| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find eigenvectors given eigenvalue |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing basic matrix operations and eigenvalue concepts. Part (a) requires routine matrix multiplication to verify associativity, part (b) is a simple demonstration that AB ≠ BA, and part (c) involves solving a standard eigenvalue equation. All parts use direct application of learned techniques with no novel problem-solving required, making it slightly easier than average even for Further Maths. |
| Spec | 1.01c Disproof by counter example4.03c Matrix multiplication: properties (associative, not commutative)4.03l Singular/non-singular matrices |
Three matrices, A, B and C, are given by $\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}$, $\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}$ and $\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}$ where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
\item Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]
\end{enumerate}
For a certain value of $a$, $\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find
\begin{itemize}
\item $y$ in terms of $x$,
\item the value of $a$. [3]
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q3 [9]}}