| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Matrix satisfying given equation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring matrix inversion using the standard formula, then solving a matrix equation by equating entries. The algebra is routine and the techniques are standard FP1 content, making it slightly easier than average overall but typical for Further Maths. |
| Spec | 4.03a Matrix language: terminology and notation4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Determinant is \(a - 2\) | M1 | Attempt \(ad - bc\) |
| \(\mathbf{X}^{-1} = \dfrac{1}{a-2}\begin{pmatrix}-1 & -a \\ 1 & 2\end{pmatrix}\) | M1 A1 (3) | \(\dfrac{1}{\det}\begin{pmatrix}-1 & -a \\ 1 & 2\end{pmatrix}\) for second M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{I} = \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}\) | B1 | Must identify I |
| Attempt to solve \(2 - \dfrac{1}{a-2} = 1\), or \(a - \dfrac{a}{a-2} = 0\), or \(-1 + \dfrac{1}{a-2} = 0\), or \(-1 + \dfrac{2}{a-2} = 1\) | M1 | |
| \(a = 3\) only | A1 cso (3) [6] | Final A1 for correct solution only |
## Question 7:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Determinant is $a - 2$ | M1 | Attempt $ad - bc$ |
| $\mathbf{X}^{-1} = \dfrac{1}{a-2}\begin{pmatrix}-1 & -a \\ 1 & 2\end{pmatrix}$ | M1 A1 (3) | $\dfrac{1}{\det}\begin{pmatrix}-1 & -a \\ 1 & 2\end{pmatrix}$ for second M1 |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{I} = \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$ | B1 | Must identify **I** |
| Attempt to solve $2 - \dfrac{1}{a-2} = 1$, or $a - \dfrac{a}{a-2} = 0$, or $-1 + \dfrac{1}{a-2} = 0$, or $-1 + \dfrac{2}{a-2} = 1$ | M1 | |
| $a = 3$ only | A1 cso (3) [6] | Final A1 for correct solution only |
---7. Given that $\mathbf { X } = \left( \begin{array} { c c } 2 & a \\ - 1 & - 1 \end{array} \right)$, where $a$ is a constant, and $a \neq 2$,
\begin{enumerate}[label=(\alph*)]
\item find $\mathbf { X } ^ { - 1 }$ in terms of $a$.
Given that $\mathbf { X } + \mathbf { X } ^ { - 1 } = \mathbf { I }$, where $\mathbf { I }$ is the $2 \times 2$ identity matrix,
\item find the value of $a$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2009 Q7 [6]}}