2025 AHS PRELIMS P2 Q4

In the diagram, it is given that \(AB = 15\) cm, \(BC = 40\) cm and \(\angle OCB = \theta\). The vertical line \(OC\) is perpendicular to \(OA\) and \(AB\) is perpendicular to \(BC\).

1. Show that \(OC = 15\sin\theta + 40\cos\theta\).[2]
2. Express \(OC\) in the form \(R\sin(\theta + \alpha)\), where \(R\) is a positive constant and \(\alpha\) is an acute angle in degrees.[3]
3. Given that \(OC\) is at the maximum, determine the shortest distance of \(B\) to \(OC\).[3]