2024 FMSS PRELIMS P2 Q9

The diagram above shows two rods \(AB\) and \(BC\) of length \(4\) m and \(9\) m respectively and \(\angle ABC = 90^{\circ} \). Rollers are fixed at points \(A\) and \(B\) such that \(A\) is able to move along the wall and \(B\) is able to move along the celling. The horizontal distance of \(C\) from the vertical wall is \(d\) m.

1. Show clearly that \(d = 4\sin \theta + 9\cos \theta\).[2]
2. Express \(d\) in the form \(R\cos(\theta - \alpha)\) where \(R > 0\) and \(0^{\circ} < \alpha < 90^{\circ}\).[4]
3. Find the value of \(\theta\) for which \(d = 6\text{ m}\).[2]
4. Find the maximum value of \(d\) and the corresponding value of \(\theta\).[2]