| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Verify invariant line property |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on linear transformations and eigenvectors. Part (i) requires basic matrix multiplication to transform vertices of the unit square. Part (ii) guides students through finding an eigenvector by setting up the eigenvalue equation with k as unknown, then interpreting geometrically as an invariant line. The scaffolding makes this accessible, requiring only routine algebraic manipulation and understanding of invariant lines, with no novel problem-solving insight needed. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03g Invariant points and lines |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (i) | parallelogram vertices (0, 0), (2, 1), (5, 5), (3, 4) |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | 3 correct coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (ii) | (A) |
| Answer | Marks |
|---|---|
| Obtain k(cid:32) 1 twice | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.1a | |
| 1.1 | n | |
| 3 | (ii) | (B) |
| [1] | 2.2a | e |
| 3 | (ii) | (C) |
| [1] | m |
Question 3:
3 | (i) | parallelogram vertices (0, 0), (2, 1), (5, 5), (3, 4) | M1
A1
[2] | 1.1
1.1 | 3 correct coordinates
calculated or drawn
all correct
3 | (ii) | (A) | (cid:167)2 3(cid:183)(cid:167) x (cid:183) (cid:167) x (cid:183)
Solve (cid:168) (cid:184)(cid:168) (cid:184) (cid:32)5 (cid:168) (cid:184)
(cid:169)1 4(cid:185)(cid:169)kx(cid:185) (cid:169)kx(cid:185)
Obtain k(cid:32) 1 twice | M1
A1
[2] | 1.1a
1.1 | n
3 | (ii) | (B) | Hence y(cid:32)x is an invariant line | E1
[1] | 2.2a | e
3 | (ii) | (C) | Drawy(cid:32)x. | B1
[1] | m
1.1Transformation M is represented by matrix $\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$.
\begin{enumerate}[label=(\roman*)]
\item On the diagram in the Printed Answer Booklet draw the image of the unit square under M. [2]
\item \begin{enumerate}[label=(\Alph*)]
\item Show that there is a constant $k$ such that $\mathbf{M} \begin{pmatrix} x \\ kx \end{pmatrix} = 5 \begin{pmatrix} x \\ kx \end{pmatrix}$ for all $x$. [2]
\item Hence find the equation of an invariant line under M. [1]
\item Draw the invariant line from part (ii) (B) on your diagram for part (i). [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core Q3 [6]}}