| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Describe reflection from matrix |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure 1 question testing basic matrix operations and geometric interpretation. Part (a) requires recognizing a standard transformation (reflection in y=-x), computing A², and explaining why it's the identity. Part (b) involves routine matrix arithmetic. All parts are direct applications of standard techniques with no problem-solving or novel insight required, making it easier than average even for FP1. |
| 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 |
|---|---|---|
| (a)(i) Reflection ... | M1 | |
| ... in \(y = -x\) | A1 | |
| (ii) \(A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) | M1A1 | M1A0 for three correct entries |
| (iii) \(A^2 = I\) or geometrical reasoning | E1; M1A1 | M1A0 for three correct entries |
| (b)(i) \(B^2 = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\) | A1√ | ft errors, dependent on both M marks |
| (ii) \((B + A)(B - A) = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}\) | B1 | |
| \(... = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix}\) | M1; A1√ | ft one error; M1A0 for three correct (ft) entries |
**(a)(i)** Reflection ... | M1
... in $y = -x$ | A1 |
**(ii)** $A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ | M1A1 | M1A0 for three correct entries
**(iii)** $A^2 = I$ or geometrical reasoning | E1; M1A1 | M1A0 for three correct entries
**(b)(i)** $B^2 = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ | A1√ | ft errors, dependent on both M marks
**(ii)** $(B + A)(B - A) = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}$ | B1
$... = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix}$ | M1; A1√ | ft one error; M1A0 for three correct (ft) entries
**Total for Q7: 11 marks**
---7
\begin{enumerate}[label=(\alph*)]
\item The transformation T is defined by the matrix $\mathbf { A }$, where
$$\mathbf { A } = \left[ \begin{array} { r r }
0 & - 1 \\
- 1 & 0
\end{array} \right]$$
\begin{enumerate}[label=(\roman*)]
\item Describe the transformation T geometrically.
\item Calculate the matrix product $\mathbf { A } ^ { 2 }$.
\item Explain briefly why the transformation T followed by T is the identity transformation.
\end{enumerate}\item The matrix $\mathbf { B }$ is defined by
$$\mathbf { B } = \left[ \begin{array} { l l }
1 & 1 \\
0 & 1
\end{array} \right]$$
\begin{enumerate}[label=(\roman*)]
\item Calculate $\mathbf { B } ^ { 2 } - \mathbf { A } ^ { 2 }$.
\item Calculate $( \mathbf { B } + \mathbf { A } ) ( \mathbf { B } - \mathbf { A } )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2006 Q7 [11]}}