C3 June 2008 Q2
2.\[\mathrm{f}(x) = 5\cos x + 12\sin x\]Given that \(\mathrm{f}(x) = R\cos(x - \alpha)\), where \(R > 0\) and \(0 < \alpha < \dfrac{\pi}{2}\),
(a) find the value of \(R\) and the value of \(\alpha\) to 3 decimal places. (4)
(b) Hence solve the equation\[5\cos x + 12\sin x = 6\]for \(0 \leqslant x < 2\pi\). (5)
(c)
(i) Write down the maximum value of \(5\cos x + 12\sin x\). (1)
(ii) Find the smallest positive value of \(x\) for which this maximum value occurs. (2)
| Scheme | Marks |
|---|---|
| \(R^2 = 5^2 + 12^2\) | M1 |
| \(R = 13\) | A1 |
| \(\tan\alpha = \dfrac{12}{5}\) | M1 |
| \(\alpha \approx 1.176\) cao | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\cos(x - \alpha) = \dfrac{6}{13}\) | M1 |
| \(x - \alpha = \arccos\dfrac{6}{13} = 1.091\ \ldots\) | A1 |
| \(x = 1.091\ \ldots + 1.176\ \ldots \approx 2.267\ldots\) awrt 2.3 | A1 |
| \(x - \alpha = -1.091\ \ldots\) accept \(\ldots = 5.19\ \ldots\) for M | M1 |
| \(x = -1.091\ \ldots + 1.176\ \ldots \approx 0.0849\ \ldots\) awrt 0.084 or 0.085 | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| (i) \(R_{\max} = 13\) ft their \(R\) | B1 ft |
| (ii) At the maximum, \(\cos(x - \alpha) = 1\) or \(x - \alpha = 0\) | M1 |
| \(x = \alpha = 1.176\ \ldots\) awrt 1.2, ft their \(\alpha\) | A1ft |
| (3) | |
| (12 marks) |