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7. [Maximum mark: 7]
   A function $g(x)$ is defined by $g(x) = 2x^3 - 7x^2 + dx - e$, where $d, e \in \mathbb{R}$. $\alpha, \beta$ and $\gamma$ are the three roots of the equation $g(x) = 0$ where $\alpha, \beta, \gamma \in \mathbb{R}$.

(a) Write down the value of $\alpha + \beta + \gamma$.

A function $h(z)$ is defined by $h(z) = 2z^5 - 11z^4 + rz^3 + sz^2 + tz - 20$, where $r, s, t \in \mathbb{R}$. $\alpha, \beta$ and $\gamma$ are also roots of the equation $h(z) = 0$. It is given that $h(z) = 0$ is satisfied by the complex number $z = p + 3i$.

(b) Show that $p = 1$.

It is now given that $h\!\left(\frac{1}{2}\right) = 0$, and $\alpha, \beta \in \mathbb{Z}^+, \alpha < \beta$ and $\gamma \in \mathbb{Q}$.

(c) (i) Find the value of the product $\alpha\beta$.

(ii) Write down the value of $\alpha$ and the value of $\beta$.