A-LevelMathematicsNumerical methods9709/22 · Feb/Mar 2025 · Q58 Marks

Sketch on the same diagram the graphs of y = |2x-3| and y = ln(x+1).

(a)Sketch on the same diagram the graphs of y = |2x-3| and y = ln(x+1).[2]
(b)The x-coordinates of the points where the graphs intersect are denoted by α and β, where α < β. Show that α = 1.5-0.5 ln(α+1).[1]
(c)Use an iterative formula, based on the equation in part (b), to find the value of α correct to 3 significant figures. Give the result of each iteration to 5 significant figures.[3]
(d)Show by calculation that 2.055 < β < 2.065.[2]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

[B1] Show an increasing curve through the origin for y = ln(x+1) Appearing in first and third quadrants.
[B1] Show V-shaped graph with vertex on positive x-axis and showing two intersections
[B1] Equate –(2x-3) and ln(x+1) or equivalent and confirm result (using x or α) AG Necessary detail needed.
[M1] Use iterative process correctly at least once
[A1] Obtain final answer 1.12 Answer required to 3 significant figures only.
[A1] Show sufficient iterations to justify answer or show a sign change in the interval [1.115, 1.125]
[M1] Consider sign of 2x-3-ln(x+1) or equivalent for 2.055 and 2.065 But not for –(2x-3)-ln(x+1), nor for calculations based on equation in part (b).
[A1] Obtain –0.006... and 0.009... or equivalents and justify conclusion

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