| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Find invariant lines through origin |
| Difficulty | Standard +0.3 This is a standard FP1 linear transformations question testing routine techniques: finding invariant lines (substitution), computing determinants, finding matrix inverses, and applying inverse transformations. All parts follow textbook methods with no novel insight required, making it slightly easier than average even for Further Maths. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation4.03n Inverse 2x2 matrix |
$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}, \text{ where } k \text{ is constant.}$$
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}
A point $P$ is mapped onto a point $Q$ under $T$.
The point $Q$ has position vector $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$ relative to an origin $O$.
Given that $k = 3$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item find the position vector of $P$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q47 [11]}}