If $x^{3} - a x^{2} + b x + c = 0$ has the roots $α^{2} + β^{3} + γ^{4}, β^{2} + γ^{3} + α^{4}$ and $γ^{2} + α^{3} + β^{4}$ where $α, β, γ$ are the roots of $x^{3} - x^{2} - 1 = 0$ then the value of $2 a + b + c$ is equal to __

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answer is D.

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Detailed Solution

$x^{3} - x^{2} - 1 = 0 \to α, β, γ \Rightarrow α^{2} + β^{2} + γ^{2} = 1$
$α^{2} + β^{3} + γ^{4} = α^{2} + β^{2} + 1 + γ^{3} + γ = 3 + γ$
In given equation replace $x b y x - 3 \Rightarrow x^{3} - 10 x^{2} + 33 x - 37 = 0$

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