A-LevelMathematicsNumerical methods9709/32 · Feb/Mar 2022 · Q77 Marks

By sketching a suitable pair of graphs, show that the equation 4 – x² = sec ½x has exactly one root in the interval 0 < x < π.

(a)By sketching a suitable pair of graphs, show that the equation 4 – x² = sec ½x has exactly one root in the interval 0 < x < π.[2]
(b)Verify by calculation that this root lies between 1 and 2.[2]
(c)Use the iterative formula xₙ₊₁ = √4 – sec ½xₙ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]

Mark scheme

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

[1] Sketch a relevant graph, e.g. y = 4 – x²
[1] Sketch a second relevant graph, e.g. y = sec ½x, and justify the given statement
[1] Calculate the value of a relevant expression or values of a pair of relevant expressions at x = 1 and x = 2.
[1] Complete the argument with correct calculated values
[1] Use the iterative process correctly at least twice
[1] Obtain final answer 1.60
[1] Show sufficient iterations to 4 d.p. to justify 1.60 to 2 d.p. or show there is a sign change in the interval (1.595, 1.605)

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