| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Matrix inverse calculation |
| Difficulty | Standard +0.3 This is a standard Further Pure 1 matrix question requiring determinant calculation (using cofactor expansion or row operations), solving a quadratic equation for singularity, and finding the inverse using the adjugate method. While it involves algebraic manipulation with parameter 'a', these are routine FP1 techniques with no novel problem-solving required, making it slightly easier than average overall but typical for further maths content. |
| Spec | 4.03j Determinant 3x3: calculation4.03l Singular/non-singular matrices4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\det\mathbf{X} = \Delta = 10 - 9a - a^2\) | M1, M1, A1 | Show correct expansion process for \(3\times3\); Correct evaluation of any \(2\times2\); Obtain correct answer |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = 1\) or \(-10\) | M1, A1FT, A1FT | Their \(\det\mathbf{X}=0\); Obtain correct answers from their (i) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{\Delta}\begin{pmatrix}-a&2&6-9a\\5&-a-9&18-3a\\-a&2&a^2-4\end{pmatrix}\) | M1, A1, A1, B1ft | Show correct process for adjoint entries; Obtain at least four correct entries; Obtain completely correct adjoint; Divide by their determinant |
| [4] |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\det\mathbf{X} = \Delta = 10 - 9a - a^2$ | M1, M1, A1 | Show correct expansion process for $3\times3$; Correct evaluation of any $2\times2$; Obtain correct answer |
| **[3]** | | |
---
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = 1$ or $-10$ | M1, A1FT, A1FT | Their $\det\mathbf{X}=0$; Obtain correct answers from their (i) |
| **[3]** | | |
---
## Question 9(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{\Delta}\begin{pmatrix}-a&2&6-9a\\5&-a-9&18-3a\\-a&2&a^2-4\end{pmatrix}$ | M1, A1, A1, B1ft | Show correct process for adjoint entries; Obtain at least four correct entries; Obtain completely correct adjoint; Divide by their determinant |
| **[4]** | | |
---$\mathbf { 9 }$ The matrix $\mathbf { X }$ is given by $\mathbf { X } = \left( \begin{array} { r r r } a & 2 & 9 \\ 2 & a & 3 \\ 1 & 0 & - 1 \end{array} \right)$.\\
(i) Find the determinant of $\mathbf { X }$ in terms of $a$.\\
(ii) Hence find the values of $a$ for which $\mathbf { X }$ is singular.\\
(iii) Given that $\mathbf { X }$ is non-singular, find $\mathbf { X } ^ { - 1 }$ in terms of $a$.
\hfill \mbox{\textit{OCR FP1 2012 Q9 [10]}}