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Question

If a line makes angles α,β,γ and δ with the diagonals of a cube, Then, cos2α+cos2β+cos2γ+cos2δ=ab, where a and b are in lowest form, find a+b

A
7
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B
6
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C
8
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D
None of these
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Solution

The correct option is A 7
A cube is a rectangular parallelopiped having equal length, breadth and height. Let OADBFEGC be the cube with each side of length a units.
The four diagonals are OE, AF, BG and CD.
The direction \cos ines of the diagonal OE which is the line joining two points O and E are
a0a2+a2+a2,a0a2+a2+a2,a0a2+a2+a2
i.e. 13,13,13
Similarly, the direction \cos ines of AF, BG and CD are (13,13,13);(1313,13);(13,13,13), respectively.
Let l, m, n be the direction \cos ines of the given line which makes angles α,β,γ,δ with OE, AF, BG, CD, respectively. Then
cosα=13(l+m+n);cosβ=13(l+m+n);
cosγ=13(lm+n);cosδ=13(l+m+n);
Squaring and adding, we get
cso2α+cos2β+cos2γ+cos2δ=13[(1+m+n)2+(1+m+n)2+(lm+n)2+(l+m+n)2]=13[4(l2+m2+n2)]=43
(as l2+m2+n2=1)

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