2015年高考全国卷2理科数学试题及答案解析（word精校版）

注意事项：

1．本试卷分第I卷（选择题）和第II卷（非选择题）两部分。答卷前，考生务必先将自己的姓名、准考证号码填写在答题卡上。

2．回答第I卷时，选出每小题的答案后，用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，再选涂其他答案标号。写在本试卷上无效。

3．回答第II卷时，将答案写在答题卡上，写在本试卷上无效。

4．考试结束后，将本试卷和答题卡一并交回。

一、选择题：本大题共12小题，每小题5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）已知集合A=｛-2，-1,0，1,2｝，B=｛x|（X-1）（x+2）＜0｝,则A∩B=（）

（A）｛--1,0｝（B）｛0,1｝（C）｛-1,0,1｝（D）｛,0,，1，2｝

答案A

解析由已知得

，故 ，故选A

（2）若a为实数且（2+ai）（a-2i）=-4i,则a=（）

（A）-1（B）0（C）1（D）2

答案B

（3）根据下面给出的2004年至2013年我国二氧化硫排放量（单位：万吨）柱形图。以下结论不正确的是()

（A）逐年比较，2008年减少二氧化硫排放量的效果最显著

（B）2007年我国治理二氧化硫排放显现

（C）2006年以来我国二氧化硫年排放量呈减少趋势

（D）2006年以来我国二氧化硫年排放量与年份正相关

答案D

解析由柱形图得，从2006年以来，我国二氧化硫排放量呈下降趋势，故年排放量与年份负相关．

（4）等比数列｛an｝满足a1=3，

=21，则 ()

（A）21（B）42（C）63（D）84

答案B

（5）设函数

, ()

（A）3（B）6（C）9（D）12

答案C

解析由已知得

，又 ，所以 ，故 ．

（6）一个正方体被一个平面截去一部分后，剩余部分的三视图如右图，则截去部分体积与剩余部分体积的比值为

（A）

（B） （C） （D）

答案D

解析由三视图得，在正方体

中，截去四面体 ，如图所示，，设正方体棱长为 ，则 ，故剩余几何体体积为 ，所以截去部分体积与剩余部分体积的比值为 ．

（7）过三点A（1,3），B（4,2），C（1,-7）的圆交于y轴于M、N两点，则

=

（A）2

（B）8（C）4 （D）10

答案C

（8）右边程序抗土的算法思路源于我国古代数学名著《九章算术》中的“更相减损术”。执行该程序框图，若输入a,b分别为14,18，则输出的a=

A.0B.2C.4D.14

答案B

解析程序在执行过程中，

， 的值依次为 ， ； ； ； ； ； ，此时 程序结束，输出 的值为2，故选B．

（9）已知A,B是球O的球面上两点，∠AOB=90,C为该球面上的动点，若三棱锥O-ABC体积的最大值为36，则球O的表面积为

A．36πB.64πC.144πD.256π

答案C

解析如图所示，当点C位于垂直于面

的直径端点时，三棱锥 的体积最大，设球 的半径为 ，此时 ，故 ，则球 的表面积为 ，故选C．

10.如图，长方形ABCD的边AB=2，BC=1，O是AB的中点，点P沿着边BC，CD与DA运动，记∠BOP=x．将动点P到A、B两点距离之和表示为x的函数f（x），则f（x）的图像大致为

答案B

的运动过程可以看出，轨迹关于直线 对称，且 ，且轨迹非线型，故选B．

（11）已知A，B为双曲线E的左，右顶点，点M在E上，?ABM为等腰三角形，且顶角为120°，则E的离心率为

（A）√5（B）2（C）√3（D）√2

答案D

（12）设函数f’(x)是奇函数

的导函数，f（-1）=0，当 时， ，则使得 成立的x的取值范围是

（A）

（B）

（C）

（D）

答案A

解析

记函数

，则 ，因为当 时， ，故当 时， ，所以 在 单调递减；又因为函数 是奇函数，故函数 是偶函数，所以 在 单调递减，且 ．当 时， ，则 ；当 时， ，则 ，综上所述，使得 成立的 的取值范围是 ，故选A．

