AS June 2025 Paper 1 Q4
4 Two transformations, \(\mathrm{T_A}\) and \(\mathrm{T_B}\), are represented by matrices \(\mathbf{A}\) and \(\mathbf{B}\) respectively.
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\).
The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \dfrac{1}{2}\begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{pmatrix}\).
The transformation \(\mathrm{T_C}\) is equivalent to \(\mathrm{T_A}\) followed by \(\mathrm{T_B}\).
| Scheme | Marks | AO |
|---|---|---|
| (i) Reflection in the line \(y = x\). | B1 | 1.2 |
| [1] | ||
| (ii) If you carry out the same reflection twice you get back to where you started. | B1 | 2.4 |
| [1] |
Notes
(a)(i)
B1: Not “mirrored”
(a)(ii)
B1: The idea that the second identical reflection undoes the first.
BOD reference to mirror here
Must be a geometrical argument
B0 if only argument is det(M) = -1
| Scheme | Marks | AO |
|---|---|---|
| Rotation | M1 | 2.2a |
| By \((+)\dfrac{1}{3}\pi\) (radians) anticlockwise about \(O\). | A1 | 1.1 |
| [2] |
Notes
M1: Recognising the matrix as a rotation.
A1: Angle and sense given.
Allow \(60^\circ\) for angle
Allow omissions of \(O\)
A0 if includes another transformation as well
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{T_C}\) is represented by \(\mathbf{BA}\) | M1 | 2.2a |
| \(= \dfrac{1}{2}\begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \dfrac{1}{2}\begin{pmatrix} -\sqrt{3} & 1 \\ 1 & \sqrt{3} \end{pmatrix}\) | A1 | 1.1 |
| [2] |
Notes
M1: Recognising the correct order for the multiplication
Determine: Needs some evidence. Could see the correct order of matrices before multiplication
A1: If M0 then SCB1 for correct answer with no working.