If - - and - are the roots of the equation x3-3 x-2-0 and -Y-Y- are the roots of equation y 3-9 y 2-216-0- Express y in terms of -A- y-2-1- y-2 -2-2-0C- y-2-2-y-2 -2-1-

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Inequalities Involving Modulus Function

If α, β and γ...

Question

If $α, β a n d γ$ are the roots of the equation $x^{3} + 3 x + 2 = 0 a n d (α - β) (α - γ), (β - γ) (β - α), (γ - α) (γ - β)$ are the roots of equation $y^{3} - 9 y^{2} - 216 = 0$ . Express y in terms of α.

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Solution

The correct option is B

$α, β, γ$ are the roots of equation $x^{3} + 3 x + 2 = 0$ ___________(1)

Sum of the roots $α + β + γ = \frac{- b}{a} = 0 \to β + γ = - α$

Sum of the roots taking two at a time $α β + β γ + γ α = \frac{c}{a} = 3$

Product of the roots $α β γ = \frac{- d}{a} = - 2 \to β γ = \frac{- 2}{α}$

We can take y = $(α - β) (α - γ) =$ $α^{2} - α β - α γ + β γ$

$= α^{2} - α (β + γ) + β γ β + γ = - α, β γ = \frac{- 2}{α}$

= $α^{2} - α (- α) + \frac{- 2}{α}$

$y = 2 α^{2} - \frac{2}{α}$

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If α,β and γ are the roots of the equation x3+3x+2=0 and (α−β)(α−γ),(β−γ)(β−α),(γ−α)(γ−β) are the roots of equation y3−9y2−216=0 . Express y in terms of α.

If $α, β a n d γ$ are the roots of the equation $x^{3} + 3 x + 2 = 0 a n d (α - β) (α - γ), (β - γ) (β - α), (γ - α) (γ - β)$ are the roots of equation $y^{3} - 9 y^{2} - 216 = 0$ . Express y in terms of α.