| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | January |
| Marks | 12 |
| Topic | Linear transformations |
| Type | Find image coordinates under transformation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths matrices question testing standard transformations (scaling, reflection, rotation) and matrix multiplication. Part (a) requires recognition of standard transformation matrices, part (c) is routine verification, parts (d-e) involve basic matrix-vector multiplication and using determinants for area scaling. All techniques are standard FM content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03i Determinant: area scale factor and orientation |
$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the transformations described by each of the matrices $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$. [4]
\end{enumerate}
It is given that the matrix $\mathbf{D} = \mathbf{CA}$, and that the matrix $\mathbf{E} = \mathbf{DB}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Show that $\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}$. [1]
\end{enumerate}
The triangle $ORS$ has vertices at the points with coordinates $(0, 0)$, $(-15, 15)$ and $(4, 21)$. This triangle is transformed onto the triangle $OR'S'$ by the transformation described by $\mathbf{E}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the coordinates of the vertices of triangle $OR'S'$. [4]
\item Find the area of triangle $OR'S'$ and deduce the area of triangle $ORS$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q4 [12]}}