The diagram shows the circle with centre C(-4,5) and radius √20 units. The circle intersects the y-axis at the points A and B. The size of angle ACB is θ radians. [Figure 10]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

1. [B1] Obtain gradient of relevant radius is –2
2. [M1] Using m₁m₂ =-1 obtain the gradient of the tangent and use it to form a straight line equation for a line containing (–6, 9) m₁ must be from an attempt to find the gradient of the radius using the centre and the given point.
3. [A1] Obtain y=½x+12 OE e.g. y-9=½(x+6).
4. [B1] State or imply (x + 4)² + (y − 5)² = 20 If x² + y²−2gx−2fy+c=0 is used correctly with (–g,–f)=(–4,5) and c=g²+f²−r² then M1.
5. [B1] Obtain x² + y² + 8x-10y+21=0 A1 if above method used.
6. [B1] Substitute x=0 in equation of circle to find y-values 3 and 7 or state C to AB = 4 May be implied by AB = 4 or use of |x-coordinate of C|.
7. [M1] Attempt value of θ either using cosine rule or via ½θ using right-angled triangle Using their AB. If θ/2 used, must be multiplied by 2.
8. [A1] Obtain θ=0.9273 Or greater accuracy. A correct answer implies the M1.
9. [M1] Attempt arc length using rθ formula with their θ (not their θ/2) and r = √20 Expect 4.15.
10. [A1] Obtain perimeter = 8.15 or greater accuracy Condone missing units or incorrect units.
11. [M1] Attempt area using ½r²(θ-sinθ) formula or equivalent with their θ and r = √20 If sector – triangle used, both formulae must be correct. If triangle ACM used, area must be multiplied by 2.
12. [A1] Obtain area = 1.27 or greater accuracy Condone missing units or incorrect units.