| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Transformation mapping problems |
| Difficulty | Standard +0.8 This FP3 question involves multiple standard techniques (orthogonality check, eigenvalue/eigenvector computation, line transformation) but part (d) requires non-trivial geometric insight to parametrize the line, apply the matrix transformation, and eliminate parameters to find the Cartesian equation of the image—going beyond routine calculation to require problem-solving across different areas of further maths. |
| Spec | 4.03a Matrix language: terminology and notation4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation4.03j Determinant 3x3: calculation |
$$\mathbf{M} = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 4 & 1 \\ 0 & 5 & 0 \end{pmatrix}$$
\begin{enumerate}[label=(\alph*)]
\item Show that matrix $\mathbf{M}$ is not orthogonal. [2]
\item Using algebra, show that $1$ is an eigenvalue of $\mathbf{M}$ and find the other two eigenvalues of $\mathbf{M}$. [5]
\item Find an eigenvector of $\mathbf{M}$ which corresponds to the eigenvalue $1$ [2]
\end{enumerate}
The transformation $M : \mathbb{R}^3 \to \mathbb{R}^3$ is represented by the matrix $\mathbf{M}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find a cartesian equation of the image, under this transformation, of the line
$$x = \frac{y}{2} = \frac{z}{-1}$$
[4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 2014 Q2 [13]}}