| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on matrix transformations requiring routine application of stretch matrices, rotation matrices, and matrix multiplication. While it involves several steps, each component is standard FP1 material with no novel insight required—students apply a given transformation, sketch results, recall the 90° clockwise rotation matrix, and multiply matrices in the correct order. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| 6(a)(i) | Coords \((3, 2), (9, 2), (9, 4), (3, 4)\) | M1A1 |
| 6(a)(ii) | \(R_2\) shown correctly on insert | B1 |
| 6(b)(i) | \(R_3\) shown correctly on insert | B2, 1F |
| 6(b)(ii) | Matrix of rotation is \(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\) | B1 |
| 6(c) | Multiplication of matrices | M1 |
| Required matrix is \(\begin{bmatrix} 0 & 2 \\ -3 & 0 \end{bmatrix}\) | A1 | 2 marks |
| Total for Q6 | 8 marks |
6(a)(i) | Coords $(3, 2), (9, 2), (9, 4), (3, 4)$ | M1A1 | 2 marks | M1 for multn of $x$ by 3 or $y$ by 2 (PI) |
6(a)(ii) | $R_2$ shown correctly on insert | B1 | 1 mark |
6(b)(i) | $R_3$ shown correctly on insert | B2, 1F | 2 marks | B1 for rectangle with 2 vertices correct; ft if $c$'s $R_2$ is a rectangle in 1st quad |
6(b)(ii) | Matrix of rotation is $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ | B1 | 1 mark |
6(c) | Multiplication of matrices | M1 | (either way) or other complete method |
| Required matrix is $\begin{bmatrix} 0 & 2 \\ -3 & 0 \end{bmatrix}$ | A1 | 2 marks |
| **Total for Q6** | **8 marks** |
---6 [Figure 1, printed on the insert, is provided for use in this question.]\\
The diagram shows a rectangle $R _ { 1 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-4_652_1136_470_429}
\begin{enumerate}[label=(\alph*)]
\item The rectangle $R _ { 1 }$ is mapped onto a second rectangle, $R _ { 2 }$, by a transformation with matrix $\left[ \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right]$.
\begin{enumerate}[label=(\roman*)]
\item Calculate the coordinates of the vertices of the rectangle $R _ { 2 }$.
\item On Figure 1, draw the rectangle $R _ { 2 }$.
\end{enumerate}\item The rectangle $R _ { 2 }$ is rotated through $90 ^ { \circ }$ clockwise about the origin to give a third rectangle, $R _ { 3 }$.
\begin{enumerate}[label=(\roman*)]
\item On Figure 1, draw the rectangle $R _ { 3 }$.
\item Write down the matrix of the rotation which maps $R _ { 2 }$ onto $R _ { 3 }$.
\end{enumerate}\item Find the matrix of the transformation which maps $R _ { 1 }$ onto $R _ { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q6 [8]}}