| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Inverse given/derived then solve system |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question with two routine parts: (i) finding p using the property NN^(-1)=I (simple matrix multiplication and equation solving), and (ii) solving a 3×3 system by multiplying the given inverse by the constant vector. Both parts are mechanical applications of standard techniques with no conceptual challenges, making it slightly easier than average even for Further Maths. |
| Spec | 4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| \(-2 - 4p = 0\) | M1 | Any valid row × column leading to \(p\) |
| \(\Rightarrow p = -\frac{1}{2}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \mathbf{N}^{-1}\begin{pmatrix} -39 \\ 5 \\ 22 \end{pmatrix}\) | M1 | Attempt to use \(\mathbf{N}^{-1}\); correct solution by means of simultaneous equations can earn full marks |
| \(= \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ \frac{7}{2} & \frac{-1}{2} & -6 \end{pmatrix}\begin{pmatrix} -39 \\ 5 \\ 22 \end{pmatrix}\) | M1 | Attempt to multiply matrices (implied by 3×1 result); M1 elimination of one unknown, M1 solution for one unknown |
| \(= \begin{pmatrix} 5 \\ -7 \\ 2 \end{pmatrix}\) | A1 | One element correct; A1 one correct |
| A1 | All 3 correct, FT their \(p\); A1 all correct |
## Question 3:
### Part (i):
$-2 - 4p = 0$ | M1 | Any valid row × column leading to $p$
$\Rightarrow p = -\frac{1}{2}$ | B1 |
**[2]**
### Part (ii):
$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \mathbf{N}^{-1}\begin{pmatrix} -39 \\ 5 \\ 22 \end{pmatrix}$ | M1 | Attempt to use $\mathbf{N}^{-1}$; correct solution by means of simultaneous equations can earn full marks
$= \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ \frac{7}{2} & \frac{-1}{2} & -6 \end{pmatrix}\begin{pmatrix} -39 \\ 5 \\ 22 \end{pmatrix}$ | M1 | Attempt to multiply matrices (implied by 3×1 result); M1 elimination of one unknown, M1 solution for one unknown
$= \begin{pmatrix} 5 \\ -7 \\ 2 \end{pmatrix}$ | A1 | One element correct; A1 one correct
| A1 | All 3 correct, FT their $p$; A1 all correct
**[4]**
---3 You are given that $\mathbf { N } = \left( \begin{array} { r r r } - 9 & - 2 & - 4 \\ 3 & 2 & 2 \\ 5 & 1 & 2 \end{array} \right)$ and $\mathbf { N } ^ { - 1 } = \left( \begin{array} { r r r } 1 & 0 & 2 \\ 2 & 1 & 3 \\ - \frac { 7 } { 2 } & p & - 6 \end{array} \right)$.\\
(i) Find the value of $p$.\\
(ii) Solve the equation $\mathbf { N } \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } - 39 \\ 5 \\ 22 \end{array} \right)$.
\hfill \mbox{\textit{OCR MEI FP1 2013 Q3 [6]}}