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【334290】2019年全国统一高考数学试卷理科新课标Ⅱ

时间:2025-01-21 17:54:18 作者: 字数:23226字
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绝密★启用前

2019年全国统一高考数学试卷(理科)(新课标Ⅱ)

试卷副标题

考试范围:xxx;考试时间:100分钟;命题人:xxx

题号

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一、单选题

1设集合A={x|x2-5x+6>0}B={ x|x-1<0},则AB=

A(-1) B(-21)

C(-3-1) D(3+)

【答案】A

【解析】

【分析】

先求出集合A,再求出交集.

【详解】

由题意得, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .故选A

【点睛】

本题考点为集合的运算,为基础题目.

2z=-3+2i,则在复平面内 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 对应的点位于

A第一象限 B第二象限

C第三象限 D第四象限

【答案】C

【解析】

【分析】

先求出共轭复数再判断结果.

【详解】

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 对应点(-3-2)位于第三象限.故选C

【点睛】

本题考点为共轭复数,为基础题目.

3已知 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> =(2,3) <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> =(3t) <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> =1,则 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> =

A-3 B-2

C2 D3

【答案】C

【解析】

【分析】

根据向量三角形法则求出t,再求出向量的数量积.

【详解】

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .故选C

【点睛】

本题考点为平面向量的数量积,侧重基础知识和基本技能,难度不大.

4201913日嫦娥四号探测器成功实现人类历史上首次月球背面软着陆,我国航天事业取得又一重大成就,实现月球背面软着陆需要解决的一个关键技术问题是地面与探测器的通讯联系.为解决这个问题,发射了嫦娥四号中继星“鹊桥”,鹊桥沿着围绕地月拉格朗日 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 点的轨道运行. <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 点是平衡点,位于地月连线的延长线上.设地球质量为M,月球质量为M,地月距离为R <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 点到月球的距离为r,根据牛顿运动定律和万有引力定律,r满足方程:

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,由于 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的值很小,因此在近似计算中 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则r的近似值为

A <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> B <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

C <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> D <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【答案】D

【解析】

【分析】

本题在正确理解题意的基础上,将有关式子代入给定公式,建立 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程,解方程、近似计算.题目所处位置应是“解答题”,但由于题干较长,易使考生“望而生畏”,注重了阅读理解、数学式子的变形及运算求解能力的考查.

【详解】

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

解得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

由于本题题干较长,所以,易错点之一就是能否静心读题,正确理解题意;易错点之二是复杂式子的变形出错.

5演讲比赛共有9位评委分别给出某选手的原始评分,评定该选手的成绩时,从9个原始评分中去掉1个最高分、1个最低分,得到7个有效评分.7个有效评分与9个原始评分相比,不变的数字特征是

A中位数 B平均数

C方差 D极差

【答案】A

【解析】

【分析】

可不用动笔,直接得到答案,亦可采用特殊数据,特值法筛选答案.

【详解】

9位评委评分按从小到大排列为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

则①原始中位数为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,去掉最低分 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,最高分 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,后剩余 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

中位数仍为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> A正确.

原始平均数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,后来平均数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

平均数受极端值影响较大, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 不一定相同,B不正确

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 由②易知,C不正确.

原极差 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,后来极差 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 可能相等可能变小,D不正确.

【点睛】

本题旨在考查学生对中位数、平均数、方差、极差本质的理解.

6a>b,则

Aln(ab)>0 B3a<3b

Ca3b3>0 Da│>│b

【答案】C

【解析】

【分析】

本题也可用直接法,因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,知A错,因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是增函数,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,故B错;因为幂函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是增函数, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,知C正确;取 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,满足 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,知D错.

【详解】

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,满足 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,知A错,排除A;因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,知B错,排除B;取 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,满足 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,知D错,排除D,因为幂函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是增函数, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,故选C

【点睛】

本题主要考查对数函数性质、指数函数性质、幂函数性质及绝对值意义,渗透了逻辑推理和运算能力素养,利用特殊值排除即可判断.

