2026 MSHS P2 Q12

In the diagram, \(ABCFD\) shows the design of the front of a handbag. \(ABCFD\) is a rectangle, triangle \(ABE\) is isosceles with \(AE=BE=14\) cm and \(\angle AEB=2\theta\) radians. \(EF\) is parallel to \(AD\) and \(EF=10\) cm. The perimeter of the handbag is \(P\) cm.

1. Show that \(P\) can be expressed as \(20+28\cos\theta+56\sin\theta\).[2]
2. Express \(P\) in the form \(20+R\cos(\theta-\alpha)\), where \(R>0\) and \(0<\alpha<\dfrac\pi2\).[3]
3. The perimeter of the handbag is lined with a chain of pearls. Given that \(\theta\) can vary and that the designer paid $110 for the pearls, find the minimum cost per metre of chained pearls and the corresponding value of \(\theta\).[3]
4. Explain whether \(P\) can be equal to 20 cm.[2]