It is given that a = sin θ – 3 cos θ and b = 3 sin θ + cos θ, where 0° ≤ θ ≤ 360°.

Mark scheme

Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.

1. [1] B1 For (i): Correct squaring of a and b (e.g., a² = s² + 9c² - 6sc, b² = 9s² + c² + 6sc)
2. [1] M1 For (i): Sums a² and b² and uses the identity s² + c² = 1 to simplify to a constant value
3. [1] A1 For (i): Shows a² + b² = 10s² + 10c² = 10 (AG - answer given)
4. [1] M1 For (ii): Substitutes 2a = b and collects terms to get a single trigonometric ratio (e.g., s = -7c)
5. [1] A1 For (ii): Correct value of tan θ = -7
6. [1] A1 For (ii): First correct angle θ = 98.1° (1 d.p.)
7. [1] A1 For (ii): Second correct angle θ = 278.1° (1 d.p.)