Solve the equation 3 cos 2θ = 3 cos θ + 2, for 0° ≤ θ ≤ 360°.

Mark scheme

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

1. M1 Use correct double-angle formula to obtain an equation in cosθ e.g. 3(2cos²θ-1) = 3cosθ + 2
2. A1 Obtain 6cos²θ – 3cosθ – 5 = 0, or 3-term equivalent M1 A0 is scored if they use any correct formula for cos 2θ and make a subsequent error.
3. M1 Solve a 3-term quadratic in cosθ for θ As far as θ = cos⁻¹ ((3-√129)/12) if quadratic correct.
4. A1 Obtain a correct answer, e.g. 134.1° Accept greater accuracy e.g. 134.1456, 225.8544.
5. A1 FT Obtain a second answer, e.g. 225.9° and no other in [0°, 360°] Treat answers in radians (2.34 and 3.94) as a misread. Ignore answers outside [0°, 360°]. The FT is for 360° minus the first answer.