JEE Main 2025 — Complex Number Question with Solution
JEE Main 2025 (23 Jan Shift 2)
Question
Let be the roots of the equation with . Let . If and , then is equal to .
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Show full solutionCorrect answer: 31
Correct answer
31
Step-by-step explanation
$\begin{aligned}
& \alpha+\beta=\mathrm{a} \quad \alpha \beta=-\mathrm{b} \\
& \mathrm{P}_6=\mathrm{aP}_5+\mathrm{bP}_4 \\
& 45 \sqrt{7} \mathrm{i}=\mathrm{a} \times 11 \sqrt{7} \mathrm{i}+\mathrm{b}(-3 \sqrt{7}) \mathrm{i} \\
& 45=11 \mathrm{a}-3 \mathrm{~b}
\end{aligned}$
and
$\begin{aligned}
& P_5=\mathrm{aP}_4+b P_3 \\
& 11 \sqrt{7} \mathrm{i}=\mathrm{a}(-3 \sqrt{7} \mathrm{i})+\mathrm{b}(-5 \sqrt{7} \mathrm{i}) \\
& 11=-3 \mathrm{a}-5 \mathrm{~b} \\
& \mathrm{a}=3, \mathrm{~b}=-4 \\
& \left|\alpha^4+\beta^4\right|=\sqrt{\left(\alpha^4-\beta^4\right)^2+4 \alpha^4 \beta^4} \\
& =\sqrt{-63+4.4^4} \\
& =\sqrt{-63+1024}=\sqrt{961}=31
\end{aligned}$
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This is a previous-year question from JEE Main 2025, covering the Complex Number chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.