JEE Advanced 2013 · Paper 2 · Mathematics Q41

← JEE Advanced 2013 · Paper 2

MathematicsMultiple correct+3 / -1Question 41 of 60

Let $w=\dfrac{\sqrt3+i}2$ and $P=\{w^n:n=1,2,3,\ldots\}$ . Further $H_1=\left\{z\in\mathbb C:\operatorname{Re}z>\frac12\right\}$ and $H_2=\left\{z\in\mathbb C:\operatorname{Re}z<\frac{-1}2\right\}$ , where $\mathbb C$ is the set of all complex numbers. If $z_1\in P\cap H_1$ , $z_2\in P\cap H_2$ and $O$ represents the origin, then $\angle z_1Oz_2=$

\dfrac\pi2

\dfrac\pi6

\dfrac{2\pi}3

\dfrac{5\pi}6

Answer and worked solution

Official answer: C, D

The set P. $w=e^{i\pi/6}$ , so $P$ is the 12 points $e^{ik\pi/6}$ on the unit circle.

$P\cap H_1$ ( $\cos\theta>\frac12$ ): $\theta=0,\pm\frac\pi6$ .
$P\cap H_2$ ( $\cos\theta<-\frac12$ ): $\theta=\frac{5\pi}6,\pi,\frac{7\pi}6$ .

Angles between them are $\frac{2\pi}3$ , $\frac{5\pi}6$ or $\pi$ . Of the options, $\frac{2\pi}3$ and $\frac{5\pi}6$ occur.

(C), (D)

Topics:Polar and Euler form Geometry in the complex plane

← Previous question Next question →

JEE Advanced 2013 · Paper 2, Mathematics question 41