| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Moderate -0.3 This is a straightforward Further Pure 1 question testing standard matrix operations and transformation recognition. Part (a) requires recall of standard rotation matrices, part (b) involves routine matrix multiplication and algebra, and part (c) requires identifying transformations from matrices—all textbook exercises with no novel problem-solving required. While it's Further Maths content, the techniques are mechanical and well-practiced. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a)(i) \(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\) | B1 | 1 mark |
| 3(a)(ii) \(\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\) | B1 | 1 mark |
| 3(b)(i) \(AB = \begin{bmatrix} -20 & 14 \\ 14 & -10 \end{bmatrix}\) | M1A1 | M1A0 if 3 entries correct |
| 3(b)(ii) \(A + B = \begin{bmatrix} 0 & 5 \\ -5 & 0 \end{bmatrix}\) | B1 | |
| \((A + B)^2 = \begin{bmatrix} -25 & 0 \\ 0 & -25 \end{bmatrix}\) | B1 | |
| \(\ldots = -25I\) | B1F | ft if \(c^2(A + B)^2\) is of the form \(kI\) |
| 3(c)(i) Rot'n 90° clockwise, enlargement SF 5 | B2, 1 | 2 marks |
| 3(c)(ii) Rotation 180°, enlargement SF 25 | B2, 1F | 2 marks |
| 3(c)(iii) Enlargement SF 625 | B2, 1F | 2 marks |
**3(a)(i)** $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ | B1 | 1 mark
**3(a)(ii)** $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ | B1 | 1 mark
**3(b)(i)** $AB = \begin{bmatrix} -20 & 14 \\ 14 & -10 \end{bmatrix}$ | M1A1 | M1A0 if 3 entries correct | 2 marks
**3(b)(ii)** $A + B = \begin{bmatrix} 0 & 5 \\ -5 & 0 \end{bmatrix}$ | B1 | | 3 marks
$(A + B)^2 = \begin{bmatrix} -25 & 0 \\ 0 & -25 \end{bmatrix}$ | B1 | |
$\ldots = -25I$ | B1F | ft if $c^2(A + B)^2$ is of the form $kI$
**3(c)(i)** Rot'n 90° clockwise, enlargement SF 5 | B2, 1 | 2 marks | OE
**3(c)(ii)** Rotation 180°, enlargement SF 25 | B2, 1F | 2 marks | Accept 'enlargement SF -25'; ft wrong value of $k$
**3(c)(iii)** Enlargement SF 625 | B2, 1F | 2 marks | B1 for pure enlargement; ft ditto
**Total: 13 marks**
---3
\begin{enumerate}[label=(\alph*)]
\item Write down the $2 \times 2$ matrix corresponding to each of the following transformations:
\begin{enumerate}[label=(\roman*)]
\item a rotation about the origin through $90 ^ { \circ }$ clockwise;
\item a rotation about the origin through $180 ^ { \circ }$.
\end{enumerate}\item The matrices $\mathbf { A }$ and $\mathbf { B }$ are defined by
$$\mathbf { A } = \left[ \begin{array} { r r }
2 & 4 \\
- 1 & - 3
\end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l }
- 2 & 1 \\
- 4 & 3
\end{array} \right]$$
\begin{enumerate}[label=(\roman*)]
\item Calculate the matrix $\mathbf { A B }$.
\item Show that $( \mathbf { A } + \mathbf { B } ) ^ { 2 } = k \mathbf { I }$, where $\mathbf { I }$ is the identity matrix, for some integer $k$.
\end{enumerate}\item Describe the single geometrical transformation, or combination of two geometrical transformations, represented by each of the following matrices:
\begin{enumerate}[label=(\roman*)]
\item $\mathbf { A } + \mathbf { B }$;
\item $( \mathbf { A } + \mathbf { B } ) ^ { 2 }$;
\item $( \mathbf { A } + \mathbf { B } ) ^ { 4 }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q3 [13]}}