Q50 / 75 · JEE Main 2025 (23 Jan Shift 2) · Mathematics
Let $\alpha, \beta$ be the roots of the equation $x^2-a x-b=0$ with $\operatorname{Im}(\alpha) \lt \operatorname{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $\mathrm{P}_3=-5 \sqrt{7} i, \mathrm{P}_4=-3 \sqrt{7} i, \mathrm{P}_5=11 \sqrt{7} i$ and $\mathrm{P}_6=45 \sqrt{7} i$, then $\left|\alpha^4+\beta^4\right|$ is equal to $\qquad$ .
Correct answer: 31
Solution
$\begin{aligned}
& \alpha+\beta=\mathrm{a} \quad \alpha \beta=-\mathrm{b} \\
& \mathrm{P}_6=\mathrm{aP}_5+\mathrm{bP}_4 \\
& 45 \sqrt{7} \mathrm{i}=\mathrm{a} \times 11 \sqrt{7} \mathrm{i}+\mathrm{b}(-3 \sqrt{7}) \mathrm{i} \\
& 45=11 \mathrm{a}-3 \mathrm{~b}
\end{aligned}$
and
$\begin{aligned}
& P_5=\mathrm{aP}_4+b P_3 \\
& 11 \sqrt{7} \mathrm{i}=\mathrm{a}(-3 \sqrt{7} \mathrm{i})+\mathrm{b}(-5 \sqrt{7} \mathrm{i}) \\
& 11=-3 \mathrm{a}-5 \mathrm{~b} \\
& \mathrm{a}=3, \mathrm{~b}=-4 \\
& \left|\alpha^4+\beta^4\right|=\sqrt{\left(\alpha^4-\beta^4\right)^2+4 \alpha^4 \beta^4} \\
& =\sqrt{-63+4.4^4} \\
& =\sqrt{-63+1024}=\sqrt{961}=31
\end{aligned}$
Hence, the correct answer is 31.