A-LevelMathematicsTrigonometry9709/21 · Oct/Nov 2018 · Q710 Marks

The diagram shows the curve with equation y = sin 2x + 3 cos 2x for 0 ≤ x ≤ π. At the points P and Q on the curve, the gradient of the curve is 3. [Figure 12.1]

(i)Find an expression for dy/dx.[2]
(ii)By first expressing dy/dx in the form Rcos(2x + α), where R > 0 and 0 < α < ½π, find the x-coordinates of P and Q, giving your answers correct to 4 significant figures.[8]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

[M1] State expression of form k₁ cos 2x + k₂ sin 2x
[A1] State correct 2cos 2x – 6 sin 2x
[B1] State R = √40 or 6.324... FT Following their derivative
[M1] Use appropriate trigonometry to find α
[A1] Obtain 1.249... Allow α in degrees at this point
[*M1] Equate their Rcos(2x + α) to 3 and find cos⁻¹(3÷R)
[M1] Carry out correct process to find one value of x Dependent on *M1, allow for -0.086....
[A1] Obtain 1.979
[M1] Carry out correct process to find second value of x within the range Dependent on *M1
[A1] Obtain 3.055 Allow 3.056

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