| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Describe rotation from matrix |
| Difficulty | Moderate -0.3 This is a standard FP1 matrix transformations question requiring recognition of rotation and reflection matrices. Parts (a)-(b) involve identifying transformations from standard forms (rotation by -45° and reflection), (c)-(d) require matrix multiplication then identification, and (e) combines two transformations. While it has multiple parts, each involves routine pattern recognition and calculation without requiring novel geometric insight or complex reasoning. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| Rotation of 45° (anticlockwise) about the origin | B1, B1 | B1 for rotation, B1 for 45° anticlockwise about origin |
| Answer | Marks | Guidance |
|---|---|---|
| Reflection in the line \(y = x\tan(22.5°)\) or reflection in \(y = x\) at 22.5° — actually: Reflection in the line \(y = x\) | B1, B1 | B1 for reflection, B1 for line \(y=x\) — Note: B is \(\begin{pmatrix}\cos(-45°) & \sin(-45°)\\ \sin(-45°) & -\cos(-45°)\end{pmatrix}\), so reflection in line \(y = x\tan(-22.5°)\), i.e. reflection in \(y = -x\tan(22.5°)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Rotation of 90° anticlockwise about the origin | M1, A1 | M1 for attempting \(A^2\), A1 correct description |
| Answer | Marks | Guidance |
|---|---|---|
| Identity transformation (or every point maps to itself) | M1, A1 | M1 for attempting \(B^2\), A1 for identity |
| Answer | Marks | Guidance |
|---|---|---|
| \(AB = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}\); Reflection in the line \(y = x\) | M1, A1, A1 | M1 for attempting product, A1 correct matrix, A1 correct description |
# Question 6:
**(a) Matrix A**
| Rotation of 45° (anticlockwise) about the origin | B1, B1 | B1 for rotation, B1 for 45° anticlockwise about origin |
**(b) Matrix B**
| Reflection in the line $y = x\tan(22.5°)$ or reflection in $y = x$ at 22.5° — actually: Reflection in the line $y = x$ | B1, B1 | B1 for reflection, B1 for line $y=x$ — Note: B is $\begin{pmatrix}\cos(-45°) & \sin(-45°)\\ \sin(-45°) & -\cos(-45°)\end{pmatrix}$, so reflection in line $y = x\tan(-22.5°)$, i.e. reflection in $y = -x\tan(22.5°)$ |
**(c) $A^2$**
| Rotation of 90° anticlockwise about the origin | M1, A1 | M1 for attempting $A^2$, A1 correct description |
**(d) $B^2$**
| Identity transformation (or every point maps to itself) | M1, A1 | M1 for attempting $B^2$, A1 for identity |
**(e) AB**
| $AB = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}$; Reflection in the line $y = x$ | M1, A1, A1 | M1 for attempting product, A1 correct matrix, A1 correct description |
---6 The matrices $\mathbf { A }$ and $\mathbf { B }$ are defined by
$$\mathbf { A } = \left[ \begin{array} { c c }
\frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\
\frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } }
\end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c }
\frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\
\frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } }
\end{array} \right]$$
Describe fully the geometrical transformation represented by each of the following matrices:
\begin{enumerate}[label=(\alph*)]
\item A ;
\item B ;
\item $\quad \mathbf { A } ^ { 2 }$;
\item $\quad \mathbf { B } ^ { 2 }$;
\item AB.\\
□
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q6 [11]}}