二、填空题

（13）设向量

， 不平行，向量 与 平行，则实数 _________．

答案

解析因为向量

与 平行，所以 ，则 所以 ．

（14）若x，y满足约束条件

，则 的最大值为____________．

答案

（15）

的展开式中x的奇数次幂项的系数之和为32，则 __________．

答案

解析由已知得

，故 的展开式中x的奇数次幂项分别为 ， ， ， ， ，其系数之和为 ，解得 ．

（16）设

是数列 的前n项和，且 ， ，则 ________．

答案

解析由已知得

，两边同时除以 ，得 ，故数列 是以 为首项， 为公差

的等差数列，则 ，所以 ．

三．解答题

（17）?ABC中，D是BC上的点，AD平分∠BAC，?ABD是?ADC面积的2倍。

(Ⅰ)求

(Ⅱ)若

=1，

=

求

和

的长.

(18)某公司为了解用户对其产品的满意度，从A，B两地区分别随机调查了20个用户，得到用户对产品的满意度评分如下：

A地区：62738192958574645376

78869566977888827689

B地区：73836251914653736482

93486581745654766579

（Ⅰ）根据两组数据完成两地区用户满意度评分的茎叶图，并通过茎叶图比较两地区满意度评分的平均值及分散程度（不要求计算出具体值，得出结论即可）；

（Ⅱ）根据用户满意度评分，将用户的满意度从低到高分为三个不等级：

满意度评分

低于70分

70分到89分

不低于90分

满意度等级

不满意

满意

非常满意

记时间C：“A地区用户的满意度等级高于B地区用户的满意度等级”。假设两地区用户的评价结果相互独立。根据所给数据，以事件发生的频率作为相应事件发生的概率，求C的概率

19．（本小题满分12分）

如图，长方体ABCD—A1B1C1D1中，AB=16，BC=10，AA1=8，点E，F分别在A1B1，D1C1上，A1E=D1F=4，过点E，F的平面α与此长方体的面相交，交线围成一个正方形。

