| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Extract enlargement and rotation parameters |
| Difficulty | Standard +0.3 This is a straightforward application of matrix transformations requiring multiplication of a 2×2 matrix by coordinate vectors, followed by extracting the scale factor (determinant square root) and rotation angle (arctan of matrix elements) from a transformation matrix. The calculations are routine for FP1 students who have learned the standard formulas for decomposing transformation matrices. |
| Spec | 4.03a Matrix language: terminology and notation4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| \(D(4, 0)\); \(E(8, -4), F(10, -2)\); Correct sketch | M1A1, A1A1, m1A1F | 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| — | — | — |
| Answer | Marks | Guidance |
|---|---|---|
| Scale factor is \(2\sqrt{2}\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Angle \(45°\) | M1A1 | 2 marks |
**Part (a)(i)**
| $D(4, 0)$; $E(8, -4), F(10, -2)$; Correct sketch | M1A1, A1A1, m1A1F | 4 marks | M1 if at least one point correct; — |
**Part (a)(ii)**
| — | — | — | — |
**Part (b)(i)**
| Scale factor is $2\sqrt{2}$ | M1A1 | 2 marks | NMS 2/2; 1/2 for AWRT 2.8 |
**Part (b)(ii)**
| Angle $45°$ | M1A1 | 2 marks | NMS 2/2; condone $315°$; 1/2 for AWRT $44-46°$ OE |
**Total: 10 marks**
---7 [Figure 1, printed on the insert, is provided for use in this question.]\\
The diagram shows a triangle with vertices $A ( 1,1 ) , B ( 3,1 )$ and $C ( 3,2 )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-04_1114_1141_552_360}
\begin{enumerate}[label=(\alph*)]
\item The triangle $D E F$ is obtained by applying to triangle $A B C$ the transformation T represented by the matrix
$$\left[ \begin{array} { r r }
2 & 2 \\
- 2 & 2
\end{array} \right]$$
\begin{enumerate}[label=(\roman*)]
\item Calculate the coordinates of $D , E$ and $F$.
\item Draw the triangle $D E F$ on Figure 1.
\end{enumerate}\item Given that T is a combination of an enlargement and a rotation, find the exact value of:
\begin{enumerate}[label=(\roman*)]
\item the scale factor of the enlargement;
\item the magnitude of the angle of the rotation.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q7 [11]}}