Answer only one of the following two alternatives. EITHER The diagram shows an isosceles triangle AOB and a sector OCDEO of a circle with centre O. The line AB is a tangent to the circle. Angle AOB = 1.8 radians and the radius of the circle is 12 cm. OR The diagram shows part of the curve y=xsinx and the normal to the curve at the point P(π/2, π/2). The curve passes through the point Q(π, 0).

Mark scheme

Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.

1. [1] (EITHER)(i) OA = 12/cos0.9
2. [1] (EITHER)(i) AC = 19.3 - 12 = 7.3
3. [1] (EITHER)(ii) Complete method for major arc (2π− 1.8) × 12
4. [1] (EITHER)(ii) 53.8
5. [1] (EITHER)(ii) AB = 2 × 12tan0.9 or cosine rule
6. [1] (EITHER)(ii) 30.2
7. [1] (EITHER)(ii) Complete plan (53.8 + 30.2 + 2 × 7.3)
8. [1] (EITHER)(ii) 98.6
9. [1] (EITHER)(iii) Complete method for major sector (1/2) × 12² × (2π−1.8)
10. [1] (EITHER)(iii) (1/2) × 19.3² × sin1.8 or (1/2) × 30.2 × 12
11. [1] (EITHER)(iii) 323 or 181
12. [1] (EITHER)(iii) 504
13. [1] (OR)(i) Uses product rule dy/dx = sinx + x cos x
14. [1] (OR)(i) At x = π/2 gradient = 1
15. [1] (OR)(i) Uses m₁m₂ = -1
16. [1] (OR)(i) Correctly reaches conclusion. e.g. y - π/2 = -1 (x - π/2) with y = 0
17. [1] (OR)(ii) ∫cosx dx – ∫x sinx dx = x cos x
18. [1] (OR)(ii) ∫x sinx dx = ∫cos x dx - x cos x
19. [1] (OR)(ii) sinx - xcosx
20. [1] (OR)(iii) Uses limits of π and π/2
21. [1] (OR)(iii) 2.14 or π – 1
22. [1] (OR)(iii) Area triangle = (1/2) × (π/2) × (π/2) = π²/8 or 1.23
23. [1] (OR)(iii) Subtracts area of triangle
24. [1] (OR)(iii) 0.908 (allow 0.906 or 0.907) or 0.91 or π - 1 - π²/8