The variables x and θ satisfy the differential equation x cos² θ (dx/dθ) = 2 tan θ + 1, for 0 ≤ θ < ½π and x > 0. It is given that x = 1 when θ = ¼π.

Mark scheme

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

1. [B1] Show sufficient working to justify the given statement AG
2. [B1] Separate variables correctly and attempt integration of at least one side
3. [B1] Obtain term ½x²
4. [B1 + B1] Obtain terms tan²θ + tanθ, or sec²θ + tanθ
5. [M1] Evaluate a constant, or use limits x = 1, θ = ¼π, in a solution with two terms of the form ax² and b tanθ, where ab ≠ 0
6. [A1] State correct answer in any form, e.g. ½x² = tan²θ + tanθ – ¾
7. [A1] Substitute θ = ⅓π and obtain x = 2.54