2025 IGCSE Additional Mathematics May/June 0606/12 Q11

The diagram shows the triangle \(OAB\), where \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\).

The point \(P\) lies on \(OA\) such that \(\overrightarrow{OP}=\dfrac34\overrightarrow{OA}\).

The point \(Q\) lies on \(AB\) such that \(\overrightarrow{AQ}=\dfrac13\overrightarrow{AB}\).

The straight line through \(P\) and \(Q\) meets the straight line through \(O\) and \(B\) at the point \(R\).

It is given that \(\overrightarrow{OR}=\lambda\mathbf b\) and \(\overrightarrow{PR}=\mu\overrightarrow{PQ}\), where \(\lambda\) and \(\mu\) are constants.

1. Find \(\overrightarrow{OR}\) in terms of \(\mathbf a\), \(\mathbf b\) and \(\mu\).[6]
2. Hence find the values of \(\lambda\) and \(\mu\).[3]