如圖,四棱錐

的底面是正方形,每條側(cè)棱的長都是底面邊長的

倍,
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為側(cè)棱

上的點(diǎn)。
(Ⅰ)求證:

;
(Ⅱ)若
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平面

,求二面角

的大��;
(Ⅲ)在(Ⅱ)的條件下,側(cè)棱

上是否存在一點(diǎn)

, 使得

平面

。若存在,求
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的值;若不存在,試說明理由。

解法一:
(Ⅰ);連
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,設(shè)
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交于

于
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,由題意知
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.以O(shè)為坐標(biāo)原點(diǎn),
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分別為
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軸、
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軸、
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軸正方向,建立坐標(biāo)系
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如圖。
設(shè)底面邊長為

,則高

。 于是
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

故

從而
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(Ⅱ)由題設(shè)知,平面

的一個(gè)法向量

,平面

的一個(gè)法向量

,設(shè)所求二面角為
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,則
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,所求二面角的大小為
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(Ⅲ)在棱
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上存在一點(diǎn)

使
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.由(Ⅱ)知

是平面

的一個(gè)法向量,
且

設(shè)
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則
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而
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即當(dāng)
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時(shí),
而
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不在平面
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內(nèi),故
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解法二:(Ⅰ)連BD,設(shè)AC交BD于O,由題意
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。在正方形ABCD中,
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,所以
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,得
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.
(Ⅱ)設(shè)正方形邊長
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,則
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。
又
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,所以

,
連
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,由(Ⅰ)知

,所以

,
且

,所以

是二面角

的平面角。
由

,知

,所以

,
即二面角

的大小為

。
Ⅲ)在棱SC上存在一點(diǎn)E,使

由(Ⅱ)可得

,故可在

上取一點(diǎn)

,使
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,過
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作

的平行線與

的交點(diǎn)即為

。連BN。在

中知

,又由于

,故平面

,得

,由于

,故

.
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三棱錐被平行于底面

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,

,
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
平面
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,
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,
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,
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為
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中點(diǎn).
(Ⅰ)證明:平面
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平面
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;
(Ⅱ)求二面角
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的正弦值.
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科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
如圖,矩形
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中,
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,沿對(duì)角線
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將
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折起,使
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點(diǎn)在平面
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內(nèi)的射影落在
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邊上,若二面角
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的平面角大小為
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,則
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的值為_______________▲_______________
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科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
a,b,c分別表示三條直線,M表示平面,給出下列四個(gè)命題:①若a∥M,b∥M,則a∥b;②若b
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M,a∥b,則a∥M;③若a⊥c,b⊥c,則a∥b;④若a⊥M,b⊥M,則a∥b.其中不正確命題的有
(填序號(hào))
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科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
如圖,已知平面

平面
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,
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、
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是平面
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與平面
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的交線上的兩個(gè)定點(diǎn),
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,且
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,
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,
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,
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,
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,在平面

上有一個(gè)動(dòng)點(diǎn)
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,使得
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,則
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的面積的最大值是( )
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