A-LevelMathematicsNumerical methods9709/31 · May/June 2022 · Q1011 Marks

The curve y = x√sinx has one stationary point in the interval 0 < x < π, where x = a (see diagram). [Figure 10.1]

(a)Show that tan a = −½a.[4]
(b)Verify by calculation that a lies between 2 and 2.5.[2]
(c)Show that if a sequence of values in the interval 0 < x < π given by the iterative formula Xn+1 = π - tan¯¹(2x) converges, then it converges to a, the root of the equation in part (a).[2]
(d)Use the iterative formula given in part (c) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[M1] Use correct product rule Condone incorrect / missing chain rule
[A1] Obtain correct derivative in any form e.g. dy/dx = √sinx + (x cosx)/(2√sinx) or 2y dy/dx = 2x sinx + x² cosx
[M1] Equate derivative to zero and obtain an equation in tan x or tan a
[A1] Obtain tana=-a correctly AG
[M1] Must be working in radians Calculate the value of a relevant expression or pair of expressions at a = 2 and a = 2.5 At least one correct
[A1] Complete the argument correctly with correct calculated values e.g. -1 > -2.18 and -1.25 < -0.747
[B1] State a suitable equation, e.g. x = π – tan⁻¹(2/x) A correct equation without subscripts or quote tan θ = - tan (π – θ)
[B1] Using tan(A+B) formula, or otherwise, rearrange this as tan x = -x/2
[M1] Use the iterative process correctly at least once Must be working in radians
[A1] Obtain answer a = 2.29
[A1] Show sufficient iterations to 4 dp to justify 2.29 to 2 dp or show there is a sign change in the interval (2.285, 2.295) e.g. 2.25, 2.2974, 2.2871, 2.2893, 2.2888, ...

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