A-LevelMathematicsDifferential equations9709/31 · May/June 2019 · Q58 Marks

Differentiate 1 with respect to θ. sin²θ

(i)Differentiate 1 with respect to θ. sin²θ[2]
(ii)The variables x and θ satisfy the differential equation dx x tan θ - dθ + cosec² θ = 0, for 0 < θ < ½π and x > 0. It is given that x = 4 when θ = ⅙π. Solve the differential equation, obtaining an expression for x in terms of θ.[6]

Mark scheme

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

[M1] (i) Use chain rule
[A1] (i) Obtain correct answer in any form e.g. -2cosec²θcot θ
[B1] (ii) Separate variables correctly and integrate at least one side [∫x dx = ∫-cosec²θcotθ dθ]
[B1] (ii) Obtain term ½x²
[M1*] (ii) Obtain term of the form k/sin²θ or equivalent
[A1] (ii) Obtain term 1/(2sin²θ) or equivalent
[DM1] (ii) Use x = 4, θ=⅙π to evaluate a constant, or as limits, in a solution with terms ax² and b/sin²θ, where ab ≠ 0
[A1] (ii) Obtain solution x = √(cosec²θ+12) or equivalent

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About this question

Topic

Differential equations, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.

Source

A-Level Mathematics, May/June 2019 Paper 3 Variant 1, question 5.

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