| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Verify invariant line property |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing basic matrix properties and invariant lines. Part (a) requires computing a determinant and showing it's non-zero, part (b) involves solving simultaneous equations from matrix multiplication, and part (c) requires understanding that invariant lines correspond to eigenvectors. All parts use standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.03h Determinant 2x2: calculation4.03l Singular/non-singular matrices4.03n Inverse 2x2 matrix |
9.
$$\mathbf { A } = \left( \begin{array} { c c }
k & - 2 \\
1 - k & k
\end{array} \right) \quad \text { where } k \text { is a constant }$$
\begin{enumerate}[label=(\alph*)]
\item Show that the matrix $\mathbf { A }$ is non-singular for all values of $k$.
A transformation $T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ is represented by the matrix $\mathbf { A }$.\\
The point $P$ has position vector $\binom { a } { 2 a }$ relative to an origin $O$.\\
The point $Q$ has position vector $\binom { 7 } { - 3 }$ relative to $O$.\\
Given that the point $P$ is mapped onto the point $Q$ under $T$,
\item determine the value of $a$ and the value of $k$.
Given that, for a different value of $k , T$ maps the line $y = 2 x$ onto itself,
\item determine this value of $k$.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q9 [8]}}