By writing y = sec³θ as (1/cos³θ) show that dy/dθ = 3 sin θ sec⁴θ.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [M1] Use of correct chain rule (and correct quotient rule) and cos⁻³θ. Obtain k × (cosθ)⁻⁴ × sinθ or equivalent.
2. [A1] Obtain given answer from full and correct working (signs must be shown), but condone (sec³θ)' = .... and y'(θ). dy/dθ = -(-3)(cosθ)⁻⁴(-sinθ) = 3sinθsec⁴θ.
3. [B1] Separate variables: ∫(sinθ / cos⁴θ) dθ = ∫((x+3) / (x²+9)) dx. Or ∫(3sinθ / cos⁴θ) dθ = ∫((3x+9) / (x²+9)) dx. Condone missing integral signs or missing dx or dθ, but not both.
4. [B1] Obtain p sec³θ (+A). Correct form, p any constant but not 0.
5. [*M1] Obtain q ln(x²+9) and r tan⁻¹(x/3) (+C). Might have one third of both sides. Alt: substitute x = 3 tanφ to obtain q∫1+tanφdφ; condone if have θ in place of φ in this method.
6. [DM1] Obtain q(=(1/2)ln(cos)) OE.
7. [A1] Obtain (1/3)sec³θ = (1/2)ln(x²+9) + tan⁻¹(x/3) (+C) or equivalent. Or might see a third of both sides. Must have 2 different variables.
8. [M1] Use θ = (1/3)π, x = 3 in an equation including psec³θ, qln(x²+9) and r tan⁻¹(x/3) to evaluate the constant of integration. Or as limits in a definite integral. Limits for θ are 0 and π/4.
9. [A1] Obtain constant = 8 - ln 18 - π. OE, e.g. 1.308... to at least 3sf.
10. [A1] Obtain cos θ = 0.601. Accept AWRT 0.601.