| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | January |
| Marks | 10 |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find invariant lines through origin |
| Difficulty | Standard +0.8 This is a Further Maths linear algebra question requiring matrix multiplication, solving simultaneous equations from matrix equality, finding eigenvalues/eigenvectors for invariant lines, and understanding the distinction between invariant lines and fixed points. While systematic, it demands multiple techniques and conceptual understanding beyond standard A-level, placing it moderately above average difficulty. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03g Invariant points and lines4.03n Inverse 2x2 matrix |
$$\mathbf{M} = \begin{pmatrix} -2 & 5 \\ 6 & k \end{pmatrix}$$
where $k$ is a constant.
Given that
$$\mathbf{M}^2 + 11\mathbf{M} = a\mathbf{I}$$
where $a$ is a constant and $\mathbf{I}$ is the $2 \times 2$ identity matrix,
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item determine the value of $a$
\item show that $k = -9$ [3]
\end{enumerate}
\item Determine the equations of the invariant lines of the transformation represented by $\mathbf{M}$. [6]
\item State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer. [1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q6 [10]}}