| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Find inverse then solve system |
| Difficulty | Standard +0.3 This is a standard FP1 matrix question requiring determinant calculation for singularity, then finding the inverse using cofactors/adjugate method, and finally solving a system. While it involves multiple steps and 3×3 matrices (making it harder than basic A-level), these are routine algorithmic procedures taught directly in FP1 with no novel insight required, placing it slightly above average difficulty. |
| Spec | 4.03j Determinant 3x3: calculation4.03l Singular/non-singular matrices4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| M1 | Show correct expansion process for \(3 \times 3\) | |
| M1 | Correct evaluation of any \(2 \times 2\) | |
| \(a + 3\) | A1 | Obtain correct answer |
| M1 | Use \(\det \mathbf{A} = 0\) | |
| \(a = -3\) | A1FT | Obtain correct answer from their det A |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{a+3}\begin{pmatrix} 1 & -1 & 1 \\ 7 & a-4 & 1-2a \\ -11 & 8-a & 3a-2 \end{pmatrix}\) | M1 | Show correct processes for adjoint entries |
| A1 | Obtain at least 4 correct entries in adjoint | |
| A1 | Obtain completely correct adjoint | |
| B1 | Divide adjoint by their det A | |
| \(\frac{1}{a+3}\begin{pmatrix} 2 \\ 2-4a \\ 7a-1 \end{pmatrix}\) | M1 | Pre-multiply column matrix by their \(\mathbf{A}^{-1}\) |
| A2 | Obtain correct answer, A1 for 1 element correct | |
| [7] |
## Question 10(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| | M1 | Show correct expansion process for $3 \times 3$ |
| | M1 | Correct evaluation of any $2 \times 2$ |
| $a + 3$ | A1 | Obtain correct answer |
| | M1 | Use $\det \mathbf{A} = 0$ |
| $a = -3$ | A1FT | Obtain correct answer from their det **A** |
| **[5]** | | |
---
## Question 10(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{a+3}\begin{pmatrix} 1 & -1 & 1 \\ 7 & a-4 & 1-2a \\ -11 & 8-a & 3a-2 \end{pmatrix}$ | M1 | Show correct processes for adjoint entries |
| | A1 | Obtain at least 4 correct entries in adjoint |
| | A1 | Obtain completely correct adjoint |
| | B1 | Divide adjoint by their det **A** |
| $\frac{1}{a+3}\begin{pmatrix} 2 \\ 2-4a \\ 7a-1 \end{pmatrix}$ | M1 | Pre-multiply column matrix by their $\mathbf{A}^{-1}$ |
| | A2 | Obtain correct answer, A1 for 1 element correct |
| **[7]** | | |10 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 1 \\ 1 & 3 & 2 \\ 4 & 1 & 1 \end{array} \right)$.\\
(i) Find the value of $a$ for which $\mathbf { A }$ is singular.\\
(ii) Given that $\mathbf { A }$ is non-singular, find $\mathbf { A } ^ { - 1 }$ and hence solve the equations
$$\begin{aligned}
a x + 2 y + z & = 1 \\
x + 3 y + 2 z & = 2 \\
4 x + y + z & = 3
\end{aligned}$$
\hfill \mbox{\textit{OCR FP1 2013 Q10 [12]}}