如图,三棱柱ABC-A
1B
1C
1的底面是边长为2的正三角形且侧棱垂直于底面,侧棱长是
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238018_ST/0.png)
,D是AC的中点.
(1)求证:B
1C∥平面A
1BD;
(2)求二面角A
1-BD-A的大小;
(3)求直线AB
1与平面A
1BD所成的角的正弦值.
考点分析:
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设椭圆C:
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238017_ST/0.png)
的左、右焦点分别为F
1,F
2,上顶点为A,过点A与AF
2垂直的直线交x轴负半轴于点Q,且
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238017_ST/1.png)
.
(1)求椭圆C的离心率;
(2)若过A、Q、F
2三点的圆恰好与直线l:
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238017_ST/2.png)
相切,求椭圆C的方程;
(3)在(2)的条件下,过右焦点F
2作斜率为k的直线l与椭圆C交于M、N两点,在x轴上是否存在点P(m,0)使得以PM,PN为邻边的平行四边形是菱形,如果存在,求出m的取值范围,如果不存在,说明理由.
.
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在数列{a
n}中,a
1=
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238016_ST/0.png)
,并且对于任意n∈N
*,且n>1时,都有a
n•a
n-1=a
n-1-a
n成立,令b
n=
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238016_ST/1.png)
(n∈N
*).
(I)求数列{b
n}的通项公式;
(II)求数列{
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238016_ST/2.png)
}的前n项和T
n,并证明T
n<
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238016_ST/3.png)
-
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238016_ST/4.png)
.
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已知向量
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238015_ST/0.png)
=(cosx,sinx),
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238015_ST/1.png)
=(-cosx,cosx)
(1)当x∈[
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238015_ST/2.png)
,
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238015_ST/3.png)
]时,求函数f(x)=2
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238015_ST/4.png)
•
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238015_ST/5.png)
+1的最大值.
(2)设f(x)=2
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238015_ST/6.png)
•
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238015_ST/7.png)
+1,将函数y=f(x)的图象向右平移
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238015_ST/8.png)
个单位后,再将得到的图象上各点的横坐标伸长到原来的4倍,纵坐标不变,得到函数y=g(x)的图象,求y=g(x)的单调递减区间.
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给出下列四个命题:
①若|x-lgx|<x+|lgx|成立,则x>1;
②若p=a+
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/0.png)
(a>2),q=
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/1.png)
(x∈R),则p>q,
③已知
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/2.png)
=|
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/3.png)
|=2,
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/4.png)
与
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/5.png)
的夹角为
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/6.png)
,则
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/7.png)
+
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/8.png)
在
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/9.png)
上的投影为3;
④已知f(x)=asinx-bcosx,(a,b∈R)在x=
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131101222837800723835/SYS201311012228378007238014_ST/10.png)
处取得最小值,则f(
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-x)=-f(x).
其中正确命题的序号是
.(把你认为正确的命题的序号都填上)
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设曲线y=x
n+1(n∈N
*)在点(1,1)处的切线与x轴的交点的横坐标为x
n,令a
n=lgx
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