Complex numbers in algebraic and polar form

9000039110

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

9000039108

Level: 
C
Assuming \(z\in \mathbb{C}\), solve the following equation. By \(\overline{z }\) the complex conjugate of \(z \) is denoted. \[ 2z -\mathrm{i}\, \overline{z} = 1 -\mathrm{i} \]
\(z = \frac{1} {3} -\frac{1} {3}\mathrm{i}\)
\(z = 1 + \mathrm{i}\)
\(z = -\frac{3} {5} + \frac{6} {5}\mathrm{i}\)
\(z = -\frac{1} {5} -\frac{3} {5}\mathrm{i}\)

9000039101

Level: 
B
Find the polar form of the complex number \(z=\frac{\mathrm{i}^{14}-1} {\mathrm{i}^{9}+1} \).
\(\sqrt{2}\left (\cos \frac{3\pi } {4} + \mathrm{i}\sin \frac{3\pi } {4}\right )\)
\(\sqrt{2}\left (\cos \frac{5\pi } {4} + \mathrm{i}\sin \frac{5\pi } {4}\right )\)
\(\sqrt{2}\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
\(\sqrt{2}\left (\cos \frac{7\pi } {4} + \mathrm{i}\sin \frac{7\pi } {4}\right )\)

9000037502

Level: 
A
Find the total sum of the complex numbers \(a\), \(b\) and \(c\). \[ a = 3 + \sqrt{2}\mathrm{i},\quad b = 1 - 4\mathrm{i},\quad c = \sqrt{3} - 3\mathrm{i} \]
\(4 + \sqrt{3} + \mathrm{i}(\sqrt{2} - 7)\)
\(4 + \mathrm{i}\sqrt{3}\)
\(4 + \sqrt{2} + \mathrm{i}(\sqrt{3} - 3)\)
\(4 + \sqrt{3} -\mathrm{i}(\sqrt{2} - 7)\)

9000037506

Level: 
A
Given complex numbers \[ a = 3 + 5\mathrm{i}\text{, }\quad b = 2 -\mathrm{i}\text{, } \] find the quotient \(\frac{a} {b}\).
\(\frac{1} {5} + \mathrm{i}\frac{13} {5} \)
\(\frac{1} {3} + \mathrm{i}\frac{13} {3} \)
\(\frac{1} {5} + \mathrm{i}\frac{7} {5}\)
\(\frac{1} {3} + \mathrm{i}\frac{7} {3}\)