With reference to the origin \(O\), the position vectors of the points \(A\) and \(B\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively.
Given that \(\overrightarrow{OP}=3\mathbf{b}-2\mathbf{a}\), show that the points \(A\), \(B\) and \(P\) are collinear.[2]
The point \(D\) lies on \(OP\) such that \(DP:OP=2:3\).
Find the \(\overrightarrow{OD}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[2]
Show that the quadrilateral \(OABD\) is a trapezium but not a parallelogram.[2]
Given further that \(|\mathbf{a}|=3\), \(|\mathbf{b}|=2\) and angle \(AOB=30^\circ\), find the exact area of triangle \(OAD\).[3]