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  4. Let α and β be the roots of the equation x 2+(2 i - 1)=0. Then, the value of |α8+β8| is equal to :

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Q. Let $\alpha$ and $\beta$ be the roots of the equation $x ^{2}+(2 i - 1)=0$. Then, the value of $\left|\alpha^{8}+\beta^{8}\right|$ is equal to :

JEE MainJEE Main 2022Complex Numbers and Quadratic Equations

Solution:

$X ^{2}=1-2 i \Rightarrow \alpha^{2}=1-2 i , \beta^{2}=1-2 i$
Hence $\alpha^{8}=\beta^{8}$
$\left|\alpha^{8}+\beta^{8}\right|=\left|2\alpha^{8}\right|=2\left|\alpha^{2}\right|^{4} $
$=2 \sqrt{5}^{4}=50$