| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Moderate -0.8 This is a standard Further Maths question testing recall of standard transformation matrices and matrix multiplication. Parts (a) and (b) require memorized matrices, (c) tests understanding that composition means QP, (d) is routine multiplication, and (e) requires recognizing the result as a reflection. All steps are textbook exercises with no problem-solving or novel insight required. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) | B1 | (1) |
| (b) \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\) | B1 | (1) |
| (c) \(\mathbf{R} = \mathbf{QP}\) | B1 | (1) |
| (d) \(\mathbf{R} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} - \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) | M1 A1 cao | (2) |
| (e) Reflection in the \(y\) axis | B1; B1 | (2) [7] |
(a) $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ | B1 | (1)
(b) $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ | B1 | (1)
(c) $\mathbf{R} = \mathbf{QP}$ | B1 | (1)
(d) $\mathbf{R} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} - \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ | M1 A1 cao | (2)
(e) Reflection in the $y$ axis | B1; B1 | (2) [7]
**Notes:**
(a) and (b) Signs must be clear for B marks.
(c) Accept $\mathbf{QP}$ or their 2×2 matrices in the correct order only for B1.
(d) M for their $\mathbf{QP}$ where answer involves ±1 and 0 in a 2×2 matrix; A for correct answer only.
(e) First B for Reflection, Second B for 'y axis' or 'x'=0'. Must be single transformation. Ignore any superfluous information.
---4. The transformation $U$, represented by the $2 \times 2$ matrix $\mathbf { P }$, is a rotation through $90 ^ { \circ }$ anticlockwise about the origin.
\begin{enumerate}[label=(\alph*)]
\item Write down the matrix $\mathbf { P }$.
The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a reflection in the line $y = - x$.
\item Write down the matrix $\mathbf { Q }$.
Given that $U$ followed by $V$ is transformation $T$, which is represented by the matrix $\mathbf { R }$, (c) express $\mathbf { R }$ in terms of $\mathbf { P }$ and $\mathbf { Q }$,\\
(d) find the matrix $\mathbf { R }$,\\
(e) give a full geometrical description of $T$ as a single transformation.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2013 Q4 [7]}}