A-LevelMathematicsTrigonometry9709/22 · Oct/Nov 2023 · Q711 Marks

Prove that sin 2x(cot x + 3 tan x) = 4 – 2 cos 2x.

(a)Prove that sin 2x(cot x + 3 tan x) = 4 – 2 cos 2x.[4]
(b)Hence find the exact value of cot (π/12) + 3 tan (π/12).[2]
(c)The diagram shows the curve with equation y = 4 – 2 cos 2x, for 0 < x < 2π. At the point A, the gradient of the curve is 4. The point B is a minimum point. The x-coordinates of A and B are a and b respectively. [Figure 7.1] Show that ∫[from a to b] (4-2 cos 2x) dx = 3π + 1.[5]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

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

[*M1] (a) Attempt to express LHS in terms of sinx and cosx only
[A1] (a) Obtain 2cos²x + 6sin² x OE
[DM1] (a) Use relevant identities to express in terms of cos2x only
[A1] (a) Confirm given result 4-2cos2x with sufficient detail AG
[M1] (b) Substitute x = π/12 and attempt exact value of sin 2(π/12)
[A1] (b) Obtain 2(4-2cos(π/6)) or equivalent and hence 8-2√3
[B1] (c) Differentiate to obtain 4 sin 2x
[B2] (c) Equate to 4 and obtain limits π/12 and 7π/12 if both limits obtained by symmetry or observation. B1 for one limit.
[B1] (c) Integrate to obtain 4x-sin2x
[M1] (c) Use limits their a and b to find exact value for integral of form k₁x+k₂ sin 2x Angles must be in radians in terms of π.
[A1] (c) Obtain 4π – sin 2π - (π - sin π) and confirm given result 3π+1 with sufficient detail AG

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.