| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Standard +0.3 This is a straightforward FP1 question on linear transformations requiring identification of a stretch matrix from a diagram, performing a reflection in y=x, and finding a combined transformation matrix product. All components are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Matrix is \(\begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}\) | M1A1 | 2 marks |
| (b) Third triangle shown correctly | M1A1 | 2 marks |
| (c) Matrix of reflection is \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\); Multiplication of above matrices; Answer is \(\begin{bmatrix} 0 & 1 \\ 3 & 0 \end{bmatrix}\) | B1, M1, A1F | 3 marks |
**(a)** Matrix is $\begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}$ | M1A1 | 2 marks | M1 if zeros in correct positions; allow NMS
**(b)** Third triangle shown correctly | M1A1 | 2 marks | M1A0 if one point wrong
**(c)** Matrix of reflection is $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$; Multiplication of above matrices; Answer is $\begin{bmatrix} 0 & 1 \\ 3 & 0 \end{bmatrix}$ | B1, M1, A1F | 3 marks | Alt: calculating matrix from the coordinates: M1 A2,1; in incorrect order; ft wrong answer to (a); NMS 1/3
**Total: 7 marks**8 [Figure 3, printed on the insert, is provided for use in this question.]\\
The diagram shows two triangles, $T _ { 1 }$ and $T _ { 2 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-04_866_883_1318_550}
\begin{enumerate}[label=(\alph*)]
\item Find the matrix of the stretch which maps $T _ { 1 }$ to $T _ { 2 }$.
\item The triangle $T _ { 2 }$ is reflected in the line $y = x$ to give a third triangle, $T _ { 3 }$.
On Figure 3, draw the triangle $T _ { 3 }$.
\item Find the matrix of the transformation which maps $T _ { 1 }$ to $T _ { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2008 Q8 [7]}}