2023 DHS P1 Q5

With respect to an origin \(O\), the points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively where \(\mathbf{a}\) and \(\mathbf{b}\) are non-parallel. It is given that \(B\) lies on the line segment \(AC\) such that \(\overrightarrow{BC}=3\mathbf{b}-\mu \mathbf{a}\).

1. Find the value of \(\mu \). Hence find \(\overrightarrow{OC}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[3]
2. \(Q\) is a point on line segment \(OC\) where \(\overrightarrow{OQ}=t\overrightarrow{OC}\). The line segment \(AQ\) meets line segment \(OB\) at point \(P\). Given that \(AP:PQ=\lambda :1\), deduce the value of \(\lambda t\).[4]
3. By using your result in part (b), find the ratio \(OP:PB\) when \(\lambda =5\).[2]