7αβ为两个平面,则αβ的充要条件是

Aα内有无数条直线与β平行

Bα内有两条相交直线与β平行

Cαβ平行于同一条直线

Dαβ垂直于同一平面

【答案】B

【解析】

【分析】

本题考查了空间两个平面的判定与性质及充要条件,渗透直观想象、逻辑推理素养,利用面面平行的判定定理与性质定理即可作出判断.

【详解】

由面面平行的判定定理知: <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 内两条相交直线都与 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 平行是 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的充分条件,由面面平行性质定理知,若 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 内任意一条直线都与 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 平行,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 内两条相交直线都与 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 平行是 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的必要条件,故选B

【点睛】

面面平行的判定问题要紧扣面面平行判定定理,最容易犯的错误为定理记不住,凭主观臆断,如:“若 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ”此类的错误.

8若抛物线y2=2pxp>0)的焦点是椭圆 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的一个焦点,则p=

A2 B3

C4 D8

【答案】D

【解析】

【分析】

利用抛物线与椭圆有共同的焦点即可列出关于 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程,即可解出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,或者利用检验排除的方法,如 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,抛物线焦点为(10),椭圆焦点为(±20),排除A,同样可排除BC,故选D

【详解】

因为抛物线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的焦点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是椭圆 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的一个焦点,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,解得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,故选D

【点睛】

本题主要考查抛物线与椭圆的几何性质,渗透逻辑推理、运算能力素养.

9下列函数中,以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 为周期且在区间( <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> )单调递增的是

Af(x)=│cos 2x Bf(x)=│sin 2x

Cf(x)=cos│x Df(x)= sin│x

【答案】A

【解析】

【分析】

本题主要考查三角函数图象与性质,渗透直观想象、逻辑推理等数学素养.画出各函数图象,即可做出选择.

【详解】

因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 图象如下图,知其不是周期函数,排除D;因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,周期为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,排除C,作出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 图象,由图象知,其周期为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,在区间 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 单调递增,A正确;作出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的图象,由图象知,其周期为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,在区间 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 单调递减,排除B,故选A

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

利用二级结论:①函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的周期是函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 周期的一半;② <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 不是周期函数;

10已知a0 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ),2sin2α=cos2α+1,则sinα=

A <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> B <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

C <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> D <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【答案】B

【解析】

【分析】

利用二倍角公式得到正余弦关系,利用角范围及正余弦平方和为1关系得出答案.

【详解】

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,又 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,又 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,故选B

【点睛】

本题为三角函数中二倍角公式、同角三角函数基本关系式的考查,中等难度,判断正余弦正负,运算准确性是关键,题目不难,需细心,解决三角函数问题,研究角的范围后得出三角函数值的正负,很关键,切记不能凭感觉.

11F为双曲线C <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> a>0b>0)的右焦点,O为坐标原点,以OF为直径的圆与圆x2+y2=a2交于PQ两点.若|PQ|=|OF|,则C的离心率为

A <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> B <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

C2 D <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【答案】A

【解析】

【分析】

准确画图,由图形对称性得出P点坐标,代入圆的方程得到ca关系,可求双曲线的离心率.

【详解】

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 轴交于点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,由对称性可知 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 轴,

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 为以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 为直径的圆的半径,

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 为圆心 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,又 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 点在圆 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 上,

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,故选A

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

本题为圆锥曲线离心率的求解,难度适中,审题时注意半径还是直径,优先考虑几何法,避免代数法从头至尾,运算繁琐,准确率大大降低,双曲线离心率问题是圆锥曲线中的重点问题,需强化练习,才能在解决此类问题时事半功倍,信手拈来.

12设函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的定义域为R,满足 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,且当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .若对任意 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,都有 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则m的取值范围是

A. <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> B. <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

C. <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> D. <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【答案】B

【解析】

【分析】

本题为选择压轴题,考查函数平移伸缩,恒成立问题,需准确求出函数每一段解析式,分析出临界点位置,精准运算得到解决.

【详解】

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 右移1个单位,图像变为原来的2倍.

如图所示:当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,令 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,整理得: <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> (舍), <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 成立,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,故选B

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

易错警示:图像解析式求解过程容易求反,画错示意图,画成向左侧扩大到2倍,导致题目出错,需加深对抽象函数表达式的理解,平时应加强这方面练习,提高抽象概括、数学建模能力.



