A-LevelMathematicsIntegration9709/33 · Oct/Nov 2023 · Q109 Marks

The diagram shows the curve y = x cos 2x, for x ≥ 0. [Figure X.X]

(a)Find the equation of the tangent to the curve at the point where x = 1/2 π.[4]
(b)Find the exact area of the shaded region shown in the diagram, bounded by the curve and the x-axis.[5]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[M1] (a) Use the product rule correctly on y = x cos 2x. dx/dx cos 2x + x d/dx(cos 2x) attempted.
[A1] (a) Obtain the correct derivative in any form, e.g. cos 2x – 2x sin 2x.
[A1 FT] (a) Obtain y = π/2 and dy/dx = -1 when x = π/2. FT their dy/dx with x = π/2 substituted.
[A1] (a) Obtain answer x + y = 0. OE. CWO. Need to see y and dy/dx at x = π/2.
[M1] (b) Integrate by parts and reach axsin 2x + b∫ sin 2xdx.
[A1] (b) Obtain -1/2 x sin 2x - ∫ -1/2 sin 2xdx. OE.
[A1] (b) Complete integration and obtain -1/2 x sin 2x + 1/4 cos 2x. OE.
[DM1] (b) Use limits of x = 0 and x = π/4 in the correct order, having integrated twice to obtain ax sin 2x + ccos 2x.
[A1] (b) Obtain answer π/8 - 1/4 or exact simplified two term equivalent. ISW. Accept (π-2)/8. Accept -1/2 x sin 2x + 1/4 cos 2x then final answer.

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