A-LevelMathematicsIntegration9709/21 · May/June 2025 · Q47 Marks

The diagram shows the curve with equation y = 6e^2x – e^3x. The shaded region is bounded by the axes and the curve.

(a)Find the exact x-coordinate of the maximum point.[3]
(b)Find the area of the shaded region. Give your answer in the form p/q, where p and q are integers.[4]

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Mark scheme

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

[B1] Differentiate to obtain 12e^2x – 3e^3x.
[M1] Attempt solution for x of equation of form k1e^2x + k2e^3x = 0 (but not for y = 0 ), to obtain e^x = k.
[A1] Obtain x = ln 4 or x = 2ln 2.
[B1] Identify x = ln6 as point where curve meets x-axis (May see in part (a) but must be used here).
[M1] Integrate to obtain the form k3e^2x + k4e^3x (Where k3 ≠ 6 and k4 = −1).
[A1] Obtain 3e^2x - e^3x (May see inclusion of + c).
[A1] Apply limits to obtain answer 100/3 only.

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