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 = angle EOF. OCD is a sector of a circle with centre O and radius 4r/3. Angle COD is θ 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.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [B1] Area of sector OCD = (1/2)(4r/3)²θ
2. [B1] Area of triangle OBE = (1/2)r²sinθ oe (or r²sin(θ/2)cos(θ/2) oe)
3. [B1] Final area = (1/2)(4r/3)²θ - (1/2)r²sinθ oe (Allow final answer unsimplified)
4. [B1] Arc length AB (or EF) = r(π-θ)/2
5. [M1] BE = √2r²(1-cosθ) or 2r sin(θ/2) or r sin(π-θ)/2 oe (for complete method to find BE or BE², using either cosine rule, sine rule or basic trig using r and a correct angle. A0 if an error in rearranging to obtain BE is made. Allow unsimplified)
6. [M1] Perimeter = r(π−θ)+2r + √2r²(1-cosθ) oe (for a correct plan using their lengths, i.e., 2 arc lengths + 2r + their BE)
7. [A1] Allow A1 for an unsimplified answer