A-LevelMathematicsDifferentiation9709/32 · Oct/Nov 2018 · Q710 Marks

A curve has equation y = (3 cos x)/(2 + sin x) for −½π < x < ½π.

(i)Find the exact coordinates of the stationary point of the curve.[6]
(ii)The constant a is such that ∫₀ᵃ (3 cos x)/(2 + sin x) dx = 1. Find the value of a, giving your answer correct to 3 significant figures.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

[M1] Use correct quotient or product rule
[A1] Obtain correct derivative in any form
[M1] Equate numerator to zero
[M1] Use cos²x+sin²x=1 and solve for sin x
[A1+A1] Obtain coordinates x=-π/6 and y = √3 ISW
[M1*] State indefinite integral of the form k ln (2 + sin x)
[A1] Substitute limits correctly, equate result to 1 and obtain 3 ln (2 + sin a) – 3 ln 2 = 1 or equivalent
[M1(dep*)] Use correct method to solve for a
[A1] Obtain answer a = 0.913 or better

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Topic: Differentiation, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, Oct/Nov 2018 Paper 3 Variant 2, question 7.

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