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  4. If α , β ∈ C are the distinct roots, of the equation x2 - x + 1 = 0 , then α101 + β107 is equal to

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Q. If $\alpha , \beta \in C$ are the distinct roots, of the equation $x^2 - x + 1 = 0 $ , then $\alpha^{101} + \beta^{107} $ is equal to

JEE MainJEE Main 2018Complex Numbers and Quadratic Equations

Solution:

$x^{2}-x+1=0$
Roots are $-\omega,-\omega^{2}$
Let $\alpha=-\omega, \beta=-\omega^{2}$
$\alpha^{101}+\beta^{107}=(-\omega)^{101}+\left(-\omega^{2}\right)^{107}$
$=-\left(\omega^{101}+\omega^{214}\right)$
$=-\left(\omega^{2}+\omega\right)$
$=1$