| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Inverse given/derived then solve system |
| Difficulty | Standard +0.8 This is a Further Maths FP1 question involving matrix multiplication, inverse matrices, and solving simultaneous equations. While it requires multiple steps and understanding of the relationship AB = (k-n)I, the techniques are standard for FP1: multiply matrices to find n, recognize that B is a scalar multiple of A^(-1), and apply matrix methods to solve equations. The algebraic manipulation is moderate but systematic rather than requiring novel insight. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix |
You are given that $\mathbf{A} = \begin{pmatrix} 1 & -2 & k \\ 2 & 1 & 2 \\ 3 & 2 & -1 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} -5 & -2+2k & -4-k \\ 8 & -1-3k & -2+2k \\ 1 & -8 & 5 \end{pmatrix}$ and that $\mathbf{AB}$ is of the form $\mathbf{AB} = \begin{pmatrix} k-n & 0 & 0 \\ 0 & k-n & 0 \\ 0 & 0 & k-n \end{pmatrix}$.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $n$. [2]
\item Write down the inverse matrix $\mathbf{A}^{-1}$ and state the condition on $k$ for this inverse to exist. [4]
\item Using the result from part (ii), or otherwise, solve the following simultaneous equations.
\begin{align}
x - 2y + z &= 1 \\
2x + y + 2z &= 12 \\
3x + 2y - z &= 3
\end{align} [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2007 Q10 [11]}}