| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Solving linear systems using matrices |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths question testing basic matrix inversion (2×2 formula) and solving linear systems. While it's Further Maths content, the techniques are routine and mechanical with no problem-solving required—students simply apply the standard 2×2 inverse formula then multiply matrices. It's easier than average even for Further Maths students. |
| Spec | 4.03n Inverse 2x2 matrix4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{A}^{-1} = \frac{1}{5}\begin{pmatrix} 2 & -3 \\ -1 & 4 \end{pmatrix}\) | M1, A1 | M1 for dividing by determinant |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{5}\begin{pmatrix} 2 & -3 \\ -1 & 4 \end{pmatrix}\begin{pmatrix} 5 \\ -4 \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{5}\begin{pmatrix} 22 \\ -21 \end{pmatrix}\) | M1 | Pre-multiplying by their inverse |
| \(\Rightarrow x = \frac{22}{5},\ y = \frac{-21}{5}\) | A1(ft), A1(ft) [5] | Follow through use of their inverse. No marks for solving without using inverse matrix |
## Question 1:
### Part (i)
$\mathbf{A}^{-1} = \frac{1}{5}\begin{pmatrix} 2 & -3 \\ -1 & 4 \end{pmatrix}$ | M1, A1 | M1 for dividing by determinant
### Part (ii)
$\frac{1}{5}\begin{pmatrix} 2 & -3 \\ -1 & 4 \end{pmatrix}\begin{pmatrix} 5 \\ -4 \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{5}\begin{pmatrix} 22 \\ -21 \end{pmatrix}$ | M1 | Pre-multiplying by their inverse
$\Rightarrow x = \frac{22}{5},\ y = \frac{-21}{5}$ | A1(ft), A1(ft) **[5]** | Follow through use of their inverse. No marks for solving without using inverse matrix
---1 (i) Find the inverse of the matrix $\mathbf { A } = \left( \begin{array} { l l } 4 & 3 \\ 1 & 2 \end{array} \right)$.\\
(ii) Use this inverse to solve the simultaneous equations
$$\begin{aligned}
4 x + 3 y & = 5 \\
x + 2 y & = - 4
\end{aligned}$$
showing your working clearly.
\hfill \mbox{\textit{OCR MEI FP1 2005 Q1 [5]}}