2023 SCGS 4AM CA2 Q6

The diagram shows a trapezium \(OPQR\), where \(OR=6\) cm and \(\angle OPQ=\angle PQR=\frac{\pi}{2}\) radians. \(\angle ROP=\theta\) radians, where \(\theta\) is acute. \(T\) is a point on \(PQ\) such that \(\angle ORT=\frac{\pi}{2}\) radians and \(RT=3\) cm.

1. Show that the area of the trapezium \(OPQR\) is \((9\sin2\theta+18\sin^2\theta)\text{ cm}^2\).[3]
2. Given that \(\theta\) can vary, find the exact value of \(\theta\) for which the area of trapezium \(OPQR\) is stationary.[3]
3. Determine whether this value of \(\theta\) gives a maximum or minimum value of the area.[2]