| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Describe reflection from matrix |
| Difficulty | Easy -1.2 This is a straightforward Further Maths question testing basic matrix operations and recognition of a standard transformation. Part (a) is routine matrix multiplication, part (b) requires recognizing reflection in the y-axis from a standard matrix form, and part (c) follows immediately from understanding the transformation. While it's from FP1, the question requires only direct recall and simple computation with no problem-solving or insight needed. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{AB} = \begin{pmatrix}2(-3)+0(5) & 2(-1)+0(2)\\ 5(-3)+3(5) & 5(-1)+3(2)\end{pmatrix}\) | M1 | Correct method to multiply two matrices; implied by two out of four correct elements |
| \(= \begin{pmatrix}-6 & -2\\ 0 & 1\end{pmatrix}\) | A1 | Any three elements correct |
| A1 | Correct answer; correct answer only 3/3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Reflection; about the \(y\)-axis | M1 | Reflection |
| A1 | \(y\)-axis (or \(x=0\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{C}^{100} = \mathbf{I} = \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\) | B1 | \(\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\) or \(\mathbf{I}\) |
## Question 2:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{AB} = \begin{pmatrix}2(-3)+0(5) & 2(-1)+0(2)\\ 5(-3)+3(5) & 5(-1)+3(2)\end{pmatrix}$ | M1 | Correct method to multiply two matrices; implied by two out of four correct elements |
| $= \begin{pmatrix}-6 & -2\\ 0 & 1\end{pmatrix}$ | A1 | Any three elements correct |
| | A1 | Correct answer; correct answer only 3/3 |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Reflection; about the $y$-axis | M1 | Reflection |
| | A1 | $y$-axis (or $x=0$) |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{C}^{100} = \mathbf{I} = \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$ | B1 | $\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$ or $\mathbf{I}$ |
---2.
$$\mathbf { A } = \left( \begin{array} { l l }
2 & 0 \\
5 & 3
\end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r }
- 3 & - 1 \\
5 & 2
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Find $\mathbf { A B }$.
Given that
$$\mathbf { C } = \left( \begin{array} { r r }
- 1 & 0 \\
0 & 1
\end{array} \right)$$
\item describe fully the geometrical transformation represented by $\mathbf { C }$,
\item write down $\mathbf { C } ^ { 100 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-03_99_97_2631_1784}
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2011 Q2 [6]}}