| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Find inverse then solve system |
| Difficulty | Standard +0.3 This is a straightforward Further Pure 1 question requiring standard matrix inversion of a 3×3 matrix with a parameter, followed by routine application to solve a linear system. While 3×3 matrix inversion involves more computation than 2×2, the technique is mechanical (cofactor method or row reduction), and the system-solving is direct matrix multiplication. The parameter 'a' adds minimal complexity since it's carried through symbolically. This is slightly easier than average even for FP1 standards, as it's a textbook application with no conceptual challenges. |
| Spec | 4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\Delta = \det \mathbf{D} = 3a - 6\) | M1 | Show correct expansion process for 3 x 3 |
| M1 | Correct evaluation of any 2 x 2 det | |
| A1 | Obtain correct answer | |
| \(\mathbf{D}^{-1} = \frac{1}{3}\begin{pmatrix} 3 & -2 & 4 \\ -3 & a & -2a \\ -3 & a & a - 6 \end{pmatrix}\) | A1 | Show correct process for adjoint entries |
| B1 | entries | |
| A1 | Obtain at least 4 correct entries in adjoint | |
| (ii) \(\frac{1}{\Delta}\begin{pmatrix} 5 \\ 2a - 9 \\ 5a - 15 \end{pmatrix}\) | M1 | Divide by their determinant |
| A1 A1 A1 A1 (ft all 3) | Obtain completely correct answer | (4 marks, 11 total) |
(i) $\Delta = \det \mathbf{D} = 3a - 6$ | M1 | Show correct expansion process for 3 x 3
| M1 | Correct evaluation of any 2 x 2 det
| A1 | Obtain correct answer
$\mathbf{D}^{-1} = \frac{1}{3}\begin{pmatrix} 3 & -2 & 4 \\ -3 & a & -2a \\ -3 & a & a - 6 \end{pmatrix}$ | A1 | Show correct process for adjoint entries
| B1 | entries
| A1 | Obtain at least 4 correct entries in adjoint
(ii) $\frac{1}{\Delta}\begin{pmatrix} 5 \\ 2a - 9 \\ 5a - 15 \end{pmatrix}$ | M1 | Divide by their determinant
| A1 A1 A1 A1 (ft all 3) | Obtain completely correct answer | (4 marks, 11 total)10 The matrix $\mathbf { D }$ is given by $\mathbf { D } = \left( \begin{array} { r r r } a & 2 & 0 \\ 3 & 1 & 2 \\ 0 & - 1 & 1 \end{array} \right)$, where $a \neq 2$.\\
(i) Find $\mathbf { D } ^ { - 1 }$.\\
(ii) Hence, or otherwise, solve the equations
$$\begin{aligned}
a x + 2 y & = 3 \\
3 x + y + 2 z & = 4 \\
- y + z & = 1
\end{aligned}$$
\hfill \mbox{\textit{OCR FP1 2007 Q10 [11]}}