The parametric equations of a curve are x = (cos θ) / (2 - sin θ), y = θ + 2 cos θ. Show that dy/dx = (2 − sin θ)².

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [B1] State dy/dθ = 1-2 sin θ; Ignore left side throughout dx/dt, dy/dt, dx, dy but must see dy/dx for final A1.
2. [M1] Use correct quotient rule, or product rule if rewrite x as cos θ( 2 – sin θ)^-1; Incorrect formula seen M0 A0 otherwise BOD.
3. [A1] Obtain dx/dθ = (-sin θ(2-sin θ) + cos² θ) / (2-sin θ)² o.e. ; - sin θ(2 – sin θ)⁻¹ – cos θ(2 – sin θ)⁻²(− cos θ) or equivalent.
4. [M1] Use dy/dx = dy/dθ ÷ dx/dθ; dy/dx = (1-2sinθ) ÷ ( (1-2sinθ) / (2-sinθ)² ) ; Allow M1 even if errors in both derivatives.
5. [A1] Obtain dy/dx = (2-sin θ)².; AG - must see working in above cell to gain final A1. Allow cos² θ + sin² θ = 1 to be implied. x instead of θ or missing θ more than twice on right side then A0 final mark.