| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| 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 standard FP1 matrix transformation question requiring matrix multiplication and recognition of rotation matrices. Part (a) involves routine calculation, while part (b) requires identifying that A represents a rotation by 120° (or reflection composition), which is a well-practiced skill at this level. The presence of √3 is a clear hint toward 60°/120° angles, making this slightly easier than average. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{A}^2 = \begin{pmatrix} -2 & 2\sqrt{3} \\ -2\sqrt{3} & -2 \end{pmatrix}\) | M1A1 | M1 if at least two entries correct |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{A}^3 = \begin{pmatrix} 8 & 0 \\ 0 & 8 \end{pmatrix}\) | M1 | if at least two entries correct |
| \(\ldots = 8\mathbf{I}\) | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(A^3\) gives enlargement with SF 8 (centre the origin) | M1A1F | M1 for enlargement (only); ft wrong value for \(k\) |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Enlargement and rotation | M1 | Some detail needed |
| Enlargement scale factor 2 | A1 | |
| Rotation through \(120°\) (antic'wise) | A1 | |
| Total | 3 |
## Question 7:
### Part (a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{A}^2 = \begin{pmatrix} -2 & 2\sqrt{3} \\ -2\sqrt{3} & -2 \end{pmatrix}$ | M1A1 | M1 if at least two entries correct |
| **Total** | **2** | |
### Part (a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{A}^3 = \begin{pmatrix} 8 & 0 \\ 0 & 8 \end{pmatrix}$ | M1 | if at least two entries correct |
| $\ldots = 8\mathbf{I}$ | A1 | |
| **Total** | **2** | |
### Part (b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $A^3$ gives enlargement with SF 8 (centre the origin) | M1A1F | M1 for enlargement (only); ft wrong value for $k$ |
| **Total** | **2** | |
### Part (b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Enlargement and rotation | M1 | Some detail needed |
| Enlargement scale factor 2 | A1 | |
| Rotation through $120°$ (antic'wise) | A1 | |
| **Total** | **3** | |
---7 The matrix $\mathbf { A }$ is defined by
$$\mathbf { A } = \left[ \begin{array} { c c }
- 1 & - \sqrt { 3 } \\
\sqrt { 3 } & - 1
\end{array} \right]$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate the matrix $\mathbf { A } ^ { 2 }$.
\item Show that $\mathbf { A } ^ { 3 } = k \mathbf { I }$, where $k$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
\end{enumerate}\item Describe the single geometrical transformation, or combination of two geometrical transformations, corresponding to each of the matrices:
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { A } ^ { 3 }$;
\item A.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q7 [9]}}