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Question

A line makes angles a,b,c,d with the four diagonals of a cube, then cos2a+cos2b+cos2c+cos2d=

A
13
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B
43
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C
23
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D
53
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Solution

The correct option is C 43
Four diagonal of the cube are AL,BM,CN and OP
(putting a=b=c)

Direction cosine of the four diagonals are
OP(13,13,13), AL(13,13,13)BM(13,13,13), CN(13,13,13)
Let the D.C.'s of the line is l,m,n then
cosa=l3+m3+n3=l+m+n3

Similarly,
cosb=l+m+n3cosc=lm+n3cosd=l+mn3

Squaring and adding
cos2a+cos2b+cos2c+cos2d=13[(l+m+n)2+(l+mn)2+(lm+n)2+(l+m+n)2]=43(l2+m2+n2)=43

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