2025 AISS PRELIMS P2 Q3

The diagram shows a \(L\)-shaped rod \(ABC\) where \(AB\) and \(BC\) have length 18 m and 8 m respectively and angle \(ABC\) is \(90^\circ\). The rod is hinged to a wall at \(A\) so as to rotate in a vertical plane. The rod \(AB\) makes an acute angle \(\theta\) with the vertical wall surface \(OA\).

1. Given that \(G\) is a point directly below \(C\), show that \(OG=p\cos\theta+q\sin\theta\), where \(p\) and \(q\) are constants to be found.[2]
2. Express \(OG\) in the form \(R\cos(\theta-\alpha)\) where \(R>0\) and \(0^\circ\leq\alpha\leq90^\circ\).[3]
3. Find the length of \(OG\) and the corresponding value of \(\theta\) if \(G\) is at maximum displacement from \(O\).[3]