A-LevelMathematicsNumerical methods9709/31 · May/June 2021 · Q79 Marks

The diagram shows the curve y = (tan⁻¹ x) / √x and its maximum point M where x = a. [Figure 7.1]

(a)Show that a satisfies the equation a = tan((2a) / (1 + a²)).[4]
(b)Verify by calculation that a lies between 1.3 and 1.5.[2]
(c)Use an iterative formula based on the equation in part (a) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[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 (May/June 2021). The mark or mark type is shown in brackets.

[M1] Use correct quotient rule or correct product rule
[A1] Obtain correct derivative in any form
[M1] Equate derivative to zero and remove inverse tangent
[A1 AG] Obtain a = tan( (2a) / (1+a²) ) from correct working
[M1] Calculate the value of a relevant expression or pair of expressions at a = 1.3 and a = 1.5
[A1] Complete the argument correctly with correct calculated values
[M1] Use the iterative process a_n+1 = tan( (2a_n) / (1+a_n²) ) correctly at least twice
[A1] Obtain final answer 1.39
[A1] Show sufficient iterations to at least 4 d.p. to justify 1.39 to 2 d.p. or show there is a sign change in the interval (1.385, 1.395)

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