A-LevelMathematicsDifferential equations9709/32 · Oct/Nov 2022 · Q78 Marks

The variables x and θ satisfy the differential equation x sin² θ (dx/dθ) = tan² θ – 2 cot θ, for 0 < θ < ½π and x > 0. It is given that x = 2 when θ = ¼π.

(a)Show that (d/dθ)(cot²θ) = -2cotθ / sin²θ. (You may assume without proof that the derivative of cot θ with respect to θ is –cosec²θ.)[1]
(b)Solve the differential equation and find the value of x when θ = ⅙π.[7]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[B1] (a) Show sufficient working to justify the given statement
[B1] (b) Separate variables correctly
[B1] (b) Check for relevant working in (a)
[B1] (b) Obtain term ½x²
[B1 + B1] (b) Obtain terms tanθ + cot²θ
[M1] (b) Form an equation for the constant of integration, or use limits x = 2, θ = ¼π, in a solution with at least two correctly obtained terms of the form ax², btanθ and ccot²θ, where abc ≠ 0
[A1] (b) State correct solution in any form, e.g. ½x² = tanθ + cot²θ
[A1] (b) Substitute θ = ⅙π and obtain answer x = 2.67

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Topic: Differential equations, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source: A-Level Mathematics, Oct/Nov 2022 Paper 3 Variant 2, question 7.

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