| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| 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 standard FP1 matrix transformation question requiring application of a 2×2 matrix to vertices (routine calculation) and decomposition into two simpler transformations. While it requires understanding of geometric transformations, the decomposition into rotation and enlargement (or reflection and stretch) follows predictable patterns taught in FP1. Slightly above average due to the decomposition requiring some geometric insight, but well within standard FP1 scope. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(A'(0,-1)\ \ B'(2,-1)\ \ C'(2,0)\) | B1 | Coordinates of any 2 images seen; might be columns |
| B1 | Coordinates of 3rd image seen | |
| B1 | Completely correct labelled diagram, must include indication of coordinates | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}\) and \(\begin{pmatrix}2 & 0\\0 & 1\end{pmatrix}\) | B1 | Rotation and stretch or vice versa |
| B1 | Rotation \(90°\) clockwise, then Stretch s.f. 2 parallel to \(x\)-axis; Or Stretch s.f. 2 parallel to \(y\)-axis & Rotation \(90°\) clockwise. Must be a correct pair in correct order, consistent with their pair of transformations | |
| B1ft | Correct matrix | |
| B1ft | Correct matrix; Just a trig form for rotation not acceptable | |
| Or \(\begin{pmatrix}1 & 0\\0 & 2\end{pmatrix}\) and \(\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}\) | S.C. If 1 matrix correct, correct 2nd matrix can be found by matrix multiplication and not be necessarily consistent with their transformation, but not ft. | |
| [4] |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $A'(0,-1)\ \ B'(2,-1)\ \ C'(2,0)$ | B1 | Coordinates of any 2 images seen; might be columns |
| | B1 | Coordinates of 3rd image seen |
| | B1 | Completely correct labelled diagram, must include indication of coordinates |
| **[3]** | | |
---
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}$ and $\begin{pmatrix}2 & 0\\0 & 1\end{pmatrix}$ | B1 | Rotation and stretch or vice versa |
| | B1 | Rotation $90°$ clockwise, then Stretch s.f. 2 parallel to $x$-axis; Or Stretch s.f. 2 parallel to $y$-axis & Rotation $90°$ clockwise. Must be a correct pair in correct order, consistent with their pair of transformations |
| | B1ft | Correct matrix |
| | B1ft | Correct matrix; Just a trig form for rotation not acceptable |
| Or $\begin{pmatrix}1 & 0\\0 & 2\end{pmatrix}$ and $\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}$ | | **S.C. If 1 matrix correct, correct 2nd matrix can be found by matrix multiplication and not be necessarily consistent with their transformation, but not ft.** |
| **[4]** | | |
---6 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r } 0 & 2 \\ - 1 & 0 \end{array} \right)$.\\
(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 $\mathbf { M }$ is $O A ^ { \prime } B ^ { \prime } C ^ { \prime }$. Draw and label $O A ^ { \prime } B ^ { \prime } C ^ { \prime }$, indicating clearly the coordinates of $A ^ { \prime } , B ^ { \prime }$ and $C ^ { \prime }$.\\
(ii) The transformation represented by $\mathbf { M }$ is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a possible pair of transformations P and Q and state the matrices that represent them.
\hfill \mbox{\textit{OCR FP1 2015 Q6 [7]}}