| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Topic | Linear transformations |
| Type | Extract enlargement and rotation parameters |
| Difficulty | Standard +0.3 This is a straightforward Further Maths matrix question testing standard techniques: computing a determinant (routine), using |det M| for area scaling (direct application), and decomposing a transformation matrix into enlargement and rotation (standard FM topic with clear method). All parts are bookwork-style with no novel problem-solving required, making it slightly easier than average even for FM. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
You are given that $M = \begin{pmatrix} \frac{1}{\sqrt{3}} & -\sqrt{3} \\ \frac{1}{\sqrt{3}} & 1 \end{pmatrix}$.
\begin{enumerate}[label=(\roman*)]
\item Show that $M$ is non-singular. [2]
\end{enumerate}
The hexagon $R$ is transformed to the hexagon $S$ by the transformation represented by the matrix $M$.
Given that the area of hexagon $R$ is 5 square units,
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item find the area of hexagon $S$. [1]
\end{enumerate}
The matrix $M$ represents an enlargement, with centre $(0,0)$ and scale factor $k$, where $k > 0$, followed by a rotation anti-clockwise through an angle $\theta$ about $(0,0)$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find the value of $k$. [2]
\item Find the value of $\theta$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q4 [7]}}