## Question 8(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(\theta) = 9\cos^2\theta + \sin^2\theta = 9\cos^2\theta + 1 - \cos^2\theta$ | M1 | |
| $= 8\cos^2\theta + 1 = 8\cdot\frac{(\cos 2\theta +1)}{2} + 1$ | M1 | |
| $= 5 + 4\cos 2\theta$ | A1 | |
| **Or:** $f(\theta) = 9\cdot\frac{(\cos 2\theta+1)}{2} + 1\cdot\frac{(1-\cos 2\theta)}{2}$ | M1 M1 | |
| $= 5 + 4\cos 2\theta$ | A1 | |

## Question 8(b) Way 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Splits as $\int_0^{\frac{\pi}{2}} a\theta^2\, d\theta + \int_0^{\frac{\pi}{2}} b\theta^2\cos 2\theta\, d\theta$ | | |
| $\int b\theta^2\cos 2\theta\, d\theta = \ldots\theta^2\sin 2\theta \pm \int \ldots\theta\sin 2\theta\, d\theta$ | M1 | Integration by parts |
| $= \ldots\theta^2\sin 2\theta \pm \ldots\theta\cos 2\theta \pm \int\ldots\cos 2\theta\, d\theta$ | dM1 | Second application of parts |
| Integral $= \left[2\theta^2\sin 2\theta + 2\theta\cos 2\theta - \sin 2\theta\right] + \frac{5}{3}\theta^3$ | $\underline{\text{A1}}$ **B1ft** | |
| Use limits: $\left[\frac{5\left(\frac{\pi}{2}\right)^3}{3} - \pi\right] - [0] = \frac{5\pi^3}{24} - \pi$ | ddM1 A1 | |

## Question 8(b) Way 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_0^{\frac{\pi}{2}}\theta^2(a + b\cos 2\theta)\,d\theta = \theta^2(\ldots\theta \pm \ldots\sin 2\theta) - \int\ldots\theta(\ldots\theta \pm \ldots\sin 2\theta)\,d\theta$ | M1 | |
| $= \theta^2(\ldots\theta \pm \ldots\sin 2\theta) - \ldots\theta\left(\ldots\theta^2 \pm \ldots\cos 2\theta\right) + \int\left(\ldots\theta^2 \pm \ldots\cos 2\theta\right)d\theta$ | dM1 | |
| $= \boldsymbol{\theta}^2(5\boldsymbol{\theta} + 2\sin 2\theta) - 2\boldsymbol{\theta}\!\left(\frac{5\boldsymbol{\theta}^2}{2} - \cos 2\theta\right) + \left(\frac{5\boldsymbol{\theta}^3}{3} - \sin 2\theta\right)$ | A1 **B1ft** | |

# Question (a) - Trigonometric Identity

| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses $\sin^2\theta = 1-\cos^2\theta$ or $\cos^2\theta = 1-\sin^2\theta$ | M1 | To reach expression in $\sin^2\theta$ or $\cos^2\theta$ |
| Attempts double angle formula $\cos2\theta = \pm1\pm2\sin^2\theta$ or $\cos2\theta = \pm2\cos^2\theta\pm1$ | M1 | To convert to form $a+b\cos2\theta$ |
| $\text{cao} = 5+4\cos2\theta$ | A1 | |

**Alternative:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| One attempted application of double angle formula on $\sin^2\theta$ or $\cos^2\theta$ | M1 | |
| Second attempted application to form $a+b\cos2\theta$ | M1 | |
| $\text{cao} = 5+4\cos2\theta$ | A1 | |

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# Question (b) - Integration by Parts

| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt integration by parts correct way around | M1 | Accept if no incorrect formula stated. Way One: $\int b\theta^2\cos2\theta\,d\theta \to \pm..\theta^2\sin2\theta\pm\int..\theta\sin2\theta\,d\theta$ |
| Way Two: $\int\theta^2(a+b\cos2\theta)d\theta \to \left[\theta^2(..\theta\pm..\sin2\theta)\right]-\int..\theta(..\theta\pm..\sin2\theta)d\theta$ | | |
| Second integration by parts correct way | dM1 | Dependent on M1. Way One: $\to\pm..\theta^2\sin2\theta\pm..\theta\cos2\theta\pm\int..\cos2\theta\,d\theta$ |
| $\left[2\theta^2\sin2\theta+2\theta\cos2\theta-\sin2\theta\right]$ | A1 | Accept unsimplified |
| $\int a\theta^2\,d\theta \to a\dfrac{\theta^3}{3}$ | B1ft | Scored for term independent of trig terms |
| Apply both limits | ddM1 | Dependent on both previous M's; decimal 3.318 implies this mark |
| Final answer | A1 | cso; correct answer does not necessarily imply correct solution |

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