A-LevelMathematicsNumerical methods9709/23 · Oct/Nov 2022 · Q79 Marks

The diagram shows the curve with equation y = (2 ln x) / (3x + 1). The curve crosses the x-axis at the point A and has a maximum point B. The shaded region is bounded by the curve and the lines x = 3 and y = 0.

(a)Find the gradient of the curve at A.[3]
(b)Show by calculation that the x-coordinate of B lies between 3.0 and 3.1.[3]
(c)Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 decimal places.[3]

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Mark scheme

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

[M1] (7a) Differentiate using quotient rule (or product rule)
[A1 OE] (7a) Obtain ( (3x+1)(2/x) - 6lnx ) / (3x+1)²
[A1 OE] (7a) Substitute x=1 to obtain 2/25
[M1] (7b) Equate numerator of first derivative to zero (May be implied)
[M1 OE] (7b) Consider sign of (2/x)(3x+1)-6lnx for 3.0 and 3.1
[A1 AG] (7b) Obtain 0.074... and –0.14... or equivalents and justify conclusion (AG - necessary detail needed. 0.00075 and – 0.001275)
[B1] (7c) Use y-values [0], 2ln2/(3(2)+1) or 0.1980 and 2ln3/(3(3)+1) or 0.2197
[M1] (7c) Use correct formula, or equivalent, with h=1
[A1] (7c) Obtain 0.31

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