Problem

The complex number $z$ is defined by

$z = -\lambda + 3i \quad \text{where } \lambda \text{ is a positive real constant}$

Given that the modulus of $z$ is $5$

(a) write down the value of $\lambda$

(b) determine the argument of $z$, giving your answer in radians to one decimal place.

In part (c) you must show detailed reasoning. Solutions relying on calculator technology are not acceptable.

(c) Express in the form $a + ib$ where $a$ and $b$ are real,

(i) $\dfrac{z + 3i}{2 - 4i}$

(ii) $z^2$

(d) Show on a single Argand diagram the points $A$, $B$, $C$ and $D$ that represent the complex numbers

$z, \quad z^*, \quad \frac{z + 3i}{2 - 4i} \quad \text{and} \quad z^2$