A-LevelMathematicsIntegration9709/33 · May/June 2021 · Q810 Marks
The diagram shows the curve y = (ln x) / x³ and its maximum point M. [Figure X.X]
- (a)Find the exact coordinates of M.[4]
- (b)By using integration by parts, show that for all a > 1, ∫₁ᵃ (ln x) / x⁴ dx < ⅓.[6]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.
- [M1] Use quotient or product rule
- [A1] Obtain correct derivative in any form
- [M1] Equate derivative to zero and solve for x
- [A1] Obtain x = ³√e and y = 1/(3e), or exact equivalents
- [*M1] Commence integration and reach ax⁻³ lnx+b ∫x⁻³. 1/x dx
- [A1] Obtain -1/3 x⁻³ lnx + ∫1/3 x⁻⁴ dx OE
- [A1] Complete integration and obtain -1/3 x⁻³ lnx - 1/9 x⁻³
- [DM1] Substitute limits correctly, having integrated twice
- [A1] Obtain answer 1/9 - 1/3 a⁻³ lna - 1/9 a⁻³
- [A1] Justify the given statement
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