5 The time, \(T\) minutes, taken by people to complete a test has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 10 t - t ^ { 2 } \right) & 5 \leqslant t \leqslant 10
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 250 }\).
2. Find \(\mathrm { E } ( T )\).
3. Find the probability that a randomly chosen value of \(T\) lies between \(\mathrm { E } ( T )\) and the median of \(T\). [3]
4. State the greatest possible length of time taken to complete the test.
   \(6 X\) and \(Y\) are independent random variables with distributions \(\operatorname { Po } ( 1.6 )\) and \(\operatorname { Po } ( 2.3 )\) respectively.