| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Matrix powers and repeated transformations |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question on matrix transformations. Part (a) involves routine matrix multiplication, part (b) requires recognizing a standard rotation matrix (45° anticlockwise), and part (c) uses the pattern from parts (a)(i) and (ii) to find M^2006. While it's Further Maths content, the techniques are mechanical and the pattern recognition is guided by the earlier parts, making it slightly easier than an average A-level question overall. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\mathbf{M}^2 = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\) | M1, A2,1 (3) | M1 if 2 entries correct; M1A1 if 3 entries correct |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\mathbf{M}^4 = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\) | B1\(\checkmark\) (1) | ft error in \(\mathbf{M}^2\) provided no surds in \(\mathbf{M}^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Rotation (about the origin) | M1 | |
| through \(45°\) clockwise | A1 (2) | OE; NMS 2/3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Awareness of \(\mathbf{M}^8 = \mathbf{I}\) | M1 | |
| complete valid method | m1 | |
| \(\mathbf{M}^{2006} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) | A1\(\checkmark\) (3) | ft error in \(\mathbf{M}^2\) as above |
## Question 5:
### Part (a)(i)
| Working | Marks | Guidance |
|---------|-------|----------|
| $\mathbf{M}^2 = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ | M1, A2,1 (3) | M1 if 2 entries correct; M1A1 if 3 entries correct |
### Part (a)(ii)
| Working | Marks | Guidance |
|---------|-------|----------|
| $\mathbf{M}^4 = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ | B1$\checkmark$ (1) | ft error in $\mathbf{M}^2$ provided no surds in $\mathbf{M}^2$ |
### Part (b)
| Working | Marks | Guidance |
|---------|-------|----------|
| Rotation (about the origin) | M1 | |
| through $45°$ clockwise | A1 (2) | OE; NMS 2/3 |
### Part (c)
| Working | Marks | Guidance |
|---------|-------|----------|
| Awareness of $\mathbf{M}^8 = \mathbf{I}$ | M1 | |
| complete valid method | m1 | |
| $\mathbf{M}^{2006} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ | A1$\checkmark$ (3) | ft error in $\mathbf{M}^2$ as above |
---5 The matrix $\mathbf { M }$ is defined by
$$\mathbf { M } = \left[ \begin{array} { c c }
\frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\
- \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } }
\end{array} \right]$$
\begin{enumerate}[label=(\alph*)]
\item Find the matrix:
\begin{enumerate}[label=(\roman*)]
\item $\mathbf { M } ^ { 2 }$;
\item $\mathbf { M } ^ { 4 }$.
\end{enumerate}\item Describe fully the geometrical transformation represented by $\mathbf { M }$.
\item Find the matrix $\mathbf { M } ^ { 2006 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2006 Q5 [9]}}