Complex Numbers

Find two inequalities in terms of $z$ that define the shaded region.

Find the greatest value of $|z|$ for points in this region.

State the values of $\omega z_1$ and $\omega z_2$. Give your answers in the form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \leq \pi$.

On a sketch of an Argand diagram with origin $O$, show the points $A, B, C$ and $D$ representing the complex numbers $z_1, z_2, \omega z_1$ and $\omega z_2$ respectively.

State the geometric effects of multiplying $z_1$ and $z_2$ by $\omega$.

It is given that $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$.

Show that $(z_1 z_2)^* = z_1^* z_2^*$.

$z = 3e^{\frac{1}{4}\pi i}$ is a root of the equation $z^2 + bz + c = 0$, where $b$ and $c$ are real.

State the other root and hence find the values of $b$ and $c$.

Express $\frac{s}{t}$ in the form $r\text{e}^{\text{i}\theta}$, where $-\pi < \theta \leq \pi$ and $r > 0$.

In an Argand diagram with origin $O$, the points $A$ and $B$ represent the complex numbers $s$ and $\frac{s}{t}$ respectively.

By considering the line segments $OA$ and $OB$, or otherwise, state the two geometric effects of dividing a complex number by $6\text{e}^{3\text{i}}$.

For the points on this locus, determine the maximum and minimum possible values of $|z|$.

For the points on this locus, determine the minimum possible value of $\arg z$.