AS June 2018 Paper 1 Q5
5 Describe fully the transformation given by the matrix
\[\begin{bmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{bmatrix}\][3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Identifies correct values of sine and cosine. | M1 | 1.1a |
| Selects correct angle. Accept \(\frac{2\pi}{3}\) or \(-240^\circ\) or \(-\frac{4\pi}{3}\) | A1 | 1.1b |
| Deduces the transformation giving a full description. FT their angle. Accept \(\frac{2\pi}{3}\) or \(-240^\circ\) or \(-\frac{4\pi}{3}\) Condone missing degree sign. Condone missing ‘anticlockwise’. NMS scores 3/3 | A1F | 2.2a |
| (3 marks) |
Typical solution
\[\cos\theta = -\frac{1}{2} \quad \text{and} \quad \sin\theta = \frac{\sqrt{3}}{2}\]\[\theta = 120^\circ\]Rotation about the \(z\)-axis through \(120^\circ\) anti-clockwise.