| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.3 This is a straightforward FP1 matrix transformation question requiring students to (i) apply a 2×2 matrix to vertices of a unit square (routine calculation) and (ii) decompose the transformation into two simpler transformations. The matrix clearly represents a reflection in y=x followed by an enlargement scale factor 3 (or vice versa), which is a standard decomposition exercise. While it requires understanding of transformation composition, it's a textbook-style question with no novel insight needed, making it slightly easier than average overall. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| B1 | \((0,3)\) seen | |
| B1 | \((3,0)\) seen | |
| B1 [3] | Square with A'B' and C' positioned correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) or \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) | B1* | Reflection in \(y=x\) or \(y=-x\) |
| DB1 | Correct matrix, dep on stating reflection | |
| \(\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}\) or \(\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}\) | B1* | Enlargement scale factor 3 or s.f. \(-3\) |
| DB1 [4] | Correct matrix, dep on stating enlargement. S.C. B2 for a pair of transformations consistent with their diagram. |
## Question 8:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1 | $(0,3)$ seen |
| | B1 | $(3,0)$ seen |
| | B1 **[3]** | Square with A'B' and C' positioned correctly |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ or $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ | B1* | Reflection in $y=x$ or $y=-x$ |
| | DB1 | Correct matrix, dep on stating reflection |
| $\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$ or $\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}$ | B1* | Enlargement scale factor 3 or s.f. $-3$ |
| | DB1 **[4]** | Correct matrix, dep on stating enlargement. **S.C. B2 for a pair of transformations consistent with their diagram.** |
---8 The matrix $\mathbf { X }$ is given by $\mathbf { X } = \left( \begin{array} { l l } 0 & 3 \\ 3 & 0 \end{array} \right)$.\\
(i) The diagram in the printed answer book shows the unit square $O A B C$. The image of the unit square under the transformation represented by $\mathbf { X }$ is $O A ^ { \prime } B ^ { \prime } C ^ { \prime }$. Draw and label $O A ^ { \prime } B ^ { \prime } C ^ { \prime }$.\\
(ii) The transformation represented by $\mathbf { X }$ 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 2011 Q8 [7]}}