2025 CGSS PRELIMS P2 Q9

A farmer owns two triangular plots of land \(ABC\) and \(ADC\) as shown in the diagram above. It is given that \(AB = 7\) m, \(AC = 8\) m and \(AD = 6\) m, angle \(BAD\) is a right angle and angle \(BAC = \theta\) radians.

1. 1. Show the total area of the two plots of land, \(A\) m\(^2\), can be expressed in the form \(a\sin\theta + b\cos\theta\), where \(a\) and \(b\) are integers.[3]
   2. Express \(A\) in the form \(R\sin(\theta + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\).[3]
   3. Find the greatest possible value of \(A\) and the value of \(\theta\) at which this occurs.[3]
2. The farmer would like to grow roses on plot \(ABC\) and hibiscus on plot \(ADC\). Given that he wants the area of plot \(ABC\) to be twice the area of plot \(ADC\), find the value of \(\theta\).[3]