2025 PHS PRELIMS P1 Q11

The diagram shows a rectangular wooden plank \(ABCD\), 5 m by 2 m, leaning against a vertical wall. \(AD\) makes an acute angle \(\theta\) with the horizontal ground.

\(E\) is a point on the vertical wall such that \(EB\) is parallel to the ground and \(F\) is a point on the ground such that \(AF\) is perpendicular to \(FD\). The point \(B\) is \(h\) m vertically above the ground.

1. Show that \(h=5\sin\theta+2\cos\theta\).[2]
2. Express \(h\) in the form \(R\sin(\theta+\alpha)\), where \(R>0\), and \(0^\circ<\alpha<90^\circ\).[3]
3. Find the greatest possible value of \(h\) and the value of \(\theta\) at which it occurs.[2]
4. Find the value of \(\theta\) for which \(h=\sqrt{15}\) m.[2]