June 2022 Paper 1 Q8
8 Use matrix multiplication to show that, in the \(x\)-\(y\) plane,
- a rotation, \(90^\circ\) anticlockwise about the origin, followed by
- a reflection in the line \(\quad y = x\)
is equivalent to a reflection in the \(x\)-axis. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| (Reflection =) \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) or (Rotation =) \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) | M1 | no words needed but if they are labelled incorrectly then M0 correct matrices could be seen amongst many others |
| \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) | M1dep | both matrices correct and written in correct order |
| \(= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) | A1 | matrices multiplied out correctly this matrix should be at the end of the proof and should not be amongst other matrices |
Additional guidance
| \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) | SC2 |
| Condone matrix written without brackets for M marks | M2A0 |
| Correct multiplication of a unit square by both matrices will imply both M marks but won’t get the A mark unless the reflection matrix is shown. The matrix multiplications would need to be done in the correct order however | M2A0 |
| Trying to operate matrices on a single point will only gain the first M mark (as it wouldn’t necessarily be true for all points). It would still require a correct matrix though | M1A0 |
| Some candidates may multiply matrices in a grid. If \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) is the result then the M1dep mark can be awarded. If not, do not award M1dep | |
| Condone commas in matrices for M marks but will lose the A mark | M2A0 |