(a)[2]
Using the fact that the graph is symmetric about the line $x = -0.5$ and that $P(X < 0) = p$, determine $P(-1 < X < 0)$ in terms of $p$.
(b(i))[3]
The probability density function in the diagram is now defined by $f(x) = \begin{cases} a - b(x^2 + x), & -3 \le x \le 2, \\ 0, & \text{otherwise}, \end{cases}$ where $a$ and $b$ are positive constants. Prove that $30a - 55b = 6$.
(b(ii))[3]
By substituting an appropriate value of $x$ into $f(x)$, obtain a further equation relating $a$ and $b$ and hence determine the values of $a$ and $b$.
(ii)[3]
By substituting an appropriate value of $x$ into $f(x)$, obtain another equation involving $a$ and $b$ and hence determine the values of $a$ and $b$.