【题文】集合A={x|x2-ax+a2-19=0},B={x|x2-5x+6=0},C={x|x2+2x-8=0}. (1)若A∩B=A∪B,求a的值
题型:难度:来源:
【题文】集合
A={
x|
x2-
ax+
a2-19=0},
B={
x|
x2-5
x+6=0},
C={
x|
x2+2
x-8=0}.
(1)若
A∩
B=
A∪
B,求
a的值;
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044328-90288.png)
A∩
B,
A∩
C=
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044328-90288.png)
,求
a的值.
答案
【答案】(1)a=5.(2)a=-2
解析
【解析】
试题分析:由已知,得
B={2,3},
C={2,-4}.
(1)∵
A∩
B=
A∪
B,∴
A=
B于是2,3是一元二次方程
x2-
ax+
a2-19=0的两个根,由韦达定理知:
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044329-86297.png)
解之得
a=5.
(2)由
A∩
B ![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044329-33971.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044328-90288.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044329-83654.png)
∩
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044330-42294.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044330-40924.jpg)
,又
A∩
C=
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044328-90288.png)
,得3∈
A,2
A,-4
A,由3∈
A,
得3
2-3
a+
a2-19=0,解得
a=5或
a=-2
当
a=5时,
A={
x|
x2-5
x+6=0}={2,3},与2
A矛盾;
当
a=-2时,
A={
x|
x2+2
x-15=0}={3,-5},符合题意
考点:集合的混合运算
点评:主要是考查了集合之间的关系以及基本运算的综合运用,属于基础题。
举一反三
【题文】集合
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044311-57051.png)
的元素个数是 ( )
【题文】对于非空实数集
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044300-49567.png)
,记
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044300-96649.png)
.设非空实数集合
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044300-52114.png)
,满足
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044301-45789.png)
. 给出以下结论:
①
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044301-89522.png)
;
②
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044301-90311.png)
;
③
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044302-38939.png)
.
其中正确的结论是
.(写出所有正确结论的序号)
【题文】设整数
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044248-44686.png)
,集合
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044249-63571.png)
.令集合
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044249-59668.png)
若
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044249-45268.png)
和
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044249-40129.png)
都在
![](http://img.shitiku.com.cn/uploads/allimg/20200314/20200314044249-97584.png)
中,则下列选项正确的是( )
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