If α , β are complex cube roots of unity, then the value of α2β2 + α10β2 + α2 β10 equals

Question

If α , β are complex cube roots of unity, then the value of α2β2 + α10β2 + α2 β10 equals (A) 1 (B) - 1 (C) 3 (D) 0

Tardigrade Admin · Accepted Answer

As α, β are complex cube roots of unity. ∴ α = ω, β = ω2 or α = ω2, β = ω ∴ α2β2 + α10β2 + α2β10 = ω2(ω2)2 + ω10ω4 + ω2ω20 = ω6 + ω14 + ω22 = 1 + ω2 + (ω3)7. ω = 1 + ω2 + ω = 0 (∵ ω3 = 1)

Q. If $α, β$ are complex cube roots of unity, then the value of $α^{2} β^{2} + α^{10} β^{2} + α^{2} β^{10}$ equals

$1$

$- 1$

$3$

$0$

Q. If α,β are complex cube roots of unity, then the value of α2β2+α10β2+α2β10 equals

1

−1

3

0

As α,β are complex cube roots of unity. ∴α=ω,β=ω2 or α=ω2,β=ω ∴α2β2+α10β2+α2β10=ω2(ω2)2+ω10ω4+ω2ω20 =ω6+ω14+ω22 =1+ω2+(ω3)7.ω =1+ω2+ω=0(∵ω3=1)

Q. If $α, β$ are complex cube roots of unity, then the value of $α^{2} β^{2} + α^{10} β^{2} + α^{2} β^{10}$ equals

$1$

$- 1$

$3$

$0$