3
\includegraphics[max width=\textwidth, alt={}, center]{823cc2e5-e408-4b81-ac4d-7e9f584107cc-06_558_1077_260_523} The diagram shows the graph of the probability density function, f, of a random variable \(X\) that takes values between \(x = 0\) and \(x = 3\) only. The graph is symmetrical about the line \(x = 1.5\).

1. It is given that \(\mathrm { P } ( X < 0.6 ) = a\) and \(\mathrm { P } ( 0.6 < X < 1.2 ) = b\). Find \(\mathrm { P } ( 0.6 < X < 1.8 )\) in terms of \(a\) and \(b\).
2. It is now given that the equation of the probability density function of \(X\) is $$f ( x ) = \begin{cases} k x ^ { 2 } ( 3 - x ) ^ { 2 } & 0 \leqslant x \leqslant 3
   0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
   1. Show that \(k = \frac { 10 } { 81 }\).
   2. Find \(\operatorname { Var } ( X )\).