Question

Let $w = \frac{\sqrt 3 + i }{2}$ and $P = \{W^n:n=1,2,3,....\}$ Further $H_1 = \bigg \{z \in C: Re \, z >\frac{1}{2}\bigg \}$ and $H_2=\bigg[ z\in C: Re \,z

• A. $\frac{\pi}{2}$
• B. $\frac{\pi}{6}$
• C. $\frac{2\pi}{3}$
• D. $\frac{1\pi}{6}$

Answer 1

The Correct Option is D

PLAN It is the simple representation of points on Argand plane and
to find the angle between the points
Here, P = $W^n=\bigg(cos\frac{\pi}{6}+i \, sin\frac{\pi}{6}\bigg)^n=cos\frac{n\pi}{6}+i \, sin\frac{n\pi}{6}$
$H_1=\bigg \{z \in \, C: Re(z)>\frac{1}{2}\bigg \}$
$\therefore \, P \cap H_1$ represents those points for which cos$\frac{n\pi}{6} is +ve$
Hence, it belongs to I or IV quadrant
$\Rightarrow \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, z_1= P \cap H =cos\frac{\pi}{6}+ i sin\frac{\pi}{6}$
or $ \, \, \, \, \, \, \, \, \, \, \, \, \, \, cos\frac{11 \pi}{6}+ i \, sin\frac{11\pi}{6}$
$\therefore \, \, \, \, \, \, \, \, \, \, \, z_1=\frac{\sqrt 3}{2}+\frac{i}{2} \, or \, \frac{\sqrt 3}{2}-\frac{i}{2} \, \, \, \, $....(i)
Similarly
$z_2=P \, \cap \, H_2$ i.e. those points for which
$ \, \, \, \, \, \, \, \, \, \, \, cos\frac{n\pi}{6}<0$
$\therefore \, \, z_2=cos\pi+i sin \pi ,cos \frac{5\pi}{6},\frac{cos 7\pi}{6}$
$\hspace45mm +i sin \frac{7\pi}{6}$
$\Rightarrow \, \, \, \, \, \, \, z_2=-1.\frac{-\sqrt 3}{2}+\frac{i}{2},\frac{-\sqrt 3}{2}-\frac{i}{2}$
$Thus, \, \, \angle z_1Oz_2=\frac{2\pi}{3},\frac{1\pi}{6} $