Complex Numbers in Algebraic and Polar Form

9000070110

Level: 
B
Given \(z_{1} = 4\left (\cos \frac{5} {3}\pi + \mathrm{i}\sin \frac{5} {3}\pi \right )\) and \(z_{2} = 2\left (\cos \frac{1} {6}\pi + \mathrm{i}\sin \frac{1} {6}\pi \right )\), evaluate \(\frac{z_{1}} {z_{2}} \).
\(- 2\mathrm{i}\)
\(4\mathrm{i}\)
\(\mathrm{i}\)
\(-\frac{1} {2}\mathrm{i}\)

9000039110

Level: 
C
Assuming \(z\in \mathbb{C}\), solve the following equation. \[ \left (1 + \mathrm{i}\sqrt{3}\right )z = 1 -\mathrm{i}\sqrt{3} \]
\(z = -\frac{1} {2} -\frac{\sqrt{3}} {2} \mathrm{i}\)
\(z = \frac{\sqrt{3}} {2} + \frac{1} {2}\mathrm{i}\)
\(z = -\frac{1} {2} + \frac{\sqrt{3}} {2} \mathrm{i}\)
\(z = -\frac{\sqrt{3}} {2} + \frac{1} {2}\mathrm{i}\)