A-LevelMathematicsNumerical methods9709/21 · May/June 2023 · Q37 Marks

It is given that ∫[from 0 to a] (3e^(2x) - 1) dx = 12, where a is a positive constant.

(a)Show that a = ½ln(9 + ⅔a).[4]
(b)Use an iterative formula, based on the equation in (a), to find the value of a correct to 4 significant figures. Use an initial value of 1 and give the result of each iteration to 6 significant figures.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[M1] Integrate to obtain the form k₁e²ˣ + k₂x Where k₁k₂ ≠ 0 .
[A1] Obtain correct (3/2)e²ˣ - x
[DM1] Use limits correctly and attempt rearrangement at least as far as e²ᵃ = ...
[A1] Confirm given result a = ½ln(9+⅔a) with sufficient detail AG
[M1] Use iteration process correctly at least once Need to see 1.13434 and 1.13895.
[A1] Obtain final answer 1.139 Final answer needed to exactly 4 sf.
[A1] Show sufficient iterations to 6 sf to justify answer or show a sign change in interval [1.1385, 1.1395]

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Topic: Numerical methods, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source: A-Level Mathematics, May/June 2023 Paper 2 Variant 1, question 3.

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