2025 CHS PRELIMS P2 Q8

The diagram shows two rods \(AB\) and \(BC\) of length 2 m and 5 m respectively, joined together at \(B\) such that angle \(ABC = 90^\circ\). Small rings are attached to \(A\) and \(C\) so that \(A\) can move along the vertical pole \(OY\) and \(C\) can move along the horizontal pole \(OX\). The rod \(BC\) makes an angle \(\theta\) with \(OX\).

1. Show that \(OA = 5\sin\theta - 2\cos\theta\).[2]
2. Express \(OA\) in the form \(R\sin(\theta - \alpha)\) where \(R > 0\) and \(0^\circ < \alpha < 90^\circ\).[3]
3. Given that \(A\) can be no higher than 3.5 m above \(O\), calculate the greatest possible value of \(\theta\).[2]