- The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\), is given by
$$w = \frac { z + p \mathrm { i } } { \mathrm { i } z + 3 } \quad z \neq 3 \mathrm { i } \quad p \in \mathbb { Z }$$
The point representing \(\mathrm { i } ( 1 + \sqrt { 3 } )\) is invariant under \(T\).
Determine the value of \(p\).