O-LevelAdditional MathematicsDifferentiation and integration4037/12 · May/June 2019 · Q99 Marks

The diagram shows the curve y = 4+2cos 3x intersecting the line y = 5 at the points P and Q. [Figure 9.1]

(i)Find, in terms of π, the x-coordinate of P and of Q.[3]
(ii)Find the exact area of the shaded region. You must show all your working.[6]

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Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

[M1] 5 = 4 + 2cos 3x For attempt to solve trig equation to obtain one correct solution
[A1] π/9
[A1] -π/9
[M1] Either: ∫(4+2cos3x – 5) dx from -π/9 to π/9 For use of subtraction method
[M1] ∫(2/3 sin 3x - x) from -π/9 to π/9 For attempt to integrate to obtain the form asin 3x + bx
[B1] 2/3 sin 3x
[B1] -x
[M1] (√3/3 - π/9) - (-√3/3 + π/9) Dep on previous M mark for correct application of their limits in radians from (i) retaining exactness
[A1] Shaded area = 2√3/3 - 2π/9 oe isw
[M1] Or: Area of rectangle = 5 × 2π/9 For 5x the difference of their limits in exact radians
[M1] Area under curve = ∫(4x + 2/3 sin 3x) from -π/9 to π/9 For attempt to integrate to obtain the form asin 3x + bx
[B1] 2/3 sin 3x
[B1] 4x
[M1] (√3/3 + 4π/9) - (-√3/3 - 4π/9) Dep on previous M mark for correct application of their limits in exact radians from (i) retaining exactness
[A1] Shaded area = 2√3/3 - 2π/9 oe isw

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