| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Describe rotation from matrix |
| Difficulty | Moderate -0.3 This is a standard FP1 matrix transformations question requiring recognition of rotation and reflection matrices from their standard forms (cos/sin patterns), plus basic matrix multiplication. While it involves radicals and multiple parts, the techniques are routine for Further Maths students who have learned to identify transformations from matrix entries. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| \(A + B = \begin{bmatrix} \sqrt{3} & 0 \\ 1 & 0 \end{bmatrix}\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(BA = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) | B3,2,1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Rotation \(30°\) anticlockwise (abt \(O\)) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Reflection in \(y = (\tan 15°)x\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Reflection in \(x\)-axis | B2F | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer to (i) followed by answer to (ii) | M1A1F | (2) marks |
### Part (a)(i)
$A + B = \begin{bmatrix} \sqrt{3} & 0 \\ 1 & 0 \end{bmatrix}$ | M1A1 | 2 marks | M1A0 if 3 entries correct; Condone $\frac{2\sqrt{3}}{2}$ for $\sqrt{3}$
### Part (a)(ii)
$BA = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ | B3,2,1 | 3 marks | Deduct one for each error; SC B2,1 for AB
### Part (b)(i)
Rotation $30°$ anticlockwise (abt $O$) | M1A1 | 2 marks | M1 for rotation
### Part (b)(ii)
Reflection in $y = (\tan 15°)x$ | M1A1 | 2 marks | M1 for reflection
### Part (b)(iii)
Reflection in $x$-axis | B2F | 2 marks | 1/2 for reflection in $y$-axis; ft (M1A1) only for the SC; M1A0 if in wrong order or if order not made clear
### Part (b) Alt
Answer to (i) followed by answer to (ii) | M1A1F | (2) marks |
### **Total for Question 2: 11 marks**
---2 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by
$$\mathbf { A } = \left[ \begin{array} { c c }
\frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\
\frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 }
\end{array} \right] , \mathbf { B } = \left[ \begin{array} { c c }
\frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\
\frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 }
\end{array} \right]$$
\begin{enumerate}[label=(\alph*)]
\item Calculate:
\begin{enumerate}[label=(\roman*)]
\item $\mathbf { A } + \mathbf { B }$;
\item $\mathbf { B A }$.
\end{enumerate}\item Describe fully the geometrical transformation represented by each of the following matrices:
\begin{enumerate}[label=(\roman*)]
\item $\mathbf { A }$;
\item $\mathbf { B }$;
\item $\mathbf { B A }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q2 [11]}}