The diagram shows the shaded region ABCD. The lines AC and BD each have a length of 12 cm. The lines AC and BD bisect each other at the point O. The lines AD and BC are parallel and each have a length of 4 cm. The arcs AB and DC are part of a circle centre O. [Figure 2.1]

Mark scheme

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

1. [M2] (a) Complete and correct method e.g.: π - 2 sin⁻¹(1/3) or π- cos⁻¹(7/9) or 3.142-0.6796... oe
2. [M1] (a) for a correct useful angle or implicit calculation for angle AOB e.g. angle BOC = 2 sin⁻¹(1/3) or 0.6796... or cos⁻¹((36+36-16)/(2(6)(6))) or cos⁻¹(7/9) oe or half angle BOC= sin⁻¹(1/3) or 0.339... or half angle AOB = cos⁻¹(2/6) or sin⁻¹(4√2/6) or 1.230... or 128=36+36-2(6)(6) cos AOB oe
3. [A1] (a) 2.46 or 2.461 to 2.462[...]
4. [M1 FT their 2.46 providing it is less than 2π] (b)(i) 6 × their 2.46 soi
5. [A1] (b)(i) 37.5 or 37.52 to 37.55
6. [M1 FT their 2.46 providing it is less than 2π] (b)(ii) 1/2 × 6² × their 2.46 or [2 × 1/2 × (6² × their 2.46)]
7. [OR] (b)(ii) 1/2 × 6² × their 0.68[0] or [2 × 1/2 × (6² × their 0.68[0])]
8. [M1 dep previous M1] (b)(ii) Correct plan e.g. : 2 sectors + 2 triangles [2 × 1/2 × (6² × their 2.46)] + [2 × 1/2 × (6² × sin their BOC)] or [2 × 1/2 × (6² × their 2.46)] + [2 × 1/2 × (6² × sin their 2.46)] or [2 × 1/2](6² × their 2.46) + (1/2 × 4 × their AB) oe or 2 segments + 1 rectangle 2 × [1/2 × 6² (their 2.46 – sin their 2.46)] + 4 × their AB oe OR whole circle radius 6 – 2 segments cut off by AD and BC π × 6² – [2 × 1/2 × 6² (their BOC – sin their BOC)]
9. [A1] (b)(ii) 111 or 111.2 to 111.3