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 B43 Four diagonal of the cube are AL,BM,CN and OP
(putting a=b=c)
Direction cosine of the four diagonals are OP(1√3,1√3,1√3),AL(−1√3,1√3,1√3)BM(1√3,−1√3,1√3),CN(−1√3,1√3,−1√3)
Let the D.C.'s of the line is l,m,n then cosa=l√3+m√3+n√3=l+m+n√3
Similarly, cosb=−l+m+n√3cosc=l−m+n√3cosd=l+m−n√3
Squaring and adding cos2a+cos2b+cos2c+cos2d=13[(l+m+n)2+(l+m−n)2+(l−m+n)2+(−l+m+n)2]=43(l2+m2+n2)=43