Let be the roots of the equation with . Let . If , then is equal to __________. [2025]

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Solution

Q.
Let $α, β$ be the roots of the equation $x^{2} - a x - b = 0$ with $I m (α) < I m (β)$ . Let $P_{n} = α^{n} - β^{n}$ . If $P_{3} = - 5 \sqrt{7} i, P_{4} = - 3 \sqrt{7} i, P_{5} = 11 \sqrt{7} i and P_{6} = 45 \sqrt{7} i$ , then $| α^{4} + β^{4} |$ is equal to __________. [2025]

Ans.
(31)

We have, $α + β = a and α β = - b$

$∵ P_{6} = a P_{5} + b P_{4}$

$\Rightarrow 45 \sqrt{7} i = a \times 11 \sqrt{7} i + b (- 3 \sqrt{7}) i$

$\Rightarrow 45 = 11 a - 3 b$ ... (i)

and $P_{5} = a P_{4} + b P_{3}$

$\Rightarrow 11 \sqrt{7} i = a (- 3 \sqrt{7} i) + b (- 5 \sqrt{7} i)$

$\Rightarrow 11 = - 3 a - 5 b$ ... (ii)

On solving equations (i) and (ii), we get a = 3, b = –4

$∴$ $| α^{4} + β^{4} |$ $= \sqrt{{(α^{4} - β^{4})}^{2} + 4 α^{4} β^{4}}$

$= \sqrt{- 63 + 4 . 4^{4}} = \sqrt{- 63 + 1024} = \sqrt{961} = 31$ .

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