Solve the equation cot²2θ+3 cosec 2θ = 9 for -90° ≤ θ ≤ 90°.

Mark scheme

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

1. [B2] cosec² 2θ + 3cosec 2θ – 10 [= 0] or 10sin² 2θ – 3sin 2θ – 1 [=0] [B1] for correctly writing the equation in terms of one trigonometric function e.g. cosec² 2θ - 1 + 3 cosec 2θ = 9 or (1 - sin² 2θ) / sin² 2θ + 3/sin 2θ = 9
2. [M1] Solves or factorises their 3-term quadratic in cosec 2θ or sin 2θ e.g. (cosec 2θ - 2) (cosec 2θ + 5) [= 0] or (2sin 2θ - 1)(5sin 2θ + 1) [= 0]
3. [A3] θ = 15°, 75°, -5.8 or -5.76 to -5.77, -84.2 or -84.23 to -84.232 and no other angles in range; nfww
4. [A2] for any 2 correct, ignoring extras in range; nfww
5. [A1] for one correct angle or one correct double angle; nfww