| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Solving linear systems using matrices |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question testing basic matrix concepts: finding when a 2×2 matrix is singular (det=0) and using matrix inversion to solve simultaneous equations. Both parts require only direct application of standard formulas with minimal problem-solving, making it slightly easier than an average A-level question despite being from FP1. |
| Spec | 4.03l Singular/non-singular matrices4.03n Inverse 2x2 matrix4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(\det \mathbf{A} = 0\) | M1 | |
| \(a = -6\) | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\mathbf{A}^{-1} = \frac{1}{a+6}\begin{pmatrix} 1 & -3 \\ 2 & a \end{pmatrix}\) | B1 | Both diagonals correct |
| B1ft | Divide by \(\det \mathbf{A}\) | |
| Premultiply column by \(\mathbf{A}^{-1}\), no other method | M1 | |
| \(x = \frac{4}{a+6}\), \(y = \frac{2-a}{a+6}\) | A1ft | Obtain correct answers from their \(\mathbf{A}^{-1}\) |
| A1ft | 5 marks |
# Question 7:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\det \mathbf{A} = 0$ | M1 | |
| $a = -6$ | A1 | **2 marks** |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{A}^{-1} = \frac{1}{a+6}\begin{pmatrix} 1 & -3 \\ 2 & a \end{pmatrix}$ | B1 | Both diagonals correct |
| | B1ft | Divide by $\det \mathbf{A}$ |
| Premultiply column by $\mathbf{A}^{-1}$, no other method | M1 | |
| $x = \frac{4}{a+6}$, $y = \frac{2-a}{a+6}$ | A1ft | Obtain correct answers from their $\mathbf{A}^{-1}$ |
| | A1ft | **5 marks** |
---7 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { c c } a & 3 \\ - 2 & 1 \end{array} \right)$.\\
(i) Given that $\mathbf { A }$ is singular, find $a$.\\
(ii) Given instead that $\mathbf { A }$ is non-singular, find $\mathbf { A } ^ { - 1 }$ and hence solve the simultaneous equations
$$\begin{aligned}
a x + 3 y & = 1 \\
- 2 x + y & = - 1
\end{aligned}$$
\hfill \mbox{\textit{OCR FP1 2008 Q7 [7]}}