Quadratic Equations
Complex Roots
JEE Main 2020
Grade 11
Question:
Let $\alpha, \beta$ be the roots of $x^2 - x + 1 = 0$. Then $\alpha^{2019} + \beta^{2019}$ equals:
(1) $-2$
(2) $-1$
(3) $1$
(4) $2$
Step-by-Step Solution
Key Concept: The roots of $x^2 - x + 1 = 0$ are primitive 6th roots of unity: $e^{\pm i\pi/3}$. Find $2019 \pmod 6$ to reduce the exponent.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)