A-LevelMathematicsDifferentiation9709/22 · May/June 2024 · Q47 Marks

A curve is defined by the parametric equations x = 4 cos²t, y = √3 sin 2t, for values of t such that 0 < t < π/2. Find the equation of the normal to the curve at the point for which t = π/6. Give your answer in the form ax+by+c = 0 where a, b and c are integers.

4A curve is defined by the parametric equations x = 4 cos²t, y = √3 sin 2t, for values of t such that 0 < t < π/2. Find the equation of the normal to the curve at the point for which t = π/6. Give your answer in the form ax+by+c = 0 where a, b and c are integers.[7]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[*M1] Obtain forms dx/dt = k₁cos t sin t or dy/dt = k₂cos 2t
[A1] Obtain correct -8cos t sin t or –4sin 2t and 2√3 cos 2t
[*DM1] Attempt value of dy/dx when t = π/6 Need to see attempt at substitution.
[A1] Obtain dy/dx = 1/2
[**M1FT] State or imply gradient of normal is 2 Following their value of the first derivative.
[**DM1] Attempt equation of normal Not tangent and with attempt to find coordinates (3, 3/2).
[A1] Obtain 4x-2y-9 = 0 Or equivalent of requested form.

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