Question

The principal argument of the complex number
$\frac{\left[\right(1+i{)}^5(1+\sqrt{3}i{)}^2]}{[-2i(-\sqrt{3}+i\left)\right]}$ is

• A. $\frac{19\pi }{12}$
• B. $\frac{7\pi }{12}$
• C. $\frac{5\pi }{12}$
• D. $\frac{8\pi }{12}$

Answer 1

The correct option is A $\frac{19\pi }{12}$

$z=\frac{{(1+i)}^5{(1+i\sqrt{3})}^2}{-2i(-\sqrt{3}+i)}=\frac{{(1+i)}^5(1+i\sqrt{3})}{2}$
$arg\left(z\right)=arg\left[\frac{{(1+i)}^5(1+i\sqrt{3})}{2}\right]$
$⟹arg\left(z\right)=arg{(1+i)}^5+arg(1+i\sqrt{3})=5arg(1+i)+arg(1+i\sqrt{3})$
$⟹arg\left(z\right)=\frac{5\pi }{4}+\frac{\pi }{3}=\frac{19\pi }{12}$
Ans: A