2025 TKSS PRELIMS P1 Q11

The diagram shows a frame of a model of a sail. The main beams of the frame are put together to form a triangle \(OAB\). The lengths of the main beams are \(OA = 12\) cm and \(OB = 9\) cm and angle \(AOB = 90^\circ\). A support rod of length 7.2 cm is to be installed from \(O\) to \(X\) on the diagonal beam \(AB\) such that it makes an acute angle \(\theta\) with the beam \(OB\). A canvas is to be fixed on points \(O\), \(A\), \(X\) and \(B\) of the frame to form a sail.

1. Find the area of the triangle \(OBX\) in terms of \(\theta\).[1]
2. By finding the area of the canvas, show that \(4\cos\theta + 3\sin\theta = 5\).[3]
3. Express \(4\cos\theta + 3\sin\theta\) in the form \(R\cos(\theta - \alpha)\) where \(R > 0\) and \(\alpha\) is acute.[4]
4. Find the value of \(\theta\).[1]