2026 RVHS P1 Q4

With reference to the origin \(O\), the points \(A\) and \(B\) are such that \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\), where \(\mathbf a\) and \(\mathbf b\) are both non-zero and non-parallel.

The point \(X\) lies on \(OA\) such that \(OX:XA=2:1\). The point \(Y\) lies on \(OB\) such that \(OY:YB=3:2\). The lines \(AY\) and \(BX\) intersect at the point \(P\).

1. Express \(\overrightarrow{OP}\) in terms of \(\mathbf a\) and \(\mathbf b\).[5]
2. Point \(Z\) lies on \(AB\) such that \(4AZ=3BZ\). Express \(\overrightarrow{OZ}\) in terms of \(\mathbf a\) and \(\mathbf b\).[1]
3. Show that \(O\), \(P\) and \(Z\) are collinear.[2]