解:(法1)(Ⅰ)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163615818309.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163615880313.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616301390.gif)
,∴PA⊥底面ABC,∴PA⊥BC.又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616317492.gif)
,∴AC⊥BC.∴BC⊥平面PAC.(4分)
(Ⅱ)∵D为PB的中点,DE//BC,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616348526.gif)
,
又由(Ⅰ)知,BC⊥平面PAC,∴DE⊥平面PAC,垂足为点E.
∴∠DAE是AD与平面PAC所成的角,∵PA⊥底面ABC,
∴PA⊥AB,又PA=AB,∴△ABP为等腰直角三角形,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616364446.gif)
,∴在Rt△ABC中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616379491.gif)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616395510.gif)
.
∴在Rt△ADE中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616411951.gif)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616161236.gif)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616099265.gif)
所成的角的大小
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616239510.gif)
.(8分)
(Ⅲ)∵AE//BC,又由(Ⅰ)知,BC⊥平面PAC,∴DE⊥平面PAC,
又∵AE
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616489135.gif)
平面PAC,PE
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616489135.gif)
平面PAC,∴DE⊥AE,DE⊥PE,
∴∠AEP为二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616223326.gif)
的平面角,∵PA⊥底面ABC,
∴PA⊥AC,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616551494.gif)
.∴在棱PC上存在一点E,使得AE⊥PC,
这时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616567482.gif)
,故存在点E使得二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616223326.gif)
是直二面角.(12分)
(法2)如图,以A为原煤点建立空间直角坐标系
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616613379.gif)
,设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616629275.gif)
,
由已知可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616645357.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616676359.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616691676.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616723606.gif)
.
(Ⅰ)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616738522.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616754619.gif)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616785394.gif)
,
∴BC⊥AP.又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616317492.gif)
,∴BC⊥AC,∴BC⊥平面PAC.(4分)
(Ⅱ)∵D为PB的中点,DE//BC,∴E为PC的中点,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616894718.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616910646.gif)
,∴又由(Ⅰ)知,BC⊥平面PAC,
∴DE⊥平面PAC,垂足为点E.∴∠DAE是AD与平面PAC所成的角,
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231636169251611.gif)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231636169571105.gif)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616161236.gif)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616099265.gif)
所成的角的大小
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163617019435.gif)
。(8分)
(Ⅲ)∵AE//BC,又由(Ⅰ)知,BC⊥平面PAC,∴DE⊥平面PAC,
又∵AE
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616489135.gif)
平面PAC,PE
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616489135.gif)
平面PAC,∴DE⊥AE,DE⊥PE,
∴∠AEP为二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616223326.gif)
的平面角,∵PA⊥底面ABC,
∴PA⊥AC,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616551494.gif)
.∴在棱PC上存在一点E,
使得AE⊥PC,这时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616567482.gif)
,
故存在点E使得二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163616223326.gif)
是直二面角.(12分)