JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $α, β, γ$ be the three roots of the equation $x^{3} + b x + c = 0$ . If $β γ = 1 - α$ , then $b^{3} + 2 c^{3} - 3 α^{3} - 6 β^{3} - 8 γ^{3}$ is equal to

Choose the correct answer:

Explanation

Given cubic equation is :
$x^{3} + b x + c = 0$

∴ $α, β, γ$ are the roots of above equation.

And $β γ = 1 = - a$

So, product of roots $= - c$

$\Rightarrow α β γ = - c$

$\Rightarrow (- 1) (1) = - c$

$\Rightarrow c = 1$

Since, $α = - 1$ is the root. So,

$\Rightarrow - 1 - b + c = 0$

$\Rightarrow c - b = 1$

$\Rightarrow 1 - b = 1 \Rightarrow b = 0$

The given equation becomes $x^{3} + 1 = 0$

So, roots are $- 1_{l}, - ω_{l}, - ω_{l}^{2}$

∴ $b^{3} + 2 c^{3} - 3 a^{2} - 6 β^{3} - 8 γ^{3}$

$= 0 + 2 - 3 (- 1)^{3} - 6 (- ω)^{3} - 8 (- ω^{2})^{3}$

$= 2 + 3 + 6 ω^{3} + 8 ω^{6}$

$= 5 + 6 + 8 = 19$