If $α, β, γ$ are the roots of the equation $x^{3} + {px}^{2} + qx + r = 0$ then the coefficient of $x$ in cubic equation whose roots are $α (β + γ), β (γ + α)$ and $γ (α + β)$ is

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$r (pq - r)$

$p^{2} - qr$

$q^{2} + pr$

answer is B.

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Detailed Solution

$\begin{array}{l} I f α, β, γ a r e r o o t s o f x^{3} + p x^{2} + q x + r = 0 \\ T h e n α + β + γ = - p, α β + β γ + γ α = q, α β γ = - γ \\ ∴ T h e x c o e f f i c e n t i n c u b i c e q u a t i o n w h o s e r o o t s a r e α (β + γ), β (α + γ), γ (α + β) i s \\ α (β + γ) β (α + γ) + β (α + γ) γ (α + β) + α (β + γ) γ (α + β) \\ = α β (α β + β γ + γ α + γ^{2}) + β γ (α^{2} + α β + α γ + β γ) + α γ (β^{2} + β γ + α γ + β γ) \\ = {(α β)}^{2} + {(β γ)}^{2} + {(γ α)}^{2} + 3 α β γ (α + β + γ) \\ = {(α β + β γ + γ α)}^{2} - 2 a β γ (α + β + γ) + 3 α β γ (α + β + γ) \\ = 9^{2} - 2 p r + 3 p r \\ = q^{2} + p r \end{array}$