Quadratic Equations
Complex Roots
JEE Main 2024
Grade 11
Question:
If $\alpha$ and $\beta$ are the roots of $x^2 + x + 1 = 0$, then $\alpha^{2024} + \beta^{2024}$ equals:
(1) $-1$
(2) $0$
(3) $1$
(4) $2$
Step-by-Step Solution
Key Concept: The roots are the primitive cube roots of unity $\omega$ and $\omega^2$, with $\omega^3 = 1$. Find $2024 \pmod 3$ to reduce the exponent.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)