7 The numbers \(\alpha , \beta\) and \(\gamma\) satisfy the equations
$$\begin{aligned}
& \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 10 - 12 \mathrm { i }
& \alpha \beta + \beta \gamma + \gamma \alpha = 5 + 6 \mathrm { i }
\end{aligned}$$
- Show that \(\alpha + \beta + \gamma = 0\).
- The numbers \(\alpha , \beta\) and \(\gamma\) are also the roots of the equation
$$z ^ { 3 } + p z ^ { 2 } + q z + r = 0$$
Write down the value of \(p\) and the value of \(q\).
- It is also given that \(\alpha = 3 \mathrm { i }\).
- Find the value of \(r\).
- Show that \(\beta\) and \(\gamma\) are the roots of the equation
$$z ^ { 2 } + 3 \mathrm { i } z - 4 + 6 \mathrm { i } = 0$$
- Given that \(\beta\) is real, find the values of \(\beta\) and \(\gamma\).