A-LevelMathematicsTrigonometry9709/22 · Feb/Mar 2017 · Q710 Marks

The diagram shows part of the curve y = 2 cos 2x cos(2x + ⅙π). The shaded region is bounded by the curve and the two axes.

(i)Show that 2 cos 2x cos(2x + ⅙π) can be expressed in the form k₁(1 + cos 4x) + k₂ sin 4x, where the values of the constants k₁ and k₂ are to be determined.[5]
(ii)Find the exact area of the shaded region.[5]

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Mark scheme

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

[M1] Use cos(A+B) identity
[A1] Obtain 2cos2x(cos2x.½√3 − sin2x.½)
[M1] Attempt identity expressing cos² 2x in terms of cos4x
[M1] Attempt identity expressing cos2xsin 2x in terms of sin 4x
[A1] Obtain ½√3(1+cos4x) − ½ sin 4x
[M1] Attempt to find at least one intercept with x-axis
[A1] Obtain x = ⅙π at least
[M1] Integrate to obtain k₄x + k₅ sin 4x + k₆ cos 4x
[A1] Obtain ½√3x + ¼√3 sin 4x + ⅛ cos 4x (following their answer to (i) of correct form)
[A1] Apply limits 0 and ⅙π to obtain (√3/3)π or exact equivalent (following completely correct work)

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