A-LevelMathematicsNumerical methods9709/23 · May/June 2023 · Q47 Marks

The diagram shows the graph of y = 3 – e⁻½ˣ. It is given that the two roots of 3 – e⁻½ˣ = |5x – 4| are denoted by α and β, where α < β.

(a)On the diagram, sketch the graph of y = |5x – 4|, and show that the equation 3 – e⁻½ˣ = |5x − 4| has exactly two real roots.[2]
(b)Show by calculation that α lies between 0.36 and 0.37.[2]
(c)Use the iterative formula xₙ₊₁ = (7 – e⁻½ˣⁿ)/3 to find β correct to 4 significant figures. Give the result of each iteration to 6 significant figures.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[*B1] Draw (more or less) correct sketch with vertex on positive x-axis
[DB1] Indicate in some way the two roots
[M1] Consider sign of 3 – e⁻½ˣ + 5x – 4 or of 3 – e⁻½ˣ – |5x – 4| for 0.36 and 0.37
[A1] Obtain -0.035... and 0.018..., or equivalents, and justify conclusion
[M1] Use iteration process correctly at least once
[A1] Obtain final answer 1.295
[A1] Show sufficient iterations to 6 sf to justify answer or show sign change in interval [1.2945, 1.2955]

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