2017 RI MYE Q4

Referred to the origin \(O\), points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively, such that \(\mathbf{a}\) and \(\mathbf{b}\) are non-parallel vectors. Point \(C\) lies on \(OA\) produced such that \(AC=4OA\). Point \(P\) lies on \(BC\), between \(B\) and \(C\), such that \(BP:PC=2:3\). Point \(Q\) lies on \(AB\), between \(A\) and \(B\), such that \(AQ:QB=k:1\).

1. Find the position vectors \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\), giving your answers in terms of the constant \(k\) and the vectors \(\mathbf{a}\) and \(\mathbf{b}\)[3]
2. Find the area of the triangle \(OPQ\) in the form \(\lambda \left| \mathbf{a}\times \mathbf{b} \right|\), where \(\lambda \) is a constant to be determined in terms of \(k\).

Hence, or otherwise, find the value of \(k\) such that \(O\), \(P\) and \(Q\) are collinear.[4]