Complex numbers in algebraic and polar form

9000035804

Level: 
A
Find the algebraic form of the following complex number. By \(\overline{z }\) the complex conjugate of \(z \) is denoted. \[ \overline{\overline{(2 + \mathrm{i}) }\; \overline{(3 + 2\mathrm{i}) } } \]
\(4 + 7\mathrm{i}\)
\(8 + 7\mathrm{i}\)
\(8 - 7\mathrm{i}\)
\(4 - 7\mathrm{i}\)

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 \(\frac{a} {b}\).
\(\frac{1} {5} + \mathrm{i}\frac{13} {5} \)
\(\frac{1} {3} + \mathrm{i}\frac{13} {3} \)
\(\frac{1} {5} + \mathrm{i}\frac{7} {5}\)
\(\frac{1} {3} + \mathrm{i}\frac{7} {3}\)