II卷(非选择题)

请点击修改第II卷的文字说明


评卷人

得分




二、填空题

13我国高铁发展迅速,技术先进.经统计,在经停某站的高铁列车中,有10个车次的正点率为0.97,有20个车次的正点率为0.98,有10个车次的正点率为0.99,则经停该站高铁列车所有车次的平均正点率的估计值为___________.

【答案】098.

【解析】

【分析】

本题考查通过统计数据进行概率的估计,采取估算法,利用概率思想解题.

【详解】

由题意得,经停该高铁站的列车正点数约为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,其中高铁个数为10+20+10=40,所以该站所有高铁平均正点率约为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

本题考点为概率统计,渗透了数据处理和数学运算素养.侧重统计数据的概率估算,难度不大.易忽视概率的估算值不是精确值而失误,根据分类抽样的统计数据,估算出正点列车数量与列车总数的比值.

14已知 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是奇函数,且当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> . <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> __________.

【答案】-3

【解析】

【分析】

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 代入条件即可得解.

【详解】

因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是奇函数,且当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

又因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,两边取以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 为底的对数得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

本题主要考查函数奇偶性,对数的计算.渗透了数学运算、直观想象素养.使用转化思想得出答案.

15 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的内角 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的对边分别为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> . <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的面积为__________.

【答案】 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【解析】

【分析】

本题首先应用余弦定理,建立关于 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程,应用 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的关系、三角形面积公式计算求解,本题属于常见题目,难度不大,注重了基础知识、基本方法、数学式子的变形及运算求解能力的考查.

【详解】

由余弦定理得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

解得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> (舍去)

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

本题涉及正数开平方运算,易错点往往是余弦定理应用有误或是开方导致错误.解答此类问题,关键是在明确方法的基础上,准确记忆公式,细心计算.

16中国有悠久的金石文化,印信是金石文化的代表之一.印信的形状多为长方体、正方体或圆柱体,但南北朝时期的官员独孤信的印信形状是半正多面体(图1.半正多面体是由两种或两种以上的正多边形围成的多面体.半正多面体体现了数学的对称美.图2是一个棱数为48的半正多面体,它的所有顶点都在同一个正方体的表面上,且此正方体的棱长为1.则该半正多面体共有________个面,其棱长为_________

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【答案】26个面. 棱长为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .

【解析】

【分析】

第一问可按题目数出来,第二问需在正方体中简单还原出物体位置,利用对称性,平面几何解决.

【详解】

由图可知第一层与第三层各有9个面,计18个面,第二层共有8个面,所以该半正多面体共有 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 个面.

如图,设该半正多面体的棱长为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,延长 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 交于点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,延长 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 交正方体棱于 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,由半正多面体对称性可知, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 为等腰直角三角形,

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即该半正多面体棱长为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

本题立意新颖,空间想象能力要求高,物体位置还原是关键,遇到新题别慌乱,题目其实很简单,稳中求胜是关键.立体几何平面化,无论多难都不怕,强大空间想象能力,快速还原图形.


评卷人

得分




三、解答题

17

如图,长方体ABCDA1B1C1D1的底面ABCD是正方形,点E在棱AA1上,BEEC1.

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

1)证明:BE平面EB1C1

2)若AE=A1E,求二面角BECC1的正弦值.

【答案】1)证明见解析;(2 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【解析】

【分析】

1)利用长方体的性质,可以知道 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 侧面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,利用线面垂直的性质可以证明出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,这样可以利用线面垂直的判定定理,证明出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 平面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

2)以点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 坐标原点,以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 分别为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 轴,建立空间直角坐标系,设正方形 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的边长为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,求出相应点的坐标,利用 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,可以求出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 之间的关系,分别求出平面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 、平面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的法向量,利用空间向量的数量积公式求出二面角 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的余弦值的绝对值,最后利用同角的三角函数关系,求出二面角 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的正弦值.

