时,则质检费用与赔偿费用之和为www.ziyuanku.com元,因为www.ziyuanku.com,所以需要检验

答案: 1.www.ziyuanku.com

www.ziyuanku.com时,www.ziyuanku.com,此时www.ziyuanku.comwww.ziyuanku.com上单调递减;

www.ziyuanku.com时,令www.ziyuanku.com,判别式www.ziyuanku.com

www.ziyuanku.com时,此时www.ziyuanku.com,www.ziyuanku.com,从而www.ziyuanku.comwww.ziyuanku.com上单调递减

www.ziyuanku.com时,此时www.ziyuanku.com,设www.ziyuanku.com的两根为www.ziyuanku.com,且www.ziyuanku.com,利用求根公式得

www.ziyuanku.com

www.ziyuanku.com时,www.ziyuanku.com,从而www.ziyuanku.com,www.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.com单调递减

www.ziyuanku.com时,www.ziyuanku.com,从而www.ziyuanku.com,此时www.ziyuanku.comwww.ziyuanku.com上单调递增

综上所述,当www.ziyuanku.com时,www.ziyuanku.comwww.ziyuanku.com上单调递减

www.ziyuanku.com时,www.ziyuanku.comwww.ziyuanku.comwww.ziyuanku.com上单调递减,在www.ziyuanku.com上单调递增
2.由www.ziyuanku.com可知,若www.ziyuanku.com有两个极值点,则www.ziyuanku.com,且www.ziyuanku.com的两根即为www.ziyuanku.com

且满足韦达定理www.ziyuanku.com,易得www.ziyuanku.com,www.ziyuanku.com

www.ziyuanku.com,可得www.ziyuanku.com,即www.ziyuanku.com

若要证www.ziyuanku.com,只须证www.ziyuanku.com,即证www.ziyuanku.com

整理得www.ziyuanku.com

构造函数www.ziyuanku.com,求导得www.ziyuanku.com

因此www.ziyuanku.comwww.ziyuanku.com上单调递减

www.ziyuanku.com

从而www.ziyuanku.com成立,原式得证

答案: 1.www.ziyuanku.com

www.ziyuanku.com,即www.ziyuanku.com

所以www.ziyuanku.com的直角坐标方程为www.ziyuanku.com
2.由题www.ziyuanku.com可知圆心坐标为www.ziyuanku.com,半径www.ziyuanku.com

又曲线方程www.ziyuanku.com,关于www.ziyuanku.com轴对称,且曲线过圆外定点www.ziyuanku.com

∴当曲线与圆有且仅有www.ziyuanku.com个交点时,设曲线在www.ziyuanku.com轴的右半部分与圆相切于点www.ziyuanku.com,

此时,www.ziyuanku.com

www.ziyuanku.comwww.ziyuanku.com,

www.ziyuanku.com,即直线www.ziyuanku.com的方程为www.ziyuanku.com

答案: 1.当www.ziyuanku.com时,则www.ziyuanku.com

∴当www.ziyuanku.com时,www.ziyuanku.comwww.ziyuanku.com

又当www.ziyuanku.com时,www.ziyuanku.com满足www.ziyuanku.com

综上:www.ziyuanku.com
2.当www.ziyuanku.com时,www.ziyuanku.com恒成立

www.ziyuanku.com时有:www.ziyuanku.com

www.ziyuanku.com,两边平方化简可得:www.ziyuanku.com

www.ziyuanku.com,则www.ziyuanku.com成立

函数www.ziyuanku.com可看作斜率为www.ziyuanku.com的直线,且在www.ziyuanku.com处取最大值

www.ziyuanku.com

即的取值范围是www.ziyuanku.com

 

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