1 The cubic equation $2 x ^ { 3 } + x ^ { 2 } - p x - 5 = 0$, where $p$ is a positive constant, has roots $\alpha , \beta , \gamma$.\\
(a) State, in terms of $p$, the value of $\alpha \beta + \beta \gamma + \gamma \alpha$.\\

(b) Find the value of $\alpha ^ { 2 } \beta \gamma + \alpha \beta ^ { 2 } \gamma + \alpha \beta \gamma ^ { 2 }$.\\

(c) Deduce a cubic equation whose roots are $\alpha \beta , \beta \gamma , \alpha \gamma$.\\

(d) Given that $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = \frac { 1 } { 3 }$, find the value of $p$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q1}}