| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Geometric interpretation of matrices |
| Difficulty | Standard +0.3 This is a straightforward FP1 matrices question involving basic matrix operations (addition, multiplication) and geometric interpretation. Parts (a) and (b) are routine algebraic manipulation, while part (c) requires standard knowledge of transformations (recognizing enlargements and reflections from matrix form). The geometric interpretation is accessible once students compute A² = 2I, though identifying the mirror line requires some thought. Overall slightly easier than average A-level, as it's mostly procedural with well-defined steps. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{A}+\mathbf{B} = \begin{bmatrix} 0 & 2k \\ 2k & 0 \end{bmatrix}\) | B1 | |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{A}^2 = \begin{bmatrix} 2k^2 & 0 \\ 0 & 2k^2 \end{bmatrix}\) | B2,1 | B1 if three entries correct |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((\mathbf{A}+\mathbf{B})^2 = \begin{bmatrix} 4k^2 & 0 \\ 0 & 4k^2 \end{bmatrix}\) | B2,1 | B1 if three entries correct |
| \(\mathbf{B}^2 = \mathbf{A}^2\), hence result | B1B1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{A}^2\) is an enlargement (centre \(O\)) with SF \(2\) | M1, A1 | Condone \(2k^2\) |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Scale factor is now \(\sqrt{2}\) | B1 | Condone \(\sqrt{2}k\) |
| Mirror line is \(y = x\tan 22\tfrac{1}{2}°\) | M1A1 | |
| Total | 3 | Question Total: 12 |
## Question 5(a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{A}+\mathbf{B} = \begin{bmatrix} 0 & 2k \\ 2k & 0 \end{bmatrix}$ | B1 | |
| **Total** | **1** | |
## Question 5(a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{A}^2 = \begin{bmatrix} 2k^2 & 0 \\ 0 & 2k^2 \end{bmatrix}$ | B2,1 | B1 if three entries correct |
| **Total** | **2** | |
## Question 5(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(\mathbf{A}+\mathbf{B})^2 = \begin{bmatrix} 4k^2 & 0 \\ 0 & 4k^2 \end{bmatrix}$ | B2,1 | B1 if three entries correct |
| $\mathbf{B}^2 = \mathbf{A}^2$, hence result | B1B1 | |
| **Total** | **4** | |
## Question 5(c)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{A}^2$ is an enlargement (centre $O$) with SF $2$ | M1, A1 | Condone $2k^2$ |
| **Total** | **2** | |
## Question 5(c)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Scale factor is now $\sqrt{2}$ | B1 | Condone $\sqrt{2}k$ |
| Mirror line is $y = x\tan 22\tfrac{1}{2}°$ | M1A1 | |
| **Total** | **3** | **Question Total: 12** |
---5 The matrices $\mathbf { A }$ and $\mathbf { B }$ are defined by
$$\mathbf { A } = \left[ \begin{array} { c c }
k & k \\
k & - k
\end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c }
- k & k \\
k & k
\end{array} \right]$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $k$ :
\begin{enumerate}[label=(\roman*)]
\item $\mathbf { A } + \mathbf { B }$;
\item $\mathbf { A } ^ { 2 }$.
\end{enumerate}\item Show that $( \mathbf { A } + \mathbf { B } ) ^ { 2 } = \mathbf { A } ^ { 2 } + \mathbf { B } ^ { 2 }$.
\item It is now given that $k = 1$.
\begin{enumerate}[label=(\roman*)]
\item Describe the geometrical transformation represented by the matrix $\mathbf { A } ^ { 2 }$.
\item The matrix $\mathbf { A }$ represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q5 [12]}}