4. (i) The complex number \(w\) is given by
$$w = \frac { p - 4 \mathrm { i } } { 2 - 3 \mathrm { i } }$$
where \(p\) is a real constant.
- Express \(w\) in the form \(a + b i\), where \(a\) and \(b\) are real constants.
Give your answer in its simplest form in terms of \(p\).
Given that \(\arg w = \frac { \pi } { 4 }\)
- find the value of \(p\).
(ii) The complex number \(z\) is given by
$$z = ( 1 - \lambda i ) ( 4 + 3 i )$$
where \(\lambda\) is a real constant.
Given that
$$| z | = 45$$
find the possible values of \(\lambda\).
Give your answers as exact values in their simplest form.
II