A-LevelMathematicsLogarithmic and exponential functions9709/22 · Feb/Mar 2023 · Q58 Marks

It is given that ∫₁ᵃ (4/(1 + 2x) + 3/x) dx = ln 10, where a is a constant greater than 1.

(a)Show that a = ³√(90(1 + 2a)⁻²).[5]
(b)Use an iterative formula, based on the equation in (a), to find the value of a correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures.[3]

Mark scheme

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

[*M1] (a) Integrate to obtain form k₁ ln(1+2x) + k₂ ln x k₁≠0, k₂ ≠ 0 .
[A1] (a) Obtain correct 2ln(1+2x)+3ln x
[DM1] (a) Use limits correctly and equate to ln10
[DM1] (a) Apply relevant logarithm properties correctly and arrange as far as a³ = ...
[A1] (a) Confirm given result a = ³√(90(1+2a)⁻²) with sufficient detail AG
[M1] (b) Use iteration process correctly at least once Need to see 1.6848.
[A1] (b) Obtain final answer 1.68 Answer required to exactly 3 sf.
[A1] (b) Show sufficient iterations to 5 sf to justify answer or show a sign change in the interval [1.675, 1.685]

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