AS June 2020 Paper 1 Q6
6 Anna has been asked to describe the transformation given by the matrix
\[\begin{bmatrix} 1 & 0 & 0 \\ 0 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}\]She writes her answer as follows:
The transformation is a rotation about the \(x\)-axis through an angle of \(\theta\), where
\[\sin\theta = \frac{1}{2} \quad \text{and} \quad -\sin\theta = -\frac{1}{2}\]\[\theta = 30^\circ\]Identify and correct the error in Anna’s work. [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Assesses the validity of Anna’s work by identifying her error, e.g. that the angle is not \(30^\circ\) or that \(\cos\theta\) has been ignored. | B1 | 2.3 |
| States the correct angle \(150^\circ\). | B1 | 1.1b |
| (2 marks) |
Typical solution
Anna gave the wrong angle
\[\sin\theta = \tfrac{1}{2} \quad \Rightarrow \quad \theta = 30^\circ \text{ or } \theta = 150^\circ\]\[\cos\theta = -\tfrac{\sqrt{3}}{2} \quad \Rightarrow \quad \theta = 150^\circ \text{ or } \theta = -150^\circ\]\[\therefore \theta = 150^\circ\]