| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Describe enlargement or stretch from matrix |
| Difficulty | Standard +0.3 This is a standard FP1 matrix transformations question requiring identification of basic transformations (enlargement, reflection, rotation), matrix multiplication, and application to coordinate geometry. While it has multiple parts (14 marks total), each step is routine: recognizing standard transformation matrices, performing straightforward matrix multiplication, applying transformations to vertices, and using the determinant-area relationship. The calculations are mechanical with no novel insight required, making it slightly easier than average even for Further Maths. |
| 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}{1}
\item Find $\mathbf{D}$.
[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{Edexcel FP1 Q10 [14]}}