【详解】

证明(1)因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是长方体,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 侧面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,而 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 平面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 平面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,因此 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 平面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

2)以点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 坐标原点,以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 分别为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 轴,建立如下图所示的空间直角坐标系,

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是平面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的法向量,

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是平面 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的法向量,

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

二面角 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的余弦值的绝对值为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以二面角 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的正弦值为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .

【点睛】

本题考查了利用线面垂直的性质定理证明线线垂直,考查了利用空间向量求二角角的余弦值,以及同角的三角函数关系,考查了数学运算能力.

18

11分制乒乓球比赛,每赢一球得1分,当某局打成10:10平后,每球交换发球权,先多得2分的一方获胜,该局比赛结束.甲、乙两位同学进行单打比赛,假设甲发球时甲得分的概率为0.5,乙发球时甲得分的概率为0.4,各球的结果相互独立.在某局双方10:10平后,甲先发球,两人又打了X个球该局比赛结束.

1)求PX=2);

2)求事件X=4且甲获胜的概率.

【答案】1 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ;(20.1

【解析】

【分析】

(1)本题首先可以通过题意推导出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 所包含的事件为“甲连赢两球或乙连赢两球”,然后计算出每种事件的概率并求和即可得出结果;

(2)本题首先可以通过题意推导出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 所包含的事件为“前两球甲乙各得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 分,后两球均为甲得分”,然后计算出每种事件的概率并求和即可得出结果。

【详解】

(1)由题意可知, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 所包含的事件为“甲连赢两球或乙连赢两球”

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

(2)由题意可知, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 包含的事件为“前两球甲乙各得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 分,后两球均为甲得分”

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

本题考查古典概型的相关性质,能否通过题意得出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 以及 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 所包含的事件是解决本题的关键,考查推理能力,考查学生从题目中获取所需信息的能力,是中档题。

19

已知数列{an}{bn}满足a1=1b1=0 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .

1)证明:{an+bn}是等比数列,{anbn}是等差数列;

2)求{an}{bn}的通项公式.

【答案】1)见解析;(2 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【解析】

【分析】

(1)可通过题意中的 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 以及 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 对两式进行相加和相减即可推导出数列 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是等比数列以及数列 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是等差数列;

(2)可通过(1)中的结果推导出数列 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 以及数列 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的通项公式,然后利用数列 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 以及数列 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的通项公式即可得出结果。

【详解】

(1)由题意可知 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以数列 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是首项为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 、公比为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的等比数列, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,数列 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是首项 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 、公差为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的等差数列, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

(2)(1)可知, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【点睛】

本题考查了数列的相关性质,主要考查了等差数列以及等比数列的相关证明,证明数列是等差数列或者等比数列一定要结合等差数列或者等比数列的定义,考查推理能力,考查化归与转化思想,是中档题。

20

已知函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .

1)讨论f(x)的单调性,并证明f(x)有且仅有两个零点;

2)设x0f(x)的一个零点,证明曲线y=ln x 在点A(x0ln x0)处的切线也是曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的切线.

【答案】1)函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 上是单调增函数,证明见解析;

2)证明见解析.

【解析】

【分析】

1)对函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 求导,结合定义域,判断函数的单调性;

2)先求出曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 处的切线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,然后求出当曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 切线的斜率与 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 斜率相等时,证明曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 切线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 在纵轴上的截距与 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 在纵轴的截距相等即可.

【详解】

1)函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的定义域为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,因为函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的定义域为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,因此函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 上是单调增函数;

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,而 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,显然当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 有零点,而函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 上单调递增,故当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 有唯一的零点;

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 必有一零点,而函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 上是单调递增,故当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 有唯一的零点

综上所述,函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的定义域 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 内有2个零点;

2)因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的一个零点,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 处的切线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,故曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 处的切线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程为: <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,它在纵轴的截距为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .

设曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的切点为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,过切点为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 切线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以在 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 处的切线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,因此切线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

当切线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 等于直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

切线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 在纵轴的截距为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,而 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率相等,在纵轴上的截距也相等,因此直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 重合,故曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 处的切线也是曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的切线.

【点睛】

本题考查了利用导数求已知函数的单调性、考查了曲线的切线方程,考查了数学运算能力.

21

已知点A(2,0)B(2,0),动点M(x,y)满足直线AMBM的斜率之积为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .M的轨迹为曲线C.

