1. The continuous random variable \(X\) has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0
a x + b x ^ { 2 } & 0 \leqslant x \leqslant k
1 & x > k \end{array} \right.$$ where \(a , b\) and \(k\) are positive constants.

1. Show that \(a k = 1 - b k ^ { 2 }\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\)
2. show that \(5 b k ^ { 3 } = 36 - 15 k\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\) and \(\operatorname { Var } ( X ) = \frac { 22 } { 75 }\)
3. show that \(5 b k ^ { 4 } = 52 - 10 k ^ { 2 }\) Given that \(k < 3\)
4. find the value of \(k\)
5. Hence find the value of \(a\) and the value of \(b\)