3 The cubic equation $\mathrm { x } ^ { 3 } + \mathrm { cx } + 1 = 0$, where $c$ is a constant, has roots $\alpha , \beta , \gamma$.\\
(a) Find a cubic equation whose roots are $\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }$.\\

(b) Show that $\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } = 3 - 2 c ^ { 3 }$.\\

(c) Find the real value of $c$ for which the matrix $\left( \begin{array} { c c c } 1 & \alpha ^ { 3 } & \beta ^ { 3 } \\ \alpha ^ { 3 } & 1 & \gamma ^ { 3 } \\ \beta ^ { 3 } & \gamma ^ { 3 } & 1 \end{array} \right)$ is singular.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q3}}