1)求C的方程,并说明C是什么曲线;

2)过坐标原点的直线交CPQ两点,点P在第一象限,PEx轴,垂足为E,连结QE并延长交C于点G.

i)证明: <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是直角三角形;

ii)求 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 面积的最大值.



【答案】1)详见解析(2)详见解析

【解析】

【分析】

1)分别求出直线AMBM的斜率,由已知直线AMBM的斜率之积为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,可以得到等式,化简可以求出曲线C的方程,注意直线AMBM有斜率的条件;

2)(i)设出直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程,与椭圆方程联立,求出PQ两点的坐标,进而求出点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的坐标,求出直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程,与椭圆方程联立,利用根与系数关系求出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的坐标,再求出直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率,计算 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的值,就可以证明出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是直角三角形;

ii)由(i)可知 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 三点坐标, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是直角三角形,求出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的长,利用面积公式求出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的面积,利用导数求出面积的最大值.

【详解】

1)直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,由题意可知: <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以曲线C是以坐标原点为中心,焦点在 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 轴上,不包括左右两顶点的椭圆,其方程为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

2)(i)设直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,由题意可知 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的方程与椭圆方程 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 联立, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,P在第一象限,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,因此点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的坐标为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,可得直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 方程: <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,与椭圆方程联立, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,消去 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 得, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> *),设点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,显然 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 点的横坐标 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是方程(*)的解

所以有 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,代入直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 方程中,得

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的坐标为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的斜率为;  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,

因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,因此 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 是直角三角形;

ii)由(i)可知: <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的坐标为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 单调递增,当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 单调递减,因此当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,函数 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 有最大值,最大值为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .

【点睛】

本题考查了求椭圆的标准方程,以及利用直线与椭圆的位置关系,判断三角形形状以及三角形面积最大值问题,考查了数学运算能力,考查了利用导数求函数最大值问题.

22在极坐标系中,O为极点,点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 在曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 上,直线l过点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 且与 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 垂直,垂足为P.

1)当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,求 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> l的极坐标方程;

2)当MC上运动且P在线段OM上时,求P点轨迹的极坐标方程.

【答案】1 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> l的极坐标方程为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ;(2 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【解析】

【分析】

1)先由题意,将 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 代入 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 即可求出 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ;根据题意求出直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的直角坐标方程,再化为极坐标方程即可;

2)先由题意得到P点轨迹的直角坐标方程,再化为极坐标方程即可,要注意变量的取值范围.

【详解】

1)因为点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 在曲线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 上,

所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

因为直线l过点 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 且与 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 垂直,

所以直线 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的直角坐标方程为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

因此,其极坐标方程为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即l的极坐标方程为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

2)设 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,则 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

由题意, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,故 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,整理得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

因为P在线段OM上,MC上运动,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

所以,P点轨迹的极坐标方程为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .

【点睛】

本题主要考查极坐标方程与直角坐标方程的互化,熟记公式即可,属于常考题型.

23已知 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

1)当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,求不等式 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的解集;

2)若 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,求 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的取值范围.

【答案】1 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ;(2 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

【解析】

【分析】

1)根据 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,将原不等式化为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,分别讨论 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 三种情况,即可求出结果;

2)分别讨论 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 两种情况,即可得出结果.

【详解】

1)当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,原不等式可化为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,原不等式可化为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,显然成立,

此时解集为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,原不等式可化为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,解得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,此时解集为空集;

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,原不等式可化为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,即 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,显然不成立;此时解集为空集;

综上,原不等式的解集为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

2)当 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,所以由 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 可得 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a>

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,显然恒成立;所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 满足题意;

 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> ,因为 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 时,  <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 显然不能成立,所以 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 不满足题意;

综上, <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> 的取值范围是 <a href="/tags/54/" title="试卷" class="c1" target="_blank">试卷</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/881/" title="统一" class="c1" target="_blank">统一</a> <a href="/tags/1190/" title="理科" class="c1" target="_blank">理科</a> .

【点睛】

本题主要考查含绝对值的不等式,熟记分类讨论的方法求解即可,属于常考题型.


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