A-LevelMathematicsNumerical methods9709/22 · May/June 2025 · Q48 Marks

The diagram shows parts of the curves with equations y = 4e⁻²ˣ and y = 1+0.5 sin3x. Point P is a point of intersection of the curves, and the shaded region is bounded by the two curves and the y-axis. [Figure 4.1]

(a)Show that the x-coordinate of P satisfies the equation x =-0.5 ln(0.25 +0.125 sin 3x).[1]
(b)Use an iterative formula, based on the equation in part (a), to find the x-coordinate of P correct to 4 significant figures. Use an initial value of 0.5 and give the result of each iteration to 6 significant figures.[3]
(c)Hence find the area of the shaded region. Give your answer correct to 2 significant figures.[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] State 4e⁻²ˣ =1+0.5sin3x and confirm x = −0.51n(0.25 +0.125 sin 3x) AG Necessary detail needed.
[M1] Use iterative process correctly at least once
[A1] Obtain final answer 0.4912 Answer required to exactly 4 significant figures.
[A1] Show sufficient iterations to 6 significant figures to justify answer or show a sign change in the interval [0.49115, 0.49125]
[*M1] Integrate to obtain the form k₁e⁻²ˣ + k₂x + k₃ cos3x k₁, k₂, k₃ ≠ 0 OE, with separate integrals or ‘reversed’.
[A1] Obtain correct -2e⁻²ˣ − x + ⅙ cos3x OE Allow for 2e⁻²ˣ+x-⅙cos3x
[DM1] Apply limits 0 and their part (b) answer correctly and attempt evaluation
[A1] Obtain 0.61

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