A-LevelMathematicsNumerical methods9709/21 · May/June 2022 · Q59 Marks

By sketching the graphs of y = |5-2x| and y = 3 ln x on the same diagram, show that the equation |5 – 2x| = 3 ln x has exactly two roots.

(a)By sketching the graphs of y = |5-2x| and y = 3 ln x on the same diagram, show that the equation |5 – 2x| = 3 ln x has exactly two roots.[3]
(b)Show that the value of the larger root satisfies the equation x = 2.5 + 1.5 lnx.[1]
(c)Show by calculation that the value of the larger root lies between 4.5 and 5.0.[2]
(d)Use an iterative formula, based on the equation in part (b), to find the value of the larger root 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 (May/June 2022). The mark or mark type is shown in brackets.

[1] Draw correct sketch of y = |5-2x| with vertex on positive x-axis
[1] Draw correct sketch of y = 3ln x
[1] Indicate the two roots either on the diagram or by a statement DB1
[1] State 2x-5=3lnx and rearrange to confirm x = 2.5+1.5ln x AG – necessary detail needed
[1] Consider sign of x -2.5-1.5 lnx, or equivalent, for 4.5 and 5.0
[1] Obtain -0.25... and 0.08... or equivalents and justify conclusion AG-necessary detail needed
[1] Use iteration process correctly at least once
[1] Obtain final answer 4.88 Answer required to exactly 3 s.f.
[1] Show sufficient iterations to 5 s.f. to justify answer or show sign change in interval [4.875, 4.885]

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