| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Solving matrix equations for unknown matrix |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question requiring standard matrix inversion (trivial for a 2×2 upper triangular matrix) and solving a matrix equation by post-multiplying by the inverse. Both parts are routine applications of basic matrix algebra with no problem-solving insight needed, making it slightly easier than average despite being Further Maths content. |
| Spec | 4.03n Inverse 2x2 matrix4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}\begin{pmatrix}1 & -a\\0 & 2\end{pmatrix}\) or equivalent | B1 | Both diagonals correct |
| B1 | Divide by correct determinant | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{P} = \mathbf{BA}^{-1}\) | B1 | State or use correct expression for \(\mathbf{P}\) |
| \(\begin{pmatrix}1 & 0\\2 & 1-2a\end{pmatrix}\) | M1 | Multiplication attempt, 2 elements correct for any pair of matrices |
| A1ft | Obtain correct answer a.e.f. ft for their (i) | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Using \(\mathbf{PA} = \mathbf{B}\) | B1 | State or find correct 1st column of \(\mathbf{P}\) |
| M1 | Multiplication attempt to find "\(1-2a\)" | |
| A1 | Obtain completely correct answer |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}\begin{pmatrix}1 & -a\\0 & 2\end{pmatrix}$ or equivalent | B1 | Both diagonals correct |
| | B1 | Divide by correct determinant |
| **[2]** | | |
---
## Question 3(ii):
**Either:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{P} = \mathbf{BA}^{-1}$ | B1 | State or use correct expression for $\mathbf{P}$ |
| $\begin{pmatrix}1 & 0\\2 & 1-2a\end{pmatrix}$ | M1 | Multiplication attempt, 2 elements correct for any pair of matrices |
| | A1ft | Obtain correct answer a.e.f. ft for their (i) |
| **[3]** | | |
**Or:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Using $\mathbf{PA} = \mathbf{B}$ | B1 | State or find correct 1st column of $\mathbf{P}$ |
| | M1 | Multiplication attempt to find "$1-2a$" |
| | A1 | Obtain completely correct answer |
---3 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 2 & a \\ 0 & 1 \end{array} \right)$, where $a$ is a constant.\\
(i) Find $\mathbf { A } ^ { - 1 }$.
The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { l l } 2 & a \\ 4 & 1 \end{array} \right)$.\\
(ii) Given that $\mathbf { P A } = \mathbf { B }$, find the matrix $\mathbf { P }$.
\hfill \mbox{\textit{OCR FP1 2015 Q3 [5]}}