平面直角坐标系中,O为坐标原点,给定两点A(1,0),B(0,-2),点C满足
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,其中α,β∈R,且α-2β=1.
(Ⅰ)求点C的轨迹方程;
(Ⅱ)设点C的轨迹与双曲线
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727019_ST/1.png)
交于两点M,N,且以MN为直径的圆过原点,求证:
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727019_ST/2.png)
为定值.
考点分析:
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已知
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727018_ST/0.png)
,
(1)若函数
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在区间[1,+∞)上是增函数,求实数m的取值范围.
(2)若m≤2,求函数g(x)=f(x)-lnx在区间
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上的最小值.
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设数列{a
n},{b
n}满足a
1=b
1=6,a
2=b
2=4,a
3=b
3=3,且数列{a
n+1-a
n}(n∈N
+)是等差数列,数列{b
n-2}(n∈N
+)是等比数列.
(1)求数列{a
n}和{b
n}的通项公式;
(2)是否存在k∈N
+,使
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727017_ST/0.png)
,若存在,求出k,若不存在,说明理由.
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![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727016_ST/images0.png)
如图,在底面为菱形的四棱锥P-ABCD中,∠ABC=60°,PA=AC=a,PB=PD=
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727016_ST/0.png)
a,点E在PD上,且PE:ED=2:1.
(1)求证:PA⊥平面ABCD;
(2)求面EAC与面DAC所成的二面角的大小.
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在△ABC中,a、b、c分别是角A、B、C的对边,向量
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727015_ST/0.png)
=(2a+c,b),
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727015_ST/1.png)
=(cosB,cosC),且
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727015_ST/2.png)
•
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727015_ST/3.png)
=0.
(1)求∠B的大小;
(2)若b=
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103100020169872707/SYS201311031000201698727015_ST/4.png)
,求a+c的最大值.
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定义在R上的偶函数f(x)满足f(x+1)=-f(x),且在[-1,0]上是增函数,给出下列关于f(x)的判断:
①f(x)是周期函数;
②f(x)关于直线x=1对称;
③f(x)在[0,1]上是增函数;
④f(x)在[1,2]上是减函数;
⑤f(2)=f(0),
其中正确的序号是
.
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