June 2024 Paper 1 Q22
22
\[\mathrm{f}(x) = \frac{3\sin x\cos x + \sin^2 x}{12\cos^2 x + 4\sin x\cos x}\]Simplify \(\mathrm{f}(x)\) and hence solve \(\quad \mathrm{f}(x) = -\dfrac{\sqrt{3}}{4} \quad\) for \(\quad 90^\circ \lt x \lt 180^\circ\)
You must show your working. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – common factor | ||
| \(\dfrac{\sin x(3\cos x + \sin x)}{4\cos x(3\cos x + \sin x)}\) | M1 | |
| \(\dfrac{\tan x}{4}\) | M1dep | |
| \(\tan x = -\sqrt{3}\) | M1dep | could imply first M1dep mark |
| \(120^\circ\) | A1 | |
| Alternative method 2 - divide by \(\cos^2 x\) | ||
| \(\dfrac{3\tan x + \tan^2 x}{12 + 4\tan x}\) | M1 | |
| \(\dfrac{\tan x(3 + \tan x)}{4(3 + \tan x)}\) or \(\dfrac{\tan x}{4}\) | M1dep | |
| \(\tan x = -\sqrt{3}\) | M1dep | could imply first M1dep mark |
| \(120^\circ\) | A1 | |
| Alternative method 3 – setting up an equation | ||
| \(4\sin x(3\cos x + \sin x)\) \(= -4\sqrt{3}\cos x(3\cos x + \sin x)\) or \(3\tan x + \tan^2 x = -3\sqrt{3} - \sqrt{3}\tan x\) | M1 | |
| \(\tan x(3\cos x + \sin x)\) \(= -\sqrt{3}(3\cos x + \sin x)\) or \(\tan x(3 + \tan x) = -\sqrt{3}(3 + \tan x)\) | M1dep | |
| \(\tan x = -\sqrt{3}\) | M1dep | could imply first M1dep mark |
| \(120^\circ\) | A1 | |
Additional guidance
| Additional solutions | A0 |
| Untidy mathematical notation can be condoned | |
| Correct answer with no working | M0A0 |