2025 IGCSE Additional Mathematics May/June 0606/12 Q10

The diagram shows the shape \(OABCDEF\).

\(AOF\) is a straight line.

\(OAB\) and \(OEF\) are sectors of a circle with centre \(O\) and radius \(r\).

Angle \(BOA=\text{angle}\hspace{0.5em}EOF\).

\(OCD\) is a sector of a circle with centre \(O\) and radius \(\dfrac{4r}3\).

Angle \(COD\) is \(\theta\) radians.

The point \(B\) lies on the line \(OC\) and the point \(E\) lies on the line \(OD\).

The line \(BE\) is parallel to the line \(AOF\).

1. Find, in terms of \(r\) and \(\theta\), the area of the shaded region \(BCDE\).[3]
2. The diagram shows the shape from part (a) with region \(OABEF\) shaded.
   Find, in terms of \(r\) and \(\theta\), the perimeter of the shaded region.[5]