A-LevelMathematicsDifferentiation9709/31 · Oct/Nov 2020 · Q35 Marks

The parametric equations of a curve are x = 3 - cos 2θ, y = 2θ + sin 2θ, for 0 < θ < ½π. Show that dy/dx = cot θ.

The parametric equations of a curve are x = 3 - cos 2θ, y = 2θ + sin 2θ, for 0 < θ < ½π. Show that dy/dx = cot θ.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

State or imply dx/dθ = 2sin 2θ or dy/dθ = 2+2 cos 2θ
Use dy/dx = (dy/dθ) ÷ (dx/dθ)
Obtain correct answer dy/dx = (2+2cos2θ)/(2sin2θ) OE
Use correct double angle formulae
Obtain the given answer correctly dy/dx = cot θ AG. Must have simplified numerator in terms of cosθ.
Alternative method for question 3: Start by using both correct double angle formulae e.g. x = 3 - (2cos²θ – 1), y = 2θ + 2sinθcosθ
Alternative method for question 3: dx/dθ or dy/dθ
Alternative method for question 3: dy/dx = (2+2(cos²θ-sin²θ))/(4cosθsinθ)
Alternative method for question 3: Simplify to given answer correctly dy/dx = cot θ AG

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Topic: Differentiation, A-Level Mathematics (syllabus 9709). Structured question, 5 marks.
Source: A-Level Mathematics, Oct/Nov 2020 Paper 3 Variant 1, question 3.

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