A curve has parametric equations x = tan θ, y = sin θ-2 sin³θ, for 0 < θ < ½π.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

1. [M1] Attempt use of chain rule for differentiation of y or simplification of y and use of the product rule.
2. [A1] Obtain dx/dθ = sec²θ and dy/dθ = cosθ - 6sin²θ cosθ
3. [M1] Attempt expression for dy/dx in terms of cosθ only using their dy/dθ and dx/dθ
4. [A1] Obtain 6cos⁵θ - 5cos³θ AG - necessary detail needed.
5. [M1] Attempt solution of y = 0 to find value of θ Allow if degrees used
6. [A1] Obtain sin²θ = ½ and hence θ = ¼π Must be in radians here
7. [A1] Obtain gradient of curve at the point is -1/√2 or exact equivalent
8. [M1] Attempt equation of normal using negative reciprocal for their gradient
9. [A1] Obtain y = √2x - √2 or exact equivalent