IAL 2021 Jan Q1

The transformation $T$ from the $z$ -plane, where $z = x + iy$ , to the $w$ -plane, where $w = u + iv$ , is given by

\begin{align*} w = \frac{z + p\mathrm{i}}{iz + 3} \qquad z \neq 3\mathrm{i} \qquad p \in \mathbb{Z} \end{align*}

The point representing $\mathrm{i}(1 + \sqrt{3})$ is invariant under $T$ .

Determine the value of $p$ .

(3)