The diagram shows a circle with centre O and radius r cm. Points A and B lie on the circle and angle AOB = 2θ radians. The tangents to the circle at A and B meet at T. [Figure 4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

1. [B1] Arc length AB = 2rθ
2. [B1] Tan θ = AT/r or BT/r → AT or BT = r tan θ
3. [B1FT] P = 2rθ + 2r tan θ OE, FT for their arc length + 2 × their AT
4. [B1] Area ΔAOT = 1/2 × 5 × 5 tan 1.2 or Area AOBT = 2 × 1/2 × 5 × 5 tan 1.2
5. [*M1] Sector area = 1/2 × 25 × 2.4 (or 1.2) Use of ½r²θ with θ = 1.2 or 2.4.
6. [DM1] Shaded area = 2 triangles - sector Subtraction of sector, using 2.4 where appropriate, from 2 triangles
7. [A1] Area = 34.3 (cm²) AWRT
8. [B1] Area of ΔABT = 1/2 × (5 tan 1.2)² × sin(π – 2.4) (= 55.86)
9. [*M1] Segment area = 1/2 × 25 × (2.4 – sin 2.4) (= 21.56) Use of ½r²(θ – sin θ) with θ = 1.2 or 2.4
10. [DM1] Shaded area = triangle - segment Subtraction of segment from ΔABT, using 2.4 where appropriate.
11. [A1] Area = 34.3 (cm²) AWRT