A-LevelMathematicsNumerical methods9709/33 · May/June 2021 · Q67 Marks

By sketching a suitable pair of graphs, show that the equation cot ½x = 1 + e⁻ˣ has exactly one root in the interval 0 < x < π.

(a)By sketching a suitable pair of graphs, show that the equation cot ½x = 1 + e⁻ˣ has exactly one root in the interval 0 < x < π.[2]
(b)Verify by calculation that this root lies between 1 and 1.5.[2]
(c)Use the iterative formula xₙ₊₁ = 2 tan⁻¹ (1 / (1 + e⁻ˣⁿ)) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

[B1] Sketch a relevant graph, e.g. y = cot (x/2)
[B1] Sketch a second relevant graph, e.g. y=1+e⁻ˣ, and justify the given statement
[M1] Calculate values of a relevant expression or pair of expressions at x = 1 and x = 1.5
[A1] Complete the argument correctly with correct calculated values
[M1] Use the iterative formula correctly at least once
[A1] Obtain final answer 1.34
[A1] Show sufficient iterations to 4 d.p. to justify 1.34 to 2 d.p. or show there is a sign change in the interval (1.335, 1.345)

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