A-LevelMathematicsNumerical methods9709/33 · May/June 2017 · Q67 Marks

The equation cot x = 1 - x has one root in the interval 0 < x < π, denoted by α.

(i)Show by calculation that α is greater than 2.5.[2]
(ii)Show that, if a sequence of values in the interval 0 < x < π given by the iterative formula xₙ₊₁ = π + tan⁻¹(1/(1-xₙ)) converges, then it converges to α.[2]
(iii)Use this iterative formula to determine α correct to 3 decimal places. Give the result of each iteration to 5 decimal places.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

[M1] Calculate the value of a relevant expression or expressions at x = 2.5 and at another relevant value, e.g. x = 3
[A1] Complete the argument correctly with correct calculated values
[B1] State a suitable equation, e.g. x = π + tan⁻¹ (1/(1-x)) without suffices
[B1] Rearrange this as cot x = 1 - x, or commence working vice versa
[M1] Use the iterative formula correctly at least once
[A1] Obtain final answer 2.576 only
[A1] Show sufficient iterations to 5 d.p. to justify 2.576 to 3 d.p., or show there is a sign change in the interval (2.5755, 2.5765)

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