JEE Main 2022 (Online) 25th July Morning ShiftQuestion 1
If $\alpha,\beta,\gamma,\delta$ are the roots of the equation $x^4+x^3+x^2+x+1=0$, then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to :
Complex Numbers
Solution
The roots are $\omega^k$, $k=1,2,3,4$, $\omega=e^{2\pi i/5}$. $2021\equiv1\pmod5$, so $\alpha^{2021}=\alpha$ for each root. Sum $=\alpha+\beta+\gamma+\delta=-1$. Option (B).