A-LevelMathematicsIntegration9709/33 · May/June 2022 · Q1010 Marks

The constant a is such that ∫(from 1 to a) x² ln x dx = 4.

(a)Show that a = (35 / (3 ln a - 1))^(1/3).[5]
(b)Verify by calculation that a lies between 2.4 and 2.8.[2]
(c)Use an iterative formula based on the equation in part (a) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[*M1] Commence integration and reach ax³ lnx + b∫x³. (1/x) dx
[A1] Obtain 1/3 x³ lnx - 1/3 ∫x² dx
[A1] Complete integration and obtain 1/3 x³ lnx - 1/9 x³
[DM1] Use limits correctly and equate to 4, having integrated twice
[A1] Obtain given result correctly
[M1] Calculate the values of a relevant expression or pair of expressions at a = 2.4 and a = 2.8
[A1] Justify the given statement with correct calculated values
[M1] Use the iterative process aₙ₊₁ = (35 / (3 ln aₙ - 1))^(1/3) correctly at least twice
[A1] Obtain final answer a = 2.64
[A1] Show sufficient iterations to 4 dp to justify 2.64 to 2 dp, or show there is a sign change in (2.635, 2.645)

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