| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Use Cayley-Hamilton for inverse |
| Difficulty | Challenging +1.3 This is a structured Further Maths linear algebra question requiring matrix arithmetic, finding inverses via characteristic equations, and interpreting eigenvalues/eigenvectors geometrically. Part (a) involves routine computation and algebraic manipulation. Part (b) requires understanding that a stretch has one eigenvalue k and another eigenvalue 1, then finding the eigenvector for eigenvalue 1. While it requires multiple techniques and geometric insight about transformations, the question provides significant scaffolding and the calculations are straightforward once the approach is identified. Slightly above average difficulty for Further Maths. |
| Spec | 4.03a Matrix language: terminology and notation4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
Let $\mathbf{A}$ be the matrix $\begin{pmatrix} 17 & 12 \\ 12 & 10 \end{pmatrix}$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Determine the integer $n$ for which $27\mathbf{A} - \mathbf{A}^2 = n\mathbf{I}$, where $\mathbf{I}$ is the $2 \times 2$ identity matrix. [2]
\item Hence find $\mathbf{A}^{-1}$ in the form $p\mathbf{A} + q\mathbf{I}$ for rational numbers $p$ and $q$. [2]
\end{enumerate}
\item The plane transformation $T$ is defined by $T: \begin{pmatrix} x \\ y \end{pmatrix} \mapsto \mathbf{A} \begin{pmatrix} x \\ y \end{pmatrix}$. It is given that $T$ is a stretch, with scale factor $k$, parallel to the line $y = mx$, where $m > 0$.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $k$. [2]
\item By considering $\mathbf{A} \begin{pmatrix} x \\ mx \end{pmatrix}$, or otherwise, determine the value of $m$. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2018 Q11 [10]}}