（1）在图中画出这个正方形（不必说明画法和理由）；

（2）求直线AF与平面α所成的角的正弦值。

20．（本小题满分12分）

已知椭圆C：

，直线l不过原点O且不平行于坐标轴，l与C有两个交点A，B，线段AB的中点为M。

（1）证明：直线OM的斜率与l的斜率的乘积为定值；

（2）若l过点

，延长线段OM与C交于点P，四边形OAPB能否为平行四边形？若能，求此时l的斜率；若不能，说明理由。

21．（本小题满分12分）

设函数

。

（1）证明：

在 单调递减，在 单调递增；

（2）若对于任意

，都有 ，求m的取值范围。

请考生在第22、23、24题中任选一题作答，如果多做，则按所做的第一题记分。作答时请写清题号

22．（本小题满分10分）

选修4-1：几何证明选讲

如图，O为等腰三角形ABC内一点，⊙O与ΔABC的底边BC交于M，N两点，与底边上的高AD交于点G，且与AB，AC分别相切于E，F两点。

（1）证明：EF∥BC；

（2）若AG等于⊙O的半径，且

，求四边形EBCF的面积。

23．（本小题满分10分）

选修4-4：坐标系与参数方程

在直角坐标系xOy中，曲线C1：

（t为参数，t≠0），其中0≤α<π，在以O为极点，x轴正半轴为极轴的极坐标系中，曲线C2： ，C3： 。

（1）求C2与C3交点的直角坐标；

（2）若C1与C2相交于点A，C1与C3相交于点B，求

的最大值。

24．（本小题满分10分）

选修4-5：不等式选讲

设a，b，c，d均为正数，且a+b=c+d，证明：

（1）若ab>cd；则

；

（2）

是 的充要条件。

附：全部试题答案

2021年高考英语全国卷2 - 阅读理解C

　听第10段材料，回答第18至20题。

　18. What are the tourists advised to do when touring London?

　A. Take their tour schedule.

　B. Watch out for the traffic.

　C. Wear comfortable shoes.

　19. What will the tourists do in fifteen minutes?

　A. Meet the speaker.

　B. Go to their rooms.

　C. Change some money.

　20. Where probably is the speaker?

　A. In a park.

　B. In a hotel.

　C. In a shopping centre.

　听力参考答案

　1-20 BCACB ACABC BACAC BACAB

　听力原文

　Text 1

　M: Lucy, would you like to have lunch with me tomorrow?

　W: Oh, I?d really love to, but I have an appointment with my dentist at 11:30. Thanks for inviting me.

　Text 2

　W: Peter, how is the weather now? Is it still raining?

　M: No, but there?s still lots of clouds. The weatherman said the sun wouldn?t come out until next week.

　Text 3

　M: I?m sorry I was late for class today, Dr. Simpson.

　W: Well, I?ll let it go this time. But you saw it disturbed the rest of the class.

　M: Yes, I realized that. I won?t let it happen again.

　Text 4

　W: Hi, Mike. Listen, I?m coming back this afternoon, and I?ll take a bus from the railway station. So, you don?t need to come and pick me up.

　M: OK, take care, and see you soon.

　Text 5

　M: Jenny, there?s an opening for an assistant manager in our company. You should give it a try.

　W: Thank you George, but I?ve decided to travel a bit before finding another job.

　Text 6

　M: By the way, do you know what time it is?

　W: Well, it?s a quarter to two.

　M: Oh, I?ve got to go.

　W: See Linda in the library?

　M: No. Actually I?m going to meet with Professor Smith at ten past two. You may continue our project discussion with Michael.

　W: All right.

　Text 7

　W: Harry, guess what? I?ve just received an email from Pamela. She and Peter are coming down to see us this weekend.

　M: Oh, that?s good news! We haven?t seen them for ages.

　W: Yeah. The last time we met them was at our wedding three years ago.

　M: Did Pamela mention how long they?ll stay?

　W: About one week.

　M: Great! I can?t wait to show them around our new house.

　W: Me too. We haven?t had any guests since we moved in here. If the weather is fine, we can have a barbecue in the garden.

　M: Good idea. I?ll go to the market tomorrow to buy all the things we?ll need.

　Text 8

　W: Hello, everyone. Welcome to our program. Today, we are fortunate to have a special guest with us. Some of you may have heard of him before. He?s an artist. His works have received many prizes and have been shown in over one hundred exhibitions across the country ? Los Angeles, New York, Philadelphia, to name just a few. His name is Chris Cucksy. So Chris, tell us a bit about yourself.

　M: Well, I was born in Springfield, Missouri, and grew up in Kansas. I didn?t come from a family with wealth or position, but I did manage to get a master?s degree in fine arts.

　W: When did you first start to make art? And what was the turning point in your life that made you an artist?

　M: I always liked drawing as early as I can remember, so right from then, I knew what I was going to be: an artist.

　W: What is it that always inspires you to create?

　M: Nature is the biggest inspiration. I?m always inspired by things of beauty and harmony.

　Text 9

　M: You must be pretty excited about your trip to Europe, Dorothy. When are you leaving?

　W: In just two weeks, and I am excited. I?ve been looking forward to this training program for a long time. But there are still a few things I need to do before I go.

　M: Like what?

　W: Like renewing my passport and figuring out what to do with my apartment while I?m gone.

　M: You are not going to give it up, are you?

　W: No way! I?ll never find another apartment like it around here. But I don?t like the idea of paying three months for an empty apartment, either. So, I?m looking for someone to take it while I?m away.

