Let $w=\frac{\sqrt{3}+\text{i}}{2}$ and $\text{P}=\left\{w^{\text{n}}\text{: }\text{n}=\text{1, 2, 3,}\dots \right\}$.Further ${\text{H}}_1=\left\{\text{z}\in \text{C}\text{ : }\text{Re z}>\frac{1}{2}\right\}$ and ${\text{H}}_2=\left\{\text{z}\in \text{C}\text{ : }\text{Re z}<\frac{-1}{2}\right\}$, where C is the set of all complex numbers. If ${\text{z}}_1\in \text{P}\cap {\text{H}}_1\text{, }{\text{z}}_2\in \text{P}\cap {\text{H}}_2$ and O represents the origin, then $\angle {\text{z}}_1\text{O}{\text{z}}_2=$

• A

  $\frac{\pi }{2}$
• B

  $\frac{\pi }{6}$
• Correct answer
  C

  $\frac{2\pi }{3}$
• Correct answer
  D

  $\frac{5\pi }{6}$

Solution

$\text{P}={\omega }^{\text{n}}=\text{cos}\frac{\text{n}\pi }{6}+\text{i}\text{ sin}\frac{\text{n}\pi }{6}\text{,}{\text{ H}}_1=\text{Rez}>1/2$
${\text{z}}_1=\text{P}\cap {\text{H}}_1=\frac{\sqrt{3}+\text{i}}{2}\text{,}\frac{\sqrt{3}-\text{i}}{2}$
${\text{z}}_2=\text{P}\cap {\text{H}}_2=-1\text{,}\frac{-\sqrt{3}+\text{i}}{2}\text{,}\frac{-\sqrt{3}-\text{i}}{2}$
$\angle {\text{z}}_1\text{O }{\text{z}}_2=\frac{2\pi }{3}\text{,}\frac{5\pi }{6}\text{, }\pi$

Hence, the correct answer is options: 'C', 'D'.