The diagram shows a sector OAB of a circle with centre O. Angle AOB = θ radians and OP = AP = x. [Figure 6.1]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [*B1] OA = 2xcosθ (derived from AP² = (Rcosθ - x)² + (Rsinθ)² = x²) Correct expression containing OA, OA or OA² (allow p, a or r for OA) containing only terms with x and θ but not just OA = 2xcosθ. Do not condone sin π - 2θ until missing brackets recovered or cos(180-2θ) until it becomes -cos2θ etc.
2. [DB1] OA = 2xcosθ leading to Arc length = 2xθcosθ AG Complete correct method showing all necessary working. Condone 2xcosθ×θ.
3. [M1] Sector area = 1/2 (2xcosθ)² θ OE Using sector formula with a correct OA. Condone cosθ² for cos²θ and missing brackets.
4. [M1] Triangle area = 1/2 × 2xcosθ × x sinθ OR 1/2 x² sin(π-2θ) Using a correct triangle formula for the correct triangle. Condone missing brackets and 180 for π.
5. [M1] [Area APB =] Their sector area – their triangle area Both expressions must be areas involving terms with x² and θ only. Condone missing brackets and 180 for π for the triangle. Condone calling the sector a segment.
6. [A1] [Area APB =] 1/2(2xcosθ)² θ - 1/2 x² sin(π-2θ) [= x²(2θcos²θ - sin2θ)] OE A correct expression. Mark the first unsimplified result of subtraction and ISW any incorrect 'simplifications'.