　M: Um, let me think. Oh, I know just a person. An old colleague of mine, Jim Thomas. He is coming here to do some research this summer, from June to August.

　W: Well, that?s exactly when I?ll be away!

　M: Tell you what: I?ll be calling Jim late this week anyway, so I?ll mention it to him.

　W: Well, thanks, Bill.

　Text 10

　W: I hope I?ve given you a clear idea of the schedule for your London weekend. And, before I finish, let me just give you some advice which should make your stay more enjoyable. Firstly, please do remember to put on some comfortable shoes. London is a big place, and whatever you do, you?ll find yourself doing quite a lot of walking. So, comfortable shoes are really necessary. And secondly, let me ask you to please look after your money. Keep it safe at all times, and then you will not have any unpleasant accident, which could ruin your whole weekend. You?ll find a copy of your weekend?s schedule in your room. Take a look at it, and make sure you?re clear about everything. Well, that?s all from me for now. Go and leave your luggage in your rooms. I?ll be seeing you here again in fifteen minutes. Goodbye for now!

You’ve heard that plastic is polluting the oceans—between 4.8 and 12.7 million tonnes enter ocean ecosystems every year. But does one plastic straw or cup really make a difference? Artist Benjamin Von Wong wants you to know that it does. He builds massive sculptures out of plastic garbage, foreing viewers to re-examine their relationship to single-use plastic products.

你听说过塑料正在污染海洋——每年有480万到1270万吨塑料进入海洋生态系统。但是，一根塑料吸管或杯子真的能带来变化吗？艺术家本杰明·冯·黄想让你知道这是真的。他用塑料垃圾建造了巨大的雕塑，让观众重新审视他们与一次性塑料制品的关系。

At the beginning of the year, the artist built a piece called“Strawpocalypse,” a pair of 10-foot-tall plastic waves, frozen mid-crash. Made of 168,000 plastic straws collected from several volunteer beach cleanups, the sculpture made its first appearance at the Estella Place shopping center in Ho Chi Minh City, Vietnam.

今年年初，这位艺术家创作了一幅名为“Strawpocalypse”的作品，这是一对10英尺高的塑料波浪，凝结在半空，由从几次海滩清理志愿者中收集的168,000根塑料吸管制成，首次出现在越南胡志明市的埃斯黛拉广场购物中心。

Just 9% of global plastic waste is recycled. Plastic straws are by no means the biggest source of plastic pollution, but they’ve recently come under fire because most people don’t need them to drink with and, because of their small size and weight, they cannot be recycled. Every straw that' s part of Von Wong's artwork likely came from a drink that someone used for only a few minutes. Once the drink is gone, the straw will take centuries to disappear.

全球只有9%的塑料垃圾被回收利用。塑料吸管绝不是的塑料污染的最大来源，但最近它们受到了猛烈抨击，因为大多数人不需要用吸管喝饮料，而且由于它们体积小、重量轻，无法回收。冯·黄作品中的每一根吸管都可能来自一种只喝了几分钟的饮料。一旦饮料消失，吸管需要几个世纪才能消失。

In a piece from 2018, Von Wong wanted to illustrate a specific statistic: Every 60 seconds, a truckload's worth of plastic enters the ocean. For this work, titled "Truckload of Plastic, "Von Wong and a group of volunteers collected more than 10,000 pieces of plastic, which were then tied together to look like they’d been dumped from a truck all at once.

在2018年的一篇文章中，冯·黄想举例说明具体的统计数据：每60秒，一卡车的塑料进入海洋。在这项名为“一卡车塑料”的作品中，冯·黄和一群志愿者收集了1万多块塑料，然后将它们绑在一起，看起来像是突然从卡车上被倾倒了下来。

Von Wong hopes that his work will also help pressure big companies to reduce their plastic footprint.

冯·黄希望他的工作也能帮助大公司减少塑料足迹。