The complex number ${z}$ is given by ${z=-6+bi}$, where ${a}$ is a real number.

(a) Find the possible values of ${b}$ if ${\displaystyle \frac{z^2}{z^*}}$ is real.

For the rest of the question it is further given that ${b>0.}$

(b) Find the smallest integer value of ${n}$ such that ${\left | z^n \right | \geq 1000.}$

(c) For the value of ${n}$ found in (b) find the values of ${\left|z^n\right|}$ and ${\textrm{arg}(z^n)}$, where ${-\theta < \textrm{arg}(z^n) \leq \theta.}$

(d) On a single Argand diagram mark out the points ${A,B,C,D}$ representing the complex numbers ${z, \displaystyle \frac{z^2}{z^*}, z^*,\frac{72}{z}}$ respectively.