A-LevelMathematicsNumerical methods9709/22 · Oct/Nov 2024 · Q47 Marks

Sketch the graphs of y = 1+e²ˣ and y = |x-4| on the same diagram.

(a)Sketch the graphs of y = 1+e²ˣ and y = |x-4| on the same diagram.[2]
(b)The two graphs meet at the point P. Show that the x-coordinate of P satisfies the equation x = ½ ln(3-x).[2]
(c)Use an iterative formula, based on the equation in part (b), to find the x-coordinate of P correct to 3 significant figures. Use an initial value of 0.45 and give the result of each iteration to 5 significant figures.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[*B1] (a) Show increasing curve above x-axis for y = 1+ e²ˣ
[DB1] (a) Show V-shaped graph with vertex on positive x-axis and only one point of intersection with first curve
[B1] (b) State or clearly imply 1 + e²ˣ = 4 − x
[B1] (b) Arrange to confirm x = ½ ln(3 - x)
[M1] (c) Use iterative process correctly at least once
[A1] (c) Obtain final answer 0.465
[A1] (c) Show sufficient iterations to justify answer or show a sign change in the interval [0.4645, 0.4655]

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Topic: Numerical methods, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source: A-Level Mathematics, Oct/Nov 2024 Paper 2 Variant 2, question 4.

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