FP1 January 2013 Q4
4. The transformation \(U\), represented by the \(2 \times 2\) matrix \(\mathbf{P}\), is a rotation through \(90^\circ\) anticlockwise about the origin.
The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf{Q}\), is a reflection in the line \(y = -x\).
Given that \(U\) followed by \(V\) is transformation \(T\), which is represented by the matrix \(\mathbf{R}\),
| Scheme | Marks |
|---|---|
| \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) | B1 |
| (1) |
Notes
(a) and (b) Signs must be clear for B marks.
| Scheme | Marks |
|---|---|
| \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) | B1 |
| (1) |
Notes
(a) and (b) Signs must be clear for B marks.
| Scheme | Marks |
|---|---|
| \(\mathbf{R} = \mathbf{QP}\) | B1 |
| (1) |
Notes
(c) Accept \(\mathbf{QP}\) or their 2x2 matrices in the correct order only for B1.
| Scheme | Marks |
|---|---|
| \(\mathrm{R} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) | M1 A1 cao |
| (2) |
Notes
(d) M for their \(\mathbf{QP}\) where answer involves \(\pm 1\) and 0 in a 2x2 matrix, A for correct answer only.
| Scheme | Marks |
|---|---|
| Reflection in the \(y\) axis | B1 B1 |
| (2) | |
| [7] |
Notes
(e) First B for Reflection, Second B for ‘\(y\) axis’ or ‘\(x = 0\)’. Must be single transformation. Ignore any superfluous information.