3 A function \(\mathrm { f } ( \mathrm { z } )\) is defined on all complex numbers z by \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 3 \mathrm { z } ^ { 2 } + \mathrm { k } \mathrm { z } - 5\) where \(k\) is a real constant. The roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) are \(\alpha , \beta\) and \(\gamma\). You are given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5\).

1. Explain why \(f ( z ) = 0\) has only one real root.
2. Find the value of \(k\).
3. Find a cubic equation with integer coefficients that has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).