| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 8 |
| Topic | Matrices |
| Type | Non-singular matrix proof |
| Difficulty | Standard +0.3 This is a straightforward Further Maths linear transformations question requiring: (a) computing a determinant and showing it's always non-zero (simple algebra), (b) solving simultaneous equations from matrix multiplication, and (c) finding an eigenvalue condition. All parts use standard techniques with minimal problem-solving insight needed, making it slightly easier than average. |
| Spec | 4.03g Invariant points and lines4.03h Determinant 2x2: calculation4.03r Solve simultaneous equations: using inverse matrix |
$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that the matrix $\mathbf{A}$ is non-singular for all values of $k$. [2]
\end{enumerate}
A transformation $T : \mathbb{R}^2 \to \mathbb{R}^2$ is represented by the matrix $\mathbf{A}$.
The point $P$ has position vector $\begin{pmatrix} a \\ 2a \end{pmatrix}$ relative to an origin $O$.
The point $Q$ has position vector $\begin{pmatrix} 7 \\ -3 \end{pmatrix}$ relative to $O$.
Given that the point $P$ is mapped onto the point $Q$ under $T$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item determine the value of $a$ and the value of $k$. [3]
\end{enumerate}
Given that, for a different value of $k$, $T$ maps the line $y = 2x$ onto itself,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item determine this value of $k$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q6 [8]}}