| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find line of invariant points |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing basic matrix transformations. Parts (a)-(c) involve recognizing standard transformations (rotation, reflection) and matrix multiplication—routine for F1 students. Part (d) asks for invariant points, which requires solving Rx = x for a line y=kx, a standard technique. All steps are procedural with no novel insight required, making this slightly easier than average even for Further Maths. |
| Spec | 4.03c Matrix multiplication: properties (associative, not commutative)4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
7.
$$\mathbf { P } = \left( \begin{array} { c c }
\frac { 5 } { 13 } & - \frac { 12 } { 13 } \\
\frac { 12 } { 13 } & \frac { 5 } { 13 }
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Describe fully the single geometrical transformation $U$ represented by the matrix $\mathbf { P }$.
The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a reflection in the line with equation $y = x$
\item Write down the matrix $\mathbf { Q }$.
Given that the transformation $V$ followed by the transformation $U$ is the transformation $T$, which is represented by the matrix $\mathbf { R }$,
\item find the matrix $\mathbf { R }$.
\item Show that there is a value of $k$ for which the transformation $T$ maps each point on the straight line $y = k x$ onto itself, and state the value of $k$.
\section*{II}
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2016 Q7 [10]}}