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