June 2023 Paper 1 Q3
3
(a) Circle the transformation matrix that represents a reflection in the line \(\quad y = -x\) [1 mark]
- \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\)
- \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
- \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\)
- \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)
(b) Show that\[\begin{pmatrix} 2 & 4 \\ -1 & -3 \end{pmatrix}\begin{pmatrix} -3 & -4 \\ 1 & 2 \end{pmatrix} = k\,\mathbf{I} \quad \text{where } k \text{ is an integer.}\][2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(\begin{pmatrix} 2 \times -3 + 4 \times 1 & 2 \times -4 + 4 \times 2 \\ -1 \times -3 + -3 \times 1 & -1 \times -4 + -3 \times 2 \end{pmatrix}\) or \(\begin{pmatrix} -6 + 4 & -8 + 8 \\ 3 - 3 & 4 - 6 \end{pmatrix}\) or \(\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}\) | M1 | allow M1 for any three correct calculations or any three correct terms in the matrix calculations don’t need to be in a matrix condone missing brackets |
| \(-2\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) \((= -2\,\mathbf{I})\) | A1 | may be written as \(k\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) where \(k = -2\) |
Additional guidance
| Errors in identity matrix will be penalised eg \(2\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\) or \(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\) are insufficient for A1 | |
| Must be evidence of identity matrix for A mark eg \(\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}\) and \(k = -2\) | M1 A0 |