Let $α, β, γ$ be roots of $x^{3} - p x^{2} + q x - r = 0$ then

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equation whose $α^{2} - β γ, β^{2} - γ α, γ^{2} - α β$ is $x^{3} + (3 q - p^{2}) x^{2} + q (3 q - p^{2}) x + q^{3} - p^{3} r = 0$

a permutation of $α, β, γ$ is in G.P. if $q^{3} = p^{3} r$

Square of one of the roots will be additive inverse of the product of the other two if $q^{3} = p^{3} r$

None of these

answer is A, B.

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Detailed Solution

$y = α^{2} - β γ = α^{2} - \frac{r}{α} = \frac{α^{3} - r}{α}$

$\begin{aligned} = \frac{p α^{2} - q α}{α} = p α - q \\ \Rightarrow α = \frac{1}{p} (y + q) \end{aligned}$

As $α$ is a root of the given equation

$\begin{array}{l} \frac{1}{p^{3}} (y + q)^{3} - \frac{p}{p^{2}} (y + q)^{2} + \frac{q}{p} (y + q) - r = 0 \\ \Rightarrow (y + q)^{3} - p^{2} (y + q)^{2} + p^{2} q (y + q) - p^{3} r = 0 \\ \Rightarrow y^{3} + (3 q - p^{2}) y^{2} + q (3 q - p^{2}) y + q^{3} - p^{3} r = 0 \end{array}$