| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Properties of matrix operations |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic matrix operations: multiplication (part i), inverse verification (part ii), and solving a matrix equation (part iii). All parts involve routine calculations with no conceptual challenges or problem-solving insight required—students simply apply standard algorithms they've been taught. This is easier than average for A-level, though the Further Maths context prevents it from being trivial. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03n Inverse 2x2 matrix4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{AB} = \begin{pmatrix}1&2\\2&1\end{pmatrix}\begin{pmatrix}-1&2\\2&-1\end{pmatrix} = \begin{pmatrix}3&0\\0&3\end{pmatrix}\) | M1 (AO 1.1) | Or BC. OR: M1 \(A^{-1} = \frac{1}{1-4}\begin{pmatrix}1&-2\\-2&1\end{pmatrix} = \frac{1}{-3}\begin{pmatrix}1&-2\\-2&1\end{pmatrix} = \frac{1}{3}\mathbf{B}\) |
| So \(\mathbf{AB} = 3\mathbf{I}\) and \(m = 3\) | E1 (AO 1.1) | E1 So \(\mathbf{AB} = 3\mathbf{AA}^{-1} = 3\mathbf{I}\) and \(m=3\) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{CD} = \begin{pmatrix}2&0&0\\0&2&0\\0&0&2\end{pmatrix}\) | B1 (AO 1.1) | BC. OR: B1 \(C^{-1} = \begin{pmatrix}-2&4&-1\\-2.5&4.5&-0.5\\1.5&-2.5&0.5\end{pmatrix}\) |
| so \(C^{-1} = \frac{1}{2}\mathbf{D}\) and \(k = \frac{1}{2}\) | E1 (AO 2.2a) | BC. E1 so \(C^{-1} = \frac{1}{2}\mathbf{D}\) and \(k = \frac{1}{2}\) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{EF} = \begin{pmatrix}k&k^2\\3&0\end{pmatrix}\begin{pmatrix}2\\k\end{pmatrix} = \begin{pmatrix}2k+k^3\\6\end{pmatrix}\) | M1 (AO 1.1a) | Both multiplications attempted |
| \(\Rightarrow \begin{pmatrix}2k+k^3\\6\end{pmatrix} = \begin{pmatrix}-2k\\6\end{pmatrix}\) | M1 (AO 2.1) | |
| \(\Rightarrow 2k + k^3 = -2k\) | A1 (AO 2.2a) | |
| \(k = 0,\ 2i,\ -2i\) | A1 (AO 1.1) | BC |
| \(\mathbf{F} = \begin{pmatrix}2\\0\end{pmatrix},\ \begin{pmatrix}2\\2i\end{pmatrix}\ \text{or}\ \begin{pmatrix}2\\-2i\end{pmatrix}\) | A1 (AO 1.1) | www. If A0, allow SC1 for one matrix only |
| [5] |
# Question 3(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{AB} = \begin{pmatrix}1&2\\2&1\end{pmatrix}\begin{pmatrix}-1&2\\2&-1\end{pmatrix} = \begin{pmatrix}3&0\\0&3\end{pmatrix}$ | **M1** (AO 1.1) | Or BC. **OR**: **M1** $A^{-1} = \frac{1}{1-4}\begin{pmatrix}1&-2\\-2&1\end{pmatrix} = \frac{1}{-3}\begin{pmatrix}1&-2\\-2&1\end{pmatrix} = \frac{1}{3}\mathbf{B}$ |
| So $\mathbf{AB} = 3\mathbf{I}$ and $m = 3$ | **E1** (AO 1.1) | **E1** So $\mathbf{AB} = 3\mathbf{AA}^{-1} = 3\mathbf{I}$ and $m=3$ |
| **[2]** | | |
---
# Question 3(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{CD} = \begin{pmatrix}2&0&0\\0&2&0\\0&0&2\end{pmatrix}$ | **B1** (AO 1.1) | BC. **OR**: **B1** $C^{-1} = \begin{pmatrix}-2&4&-1\\-2.5&4.5&-0.5\\1.5&-2.5&0.5\end{pmatrix}$ |
| so $C^{-1} = \frac{1}{2}\mathbf{D}$ and $k = \frac{1}{2}$ | **E1** (AO 2.2a) | BC. **E1** so $C^{-1} = \frac{1}{2}\mathbf{D}$ and $k = \frac{1}{2}$ |
| **[2]** | | |
---
# Question 3(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{EF} = \begin{pmatrix}k&k^2\\3&0\end{pmatrix}\begin{pmatrix}2\\k\end{pmatrix} = \begin{pmatrix}2k+k^3\\6\end{pmatrix}$ | **M1** (AO 1.1a) | Both multiplications attempted |
| $\Rightarrow \begin{pmatrix}2k+k^3\\6\end{pmatrix} = \begin{pmatrix}-2k\\6\end{pmatrix}$ | **M1** (AO 2.1) | |
| $\Rightarrow 2k + k^3 = -2k$ | **A1** (AO 2.2a) | |
| $k = 0,\ 2i,\ -2i$ | **A1** (AO 1.1) | BC |
| $\mathbf{F} = \begin{pmatrix}2\\0\end{pmatrix},\ \begin{pmatrix}2\\2i\end{pmatrix}\ \text{or}\ \begin{pmatrix}2\\-2i\end{pmatrix}$ | **A1** (AO 1.1) | www. If **A0**, allow SC1 for one matrix only |
| **[5]** | | |3 (i) You are given two matrices, A and B, where
$$\mathbf { A } = \left( \begin{array} { l l }
1 & 2 \\
2 & 1
\end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c }
- 1 & 2 \\
2 & - 1
\end{array} \right)$$
Show that $\mathbf { A B } = m \mathbf { I }$, where $m$ is a constant to be determined.\\
(ii) You are given two matrices, $\mathbf { C }$ and $\mathbf { D }$, where
$$\mathbf { C } = \left( \begin{array} { r r r }
2 & 1 & 5 \\
1 & 1 & 3 \\
- 1 & 2 & 2
\end{array} \right) \text { and } \mathbf { D } = \left( \begin{array} { r r r }
- 4 & 8 & - 2 \\
- 5 & 9 & - 1 \\
3 & - 5 & 1
\end{array} \right)$$
Show that $\mathbf { C } ^ { - 1 } = k \mathbf { D }$ where $k$ is a constant to be determined.\\
(iii) The matrices $\mathbf { E }$ and $\mathbf { F }$ are given by $\mathbf { E } = \left( \begin{array} { c c } k & k ^ { 2 } \\ 3 & 0 \end{array} \right)$ and $\mathbf { F } = \binom { 2 } { k }$ where $k$ is a constant.
Determine any matrix $\mathbf { F }$ for which $\mathbf { E F } = \binom { - 2 k } { 6 }$.
\hfill \mbox{\textit{OCR Further Pure Core AS Q3 [9]}}