A-LevelMathematicsNumerical methods9709/33 · May/June 2020 · Q67 Marks

By sketching a suitable pair of graphs, show that the equation x⁵ = 2 + x has exactly one real root.

(a)By sketching a suitable pair of graphs, show that the equation x⁵ = 2 + x has exactly one real root.[2]
(b)Show that if a sequence of values given by the iterative formula Xn+1 = (4x⁵n + 2) / (5x⁴n - 1) converges, then it converges to the root of the equation in part (a).[2]
(c)Use the iterative formula with initial value x₁ = 1.5 to calculate 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 (May/June 2020). The mark or mark type is shown in brackets.

[B1] Sketch a relevant graph, e.g. y = x⁵
[B1] Sketch a second relevant graph, e.g. y = x + 2 and justify the given statement
[B1] State a suitable equation, e.g. x = (4x⁵ + 2) / (5x⁴ - 1)
[B1] Rearrange this as x⁵ = 2 + x or commence working vice versa
[M1] Use the iterative formula correctly at least once
[A1] Obtain final answer 1.267
[A1] Show sufficient iterations to 5 d.p. to justify 1.267 to 3 d.p., or show there is a sign change in the interval (1.2665, 1.2675)

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