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2007届清华大学保送生暨自主招生北京冬令营
数学笔试试题（2006年12月30日）
1．求![](https://wkrtcs.bdimg.com/rtcs/image?w=68&md5sum=1485efacbd00818517d16d6563f73216&sign=f070b9c479&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=192&ipr={"t":"img","r":"r_11","w":"68","h":"44","dataType":"wmf","c":"word/media/image3.png","imgOriW":136,"imgOriH":88})
的单调区间及极值．
2．设正三角形![](https://wkrtcs.bdimg.com/rtcs/image?w=15&md5sum=1485efacbd00818517d16d6563f73216&sign=f070b9c479&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=192&ipr={"t":"img","r":"r_11","w":"15","h":"24","dataType":"wmf","c":"word/media/image4.png","imgOriW":30,"imgOriH":48})
边长为，是的中点三角形，为除去后剩下三个三角形内切圆面积之和．求．
3．已知某音响设备由五个部件组成，A电视机，B影碟机，C线路，D左声道和E右声道，其中每个部件工作的概率如下图所示．能听到声音，当且仅当A与B中有一工作，C工作，D与E中有一工作；且若D和E同时工作则有立体声效果．
求：（1）能听到立体声效果的概率；
（2）听不到声音的概率．
4．（1）求三直线![](https://wkrtcs.bdimg.com/rtcs/image?w=68&md5sum=1485efacbd00818517d16d6563f73216&sign=f070b9c479&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=192&ipr={"t":"img","r":"r_11","w":"68","h":"21","dataType":"wmf","c":"word/media/image12.png","imgOriW":136,"imgOriH":42})
，，所围成三角形上的整点个数；
（2）求方程组![](https://wkrtcs.bdimg.com/rtcs/image?w=68&md5sum=1485efacbd00818517d16d6563f73216&sign=f070b9c479&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=192&ipr={"t":"img","r":"r_11","w":"68","h":"81","dataType":"wmf","c":"word/media/image15.png","imgOriW":136,"imgOriH":162})
的整数解个数．
5．已知![](https://wkrtcs.bdimg.com/rtcs/image?w=64&md5sum=1485efacbd00818517d16d6563f73216&sign=f070b9c479&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=192&ipr={"t":"img","r":"r_11","w":"64","h":"21","dataType":"wmf","c":"word/media/image16.png","imgOriW":128,"imgOriH":42})
，△ABC是正三角形，且B、C在双曲线一支上．
（1）求证B、C关于直线![](https://wkrtcs.bdimg.com/rtcs/image?w=39&md5sum=1485efacbd00818517d16d6563f73216&sign=f070b9c479&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=192&ipr={"t":"img","r":"r_11","w":"39","h":"17","dataType":"wmf","c":"word/media/image18.png","imgOriW":78,"imgOriH":34})
对称；
（2）求△ABC的周长．
6．对于集合![](https://wkrtcs.bdimg.com/rtcs/image?w=56&md5sum=1485efacbd00818517d16d6563f73216&sign=f070b9c479&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=192&ipr={"t":"img","r":"r_11","w":"56","h":"23","dataType":"wmf","c":"word/media/image19.png","imgOriW":112,"imgOriH":46})
，称M为开集，当且仅当，，使得．判断集合与是否为开集，并证明你的结论．
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