| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find line of invariant points |
| 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 (leading to a simple eigenvalue problem), identifying rotation matrix P, computing matrix products, and applying the reflection property again. While it covers several concepts, each part follows routine procedures with clear signposting, making it slightly easier than average for Further Pure 1. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03q Inverse transformations |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{M}^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \mathbf{I}\) | B1 [1] | . |
| Answer | Marks |
|---|---|
| \(\mathbf{M}^2\) gives the identity because a reflection, followed by a second reflection in the same mirror line will get you back where you started OR reflection matrices are self-inverse. | E1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow 0.8x + 0.6y = x\) | M1 | Give both marks for either equation or for a correct geometrical argument |
| and \(0.6x - 0.8y = y\) | A1 | |
| Both of these lead to \(y = \frac{1}{3}x\) as the equation of the mirror line. | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Rotation, centre origin, \(36.9°\) anticlockwise. | B1, B1 [2] | One for rotation and centre, one for angle and sense. Accept \(323.1°\) clockwise or radian equivalents (\(0.644\) or \(5.64\)). |
| Answer | Marks |
|---|---|
| \(\mathbf{MP} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) | M1, A1 [2] |
| Answer | Marks |
|---|---|
| \(y = 0\) | B1 [1] |
# Question 9:
## Part 9(i):
$\mathbf{M}^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \mathbf{I}$ | B1 [1] | .
## Part 9(ii):
$\mathbf{M}^2$ gives the identity because a reflection, followed by a second reflection in the same mirror line will get you back where you started OR reflection matrices are self-inverse. | E1 [1] |
## Part 9(iii):
$\begin{pmatrix} 0.8 & 0.6 \\ 0.8 & -0.6 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}$
$\Rightarrow 0.8x + 0.6y = x$ | M1 | Give both marks for either equation or for a correct geometrical argument
and $0.6x - 0.8y = y$ | A1 |
Both of these lead to $y = \frac{1}{3}x$ as the equation of the mirror line. | A1 [3] |
## Part 9(iv):
Rotation, centre origin, $36.9°$ anticlockwise. | B1, B1 [2] | One for rotation and centre, one for angle and sense. Accept $323.1°$ clockwise or radian equivalents ($0.644$ or $5.64$).
## Part 9(v):
$\mathbf{MP} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ | M1, A1 [2] |
## Part 9(vi):
$y = 0$ | B1 [1] |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 ( $\mathbf { v }$ ) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?
\hfill \mbox{\textit{OCR MEI FP1 2005 Q9 [10]}}