C2 June 2014 Q7
7.
You must show each step of your working. (4)
[Solutions based entirely on graphical or numerical methods are not acceptable.] (5)
| Scheme | Marks |
|---|---|
| (i) \(9\sin(\theta + 60^\circ) = 4\); \(0 \leqslant \theta \lt 360^\circ\) (ii) \(2\tan x - 3\sin x = 0\); \(-\pi \leqslant x \lt \pi\) | |
| \(\sin(\theta + 60^\circ) = \dfrac{4}{9}\), so \((\theta + 60^\circ) = 26.3877\ldots\) \((\alpha = 26.3877\ldots)\) | M1 |
| So, \(\theta + 60^\circ = \{153.6122\ldots,\ 386.3877\ldots\}\) | M1 |
| and \(\theta = \{93.6122\ldots,\ 326.3877\ldots\}\) | A1 A1 |
| Both answers are cso and must come from correct work | |
| (4) |
Notes
M1: Sight of \(\sin^{-1}\left(\dfrac{4}{9}\right)\) or awrt \(26.4^\circ\) or \(0.461^{\mathrm{c}}\)
Can also be implied for \(\theta =\) awrt \(-33.6\) (i.e. \(26.4 - 60\))
M1: \(\boldsymbol{\theta} + 60^\circ =\) either “\(180 - \text{their } \alpha\)” or “\(360^\circ + \text{their } \alpha\)” and not for \(\boldsymbol{\theta} =\) either “\(180 - \text{their } \alpha\)” or “\(360^\circ + \text{their } \alpha\)”. This can be implied by later working. The candidate’s \(\alpha\) could also be in radians but do not allow mixing of degrees and radians.
A1: At least one of awrt \(93.6^\circ\) or awrt \(326.4^\circ\)
A1: Both awrt \(93.6^\circ\) and awrt \(326.4^\circ\)
Ignore extra solutions outside the range.
In an otherwise fully correct solution deduct the final A1for any extra solutions in range
| Scheme | Marks |
|---|---|
| \(2\left(\dfrac{\sin x}{\cos x}\right) - 3\sin x = 0\) | M1 |
| Note: Applies \(\tan x = \dfrac{\sin x}{\cos x}\) can be implied by \(2\tan x - 3\sin x = 0 \Rightarrow \tan x(2 - 3\cos x)\) | |
| \(2\sin x - 3\sin x\cos x = 0\) \(\sin x(2 - 3\cos x) = 0\) | |
| \(\cos x = \dfrac{2}{3}\) | A1 |
| \(x = \text{awrt}\{0.84, -0.84\}\) | A1A1ft |
| In this part of the solution, if there are any extra answers in range in an otherwise correct solution withhold the A1ft. | |
| \(\{\sin x = 0 \Rightarrow\}\ x = 0\) and \(-\pi\) | B1 |
| Note solutions are: \(x = \{-3.1415\ldots, -0.8410\ldots, 0, 0.8410\ldots\}\) Ignore extra solutions outside the range | |
| For all answers in degrees in (ii) M1A1A0A1ftB0 is possible | |
| Allow the use of \(\theta\) in place of \(x\) in (ii) | |
| (5) | |
| Total 9 |
Notes
M1: Applies \(\tan x = \dfrac{\sin x}{\cos x}\)
A1: \(\cos x = \dfrac{2}{3}\)
A1: One of either awrt 0.84 or awrt \(-0.84\)
A1ft: You can apply ft for \(x = \pm\alpha\), where \(\alpha = \cos^{-1} k\) and \(-1 \leqslant k \leqslant 1\)
B1: Both \(x = 0\) and \(-\pi\) or awrt \(-3.14\) from \(\sin x = 0\)
In this part of the solution, ignore extra solutions in range.