A-LevelMathematicsDifferentiation9709/32 · May/June 2017 · Q1011 Marks

The diagram shows the curve y = x² cos 2x for 0 ≤ x ≤ ¼π. The curve has a maximum point at M where x = p. [Figure 10.1]

(i)Show that p satisfies the equation p = ½ tan⁻¹ (1/p).[3]
(ii)Use the iterative formula p_{n+1} = ½ tan⁻¹ (1/p_n) to determine the value of p correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]
(iii)Find, showing all necessary working, the exact area of the region bounded by the curve and the x-axis.[5]

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Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

[M1] Use correct product rule
[A1] Obtain correct derivative in any form (y' = 2x cos 2x - 2x² sin 2x)
[A1] Equate to zero and derive the given equation
[M1] Use the iterative formula correctly at least once e.g. 0.5 -> 0.55357 -> 0.53261 -> 0.54070 -> 0.53755
[A1] Obtain final answer 0.54
[A1] Show sufficient iterations to 4 d.p. to justify 0.54 to 2 d.p., or show there is a sign change in the interval (0.535, 0.545)
[*M1] Integrate by parts and reach ax² sin 2x + b∫ x sin 2x dx
[A1] Obtain ½x² sin 2x - ∫x sin 2x dx
[A1] Complete integration and obtain ½x² sin 2x + ¼x cos 2x - (1/8) sin 2x , or equivalent
[DM1] Substitute limits x = 0, x = ¼π , having integrated twice
[A1] Obtain answer (π²/32 - 1/8), or exact equivalent

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