| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Write down transformation matrix |
| Difficulty | Moderate -0.3 This is a standard Further Maths FP1 question on matrix transformations requiring routine recognition of shear and rotation matrices. Part (a) involves direct application of shear formula, part (b)(i) requires recognizing a rotation matrix from its form (cos/sin pattern), and part (b)(ii) is straightforward determinant calculation with standard interpretation. While this is Further Maths content, these are textbook exercises with no problem-solving or novel insight required, making it slightly easier than average overall. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix}1 & 0\\4 & 1\end{pmatrix}\) — each column correct | B1 B1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Rotation, \(45°\) or \(\pi/4\) clockwise or equivalent | B1 B1 | Must be rotation and no other transformation, otherwise 0/2 |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((\det \mathbf{X} =) 1\) — correct value | B1 | |
| Scale factor for area or equivalent | B1ft | e.g. area unchanged |
| [2] |
# Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix}1 & 0\\4 & 1\end{pmatrix}$ — each column correct | B1 B1 | |
| **[2]** | | |
---
# Question 4(b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Rotation, $45°$ or $\pi/4$ clockwise or equivalent | B1 B1 | Must be rotation and no other transformation, otherwise 0/2 |
| **[2]** | | |
---
# Question 4(b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(\det \mathbf{X} =) 1$ — correct value | B1 | |
| Scale factor for **area** or equivalent | B1ft | e.g. area unchanged |
| **[2]** | | |
---4
\begin{enumerate}[label=(\alph*)]
\item Find the matrix that represents a shear with the $y$-axis invariant, the image of the point $( 1,0 )$ being the point $( 1,4 )$.
\item The matrix $\mathbf { X }$ is given by $\mathbf { X } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)$.
\begin{enumerate}[label=(\roman*)]
\item Describe fully the geometrical transformation represented by $\mathbf { X }$.
\item Find the value of the determinant of $\mathbf { X }$ and describe briefly how this value relates to the transformation represented by $\mathbf { X }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2014 Q4 [6]}}