A-LevelMathematicsNumerical methods9709/32 · May/June 2020 · Q910 Marks

The diagram shows the curves y = cos x and y = k/(1 + x), where k is a constant, for 0 ≤ x ≤ ½π. The curves touch at the point where x = p. [Figure 9.1]

(a)Show that p satisfies the equation tan p = 1/(1 + p).[5]
(b)Use the iterative formula pn+1 = tan⁻¹(1/(1 + pn)) to determine the value of p correct to 3 decimal places. Give the result of each iteration to 5 decimal places.[3]
(c)Hence find the value of k correct to 2 decimal places.[2]

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Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[B1] State cos p = k/(1+p)
[M1] Differentiate both equations and equate derivatives at x = p
[A1] Obtain a correct equation in any form, e.g. –sin p = –k/(1+p)²
[M1] Eliminate k
[A1] Obtain the given answer showing sufficient working
[M1] Use the iterative formula correctly at least once
[A1] Obtain final answer p = 0.568
[A1] Show sufficient iterations to justify 0.568 to 3 d.p., or show there is a sign change in the interval (0.5675, 0.5685)
[M1] Use a correct method to find k
[A1] Obtain answer k = 1.32

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