If satisfies the equation x 2 + x + 1 = 0 and , A , B , C ? 0 , then 5 ( 3 A - 2 B - C ) is equal to _______ . [2024]

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Solution

Q.
If $α$ satisfies the equation $x^{2} + x + 1 = 0$ and ${(1 + α)}^{7} = A + B α + C α^{2}, A, B, C \geq 0,$ then $5 (3 A - 2 B - C)$ is equal to _______ . [2024]

Ans.
(5)

As $α$ satisfies $x^{2} + x + 1 = 0 \Rightarrow α = ω or ω^{2}$

Now, ${(1 + α)}^{7} = {(1 + ω)}^{7} = A + B ω + C ω^{2}$

$\Rightarrow - ω^{2} = A + B ω + C ω^{2} [∵ 1 + ω + ω^{2} = 0 and ω^{3} = 1]$

$\Rightarrow A + B ω + (C + 1) ω^{2} = 0$

As $ω = - \frac{1}{2} + \frac{\sqrt{3}}{2} i$ & $ω^{2} = - \frac{1}{2} - \frac{\sqrt{3}}{2} i$

$\Rightarrow A + B (- \frac{1}{2} + \frac{\sqrt{3}}{2} i) + (C + 1) (- \frac{1}{2} - \frac{\sqrt{3}}{2} i) = 0$

$\Rightarrow A - \frac{B}{2} - \frac{C + 1}{2} = 0 and \frac{\sqrt{3}}{2} B - \frac{\sqrt{3}}{2} (C + 1) = 0$

$\Rightarrow 2 A - B - C - 1 = 0 and B - C = 1$

On solving, we get A = B and B - C = 1

$∴$ $5 (3 A - 2 B - C) = 5 (3 B - 2 B - C) = 5 (B - C) = 5 \times 1 = 5$

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