1. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
k \left( a x + b x ^ { 3 } - x ^ { 4 } - 4 \right) & 1 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$ where \(a\), \(b\) and \(k\) are non-zero constants.
Given that the mode of \(X\) is 1.5

1. show that \(b = 3\)
2. Hence show that \(a = 2\)
3. Show that the median of \(X\) lies between 1.4 and 1.5