JEE Main 2025 Jan 23 Shift 2, Mathematics Q22: Quadratic Equations

JEE Main2025Jan 23, Shift 2Mathematics

Let $α, β$ be the roots of the equation $x^{2} - a x - b = 0$ with $Im (α) < Im (β)$ . Let $P_{n} = α^{n} - β^{n}$ . If $P_{3} = - 57 i, P_{4} = - 37 i, P_{5} = 117 i$ and $P_{6} = 457 i$ , then $∣ α^{4} + β^{4} ∣$ is equal to ________.

Your answer

Solution

$α + β = a α β = - b$

$P_{6} = a P_{5} + b P_{4}$

$457 i = a \times 117 i + b (- 37) i$

$45 = 11 a - 3 b ... (1)$

and

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

$117 i = a (- 37 i) + b (- 57 i)$

$11 = - 3 a - 5 b ... (2)$

$a = 3, b = - 4$

$α^{4} + β^{4} = (α^{4} - β^{4})^{2} + 4 α^{4} β^{4}$

$= - 63 + 4 \cdot 4^{4}$

$= - 63 + 1024 = 961 = 31$

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JEE Main 2025 Jan 23 Shift 2, Mathematics Q22: Quadratic Equations

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