A-LevelMathematicsNumerical methods9709/21 · May/June 2018 · Q48 Marks

The diagram shows the curve with equation y = 5 ln x / (2x + 1) and has a maximum point M. The curve crosses the x-axis at the point P. [Figure 4]

(i)Find the gradient of the curve at the point P.[3]
(ii)Show that the x-coordinate of the point M satisfies the equation x = (x + 0.5) / ln x.[2]
(iii)Use an iterative formula based on the equation in part (ii) to find the x-coordinate of M correct to 4 significant figures. Show the result of each iteration to 6 significant figures.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[M1] Use quotient rule or equivalent
[A1] Obtain correct (5(2x+1)⁻¹ - 10lnx(2x+1)⁻²)/(or (5(2x+1) - 10xlnx)/(2x+1)²) or equivalent, or 5x⁻¹(2x+1)⁻¹ - 10lnx(2x+1)⁻² or equivalent
[A1] Substitute x=1 to obtain 5/3 or 1.67 or equivalent, www
[M1] Equate numerator to zero and attempt relevant arrangement
[A1] Confirm x = (x + 0.5) / ln x AG; necessary detail needed
[M1] Use iteration process correctly at least once
[A1] Obtain final answer 3.181
[A1] Show sufficient iterations to 6 sf to justify answer or show sign change in interval (3.1805, 3.1815)

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