C3 January 2007 Q5
5.
Figure 1 shows an oscilloscope screen.
The curve shown on the screen satisfies the equation\[y = \sqrt{3}\cos x + \sin x.\]
(a) Express the equation of the curve in the form \(y = R\sin(x + \alpha)\), where \(R\) and \(\alpha\) are constants, \(R \gt 0\) and \(0 \lt \alpha \lt \dfrac{\pi}{2}\). (4)
(b) Find the values of \(x\), \(0 \leqslant x \lt 2\pi\), for which \(y = 1\). (4)
| Scheme | Marks |
|---|---|
| \(R^2 = (\sqrt{3})^2 + 1^2 \Rightarrow R = 2\) | M1 A1 |
| \(\tan\alpha = \sqrt{3} \Rightarrow \alpha = \dfrac{\pi}{3}\) accept awrt 1.05 | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\sin(x + \text{their } \alpha) = \dfrac{1}{2}\) | M1 |
| \(x + \text{their } \alpha = \dfrac{\pi}{6}\ \left(\dfrac{5\pi}{6}, \dfrac{13\pi}{6}\right)\) | A1 |
| \(x = \dfrac{\pi}{2}, \dfrac{11\pi}{6}\) accept awrt 1.57, 5.76 | M1 A1 |
| (4) | |
| (8 marks) |
Notes
The use of degrees loses only one mark in this question. Penalise the first time it occurs in an answer and then ignore.