C2 June 2015 Q8
8.
| Scheme | Marks | |
|---|---|---|
| Way 1: Divides by \(\cos 3\theta\) to give \(\tan 3\theta = \sqrt{3}\) so \((3\theta) = \dfrac{\pi}{3}\) | Or Way 2: Squares both sides, uses \(\cos^2 3\theta + \sin^2 3\theta = 1\), obtains \(\cos 3\theta = \pm\dfrac{1}{2}\) or \(\sin 3\theta = \pm\dfrac{\sqrt{3}}{2}\) so \((3\theta) = \dfrac{\pi}{3}\) | M1 |
| Adds \(\pi\) or \(2\pi\) to previous value of angle ( to give \(\dfrac{4\pi}{3}\) or \(\dfrac{7\pi}{3}\)) | M1 | |
| So \(\theta = \dfrac{\pi}{9}, \dfrac{4\pi}{9}, \dfrac{7\pi}{9}\) (all three, no extra in range) | A1 | |
| (3) | ||
Notes
M1: Obtains \(\dfrac{\pi}{3}\). Allow \(x = \dfrac{\pi}{3}\) or even \(\theta = \dfrac{\pi}{3}\). Need not see working here. May be implied by \(\theta = \dfrac{\pi}{9}\) in final answer ( allow \((3\theta) = 1.05\) or \(\theta = 0.349\) as decimals or \((3\theta) = 60\) or \(\theta = 20\) as degrees for this mark)
Do not allow \(\tan 3\theta = -\sqrt{3}\) nor \(\tan 3\theta = \pm\dfrac{1}{\sqrt{3}}\)
M1: Adding \(\pi\) or \(2\pi\) to a previous value however obtained. It is not dependent on the previous mark. (May be implied by final answer of \(\theta = \dfrac{4\pi}{9}\) or \(\dfrac{7\pi}{9}\)). This mark may also be given for answers as decimals [4.19 or 7.33], or degrees ( 240 or 420).
A1: Need all three correct answers in terms of \(\pi\) and no extras in range.
Three correct answers implies M1M1A1
NB : \(\theta = 20^\circ, 80^\circ, 140^\circ\) earns M1M1A0 and 0.349, 1.40 and 2.44 earns M1M1A0
| Scheme | Marks |
|---|---|
| \(4(1 - \cos^2 x) + \cos x = 4 - k\) Applies \(\sin^2 x = 1 - \cos^2 x\) | M1 |
| Attempts to solve \(4\cos^2 x - \cos x - k = 0\), to give \(\cos x =\) | dM1 |
| \(\cos x = \dfrac{1 \pm \sqrt{1 + 16k}}{8}\) or \(\cos x = \dfrac{1}{8} \pm \sqrt{\dfrac{1}{64} + \dfrac{k}{4}}\) or other correct equivalent | A1 |
| (3) |
Notes
M1: Applies \(\sin^2 x = 1 - \cos^2 x\) (allow even if brackets are missing e.g. \(4 \times 1 - \cos^2 x\) ).
This must be awarded in (ii) (a) for an expression with \(k\) not after \(k = 3\) is substituted.
dM1: Uses formula or completion of square to obtain \(\cos x =\) expression in \(k\) (Factorisation attempt is M0) A1: cao - award for their final simplified expression
| Scheme | Marks |
|---|---|
| \(\cos x = \dfrac{1 \pm \sqrt{49}}{8} = 1\) and \(-\dfrac{3}{4}\) (see the note below if errors are made) | M1 |
| Obtains two solutions from 0 , 139 , 221 (0 or 2.42 or 3.86 in radians) | dM1 |
| \(x = 0\) and 139 and 221 (allow awrt 139 and 221) must be in degrees | A1 |
| (3) | |
| [9] |
Notes
M1: Either attempts to substitute \(k = 3\) into their answer to obtain two values for \(\cos x\)
Or restarts with \(k = 3\) to find two values for \(\cos x\) (They cannot earn marks in ii(a) for this)
In both cases they need to have applied \(\sin^2 x = 1 - \cos^2 x\) (brackets may be missing) and correct method for solving their quadratic (usual rules – see notes) The values for \(\cos x\) may be >1 or < -1
dM1: Obtains two correct values for \(x\)
A1: Obtains all three correct values in degrees (allow awrt 139 and 221) including 0. Ignore excess answers outside range (including 360 degrees) Lose this mark for excess answers in the range or radian answers.