A-LevelMathematicsNumerical methods9709/23 · Oct/Nov 2022 · Q45 Marks

By sketching a suitable pair of graphs on the same diagram, show that the equation e^(2x) = x⁵ has exactly one real root.

(a)By sketching a suitable pair of graphs on the same diagram, show that the equation e^(2x) = x⁵ has exactly one real root.[2]
(b)Use the iterative formula xₙ₊₁ = ⁵√(e^(2xₙ)) to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.[3]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

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

[B1] (4a) Draw approximately correct sketch of y = e^(x/2) (with some curve in second quadrant as well as first)
[B1] (4a) Draw approximately correct sketch of y = x² and confirm one root (with some curve in third quadrant as well as first)
[B1] (4a Alternative method) Draw approximately correct sketch of y = 5lnx or y = ln x⁵
[B1] (4a Alternative method) Draw approximately correct sketch of y = x/2 and confirm one root (Must have intersection in the 4th quadrant)
[M1] (4b) Use iteration process correctly at least once
[A1] (4b) Obtain final answer 0.9128 (answer required to exactly 4 s.f.)
[A1] (4b) Show sufficient iterations to 6 s.f. to justify answer or show sign change in the interval [0.91275, 0.91285]

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