| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Topic | 3x3 Matrices |
| Type | Inverse given/derived then solve system |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing standard matrix multiplication, recognition of inverse matrices, and solving simultaneous equations using matrices. All parts are routine applications of techniques with no novel insight required. Part (a) is mechanical computation (1 mark), part (b) requires recognizing that AB=I means B=A^{-1} (1 mark), part (c)(i) asks for standard theory about determinants/invertibility (2 marks), and part (c)(ii) is direct application of the inverse matrix already found (2 marks). While this is Further Maths content, the question is entirely procedural and below average difficulty even for FP1 standards. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix |
You are given that $\mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & -2 & -1 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 1 & 0 & 1 \\ -8 & 4 & 0 \\ 19 & -8 & -1 \end{pmatrix}$.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathbf{AB}$. [1]
\item Hence write down $\mathbf{A}^{-1}$. [1]
\item You are given three simultaneous equations
$$x + 2y + z = 0$$
$$2x + 5y + 2z = 1$$
$$3x - 2y - z = 4$$
\begin{enumerate}[label=(\roman*)]
\item Explain how you can tell, without solving them, that there is a unique solution to these equations. [2]
\item Find this unique solution. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q3 [6]}}