Write down the values of k for which the line y = k is a tangent to the curve `y = 4 sin(x + π/4) + 10`.

Mark scheme

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

1. [2] B1 (for k = 14)
2. B1 (for k = 6)
3. [4] M1 Allow (1 + cos θ)(1 − cos θ) in the denominator (for (1+tan θ)(1 + cos θ) + (1 – tan θ)(1 − cos θ) / (1-cos² θ))
4. M1 Dep on previous M mark (for Expansion of numerator and simplification of denominator)
5. B1 soi (for Use of tan θ cos θ = sin θ)
6. A1 Sufficient simplification to justify obtaining the given answer (for 2(1 + sin θ) / (sin² θ))
7. [4] M1 For use of part (a) and attempt to simplify to a 3-term quadratic equation equated to zero. (for 2(1 + sin θ) = 3 sin² θ and 3 sin² θ - 2 sin θ - 2 = 0)
8. M1 M1 for attempt to solve and obtain a value for θ, may be implied by one correct solution (for sin θ = (1 - √7) / 3 or -0.5485...)
9. A2 A1 for one solution (for 213.3° and 326.7°)
10. If 0 scored SC1 for awrt 213 and 327
11. Penalise excess solutions in the range