C2 June 2014 (R) Q7
7.
(i) Solve, for \(0 \leqslant \theta \lt 180^\circ\), the equation\[\frac{\sin 2\theta}{(4\sin 2\theta - 1)} = 1\]giving your answers to 1 decimal place. (3)
(ii) Solve, for \(0 \leqslant x \lt 2\pi\), the equation\[5\sin^2 x - 2\cos x - 5 = 0\]giving your answers to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.) (5)
(Solutions based entirely on graphical or numerical methods are not acceptable.) (5)
| Scheme | Marks |
|---|---|
| \(\dfrac{\sin 2\theta}{(4\sin 2\theta - 1)} = 1\); \(0 \leqslant \theta \lt 180^\circ\) | |
| \(\sin 2\theta = \dfrac{1}{3}\) | M1 |
| \(\{2\theta = \{19.4712\ldots, 160.5288\ldots\}\}\) | |
| \(\theta = \{9.7356\ldots, 80.2644\ldots\}\) | A1 A1 |
| Do not penalise poor accuracy more than once e.g. 9.8 and 80.2 from correct work could score M1A1A0 | |
| If both answers are correct in radians award A1A0 otherwise A0A0 Correct answers are 0.2 and 1.4 | |
| Extra solutions in range in an otherwise fully correct solution deduct the last A1 | |
| (3) |
Notes
M1: \(\sin 2\theta = k\) where \(-1 \lt k \lt 1\) Must be 2θ and not θ.
A1: Either awrt 9.7 or awrt 80.3
A1: Both awrt 9.7 and awrt 80.3
| Scheme | Marks |
|---|---|
| \(5\sin^2 x - 2\cos x - 5 = 0\), \(0 \leqslant x \lt 2\pi\). | |
| \(5(1 - \cos^2 x) - 2\cos x - 5 = 0\) | M1 |
| \(5\cos^2 x + 2\cos x = 0\) \(\cos x(5\cos x + 2) = 0\) \(\Rightarrow \cos x = \ldots.\) | dM1 |
| awrt 1.98 or awrt 4.3(0) | A1 |
| Both 1.98 and 4.3(0) | A1ft |
| awrt 1.57 or \(\dfrac{\pi}{2}\) and 4.71 or \(\dfrac{3\pi}{2}\) or \(90^\circ\) and \(270^\circ\) | B1 |
| NB: \(x =\) awrt \(\left\{1.98,\ 4.3(0),\ 1.57 \text{ or } \dfrac{\pi}{2},\ 4.71 \text{ or } \dfrac{3\pi}{2}\right\}\) | |
| (5) | |
| Total 8 |
Notes
M1: Applies \(\sin^2 x = 1 - \cos^2 x\)
dM1: Cancelling out \(\cos x\) or a valid attempt at solving the quadratic in \(\cos x\) and giving \(\cos x = \ldots\) Dependent on the previous method mark.
A1: Degrees: 113.58, 246.42
A1ft: or their \(\alpha\) and their \(2\pi - \alpha\), where \(\alpha \neq \dfrac{\pi}{2}\).
If working in degrees allow 360 – their \(\alpha\)
B1: These answers only but ignore other answers outside the range
Answers in degrees: 113.58, 246.42, 90, 270
Could score M1M1A0A1ftB1 (4/5)