The curve C has equation x^2 + 2xy - 3y^2 + 16 = 0.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. Part (a): M1 for differentiating x^2 to 2x, 2xy by product rule to 2y + 2x(dy/dx), -3y^2 to -6y(dy/dx) and 16 to 0. (At least two terms correct or using correct product rule on 2xy)
2. Part (a): A1 for 2x + 2y + 2x(dy/dx) - 6y(dy/dx) = 0 (o.e. including sign errors on 2x or 2y or 16)
3. Part (a): M1 for collecting terms in dy/dx and making dy/dx the subject. For example: dy/dx(2x - 6y) = -2x - 2y
4. Part (a): A1 for dy/dx = (-2x - 2y) / (2x - 6y) or dy/dx = (2x + 2y) / (6y - 2x)
5. Part (a): A1 for dy/dx = (x + y) / (3y - x) or dy/dx = (-x - y) / (x - 3y) (fully simplified)
6. Part (b): M1 for recognising that tangents parallel to y-axis means dy/dx is undefined, so setting denominator of their dy/dx to 0. e.g., 3y - x = 0 (or equivalent)
7. Part (b): A1 for correctly substituting x = 3y (or y = x/3) into the equation of the curve.
8. Part (b): M1 for obtaining a quadratic in one variable. e.g., (3y)^2 + 2(3y)y - 3y^2 + 16 = 0 => 9y^2 + 6y^2 - 3y^2 + 16 = 0 => 12y^2 + 16 = 0 (or equivalent for x)
9. Part (b): A1 for concluding that there are no real solutions for x or y or stating that no such points exist.