The diagram shows part of the line y = 1 and one complete period of the curve y = 1+cosx, where x is in radians. The line PQ is a tangent to the curve at P and at Q. The line QR is parallel to the y-axis. Area A is enclosed by the line y = 1 and the curve. Area B is enclosed by the line y = 1, the line PQ and the curve. Given that area A: area B is 1:k find the exact value of k. [Figure 11]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [B1] for [F(x) =]] ∫(1+cosx)dx = x + sin x oe
2. [B1] for the area of an appropriate rectangle or rectangles soi
3. [M1] for correct use of correct limits to find an appropriate area under the curve
4. [A1] for the accurate area under the curve
5. [B1] for area A or 1/2 A or B or 1/2 B OR
6. [M1] for attempting to integrate sinx or +cosx
7. [M1] dep for using correct limits
8. [A1] for a correctly integrated expression with correct limits
9. [M1] for correct use of correct limits
10. [A1] for exact value A = 2
11. OR equivalent correct plan
12. [B1] for Area(A + B) or 1/2 Area(A + B) oe
13. [M1] for using Area(A + B) to find A or B or for using 1/2 Area(A + B) to find 1/2 A or 1/2 B oe
14. [A1] for exact value
15. OR
16. [B1] for 4π - ∫(2π to 3π/2) (1+cos x)dx + ∫(3π/2 to π) (1+cos x)dx oe
17. [M1] for correct use of correct limits
18. [A1] for exact value B = 2π – 2
19. OR equivalent correct plan
20. [B1] dep on all previous marks; allow k=(2π – 2)/2