C3 June 2011 Q8
8.
(a) Express \(2\cos 3x - 3\sin 3x\) in the form \(R\cos(3x + \alpha)\), where \(R\) and \(\alpha\) are constants, \(R \gt 0\) and \(0 \lt \alpha \lt \dfrac{\pi}{2}\). Give your answers to 3 significant figures. (4)
\[\mathrm{f}(x) = \mathrm{e}^{2x}\cos 3x\]
(b) Show that \(\mathrm{f}^{\prime}(x)\) can be written in the form \[\mathrm{f}^{\prime}(x) = R\mathrm{e}^{2x}\cos(3x + \alpha)\] where \(R\) and \(\alpha\) are the constants found in part (a). (5)
(c) Hence, or otherwise, find the smallest positive value of \(x\) for which the curve with equation \(y = \mathrm{f}(x)\) has a turning point. (3)
| Scheme | Marks |
|---|---|
| \(R^2 = 2^2 + 3^2\) | M1 |
| \(R = \sqrt{13}\) or \(3.61\ldots\) | A1 |
| \(\tan\alpha = \dfrac{3}{2}\) | M1 |
| \(\alpha = 0.983\ldots\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(f^{\prime}(x) = 2e^{2x}\cos 3x - 3e^{2x}\sin 3x\) | M1A1A1 |
| \(= e^{2x}(2\cos 3x - 3\sin 3x)\) | M1 |
| \(= e^{2x}(R\cos(3x + \alpha))\) | |
| \(= Re^{2x}\cos(3x + \alpha)\) | A1* cso |
| (5) |
| Scheme | Marks |
|---|---|
| \(f^{\prime}(x) = 0 \qquad \cos(3x + \alpha) = 0\) | M1 |
| \(3x + \alpha = \dfrac{\pi}{2}\) | M1 |
| \(x = 0.196\ldots\) awrt 0.20 | A1 |
| (3) | |
| (12 marks) |
Alternative to part (c)
| Scheme | Marks |
|---|---|
| \(f^{\prime}(x) = 0 \qquad 2\cos 3x - 3\sin 3x = 0\) | M1 |
| \(\tan 3x = \dfrac{2}{3}\) | M1 |
| \(x = 0.196\ldots\) awrt 0.20 | A1 |
| (3) |