CGPET 2022 Solved Paper

Let a be the length of an edge of the cube andlet one corner be at the origin.

Clearly, OP,AR,BS andCQ are the diagonals of the cube. The DR′s of OP,AR,BS and CQ are
a−0,a−0,a−0 i.e. a,a,a
0−a,a−0,a−0 i.e. −a,a,a
a−0,0−a,a−0 i.e. a,−a,a
and
a−0,a−0,0−a i.e. a,a,−a respectively.
Let α,β,γ,δ be the angle made by a line whose DR’‌s are l,m,n with OP,AR,BS and CQ. Then,
cos‌α=

a⋅l+a⋅m+a⋅n

√a2+a2+a2√l2+m2+n2

=

l+m+n

√3√l2+m2+n2

cos‌β=

−a⋅l+a⋅m+a⋅n

√a2+a2+a2√l2+m2+n2

=

−l+m+n

√3√l2+m2+n2

cos‌γ=

al−am+an

√a2+a2+a2√l2+m2+n2

=

l−m+n

√3√l2+m2+n2

and cos‌δ=

al+am−an

√a2+a2+a2√l2+m2+n2

=

l+m−n

√3√l2+m2+n2

∴cos2α+cos2β+cos2γ+cos2δ
=

(l+m+n)2

3(l2+m2+n2)

+

(−l+m+n)2

3(l2+m2+n2)

+

(l−m+n)2

3(l2+m2+n2)

+

(l+m−n)2

3(l2+m2+n2)

=

4

3

Again,
1−sin2α+1−sin2β+1−si2γ+1−sin2δ=

4

3

⇒sin2α+sin2β+sin2γ+sin2δ=4−

4

3

=

8

3