The difference between two roots of the equation $x^{3} - 13 x^{2} + 15 x + 189 = 0$ is 2. Then the roots of the equation are

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–3, –1, 9

3, 5, 7

–3, –1, –9

–3, 7, 9

answer is A.

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Detailed Solution

$\begin{array}{l} L e t α, α + 2, β a r e r o o t s o f x^{3} - 13 x^{2} + 15 x + 189 = 0 \\ ∴ α + α + 2 + β = 13 \Rightarrow 2 α + β = 11 - - (1) \\ ∴ α (α + 2) + β (α + 2) + α β = 15 \ \\ \Rightarrow α^{2} + 2 (α + β) + 2 α β = 15 - - (2) \\ ∴ α (α + 2) β = - 189 \\ \Rightarrow α^{2} β + 2 α β = - 189 - - (3) \\ f r o m (1) β = 11 - 2 α p u t i n (2) \\ \Rightarrow α^{2} + 2 α + 2 (11 - 2 α) + 2 α (11 - 2 α) = 15 \\ \Rightarrow α^{2} + 2 α + 22 - 4 α + 22 α - 4 α^{2} = 15 \\ \Rightarrow - 3 α^{2} + 20 α + 7 = 0 \\ \Rightarrow 3 α^{2} - 20 α - 7 = 0 \\ \Rightarrow (α - 7) (3 α + 1) = 0 \\ ∴ α = 7, \frac{- 1}{3} \\ I f α = 7 t h e n β = 11 - 14 = - 3 \\ ∴ T h e r o o t s a r e 7, 9, - 3 \end{array}$