C4 June 2010 Q6
6. \[\mathrm{f}(\theta) = 4\cos^2\theta - 3\sin^2\theta\]
(a) Show that \(\mathrm{f}(\theta) = \dfrac{1}{2} + \dfrac{7}{2}\cos 2\theta\). (3)
(b) Hence, using calculus, find the exact value of \(\displaystyle\int_0^{\frac{\pi}{2}} \theta\,\mathrm{f}(\theta)\,\mathrm{d}\theta\). (7)
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(\theta) = 4\cos^2\theta - 3\sin^2\theta\) | |
| \(= 4\left(\dfrac{1}{2} + \dfrac{1}{2}\cos 2\theta\right) - 3\left(\dfrac{1}{2} - \dfrac{1}{2}\cos 2\theta\right)\) | M1 M1 |
| \(= \dfrac{1}{2} + \dfrac{7}{2}\cos 2\theta\ \ \ast\) cso | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\displaystyle\int \theta\cos 2\theta\,\mathrm{d}\theta = \frac{1}{2}\theta\sin 2\theta - \frac{1}{2}\int \sin 2\theta\,\mathrm{d}\theta\) | M1 A1 |
| \(= \dfrac{1}{2}\theta\sin 2\theta + \dfrac{1}{4}\cos 2\theta\) | A1 |
| \(\displaystyle\int \theta\,\mathrm{f}(\theta)\,\mathrm{d}\theta = \frac{1}{4}\theta^2 + \frac{7}{4}\theta\sin 2\theta + \frac{7}{8}\cos 2\theta\) | M1 A1 |
| \(\Big[\ \ldots\ \Big]_0^{\frac{\pi}{2}} = \left[\dfrac{\pi^2}{16} + 0 - \dfrac{7}{8}\right] - \left[0 + 0 + \dfrac{7}{8}\right]\) | M1 |
| \(= \dfrac{\pi^2}{16} - \dfrac{7}{4}\) | A1 |
| (7) | |
| (10 marks) |