A-LevelMathematicsIntegration9709/23 · May/June 2024 · Q38 Marks

The diagram shows the curve with equation y = 8e⁻ˣ – e²ˣ. The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

(a)Find the gradient of the curve at A.[3]
(b)Show that the x-coordinate of B is ln2 and hence find the area of the shaded region.[5]

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Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[M1] Differentiate to obtain form k₁e⁻ˣ + k₂e²ˣ Where k₁k₂ ≠ 0, k₁ ≠ 8 and k₂ ≠ −1.
[A1] Obtain -8e⁻ˣ - 2e²ˣ
[A1] Substitute x = 0 to obtain -10
[M1] Attempt to find x-coordinate of B 8e⁻ˣ - e²ˣ = 0.
[A1] Obtain e³ˣ = 8 and hence x=ln2 AG so necessary detail needed. A0 if decimals used.
[B1] Integrate to obtain -8e⁻ˣ - ½e²ˣ
[M1] Use limits 0 and ln2 correctly to find area For integral of form k₃e⁻ˣ + k₄e²ˣ where k₃k₄ ≠ 0. k₃ ≠ 8 and k₄ ≠ −1.
[A1] Obtain 5/2 OE

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