【题文】(本题12分)若集合,集合,且,求实数的取值范围.
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【题文】(本题12分)若集合
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145414-51239.png)
,集合
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145414-79740.png)
,且
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145414-41228.png)
,求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145415-50622.png)
的取值范围.
答案
【答案】
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145415-87411.png)
.
解析
【解析】
试题分析:先求出集合
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145415-36073.png)
,再由
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145414-41228.png)
讨论
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145416-67851.png)
和
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145416-23545.png)
两种情况,即可求的实数
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145415-50622.png)
的取值范围.做题时学生容易漏掉
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145416-67851.png)
的情况.这类问题是:已知一个集合(含参数)是另一个集合(给定)的子集(或真子集),求参数范围(或值).解决此类问题的关键在于准确把握所给的集合,理清两个集合的关系,正确列出数学关系式,准确求解即可.学生在处理这类问题时,易漏掉空集的情况,涉及端点问题也容易在开闭上出错.
试题解析:
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145414-79740.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145416-16615.png)
, 2分
∵
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145414-51239.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145414-41228.png)
,
∴(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145416-67851.png)
时,则
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145416-55269.png)
,解得
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145417-84218.png)
; 6分
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145416-23545.png)
时,当
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145417-56295.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145417-23138.png)
,解得
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145418-25323.png)
,此时
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145418-28617.png)
,适合题意;
8分
当
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145418-30139.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145418-57393.png)
,解得
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145419-77268.png)
,此时
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145419-32765.png)
,不合题意; 10分
综上所述,实数a的取值范围为
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145415-87411.png)
. 12分
考点:①集合与集合之间的关系;②解一元二次不等式.
举一反三
【题文】(本题13分)已知集合
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145343-93992.png)
函数
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145344-24472.png)
的定义域为集合B.
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145344-81672.png)
,求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145344-16070.png)
的值;
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145345-79740.png)
,求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145344-16070.png)
的取值范围.
【题文】(本题14分)对于函数
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145320-25973.png)
,若
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145320-29554.png)
,则称
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145320-46241.png)
为
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145320-25973.png)
的“不动点”,若
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145320-21765.png)
,则称
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145320-46241.png)
为
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145320-25973.png)
的“稳定点”,函数
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145320-25973.png)
的“不动点”和“稳定点”的集合分别记为A和B,即
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145321-63040.png)
.
(1)设
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145321-76820.png)
,求集合A和B;
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145321-64963.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145321-27259.png)
,求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145322-16978.png)
的取值范围;
(3)若
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145322-13434.png)
,求证:
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145322-15210.png)
.
【题文】已知集合
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145214-52005.png)
,那么
![](http://img.shitiku.com.cn/uploads/allimg/20200322/20200322145214-84455.png)
子集的个数是:( )
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