A-LevelMathematicsNumerical methods9709/31 · Oct/Nov 2018 · Q37 Marks

By sketching a suitable pair of graphs, show that the equation x³ = 3 − x has exactly one real root.

(i)By sketching a suitable pair of graphs, show that the equation x³ = 3 − x has exactly one real root.[2]
(ii)Show that if a sequence of real values given by the iterative formula x_(n+1) = (2x_n³ + 3) / (3x_n² + 1) converges, then it converges to the root of the equation in part (i).[2]
(iii)Use this iterative formula to determine the root 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 (Oct/Nov 2018). The mark or mark type is shown in brackets.

[B1] (i) Sketch a relevant graph, e.g. y = x³
[B1] (i) Sketch a second relevant graph, e.g. y = 3 – x, and justify the given statement
[B1] (ii) State or imply the equation x = (2x³ + 3)/(3x² +1)
[B1] (ii) Rearrange this in the form x³ =3-x, or commence work vice versa
[M1] (iii) Use the iterative formula correctly at least once
[A1] (iii) Obtain final answer 1.213
[A1] (iii) Show sufficient iterations to 5 d.p. or more to justify 1.213 to 3 d.p., or show there is a sign change in the interval (1.2125, 1.2135)

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