Answer only one of the following two alternatives. EITHER The points A, B and C have position vectors i, 2j and 4k respectively, relative to an origin O. The point N is the foot of the perpendicular from O to the plane ABC. The point P on the line-segment ON is such that OP = ¼ON. The line AP meets the plane OBC at Q. Find a vector perpendicular to the plane ABC and show that the length of ON is 4/√(21). Find the position vector of the point Q. Show that the acute angle between the planes ABC and ABQ is cos⁻¹(⅓). OR The curve C has polar equation r = a(1 − cos θ) for 0 ≤ θ < 2π. Sketch C. Find the area of the region enclosed by the arc of C for which ¼π < θ < ¾π, the half-line θ = ¼π and the half-line θ = ¾π. Show that (ds/dθ)² = 4a² sin²(½θ), where s denotes arc length, and find the length of the arc of C for which ¼π < θ < ¾π.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2015). The mark or mark type is shown in brackets.

1. [M1] (EITHER)(a) Attempt to find two vectors in the plane (e.g., AB and AC)
2. [A1] (EITHER)(a) Calculate cross product AB x AC to find normal vector (e.g., (8 4 2)ᵀ or (4 2 1)ᵀ)
3. [M1] (EITHER)(a) Use equation of plane n.r = d and find d, then find N by projecting O onto plane, or using n.ON = |ON|² and n.r=d
4. [A1] (EITHER)(a) Show length of ON is 4/√(21) (AG)
5. [B1] (EITHER)(b) Find position vector of P (e.g., (4/21)P or (1/21)(4 2 1)ᵀ)
6. [M1] (EITHER)(b) Find equation of line AP
7. [A1] (EITHER)(b) Correct equation of line AP (e.g., r = (1 0 0)ᵀ + λ(-17 2 1)ᵀ)
8. [M1] (EITHER)(b) Find equation of plane OBC (x=0, or r.i=0)
9. [A1] (EITHER)(b) Solve for parameter and find position vector of Q (e.g., (0 2/17 1/17)ᵀ)
10. [B1] (EITHER)(c) Correct normal vector for plane ABC (from part a, or implied)
11. [M1] (EITHER)(c) Attempt to find normal vector for plane ABQ (e.g., AB x AQ)
12. [A1] (EITHER)(c) Correct normal vector for plane ABQ (e.g., (2 1 32)ᵀ)
13. [M1] (EITHER)(c) Use dot product formula for angle between planes
14. [A1] (EITHER)(c) Show cos φ = 2/7 (AG in MS)
15. [B1] (OR)(a) Closed curve through pole with correct orientation.
16. [B1] (OR)(a) Completely correct.
17. [M1] (OR)(b) Use area formula 1/2 ∫ r² dθ and substitute r
18. [M1] (OR)(b) Use double angle formula for cos²θ (or equivalent) for integration
19. [A1] (OR)(b) Integrate correctly all terms in r²
20. [M1] (OR)(b) Substitute limits correctly
21. [A1] (OR)(b) Correct exact area: a²(3π/8 - 1/4)
22. [B1] (OR)(c) State arc length formula ds/dθ = √(r² + (dr/dθ)²) and differentiate r to get dr/dθ = asinθ
23. [M1] (OR)(c) Square r and dr/dθ and sum them (r² + (dr/dθ)²)
24. [M1] (OR)(c) Use identity 1-cosθ = 2sin²(½θ) to simplify to 4a²sin²(½θ)
25. [A1] (OR)(c) Show (ds/dθ)² = 4a²sin²(½θ) (AG)
26. [M1] (OR)(c) Integrate ∫ √(ds/dθ)² dθ = ∫ 2a|sin(½θ)| dθ with correct limits
27. [A1] (OR)(c) Correct integration [-4a cos(½θ)]
28. [M1] (OR)(c) Substitute limits correctly
29. [A1] (OR)(c) Final exact length: 4a[cos(π/8) - cos(3π/8)]