A-LevelMathematicsIntegration9709/32 · May/June 2018 · Q89 Marks

The diagram shows the curve y = (x + 1)e^(-x/3) and its maximum point M. [Figure X.X]

(i)Find the x-coordinate of M.[4]
(ii)Find the area of the shaded region enclosed by the curve and the axes, giving your answer in terms of e.[5]

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Mark scheme

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

[M1] Use correct product or quotient rule
[A1] Obtain complete correct derivative in any form
[M1] Equate derivative to zero and solve for x
[A1] Obtain answer x = 2 with no errors seen
[M1*] Integrate by parts and reach a(x+1)e^(-x/3) + b integral e^(-x/3) dx
[A1] Obtain -3(x+1)e^(-x/3) + 3 integral e^(-x/3) dx, or equivalent
[A1] Complete integration and obtain -3(x+1)e^(-x/3) - 9e^(-x/3), or equivalent
[M1(dep*)] Use correct limits x = 1 and x = 0 in the correct order, having integrated twice
[A1] Obtain answer 9e^(1/3) – 12, or equivalent

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, May/June 2018 Paper 3 Variant 2, question 8.

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