Complex Numbers
De Moivre's Theorem
JEE Main 2019
Grade 11

Question:

Let $z = \frac{\sqrt{3} + i}{2}$. Then $z^{69}$ equals:
(1) $1$
(2) $-1$
(3) $i$
(4) $-i$

Step-by-Step Solution

Key Concept: Write $z = e^{i\pi/6}$ and compute $z^{69} = e^{i\cdot 69\pi/6}$. Reduce $69\pi/6$ modulo $2\pi$ to a standard angle.
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Correct Answer: (4)

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