| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Extract enlargement and rotation parameters |
| Difficulty | Standard +0.3 This is a standard FP1 transformations question requiring recall of rotation/reflection matrix formulas and straightforward matrix multiplication. Part (b) involves recognizing that det(A)=−4 gives the scale factor and finding the mirror line from eigenvectors, which is routine for Further Maths students. The calculations are methodical rather than requiring novel insight. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative)4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\begin{pmatrix} \cos 30° & -\sin 30° \\ \sin 30° & \cos 30° \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix}\) | M1 A1 | M1 for correct rotation matrix structure |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Line \(y = \frac{1}{\sqrt{3}}x\) makes angle \(30°\) with \(x\)-axis | M1 | |
| \(\begin{pmatrix} \cos 60° & \sin 60° \\ \sin 60° & -\cos 60° \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\det(\mathbf{A}) = -1 - 3 = -4\), scale factor \(= 2\) | B1 | Scale factor is \(\sqrt{ |
| Mirror line \(y = \frac{1}{\sqrt{3}}x\) (i.e. \(y = \tan 30° \cdot x\)) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\mathbf{BA} = \begin{pmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{pmatrix}\begin{pmatrix} 1 & \sqrt{3} \\ \sqrt{3} & -1 \end{pmatrix}\) | M1 | Correct order of multiplication |
| \(\mathbf{BA} = \begin{pmatrix} 0 & 4 \\ 4 & 0 \end{pmatrix}\) (or equivalent) | A1 | |
| \(= 4\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) | A1 | |
| Enlargement scale factor \(4\) | A1 | |
| Combined with reflection in \(y = x\) | A1 | Full description required |
# Question 7:
## Part (a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\begin{pmatrix} \cos 30° & -\sin 30° \\ \sin 30° & \cos 30° \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix}$ | M1 A1 | M1 for correct rotation matrix structure |
## Part (a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Line $y = \frac{1}{\sqrt{3}}x$ makes angle $30°$ with $x$-axis | M1 | |
| $\begin{pmatrix} \cos 60° & \sin 60° \\ \sin 60° & -\cos 60° \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix}$ | A1 | |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\det(\mathbf{A}) = -1 - 3 = -4$, scale factor $= 2$ | B1 | Scale factor is $\sqrt{|\det A|}$ or by inspection |
| Mirror line $y = \frac{1}{\sqrt{3}}x$ (i.e. $y = \tan 30° \cdot x$) | B1 | |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{BA} = \begin{pmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{pmatrix}\begin{pmatrix} 1 & \sqrt{3} \\ \sqrt{3} & -1 \end{pmatrix}$ | M1 | Correct order of multiplication |
| $\mathbf{BA} = \begin{pmatrix} 0 & 4 \\ 4 & 0 \end{pmatrix}$ (or equivalent) | A1 | |
| $= 4\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ | A1 | |
| Enlargement scale factor $4$ | A1 | |
| Combined with reflection in $y = x$ | A1 | Full description required |
---7
\begin{enumerate}[label=(\alph*)]
\item Using surd forms where appropriate, find the matrix which represents:
\begin{enumerate}[label=(\roman*)]
\item a rotation about the origin through $30 ^ { \circ }$ anticlockwise;
\item a reflection in the line $y = \frac { 1 } { \sqrt { 3 } } x$.
\end{enumerate}\item The matrix $\mathbf { A }$, where
$$\mathbf { A } = \left[ \begin{array} { c c }
1 & \sqrt { 3 } \\
\sqrt { 3 } & - 1
\end{array} \right]$$
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.
\item The transformation represented by $\mathbf { A }$ is followed by the transformation represented by $\mathbf { B }$, where
$$\mathbf { B } = \left[ \begin{array} { c c }
\sqrt { 3 } & - 1 \\
1 & \sqrt { 3 }
\end{array} \right]$$
Find the matrix of the combined transformation and give a full geometrical description of this combined transformation.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q7 [11]}}