2024 IGCSE Additional Mathematics May/June 0606/21 Q12

The diagram shows a triangle \(OBC\). \(OA:OB=4:7\) and \(OD:OC=4:7\).

\(\overrightarrow{OB}=\mathbf b\quad\text{and}\quad\overrightarrow{OC}=\mathbf c\)

The point \(P\) is the point of intersection of \(AC\) and \(BD\) such that \(\overrightarrow{AP}=\lambda\overrightarrow{AC}\) and \(\overrightarrow{BP}=\mu\overrightarrow{BD}\), where \(\lambda\) and \(\mu\) are scalars.

1. Find two expressions for \(\overrightarrow{OP}\), each in terms of \(\mathbf b\), \(\mathbf c\) and a scalar, and hence show that \(P\) divides both \(AC\) and \(DB\) in the ratio \(4:7\).[7]
2. The point \(Q\) is such that \(\overrightarrow{OQ}=\dfrac27\mathbf b+\dfrac27\mathbf c\). Use a vector method to show that \(O\), \(Q\) and \(P\) are collinear. Justify your answer.[2]