| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 8 |
| Topic | Invariant lines and eigenvalues and vectors |
| Difficulty | Challenging +1.2 This is a Further Maths linear transformations question requiring eigenvalue analysis and geometric interpretation. Part (a) involves finding when there are no real eigenvalues (discriminant < 0), which is a standard technique. Part (b) uses the determinant condition (det = -5) to find λ, then finds eigenvectors. While it requires multiple concepts (eigenvalues, determinant, orientation), these are routine Further Maths procedures without requiring novel insight. |
| Spec | 4.03g Invariant points and lines4.03i Determinant: area scale factor and orientation |
A linear transformation of the plane is represented by the matrix $\mathbf{M} = \begin{pmatrix} 1 & -2 \\ \lambda & 3 \end{pmatrix}$, where $\lambda$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find the set of values of $\lambda$ for which the linear transformation has no invariant lines through the origin. [5]
\item Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q8 [8]}}