The expression f(θ) is defined by f(θ) = 12 sin θ cos θ + 16 cos² θ.

Mark scheme

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

1. [M1] Use correct identity for sin2θ or cos2θ (or both)
2. [A1] Obtain 6sin2θ + 8cos2θ (+8)
3. [B1] State R=10 (FT their asin2θ+bcos2θ form.)
4. [M1] Use appropriate trigonometry to find α using their asin2θ+bcos2θ form (Allow 0.927.)
5. [A1] Obtain 10cos(2θ-0.6435)+8
6. [B1] State or imply cos(2θ−0.6435) = 0.9 (FT their R, α and k provided RHS less than 1.)
7. [M1] Carry out correct process to find any positive value of θ (0.547)
8. [A1] Obtain 0.0962 (AWRT)
9. [M1] Integrate trigonometry term from part (a) to obtain k₁ sin(2θ–their 0.6435) (any non-zero constant k₁.)
10. [A1] Obtain 5sin(2θ−0.6435) + 8θ (condone absence of ... + c.)
11. [M1] Integrate to obtain at least form k₂ cos2θ+k₃ sin2θ (any non-zero constants k₂, k₃.)
12. [A1] Obtain -3cos2θ+4sin2θ+8θ (condone absence of ... + c .)