Let and . Further, H_1 = z C : Re (z) > 1 2 and H_2 = z C : Re (z) < (- 1 2 ) , where…

MATHEMATICS · Single correct · IB Diploma Programme · Mathematics: Analysis and Approaches · Number and Algebra · Number and Algebra

Problem

Let

w = \frac{\sqrt{3} + i}{2}

and

P = \{w^n : n = 1, 2, 3, \dots\}

. Further,

H_1 = \left\{ z \in C : \operatorname{Re}(z) > \frac{1}{2} \right\} \text{ and } H_2 = \left\{ z \in C : \operatorname{Re}(z) < \left(-\frac{1}{2}\right) \right\},

where

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_1 O z_2

equals to (1)

\frac{\pi}{2}

(2)

\frac{\pi}{6}

(3)

\frac{2\pi}{3}

(4)

\frac{5\pi}{6}