| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | January |
| Marks | 8 |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Verify invariant line property |
| Difficulty | Standard +0.3 This is a straightforward Further Maths linear transformations question requiring standard techniques: finding an invariant line (eigenvalue problem in disguise), computing a determinant to show non-singularity, and finding a 2×2 matrix inverse. All parts are routine applications of formulas with minimal problem-solving required, making it slightly easier than average. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03h Determinant 2x2: calculation4.03n Inverse 2x2 matrix |
$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix},$$ where $k$ is constant.
A transformation $T : \mathbb{R}^2 \to \mathbb{R}^2$ is represented by the matrix $\mathbf{A}$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $k$ for which the line $y = 2x$ is mapped onto itself under $T$. [3]
\item Show that $\mathbf{A}$ is non-singular for all values of $k$. [3]
\item Find $\mathbf{A}^{-1}$ in terms of $k$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q3 [8]}}