The equation of a curve is 2y³ – 3x²y-x³ = 16.

Mark scheme

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

1. [1] (a) State or imply 6y²(dy/dx) as the derivative of 2y³.
2. [1] (a) State or imply 3x²(dy/dx) + 6xy as the derivative of 3x²y.
3. [1] (a) Complete the differentiation and equate the derivative of the LHS to zero and solve for (dy/dx). Needs to be clear how the value of dy/dx was obtained, e.g. collecting like terms and using brackets.
4. [1] (a) Obtain dy/dx = (x²+2xy) / (2y²-x²) from correct working. No incorrect statements, e.g. ‘cancel by 3' or 'divide top and bottom by 3' at the final step scores A0.
5. [1] (b) Equate derivative to 0 and solve for x or for x in terms of y. E.g., x²+2xy = 0 ⇒ x = 0 or x = −2y. Must be using the numerator.
6. [1] (b) Obtain (0,2). Allow if the values are stated separately. Do not ISW.
7. [1] (b) Use their x = −2y to form an equation in one unknown. E.g., substitute x=-2y into the curve equation: 2y³ - 3(-2y)²y - (-2y)³ = 16. This becomes 2y³ - 12y³ + 8y³ = 16.
8. [1] (b) Obtain (4,-2). Allow if the values are stated separately. Do not ISW.