| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Prove eigenvalue/eigenvector properties |
| Difficulty | Moderate -0.3 This question tests basic matrix operations (invariant points, matrix inversion) and a simple algebraic proof. Part (i) involves routine calculations: multiplying a matrix by a vector (1 mark), finding a 2×2 inverse using the standard formula (2 marks), and verification (1 mark). Part (ii) requires manipulating the equation T·v = v by multiplying both sides by T^{-1}, which is straightforward algebraic reasoning worth only 2 marks. While this is Further Maths content, the techniques are mechanical and the proof requires minimal insight—just applying T^{-1} to both sides of an equation. This is slightly easier than an average A-level question due to its routine nature and low conceptual demand. |
| Spec | 4.01a Mathematical induction: construct proofs4.03g Invariant points and lines4.03o Inverse 3x3 matrix |
\begin{enumerate}[label=(\roman*)]
\item The matrix $\mathbf{S} = \begin{pmatrix} -1 & 2 \\ -3 & 4 \end{pmatrix}$ represents a transformation.
\begin{enumerate}[label=(\Alph*)]
\item Show that the point $(1, 1)$ is invariant under this transformation. [1]
\item Calculate $\mathbf{S}^{-1}$. [2]
\item Verify that $(1, 1)$ is also invariant under the transformation represented by $\mathbf{S}^{-1}$. [1]
\end{enumerate}
\item Part (i) may be generalised as follows.
If $(x, y)$ is an invariant point under a transformation represented by the non-singular matrix $\mathbf{T}$, it is also invariant under the transformation represented by $\mathbf{T}^{-1}$.
Starting with $\mathbf{T}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}$, or otherwise, prove this result. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2006 Q5 [6]}}