The variables x and θ satisfy the differential equation sin(θ/2) dx/dθ = (x + 2) cos(θ/2) for 0 < θ < π. It is given that x = 1 when θ = π/3. Solve the differential equation and obtain an expression for x in terms of cos θ.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

1. [B1] Separate variables correctly to obtain ∫1/(x+2) dx = ∫cot(θ/2) dθ or equivalent integrands. Integral signs SOI
2. [B1] Obtain term ln(x+2)
3. [M1] Obtain term of the form klnsin(θ/2)
4. [A1] Obtain term 2lnsin(θ/2)
5. [M1] Use x = 1, θ = π/3 to evaluate a constant, or as limits, in an expression containing pln(x + 2) and qln(sin(θ/2))
6. [A1] Obtain correct solution in any form e.g. ln(x+2) = 2lnsin(θ/2) + ln12
7. [M1] Remove logarithms and use correct double angle formula
8. [A1] Obtain answer x=4-6cosθ