A-LevelMathematicsDifferentiation9709/21 · Oct/Nov 2019 · Q710 Marks

The equation of a curve is x² – 4xy – 2y² = 1.

(i)Find an expression for dy/dx and show that the gradient of the curve at the point (–1, 2) is –5/2.[5]
(ii)Show that the curve has no stationary points.[3]
(iii)Find the x-coordinate of each of the points on the curve at which the tangent is parallel to the y-axis.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[B1] Obtain -4y - 4x(dy/dx) from use of the product rule
[B1] Differentiate -2y² to obtain -4y(dy/dx)
[B1] Obtain 2x, = 0 with no extra terms
[M1] Rearrange to obtain expression for (dy/dx) and substitute x=−1, y = 2
[A1] Obtain dy/dx = (2x-4y) / (4x+4y) OE and hence -5/2
[M1] Equate numerator of derivative to zero to produce equation in x and y
[M1] Substitute into equation of curve to produce equation in x or y
[A1] Obtain -6y² = 1 or -3x² = 1 OE and conclude
[M1] Use denominator of derivative equated to zero with equation of curve to produce equation in x
[A1] Obtain 3x² = 1 and hence x = ±(1/√3) OE

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