| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring recognition that a diagonal matrix represents a stretch, and decomposition into two simple transformations (e.g., stretch factor 2 in x-direction and stretch factor -2 in y-direction, or reflection in x-axis followed by enlargement). The matrix is simple, the transformations are standard, and multiple valid answers exist, making this easier than average even for FP1. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 2 \\ 0 \end{pmatrix}, \begin{pmatrix} 2 \\ -2 \end{pmatrix}, \begin{pmatrix} 0 \\ -2 \end{pmatrix}\) | B1 B1 B1 | For correct vertex (2, -2); For all vertices correct; For correct diagram |
| (ii) Either: \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) | B1, B1 B1 | Reflection, in x-axis; Correct matrix |
| \(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\) | B1, B1 B1 | Enlargement, centre O s.f. 2; Correct matrix |
| Or: \(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) | B1, B1 B1 | Reflection, in the y-axis; Correct matrix |
| \(\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}\) | B1, B1 B1 | Enlargement, centre O s.f. -2; Correct matrix |
| Or: \(\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\) | B1, B1 B1 | Stretch, in x-direction s.f. 2; Correct matrix |
| \(\begin{pmatrix} 1 & 0 \\ 0 & -2 \end{pmatrix}\) | B1, B1 B1 | Stretch, in y-direction s.f. -2; Correct matrix |
(i) $\begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 2 \\ 0 \end{pmatrix}, \begin{pmatrix} 2 \\ -2 \end{pmatrix}, \begin{pmatrix} 0 \\ -2 \end{pmatrix}$ | B1 B1 B1 | For correct vertex (2, -2); For all vertices correct; For correct diagram
(ii) Either: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ | B1, B1 B1 | Reflection, in x-axis; Correct matrix
$\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ | B1, B1 B1 | Enlargement, centre O s.f. 2; Correct matrix
Or: $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ | B1, B1 B1 | Reflection, in the y-axis; Correct matrix
$\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}$ | B1, B1 B1 | Enlargement, centre O s.f. -2; Correct matrix
Or: $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ | B1, B1 B1 | Stretch, in x-direction s.f. 2; Correct matrix
$\begin{pmatrix} 1 & 0 \\ 0 & -2 \end{pmatrix}$ | B1, B1 B1 | Stretch, in y-direction s.f. -2; Correct matrix
**Total: 9 marks**8 The matrix $\mathbf { T }$ is given by $\mathbf { T } = \left( \begin{array} { r r } 2 & 0 \\ 0 & - 2 \end{array} \right)$.\\
(i) Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { T }$. [3]\\
(ii) The transformation represented by matrix $\mathbf { T }$ is equivalent to a transformation $A$, followed by a transformation B. Give geometrical descriptions of possible transformations A and B, and state the matrices that represent them.
\hfill \mbox{\textit{OCR FP1 2006 Q8 [9]}}