| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Extract enlargement and rotation parameters |
| Difficulty | Moderate -0.3 This is a routine Further Maths question testing standard matrix transformation techniques. Part (a) involves solving simultaneous equations from matrix multiplication (straightforward). Part (b) requires computing matrix powers, recognizing the transformation as enlargement scale factor 2 and rotation 45°, then using the pattern B^4 = 4I to find B^17. All techniques are standard FP1 material with no novel insight required, making it slightly easier than average. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\begin{bmatrix}-2 & c \\ d & 3\end{bmatrix}\begin{bmatrix}5 \\ 2\end{bmatrix} = \begin{bmatrix}-2 \\ 1\end{bmatrix}\) | M1 | Setting up matrix equation |
| \(-10 + 2c = -2 \Rightarrow c = 4\) | A1 | Correct value of \(c\) |
| \(5d + 6 = 1 \Rightarrow d = -1\) | A1 | Correct value of \(d\) |
| A1 | Both values correct with valid method shown |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\mathbf{B}^2 = \begin{bmatrix}\sqrt{2} & \sqrt{2} \\ -\sqrt{2} & \sqrt{2}\end{bmatrix}\begin{bmatrix}\sqrt{2} & \sqrt{2} \\ -\sqrt{2} & \sqrt{2}\end{bmatrix} = \begin{bmatrix}0 & 4 \\ -4 & 0\end{bmatrix}\) | M1 | Correct multiplication to find \(\mathbf{B}^2\) |
| \(\mathbf{B}^4 = \begin{bmatrix}0 & 4 \\ -4 & 0\end{bmatrix}^2 = \begin{bmatrix}-16 & 0 \\ 0 & -16\end{bmatrix} = -16\mathbf{I}\), so \(k = -16\) | A1 | Correct completion with integer \(k\) stated |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Enlargement, scale factor \(2\) | B1 | Correct enlargement/scaling component |
| Rotation \(45°\) anticlockwise about the origin | B1+B1 | Correct angle B1, correct direction/centre B1 |
| Combined transformation stated clearly | B1+B1 | Up to 5 marks for full description of both transformations |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\mathbf{B}^{17} = \mathbf{B}^{16} \cdot \mathbf{B} = (\mathbf{B}^4)^4 \cdot \mathbf{B} = (-16)^4 \mathbf{I} \cdot \mathbf{B}\) | M1 | Using \(\mathbf{B}^4 = -16\mathbf{I}\) with valid index method |
| \(= 65536\mathbf{B} = \begin{bmatrix}65536\sqrt{2} & 65536\sqrt{2} \\ -65536\sqrt{2} & 65536\sqrt{2}\end{bmatrix}\) | A1 | Correct final matrix |
## Question 5:
**Part (a):** Find $c$ and $d$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\begin{bmatrix}-2 & c \\ d & 3\end{bmatrix}\begin{bmatrix}5 \\ 2\end{bmatrix} = \begin{bmatrix}-2 \\ 1\end{bmatrix}$ | M1 | Setting up matrix equation |
| $-10 + 2c = -2 \Rightarrow c = 4$ | A1 | Correct value of $c$ |
| $5d + 6 = 1 \Rightarrow d = -1$ | A1 | Correct value of $d$ |
| | A1 | Both values correct with valid method shown |
**Part (b)(i):** Show $\mathbf{B}^4 = k\mathbf{I}$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\mathbf{B}^2 = \begin{bmatrix}\sqrt{2} & \sqrt{2} \\ -\sqrt{2} & \sqrt{2}\end{bmatrix}\begin{bmatrix}\sqrt{2} & \sqrt{2} \\ -\sqrt{2} & \sqrt{2}\end{bmatrix} = \begin{bmatrix}0 & 4 \\ -4 & 0\end{bmatrix}$ | M1 | Correct multiplication to find $\mathbf{B}^2$ |
| $\mathbf{B}^4 = \begin{bmatrix}0 & 4 \\ -4 & 0\end{bmatrix}^2 = \begin{bmatrix}-16 & 0 \\ 0 & -16\end{bmatrix} = -16\mathbf{I}$, so $k = -16$ | A1 | Correct completion with integer $k$ stated |
**Part (b)(ii):** Describe transformation of **B**
| Working/Answer | Mark | Guidance |
|---|---|---|
| Enlargement, scale factor $2$ | B1 | Correct enlargement/scaling component |
| Rotation $45°$ anticlockwise about the origin | B1+B1 | Correct angle B1, correct direction/centre B1 |
| Combined transformation stated clearly | B1+B1 | Up to 5 marks for full description of both transformations |
**Part (b)(iii):** Find $\mathbf{B}^{17}$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\mathbf{B}^{17} = \mathbf{B}^{16} \cdot \mathbf{B} = (\mathbf{B}^4)^4 \cdot \mathbf{B} = (-16)^4 \mathbf{I} \cdot \mathbf{B}$ | M1 | Using $\mathbf{B}^4 = -16\mathbf{I}$ with valid index method |
| $= 65536\mathbf{B} = \begin{bmatrix}65536\sqrt{2} & 65536\sqrt{2} \\ -65536\sqrt{2} & 65536\sqrt{2}\end{bmatrix}$ | A1 | Correct final matrix |5
\begin{enumerate}[label=(\alph*)]
\item The matrix $\mathbf { A }$ is defined by $\mathbf { A } = \left[ \begin{array} { c c } - 2 & c \\ d & 3 \end{array} \right]$.\\
Given that the image of the point $( 5,2 )$ under the transformation represented by $\mathbf { A }$ is $( - 2,1 )$, find the value of $c$ and the value of $d$.\\[0pt]
[4 marks]
\item The matrix $\mathbf { B }$ is defined by $\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 2 } & \sqrt { 2 } \\ - \sqrt { 2 } & \sqrt { 2 } \end{array} \right]$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\mathbf { B } ^ { 4 } = k \mathbf { I }$, where $k$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
\item Describe the transformation represented by the matrix $\mathbf { B }$ as a combination of two geometrical transformations.
\item Find the matrix $\mathbf { B } ^ { 17 }$.
$6 \quad \mathrm {~A}$ curve $C _ { 1 }$ has equation
$$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2015 Q5 [13]}}