Complex 1305: conditions to be real/purely imaginary in cartesian form, Argand diagram (tys)

In this question, we will investigate conditions for a complex number (in cartesian form) to be real/purely imaginary.

We also explore properties of $\left|z^n\right|$ and $\textrm{arg}(z^n).$

We then explore plotting out points representing complex numbers on an Argand diagram.

Question types

1. given $z=a+bi$ where $b$ is given, find the possible values of $a$ such that $\displaystyle \frac{z^2}{z^*}$ is purely imaginary.
2. same as above but for $z^3$ instead.
3. given $z=a+bi$ where $a$ is given, find the possible values of $b$ such that $\displaystyle \frac{z^2}{z^*}$ is real.
4. same as above but for $z^3$ instead.

App preview

Access the app at https://1305.vercel.app.