The equation of a curve is x³ + 3x²y – y³ = 3.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. B1 State or imply 6xy+3x² dy/dx as derivative of 3x²y Allow B1 B1 for (3x2dx +)6xydx + 3x2dy – 3y²dy[= 0]
2. B1 State or imply 3y² dy/dx as derivative of y³
3. M1 Equate attempted derivative of left-hand side to zero and solve to obtain an equation with dy/dx as subject Allow if zero implied by subsequent working. Allow if recover from an extra dy/dx =... at the beginning of the left-hand side.
4. A1 Obtain dy/dx = (x²+2xy)/(y²-x²) correctly AG Accept y' for dy/dx
5. *M1 Equate numerator to zero Must be using the given derivative.
6. A1 Obtain x = -2y, or equivalent An equation with x or y as the subject SOI.
7. DM1 Use x³ + 3x²y - y³ = 3 to obtain an equation in x or y -8y³ +12y³ – y³ = 3 or x³-2x³ + x³ = 3 or any equivalent form (do not need to evaluate powers).
8. A1 Obtain the point (–2, 1) and no others from solving their cubic equation Allow if each component stated separately. ISW.
9. B1 State the point (0, -3/3), or equivalent from correct work Accept (0, -√3), or (0, -1.44) (-1.44225). Allow if each component stated separately. ISW.