| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find invariant lines through origin |
| Difficulty | Standard +0.8 This Further Maths question requires finding conditions on parameters from a determinant inequality (discriminant analysis), interpreting geometric significance, and finding invariant lines through eigenvalue analysis. While the individual techniques are standard FM topics, the multi-part structure combining algebraic manipulation, geometric interpretation, and eigenvalue work with a parameter makes it moderately challenging, above average difficulty but not requiring exceptional insight. |
| Spec | 4.03g Invariant points and lines4.03h Determinant 2x2: calculation4.03l Singular/non-singular matrices |
A linear transformation T of the $x$-$y$ plane has an associated matrix M, where $\mathbf{M} = \begin{pmatrix} \lambda & k \\ 1 & \lambda - k \end{pmatrix}$, and $\lambda$ and $k$ are real constants.
\begin{enumerate}[label=(\alph*)]
\item You are given that $\det \mathbf{M} > 0$ for all values of $\lambda$.
\begin{enumerate}[label=(\roman*)]
\item Find the range of possible values of $k$. [3]
\item What is the significance of the condition $\det \mathbf{M} > 0$ for the transformation T? [1]
\end{enumerate}
\end{enumerate}
For the remainder of this question, take $k = -2$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine whether there are any lines through the origin that are invariant lines for the transformation T. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q12 [8]}}