## Question 8(i):

**METHOD 1**

| Answer | Mark | Guidance |
|--------|------|----------|
| For $R$, $\cos^2\theta + \sin^2\theta = 1 \Rightarrow \text{ad} - \text{bc} = 1 \Rightarrow R \subset M$ | B1 | For showing $R \subset M$ |
| $R(\theta)R(\phi) = R(\theta + \phi)$ since $\cos\theta\cos\phi - \sin\theta\sin\phi = \cos(\theta+\phi)$ and $\pm(\cos\theta\sin\phi + \sin\theta\cos\phi) = \pm\sin(\theta+\phi)$ | M1, A1 | For multiplying 2 distinct elements; for obtaining $R(\theta)R(\phi) \in R$. Must demonstrate use of compound angles or explain rotations |
| Identity: $\theta = 0 \Rightarrow \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix} \in R$ | B1 | For identity element related to $\theta = 0$ |
| Inverse: $R(-\theta) = \begin{pmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{pmatrix} = \begin{pmatrix}\cos(-\theta) & -\sin(-\theta)\\\sin(-\theta) & \cos(-\theta)\end{pmatrix}$ | B1, B1 | For inverse element; converted to form of elements of $R$ |

**SR** (subgroup criterion method)

| Answer | Mark | Guidance |
|--------|------|----------|
| $R(\theta)R(\phi)^{-1} = \begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}\cos(-\phi) & -\sin(-\phi)\\\sin(-\phi) & \cos(-\phi)\end{pmatrix}$ | B1, B1, B1, M1, A1 | For $R \subset M$; considering $R(\theta)R(\phi)^{-1}$; correct inverse; multiplying elements; correct product |
| $= \begin{pmatrix}\cos(\theta-\phi) & -\sin(\theta-\phi)\\\sin(\theta-\phi) & \cos(\theta-\phi)\end{pmatrix} \in R$ | | |
| Set is non-empty | B1 | Can be implied by identity element |

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## Question 8(ii):

| Answer | Mark | Guidance |
|--------|------|----------|
| For $\theta = \frac{1}{3}k\pi$, elements are: $\begin{pmatrix}1&0\\0&1\end{pmatrix}$, $\begin{pmatrix}\frac{1}{2} & -\frac{1}{2}\sqrt{3}\\\frac{1}{2}\sqrt{3} & \frac{1}{2}\end{pmatrix}$, $\begin{pmatrix}-\frac{1}{2} & -\frac{1}{2}\sqrt{3}\\\frac{1}{2}\sqrt{3} & -\frac{1}{2}\end{pmatrix}$, $\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$, $\begin{pmatrix}-\frac{1}{2} & \frac{1}{2}\sqrt{3}\\-\frac{1}{2}\sqrt{3} & -\frac{1}{2}\end{pmatrix}$, $\begin{pmatrix}\frac{1}{2} & \frac{1}{2}\sqrt{3}\\-\frac{1}{2}\sqrt{3} & \frac{1}{2}\end{pmatrix}$ | B1, M1, A1, A1, A1 | For $\theta = \frac{1}{3}\pi$; for using "their $\theta$" in $\begin{pmatrix}\cos k\theta & \sin k\theta\\-\sin k\theta & \cos k\theta\end{pmatrix}$ for at least 2 values of $k$; for identity and one other element other than $(-I)$; for 2 more elements; for all 6 elements correct. Allow degrees instead of radians |