| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question requiring basic matrix arithmetic (subtraction, scalar multiplication, matrix multiplication) and recognition of standard transformation matrices. Part (a) is simple calculation, part (b) requires knowing that the given matrix represents reflection in y=x combined with enlargement, and part (c) is routine matrix squaring. While it's Further Maths content, the techniques are mechanical with no problem-solving or novel insight required, making it slightly easier than an average A-level question. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(M = \begin{bmatrix} 0 & -3 \\ -3 & 0 \end{bmatrix}\) | B2,1 | B1 if subtracted the wrong way round |
| (b) \(p = 3\) | B1F | |
| \(L \text{ is } y = -x\) | B1 | Allow \(p = -3\), \(L\) is \(y = x\) |
| (c) \(M^2 = \begin{bmatrix} 9 & 0 \\ 0 & 9 \end{bmatrix}\) | B1F | Or by geometrical reasoning; ft as before |
| \(\ldots = 9I\) | B1F | ft as before |
**(a)** $M = \begin{bmatrix} 0 & -3 \\ -3 & 0 \end{bmatrix}$ | B2,1 | B1 if subtracted the wrong way round
**(b)** $p = 3$ | B1F |
$L \text{ is } y = -x$ | B1 | Allow $p = -3$, $L$ is $y = x$
**(c)** $M^2 = \begin{bmatrix} 9 & 0 \\ 0 & 9 \end{bmatrix}$ | B1F | Or by geometrical reasoning; ft as before
$\ldots = 9I$ | B1F | ft as before
**Total: 6 marks**
---1 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by
$$\mathbf { A } = \left[ \begin{array} { l l }
2 & 1 \\
3 & 8
\end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l }
1 & 2 \\
3 & 4
\end{array} \right]$$
The matrix $\mathbf { M } = \mathbf { A } - 2 \mathbf { B }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathbf { M } = n \left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right]$, where $n$ is a positive integer.\\
(2 marks)
\item The matrix $\mathbf { M }$ represents a combination of an enlargement of scale factor $p$ and a reflection in a line $L$. State the value of $p$ and write down the equation of $L$.
\item Show that
$$\mathbf { M } ^ { 2 } = q \mathbf { I }$$
where $q$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q1 [6]}}