The diagram shows the circle with equation x²+y² – 14x+8y+36 = 0 and the line y=-2. The line intersects the circle at the points A and B. The centre of the circle is C.

Mark scheme

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

1. [1] B1 Condone x = 2 and y = -2 if seen together. (2,-2)
2. [1] B1 Condone x = 12 and y = -2 if seen together. (12,-2)
3. [1] B1 (7,-4) If B0 B0 for the first two marks, then SC B1 available for x=2 and x=12.
4. [1] M1 Or other correct method for an isosceles triangle using their A and B of the form (,-2) and C of the form (,-d), where -d<-2. tan(1/2 θ) = 5/2 or θ/2 = 2tan⁻¹(5/2) or sin(θ/2) = 5/√29 or cos(θ/2) = 2/√29
5. [1] A1 AWRT Final answer of 136.4° scores max 1/2. Ignore degree symbol if present. [θ =] 2.38
6. [1] B1 Sight of √29. SOI. Condone 5.4. Could be found in parts (a) or (b), but do not award unless it is seen in part (c). Length AC = Length BC = √5²+2² = √29 [=5.385...]
7. [1] M1 Use of correct sector formula with their identified radius (e.g. √29) and their (2π – θ) or their θ. May be embedded as part of the segment formula. Area of large sector = 1/2 x √29 x √29 x (2π – their θ) [= 56.587..] or Area of small sector = 1/2 x √29 x √29 x their θ [= 34.518...]
8. [1] M1 Use of correct triangle formula with their identified radius (e.g. √29) and their θ. May be embedded as part of the segment formula. Area of triangle = 1/2 x √29 x √29 x sinθ or 1/2 x 10 x 2 [=10]
9. [1] A1 AWRT [Area of circle – smaller segment = π×29-34.52 +10] [Segment area = ] 66.6