IB AA HL May 2026 P2 TZA Q7

The continuous random variable \(X\) has a probability distribution \(f\) defined as \(f(x)=\begin{cases}\frac{b}{a}x,&0\leq x\leq a,\\\frac{b}{a-5}(x-5),&a<x\leq5,\end{cases}\) where \(0<a<5\) and \(b\in\mathbb{Q}^+\).

The graph of \(f\) is shown in the following diagram.

1. Show that \(b=0.4\).[2]
2. It is given that \(\mathrm{E}(X)=3.2\).
   Find the value of \(a\).[4]