已知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742276292.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742292307.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742323503.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742339790.png)
,其中
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742354422.png)
(1)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742276292.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742292307.png)
的夾角為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742401382.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742417312.png)
的值;
(2)記
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742432637.png)
,是否存在實數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742448266.png)
,使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742464629.png)
對任意的
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742479418.png)
恒成立?若存在,求出實數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742448266.png)
的取值范圍;若不存在,試說明理由.
試題分析:(1)先運(yùn)用向量的數(shù)量積公式求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742510568.png)
,對式子
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742339790.png)
兩邊平方以及結(jié)合
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742542461.png)
的模均是1得到關(guān)于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742417312.png)
的等式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742557716.png)
;(2)利用(1)中
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742339790.png)
平方求出的式子將
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742588452.png)
表示成關(guān)于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742417312.png)
的式子
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742635968.png)
,均值不等式求得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742651760.png)
,再利用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742666460.png)
解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742448266.png)
.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240517426981212.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742713793.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240517427291070.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240517427761273.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742557716.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742807702.png)
(6分)
由(1)得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240517428221271.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240517428381069.png)
,即可得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742854811.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742635968.png)
,因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742885823.png)
對于任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742666460.png)
恒成立,又因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742651760.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742932563.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742947516.png)
對于任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742666460.png)
恒成立,構(gòu)造函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742978693.png)
從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240517429941544.png)
由此可知不存在實數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824051742448266.png)
使之成立.
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