| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Combined transformation matrix product |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths question testing basic matrix operations: writing down standard transformation matrices from memory (parts a-b), multiplying two 2×2 matrices (parts c-d), and solving simultaneous equations by equating matrix entries (part e). All steps are routine with no problem-solving or insight required beyond direct application of learned procedures. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\begin{pmatrix} 8 & 0 \\ 0 & 8 \end{pmatrix}\) | B1 | (1 mark) |
| (b) \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) | B1 | (1 mark) |
| (c) \(\mathbf{T} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 8 & 0 \\ 0 & 8 \end{pmatrix} = \begin{pmatrix} 8 & 0 \\ 0 & -8 \end{pmatrix}\) | M1 A1 | M1: Accept multiplication of their matrices either way round (this can be implied by correct answer). A1: cao |
| (d) \(\mathbf{AB} = \begin{pmatrix} 6 & 1 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} k & 1 \\ c & -6 \end{pmatrix} = \begin{pmatrix} 6k + c & 0 \\ 4k + 2c & -8 \end{pmatrix}\) | M1 A1 A1 | M1: Correct matrix multiplication method implied by one or two correct terms in correct positions. A1: for three correct terms in correct positions. Second A1: for all four terms correct and simplified |
| (e) "\(6k + c = 8\)" and "\(4k + 2c = 0\)" | M1, A1 | Form equations and solve simultaneously |
| \(k = 2\) and \(c = -4\) | (2 marks) |
**(a)** $\begin{pmatrix} 8 & 0 \\ 0 & 8 \end{pmatrix}$ | **B1** | (1 mark)
**(b)** $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ | **B1** | (1 mark)
**(c)** $\mathbf{T} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 8 & 0 \\ 0 & 8 \end{pmatrix} = \begin{pmatrix} 8 & 0 \\ 0 & -8 \end{pmatrix}$ | **M1 A1** | M1: Accept multiplication of their matrices either way round (this can be implied by correct answer). A1: cao | (2 marks)
**(d)** $\mathbf{AB} = \begin{pmatrix} 6 & 1 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} k & 1 \\ c & -6 \end{pmatrix} = \begin{pmatrix} 6k + c & 0 \\ 4k + 2c & -8 \end{pmatrix}$ | **M1 A1 A1** | M1: Correct matrix multiplication method implied by one or two correct terms in correct positions. A1: for three correct terms in correct positions. Second A1: for all four terms correct and simplified | (3 marks)
**(e)** "$6k + c = 8$" and "$4k + 2c = 0$" | **M1, A1** | Form equations and solve simultaneously |
$k = 2$ and $c = -4$ | | (2 marks)
**Total: 9 marks**
**Alternative method for (e):** M1: $\mathbf{AB} = \mathbf{T} \Rightarrow \mathbf{B} = \mathbf{A}^{-1}\mathbf{T}$ and compare elements to find $k$ and $c$. Then A1 as before.
**Guidance:**
(c) M1: Accept multiplication of their matrices either way round (this can be implied by correct answer). A1: cao
(d) M1: Correct matrix multiplication method implied by one or two correct terms in correct positions. A1: for three correct terms in correct positions. Second A1: for all four terms correct and simplified
(e) M1: follows their previous work but must give two equations from which $k$ and $c$ can be found and there must be attempt at solution getting to $k = $ or $c = $. A1: is cao (but not cso – may follow error in position of $4k + 2c$ earlier).
---6. Write down the $2 \times 2$ matrix that represents
\begin{enumerate}[label=(\alph*)]
\item an enlargement with centre $( 0,0 )$ and scale factor 8 ,
\item a reflection in the $x$-axis.
Hence, or otherwise,
\item find the matrix $\mathbf { T }$ that represents an enlargement with centre ( 0,0 ) and scale factor 8, followed by a reflection in the $x$-axis.
$$\mathbf { A } = \left( \begin{array} { l l }
6 & 1 \\
4 & 2
\end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r }
k & 1 \\
c & - 6
\end{array} \right) , \text { where } k \text { and } c \text { are constants. }$$
\item Find $\mathbf { A B }$.
Given that $\mathbf { A B }$ represents the same transformation as $\mathbf { T }$,
\item find the value of $k$ and the value of $c$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2010 Q6 [9]}}