| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Matrix powers and repeated transformations |
| Difficulty | Standard +0.3 This is a structured multi-part question on matrix transformations that guides students through standard techniques: computing M², recognizing that M²=I confirms a reflection, finding invariant points by solving Mx=x, identifying matrix P as a rotation from its form, computing matrix products, and applying the reflection composition property. While it requires multiple steps and understanding of transformation properties, each part follows predictable FP1 methods with no novel problem-solving required. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03g Invariant points and lines |
| Tuesday 7 JUNE 2005 | Afternoon | 1 hour 30 minutes | ||||
|
| Answer | Marks | Guidance |
|---|---|---|
| Multiply \(\mathbf{AB}\) and compare, element in position (2,1) gives \(\beta = 0\) | M1 | Show working from matrix multiplication |
| Correct demonstration that \(\beta = 0\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| From diagonal elements: \(\gamma = \) expression involving \(\alpha\) | M1 | |
| \(\gamma = 3\alpha^2 - 3\alpha + 5\) or correct expression | A1 | ft from their multiplication |
| Answer | Marks |
|---|---|
| When \(\alpha = 2\): \(\mathbf{A}^{-1} = \frac{1}{\gamma}\mathbf{B}\) with \(\gamma = 7\) | M1 |
| \(\mathbf{A}^{-1} = \frac{1}{7}\begin{pmatrix}5 & -8 & -1 \\ 5 & 1 & 2 \\ 5 & -5 & 5\end{pmatrix}\) | A1 |
| \(\mathbf{A}^{-1}\) does not exist when \(\gamma = 0\), state value of \(\alpha\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Write as \(\mathbf{Ax} = \mathbf{b}\), so \(\mathbf{x} = \mathbf{A}^{-1}\mathbf{b}\) | M1 | |
| \(\mathbf{x} = \frac{1}{7}\begin{pmatrix}5&-8&-1\\5&1&2\\5&-5&5\end{pmatrix}\begin{pmatrix}25\\11\\-23\end{pmatrix}\) | M1 | Correct multiplication |
| \(x = 6\), \(y = 8\), \(z = -1\) | A1 A1 A1 |
# Question 9:
**Part (i):**
| Multiply $\mathbf{AB}$ and compare, element in position (2,1) gives $\beta = 0$ | M1 | Show working from matrix multiplication |
| Correct demonstration that $\beta = 0$ | A1 | |
**Part (ii):**
| From diagonal elements: $\gamma = $ expression involving $\alpha$ | M1 | |
| $\gamma = 3\alpha^2 - 3\alpha + 5$ or correct expression | A1 | ft from their multiplication |
**Part (iii):**
| When $\alpha = 2$: $\mathbf{A}^{-1} = \frac{1}{\gamma}\mathbf{B}$ with $\gamma = 7$ | M1 | |
| $\mathbf{A}^{-1} = \frac{1}{7}\begin{pmatrix}5 & -8 & -1 \\ 5 & 1 & 2 \\ 5 & -5 & 5\end{pmatrix}$ | A1 | |
| $\mathbf{A}^{-1}$ does not exist when $\gamma = 0$, state value of $\alpha$ | B1 | |
**Part (iv):**
| Write as $\mathbf{Ax} = \mathbf{b}$, so $\mathbf{x} = \mathbf{A}^{-1}\mathbf{b}$ | M1 | |
| $\mathbf{x} = \frac{1}{7}\begin{pmatrix}5&-8&-1\\5&1&2\\5&-5&5\end{pmatrix}\begin{pmatrix}25\\11\\-23\end{pmatrix}$ | M1 | Correct multiplication |
| $x = 6$, $y = 8$, $z = -1$ | A1 A1 A1 | |9 You are given the matrix $\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)$.\\
(i) Calculate $\mathbf { M } ^ { 2 }$.
You are now given that the matrix $M$ represents a reflection in a line through the origin.\\
(ii) Explain how your answer to part (i) relates to this information.\\
(iii) By investigating the invariant points of the reflection, find the equation of the mirror line.\\
(iv) Describe fully the transformation represented by the matrix $\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)$.\\
(v) A composite transformation is formed by the transformation represented by $\mathbf { P }$ followed by the transformation represented by $\mathbf { M }$. Find the single matrix that represents this composite transformation.\\
(vi) The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?
\section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS}
\section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education}
\section*{MEI STRUCTURED MATHEMATICS \\
4755 \\
\textbackslash section*\{Further Concepts For Advanced Mathematics (FP1)\}}
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Tuesday 7 JUNE 2005 & Afternoon & 1 hour 30 minutes \\
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Additional materials: \\
Answer booklet \\
Graph paper \\
MEI Examination Formulae and Tables (MF2) \\
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TIME 1 hour 30 minutes
\begin{itemize}
\item Write your name, centre number and candidate number in the spaces provided on the answer booklet.
\item Answer all the questions.
\item You are permitted to use a graphical calculator in this paper.
\end{itemize}
\begin{itemize}
\item The number of marks is given in brackets [ ] at the end of each question or part question.
\item You are advised that an answer may receive no marks unless you show sufficient defail of the working to indicate that a correct method is being used.
\item Final answers should be given to a degree of accuracy appropriate to the context.
\item The total number of marks for this paper is 72.
\end{itemize}
\hfill \mbox{\textit{OCR MEI FP1 Q9}}