| Exam Board | SPS |
|---|---|
| Module | SPS ASFM Mechanics (SPS ASFM Mechanics) |
| Year | 2021 |
| Session | May |
| Marks | 13 |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find line of invariant points |
| Difficulty | Challenging +1.2 This question requires understanding of matrix transformations including image points, determinants (area scale factor), and invariant points/lines. Part (a) involves setting up and solving a system with constraints including finding eigenvectors, which is conceptually demanding but follows standard procedures. Parts (b) and (c) are more routine applications of invariant point/line theory. The multi-step nature and need to synthesize several matrix concepts elevates this above average difficulty, but it remains within standard A-level Further Maths scope without requiring exceptional insight. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03g Invariant points and lines4.03i Determinant: area scale factor and orientation |
The $2 \times 2$ matrix $\mathbf{A}$ represents a transformation $T$ which has the following properties.
• The image of the point $(0, 1)$ is the point $(3, 4)$.
• An object shape whose area is $7$ is transformed to an image shape whose area is $35$.
• $T$ has a line of invariant points.
\begin{enumerate}[label=(\alph*)]
\item Find a possible matrix for $\mathbf{A}$. [8]
\end{enumerate}
The transformation $S$ is represented by the matrix $\mathbf{B}$ where $\mathbf{B} = \begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the equation of the line of invariant points of $S$. [2]
\item Show that any line of the form $y = x + c$ is an invariant line of $S$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS ASFM Mechanics 2021 Q3 [13]}}