A-LevelMathematicsIntegration9709/22 · Feb/Mar 2025 · Q35 Marks

The diagram shows the curves y = e^2x and y = 8e^-x. The shaded region is bounded by the two curves and the y-axis. [Figure]

(a)Show that the x-coordinate of the point of intersection of the two curves is ln2.[2]
(b)Find the area of the shaded region.[3]

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Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

[B1] State e^2x = 8e^-x and obtain 3x=ln8 or e^x = 2 or equivalent
[B1] Confirm x = ln 2 AG Necessary detail needed.
[B1] Substitute x=ln2 in either e^2x or 8e^-x and obtain value 4
[B1] Substitute in other expression, obtain value 4 and conclude appropriately AG Necessary detail needed.
[B1] Integrate 8e^-x - e^2x to obtain –8e^-x – ½e^2x OE, involving separate integrations.
[M1] Apply limits correctly and simplify to eliminate e and ln
[A1] Obtain –8×½ – ½×4 + 8 + ½ or equivalent, and hence ½ or equivalent

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 5 marks.
Source: A-Level Mathematics, Feb/Mar 2025 Paper 2 Variant 2, question 3.

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