In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random variable \(X\) with probability density function \[f(x)=\begin{cases}-k(x-1)(x-3),&1\le x\le3,\\0,&\text{otherwise},\end{cases}\] where \(k\) is a constant.

(a) Without calculation, explain why \(E(X)=2\). [1]

(b) Show that \(k=\frac34\). [3]

(c) Find \(\operatorname{Var}(X)\). [3]

One turn consists of rolling the ball 3 times and noting the largest value of \(X\) obtained. If this largest value is greater than 2.5, the player scores a point.

(d) Find the probability that on a particular turn the player scores a point. [4]