| Exam Board | OCR MEI |
|---|---|
| Module | Further Extra Pure (Further Extra Pure) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find eigenvectors given eigenvalue |
| Difficulty | Moderate -0.3 This question tests standard eigenvalue/eigenvector theory applied to reflections. Part (a) requires recall that reflections have eigenvalues ±1 (1 mark suggests this is given knowledge). Parts (b-c) involve routine calculation of eigenvectors and interpreting the eigenspace geometrically. While it's Further Maths content, the execution is mechanical with no problem-solving insight required, making it slightly easier than an average A-level question overall. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03q Inverse transformations |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | 1 and 1 |
| [1] | 1.2 | Both values correct |
| 1 | (b) | æ 0.6 0.8 öæ x öé æ 0.6x+0.8y öù æ x ö |
| Answer | Marks |
|---|---|
| è 1 ø è -2 ø | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1a |
| Answer | Marks |
|---|---|
| 1.1 | Correct matrix equation for either |
| Answer | Marks | Guidance |
|---|---|---|
| multiples) | soi | |
| 1 | (c) | y= 1x oe |
| 2 | B1 | |
| [1] | 1.1 | |
| 1 | 2 | 1 |
Question 1:
1 | (a) | 1 and 1 | B1
[1] | 1.2 | Both values correct
1 | (b) | æ 0.6 0.8 öæ x öé æ 0.6x+0.8y öù æ x ö
ê=ç ÷ú=±ç
ç ÷ç ÷ ÷
è 0.8 -0.6 øè y øê è ç 0.8x-0.6y ø ÷ú è y ø
ë û
2y = x or y = 2x
æ ö æ ö
2 1
so eigenvectors are ç ÷ and ç ÷ oe
è 1 ø è -2 ø | M1
M1
A1
[3] | 1.1a
1.1
1.1 | Correct matrix equation for either
eigenvalue
Attempt at either equation
correctly deduced
Both correct (any non-zero
multiples) | soi
1 | (c) | y= 1x oe
2 | B1
[1] | 1.1
1 | 2 | 1 | 4 | 3The matrix A is $\begin{pmatrix} 0.6 & 0.8 \\ 0.8 & -0.6 \end{pmatrix}$
\begin{enumerate}[label=(\alph*)]
\item Given that A represents a reflection, write down the eigenvalues of A. [1]
\item Hence find the eigenvectors of A. [3]
\item Write down the equation of the mirror line of the reflection represented by A. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Extra Pure 2019 Q1 [5]}}