Question

If $i = \sqrt{- 1}$, then $4 + 5\left( - \frac{1}{2} + \frac{i\sqrt{3}}{2} \right)^{334} + 3\left( - \frac{1}{2} + \frac{i\sqrt{3}}{2} \right)^{365}$ is

equal to

• A. $1 - i\sqrt{3}$
• B. $- 1 + i\sqrt{3}$
• C. $i\sqrt{3}$
• D. $- i\sqrt{3}$

Answer 1

Answer: (C)

$i\sqrt{3}$

Answer 2

Sol. Given equations $4 + 5\left( - \frac{1}{2} + i\frac{\sqrt{3}}{2} \right)^{334} + 3\left( - \frac{1}{2} + i\frac{\sqrt{3}}{2} \right)^{365} = 4 + 5\omega^{334} + 3\omega^{365}$ $= 4 + 5\omega + 3\omega^{2}$ $= 1 + 2w$

$= 1 + 2\left( \frac{- 1 + i\sqrt{3}}{2} \right)$ $= i\sqrt{3}$