7
\includegraphics[max width=\textwidth, alt={}, center]{08f5a515-2c63-4955-b9da-7000631ff012-12_433_1006_255_568} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between - 3 and 2 only.

1. Given that the graph is symmetrical about the line \(x = - 0.5\) and that \(\mathrm { P } ( X < 0 ) = p\), find \(\mathrm { P } ( - 1 < X < 0 )\) in terms of \(p\).
2. It is now given that the probability density function shown in the diagram is given by $$f ( x ) = \begin{cases} a - b \left( x ^ { 2 } + x \right) & - 3 \leqslant x \leqslant 2
   0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are positive constants.
   1. Show that \(30 a - 55 b = 6\).
   2. By substituting a suitable value of \(x\) into \(\mathrm { f } ( x )\), find another equation relating \(a\) and \(b\) and hence determine the values of \(a\) and \(b\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.