| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Solving matrix equations for unknown matrix |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing basic matrix inverse properties. Part (i) requires the standard 2×2 inverse formula (determinant and cofactor method), while part (ii) applies the rule (AB)^{-1} = B^{-1}A^{-1}. Both are direct applications of learned techniques with no problem-solving required, making it slightly easier than average even for Further Maths content. |
| Spec | 4.03n Inverse 2x2 matrix4.03p Inverse properties: (AB)^(-1) = B^(-1)*A^(-1) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{1}{5}\begin{pmatrix} 5 & -1 \\ -3 & 1 \end{pmatrix}\) | B1 B1 | Transpose leading diagonal and negate other diagonal or solve sim. eqns. to get 1st column. Divide by the determinant or solve 2nd pair to get 2nd column. |
| (ii) | M1 M1(indep) A1ft A1ft | Attempt to use \(B^{-1}A^{-1}\) or find \(B\). Attempt at matrix multiplication. One element correct, a.e.f. All elements correct, a.e.f. NB ft consistent with their (i). |
| \(\frac{1}{2}\begin{pmatrix} 2 & 0 \\ 23 & -5 \end{pmatrix}\) |
(i) $\frac{1}{5}\begin{pmatrix} 5 & -1 \\ -3 & 1 \end{pmatrix}$ | B1 B1 | Transpose leading diagonal and negate other diagonal or solve sim. eqns. to get 1st column. Divide by the determinant or solve 2nd pair to get 2nd column. |
(ii) | M1 M1(indep) A1ft A1ft | Attempt to use $B^{-1}A^{-1}$ or find $B$. Attempt at matrix multiplication. One element correct, a.e.f. All elements correct, a.e.f. NB ft consistent with their (i). |
$\frac{1}{2}\begin{pmatrix} 2 & 0 \\ 23 & -5 \end{pmatrix}$ | | |
**Total: 2 + 4 = 6 marks**4 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 1 & 1 \\ 3 & 5 \end{array} \right)$.\\
(i) Find $\mathbf { A } ^ { - 1 }$.
The matrix $\mathbf { B } ^ { - 1 }$ is given by $\mathbf { B } ^ { - 1 } = \left( \begin{array} { r r } 1 & 1 \\ 4 & - 1 \end{array} \right)$.\\
(ii) Find $( \mathbf { A B } ) ^ { - 1 }$.
\hfill \mbox{\textit{OCR FP1 2007 Q4 [6]}}