2025 ASRJC Promo Q8

Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) are such that \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\) and \(\overrightarrow{OC}=\mathbf{c}\). The point \(P\) lies on \(AB\) produced such that \(AP:BP=5:2\).

1. Find \(\overrightarrow{OP}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[1]
2. Given that \(\mathbf{c}=\dfrac{7}{2}\mathbf{b}-\dfrac{5}{2}\mathbf{a}\), prove that the points \(A\), \(C\) and \(P\) are collinear.[2]

It is known that \(\mathbf{b}\) is a unit vector, the magnitude of \(\mathbf{a}\) is \(2\sqrt{3}\) and the angle in radian between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\dfrac{5}{6}\pi\).

4. Find the exact area of triangle \(OBC\).[3]
5. Find the exact length of projection of \(\mathbf{c}\) on \(\mathbf{a}\).[4]