Question

Principal Argument of complex number $\frac{(1 + i)^{5}(1 + \sqrt{3}i)^{2}}{- 2i( - \sqrt{3} + i)}$ is–

• A. $\frac{19\pi}{12}$
• B. $\frac{- 7\pi}{12}$
• C. $\frac{- 5\pi}{12}$
• D. $\frac{5\pi}{12}$

Answer 1

Answer: (C)

$\frac{- 5\pi}{12}$

Answer 2

Sol. z = $\frac{(1 + i)^{5}(1 + \sqrt{3}i)^{2}}{2i(\sqrt{3} - i)}$

arg z = arg (1+ i)⁵ +

arg (1 + $\sqrt{3}$i)² – arg (2i) – arg ($\sqrt{3}$–i)

= $\frac{5\pi}{4}$+ $\frac{2\pi}{3}$– $\left( \frac{\pi}{2} \right)$– $\left( \frac{- \pi}{6} \right)$ = $\left( \frac{19\pi}{12} \right)$

\ Principal argument = – $\left( 2\pi - \frac{19\pi}{12} \right)$ = $\frac{- 5\pi}{12}$