JEE PYQ: Complex Numbers Question 36
Question 36 - 2019 (09 Jan Shift 1)
Let $\alpha$ and $\beta$ be two roots of the equation $x^2 + 2x + 2 = 0$, then $\alpha^{15} + \beta^{15}$ is equal to:
(1) $-256$
(2) $512$
(3) $-512$
(4) $256$
Show Answer
Answer: (1)
Solution
Roots: $-1 \pm i = \sqrt{2}e^{\pm i3\pi/4}$. $\alpha^{15} + \beta^{15} = 2(\sqrt{2})^{15}\cos\frac{45\pi}{4} = 2^{15/2} \cdot 2 \cos\frac{\pi}{4} \cdot …$ Actually: $\alpha^{15} = (\sqrt{2})^{15}e^{i45\pi/4} = 2^{15/2}e^{i\pi/4+i\cdot 11\pi}$. $\alpha^{15} + \beta^{15} = 2 \cdot 2^{15/2}\cos(45\pi/4) = -256$.
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