A-LevelMathematicsNumerical methods9709/32 · Oct/Nov 2021 · Q1111 Marks

The equation of a curve is y = √tan x, for 0 ≤ x < ½π. [Figure X.X]

(a)Express dy/dx in terms of tan x, and verify that dy/dx = 1 when x = ¼π.[4]
(b)Show that t³ + t² + 3t − 1 = 0, where t = tan a.[4]
(c)Use the iterative formula a_n+1 = tan⁻¹ (⅓(1 – tan²a_n – tan³a_n)) to determine a correct to 2 decimal places, giving the result of each iteration to 4 decimal places.[3]

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Mark scheme

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

[M1] Use chain rule
[A1] Obtain correct derivative in any form
[A1] Use correct Pythagoras to obtain correct derivative in terms of tan x
[B1] Use a correct derivative to obtain dy/dx = 1 when x = ¼π
[*M1] Equate answer to part (a) to 1 and obtain a quartic equation in t or tan x
[A1] Obtain correct answer, i.e. t⁴ + 2t² – 4t +1 = 0
[DM1] Commence division by t – 1
[A1] Obtain the given answer
[M1] Use the iterative process correctly with the given formula at least once
[A1] Obtain final answer a = 0.29
[A1] Show sufficient iterations to 4 d.p. to justify 0.29 to 2 d.p., or show there is a sign change in (0.285, 0.295)

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