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If $α, β, γ, δ$ are the roots of the equation $x^{4} + x^{2} + 1 = 0$ such that $α + β = - 1, α^{2} = β$ and $γ + δ = 1, γ^{2} = - δ$ then $α^{2023} + β^{2023} + γ^{2024} + δ^{2024} =_____$

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$1 + 3 ω$

$ω - 2 ω^{2}$

-2

answer is B.

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Detailed Solution

$(x^{2} + x + 1) (x^{2} - x + 1) = 0$
$α = ω, β = ω^{2} & γ = - ω, δ = - ω^{2}$
$∴ G . E = ω + ω^{2} + ω^{2} + ω = 2 (ω + ω^{2})$
$= 2 (- 1) = - 2$

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