2025 IGCSE Additional Mathematics February/March 0606/12 Q11

In the diagram, \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\).

The point \(M\) is the midpoint of \(OB\).

The point \(N\) is such that \(ON=3NA\).

The lines \(BN\) and \(AM\) intersect at the point \(X\).

\(\overrightarrow{BX}=\lambda\overrightarrow{BN}\), where \(\lambda\) is a constant.

\(\overrightarrow{MX}=\mu\overrightarrow{MA}\), where \(\mu\) is a constant.

1. Find \(\overrightarrow{OX}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\lambda\).[3]
2. Find \(\overrightarrow{OX}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mu\).[2]
3. Hence find the values of \(\lambda\) and \(\mu\).[4]