Functions f and g are defined by f: x → 2 – 3 cos x for 0 ≤ x ≤ 2π, g: x → ½x for 0 ≤ x ≤ 2π.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

1. [B1] Correct fg
2. [M1] 2 - 3cos(½x) = 1 → cos(½x) = 1/3 → ½x = cos⁻¹(1/3)
3. [A1] x = 2.46 awrt or 4.7π/6 (0.784π awrt)
4. [B1] One cycle of ± cos curve, evidence of turning at the ends not required at this stage. Can be a poor curve but not an inverted “V”. If horizontal axis is not labelled mark everything to the right of the vertical axis. If axis is clearly labelled mark 0 → 2π.
5. [B1] Start and finish at roughly the same negative y value. Significantly more above the x axis than below or correct range implied by labels .
6. [B1] Fully correct. Curves not lines. Must be a reasonable curve clearly turning at both ends. Labels not required but must be appropriate if present.