A-LevelMathematicsNumerical methods9709/22 · Oct/Nov 2022 · Q59 Marks

The curve with equation y = xln(4x + 1) – 3x has one stationary point P.

(a)Show that the x-coordinate of P satisfies the equation x = 2x + 0.75 / ln(4x + 1) - 0.25.[4]
(b)Show by calculation that the x-coordinate of P lies between 1.8 and 1.9.[2]
(c)Use an iterative formula, based on the equation in part (a), to find the x-coordinate of P correct to 3 significant figures. Give the result of each iteration to 5 significant figures.[3]

Mark scheme

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

[*M1] Attempt use of product rule to differentiate xln(4x+1)
[A1] Obtain ln(4x + 1) + 4x/(4x+1) - 3
[DM1] Equate first derivative to zero and attempt correct rearrangement to obtain the form 4x+1 = (ax + b) / ln(4x+1) OE
[A1] Confirm x = (2x + 0.75) / ln(4x + 1) - 0.25 AG - necessary detail needed. Allow verification.
[M1] Consider sign of x - (2x + 0.75) / ln(4x + 1) + 0.25 or equivalent for 1.8 and 1.9
[A1] Obtain -0.017... and 0.035... or equivalents and justify conclusion AG - necessary detail needed.
[M1] Use iteration process correctly at least once
[A1] Obtain final answer 1.83 answer required to exactly 3 s.f.
[A1] Show sufficient iterations to 5 sf to justify answer or show sign change in the interval [1.825, 1.835]

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