Argand Diagram & Polar Form — practice questions

Find the modulus and principal argument of $z = 5 + 5i$, giving the argument in exact radian form.

Find the modulus and principal argument of $z = -2 - 2\sqrt{3}\,i$, giving exact values and explaining how you determined the quadrant.

The complex number $z = k + 4i$, where $k$ is real and positive, has $|z| = 2\,\text{Re}(z)$. Find the exact value of $k$ and hence the exact principal argument of $z$.

Write $z = 4\,\text{cis}\,\dfrac{\pi}{6}$ in the form $a + bi$, giving exact values.

Write $z = -\sqrt{3} - i$ in polar form, giving both the modulus and the argument exactly, and explain why a calculator's $\tan^{-1}$ does not give the correct argument here.

The complex number $z = k - ki$, where $k$ is a non-zero real number, is written in polar form. Find the modulus and the principal argument of $z$ in terms of $k$, treating the two cases $k > 0$ and $k < 0$ separately.

Given $z = 5\,\text{cis}\,\dfrac{\pi}{4}$ and $w = 3\,\text{cis}\,\dfrac{\pi}{6}$, find $zw$ in polar form.

Given $z = 6\,\text{cis}\,\dfrac{5\pi}{6}$ and $w = 2\,\text{cis}\,\dfrac{3\pi}{4}$, find $\dfrac{z}{w}$ in polar form with a principal argument, and state the geometric effect of dividing by $w$.

The points $A$ and $B$ on an Argand diagram represent $z_1 = 3 + i$ and $z_2 = 1 + 4i$. The point $C$ is obtained by rotating $B$ anticlockwise about $A$ through $\dfrac{\pi}{2}$. Find the complex number represented by $C$, in the form $a + bi$.