AS October 2020 Paper 1 Q8
8
(a) The matrix \(\mathbf{M}\) is \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\).
(i) Find \(\mathbf{M}^2\). [1]
(ii) Write down the transformation represented by \(\mathbf{M}\). [1]
(iii) Hence state the geometrical significance of the result of part (i). [1]
(b) The matrix \(\mathbf{N}\) is \(\begin{pmatrix} k + 1 & 0 \\ k & k + 2 \end{pmatrix}\), where \(k\) is a constant.
Using determinants, investigate whether \(\mathbf{N}\) can represent a reflection. [4]
Using determinants, investigate whether \(\mathbf{N}\) can represent a reflection. [4]
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\mathbf{M}^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) | B1 | 1.1 |
| [1] | ||
| (ii) Reflection in \(y = -x\) | B1 | 1.1 |
| [1] | ||
| (iii) Two reflections in \(y = -x\) are equivalent to the identity transformation. | B1 | 2.2a |
| [1] |
| Scheme | Marks | AO |
|---|---|---|
| \(\det\mathbf{N} = (k + 1)(k + 2)\) | B1 | 1.1 |
| If \(\mathbf{N}\) represents a reflection then \(\det\mathbf{N} = -1\) So \(k^2 + 3k + 3 = 0\) | B1 | 2.1 |
| discriminant \(= 9 - 12 = -3 \lt 0\) | M1 | 1.1 |
| So no roots \(\Rightarrow\) can never represent a reflection. | A1 | 2.2a |
| [4] |