| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Write down transformation matrix |
| Difficulty | Moderate -0.3 This is a standard FP1 matrices question testing routine transformations and matrix properties. Parts (a)-(b) require direct recall of standard transformation matrices (reflection in y=-x and stretch), part (c) is straightforward matrix multiplication, and part (d) involves computing A² and recognizing the form of a combined transformation. While part (d)(ii) requires identifying the enlargement factor and reflection line from matrix entries, this follows a standard method taught in FP1. The question is slightly easier than average A-level difficulty due to its structured, methodical nature with clear signposting. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
\begin{enumerate}[label=(\alph*)]
\item Write down the $2 \times 2$ matrix corresponding to each of the following transformations:
\begin{enumerate}[label=(\roman*)]
\item a reflection in the line $y = -x$; [1 mark]
\item a stretch parallel to the $y$-axis of scale factor $7$. [1 mark]
\end{enumerate}
\item Hence find the matrix corresponding to the combined transformation of a reflection in the line $y = -x$ followed by a stretch parallel to the $y$-axis of scale factor $7$. [2 marks]
\item The matrix $\mathbf{A}$ is defined by $\mathbf{A} = \begin{bmatrix} -3 & -\sqrt{3} \\ -\sqrt{3} & 3 \end{bmatrix}$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\mathbf{A}^2 = k\mathbf{I}$, where $k$ is a constant and $\mathbf{I}$ is the $2 \times 2$ identity matrix. [1 mark]
\item Show that the matrix $\mathbf{A}$ corresponds to a combination of an enlargement and a reflection. State the scale factor of the enlargement and state the equation of the line of reflection in the form $y = (\tan \theta)x$. [5 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2014 Q7 [10]}}