AS June 2025 Paper 1 Q3
3 The matrices \(\mathbf{M}\) and \(\mathbf{N}\) are given by
\(\mathbf{M} = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}\) and \(\mathbf{N} = \begin{pmatrix} b & -a \\ a & b \end{pmatrix}\) where \(a\) and \(b\) are positive constants.
(a) Given that \(\mathbf{M}^2 = \mathbf{N}\), determine the exact values of \(a\) and \(b\). [4]
(b) Hence state the transformations of the plane associated with matrices \(\mathbf{M}\) and \(\mathbf{N}\). [3]
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{M}^2 = \begin{pmatrix} a^2 - b^2 & -2ab \\ 2ab & a^2 - b^2 \end{pmatrix}\) | B1 | 1.1 |
| so \(a^2 - b^2 = b\) and \(2ab = a\) | M1 | 3.1a |
| \(b = \dfrac{1}{2}\) | A1 | 1.1 |
| \(a = \dfrac{\sqrt{3}}{2}\) | A1 | 1.1 |
| [4] |
Notes
M1: Equating entries of \(\mathbf{M}^2\) and \(\mathbf{N}\)
A1: Oe but must be exact, not \(\pm\dfrac{\sqrt{3}}{2}\)
| Scheme | Marks | AO |
|---|---|---|
| B1 | 1.1 | |
| \(\mathbf{M}\): rotation of \(30^\circ\) anticlockwise [centre O] | B1 | 1.1 |
| \(\mathbf{N}\): rotation of \(60^\circ\) anticlockwise [centre O] | B1 | 1.1 |
| [3] |
Notes
B1: For stating rotation for either \(\mathbf{M}\) or \(\mathbf{N}\) (unless neither or their \(\mathbf{M}\) or \(\mathbf{N}\) are rotation matrices)
B1 B1: SC B2 if both correct but sense (anticlockwise) omitted