The equation of a curve is x³ + y² + 3x² + 3y = 4.

Mark scheme

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

1. [B1] (a) State or imply 2y dy/dx as the derivative of y². Allow for 3x²dx + 2ydy or Fx = 3x² + 6x and Fy = 2y + 3.
2. [M1] (a) Equate derivative of LHS to zero and solve for dy/dx. 3x² + 2y dy/dx + 6x + 3 dy/dx = 0 or 3x²dx + 2ydy + 6xdx + 3y = 0 or dy/dx = -Fx/Fy need evidence from B1 mark or formula must be seen. Allow errors.
3. [A1] (a) Obtain the given answer AG. dy/dx = (3x² + 6x) / (2y + 3) not -(3x² + 6x) / (2y + 3). Must factorise with dy/dx e.g. 3x²+6x+dy/dx (2y+3) = 0 or 3x²dx + 6xdx + dy(2y +3) = 0.
4. [M1] (b) Equate numerator to zero and solve for x. Allow for just one x value.
5. [A1] (b) Obtain x = 0 and x = -2 only.
6. [DM1] (b) Substitute their x, [x = 0 or x = −2] in curve equation to obtain quadratic equation in y equal to 0. y² + 3y - 4 = 0 or y² + 3y = 0.
7. [A1] (b) Obtain y = 1 and y = −4 [when x = 0].
8. [A1] (b) Obtain y = 0 and y = −3 [when x = −2]. ISW If forget x = 0 then max 3/5.