Quadratic Equations
Power Sum Ratio via Complex Roots
nta_pyq_2023_apr
Grade 11
Question:
Let $\alpha,\beta$ be roots of $x^2+\sqrt{6}x+3=0$. Then $\dfrac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is equal to
Step-by-Step Solution
Key Concept: Roots: $\alpha=\sqrt{3}e^{i3\pi/4}$, $\beta=\sqrt{3}e^{i5\pi/4}$. Use $|\alpha|=|\beta|=\sqrt{3}$ and compute power sums via De Moivre.
$=9\cdot(\sqrt{3})^4=81$.
Correct Answer: 1