| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Solving linear systems using matrices |
| Difficulty | Moderate -0.3 This is a standard FP1 matrices question covering routine techniques: writing a matrix equation, finding conditions for invertibility (determinant = 0), computing a 2×2 inverse, and interpreting geometric cases. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average A-level difficulty. |
| Spec | 4.03a Matrix language: terminology and notation4.03l Singular/non-singular matrices4.03n Inverse 2x2 matrix4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\mathbf{M} = \begin{pmatrix} 2 & -1 \\ 3 & k \end{pmatrix}\) | B2 | \(-1\) each error |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\mathbf{M}^{-1}\) does not exist for \(2k+3=0\) | M1 | May be implied |
| \(k = \frac{-3}{2}\) | A1 | |
| \(\mathbf{M}^{-1} = \frac{1}{2k+3}\begin{pmatrix} k & 1 \\ -3 & 2 \end{pmatrix}\) | B1 | Correct inverse |
| \(\frac{1}{13}\begin{pmatrix} 5 & 1 \\ -3 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 21 \end{pmatrix}\) | M1 | Attempt to pre-multiply by their inverse |
| \(= \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) | A1ft, A1 | Correct matrix multiplication; c.a.o. |
| \(\Rightarrow x = 2,\ y = 3\) | A1ft | At least one correct |
| [7] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| There are no unique solutions | B1 | |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (A) Lines intersect | B1 | |
| (B) Lines parallel | B1 | |
| (C) Lines coincident | B1 | |
| [3] |
# Question 9(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{M} = \begin{pmatrix} 2 & -1 \\ 3 & k \end{pmatrix}$ | B2 | $-1$ each error |
| | **[2]** | |
---
# Question 9(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{M}^{-1}$ does not exist for $2k+3=0$ | M1 | May be implied |
| $k = \frac{-3}{2}$ | A1 | |
| $\mathbf{M}^{-1} = \frac{1}{2k+3}\begin{pmatrix} k & 1 \\ -3 & 2 \end{pmatrix}$ | B1 | Correct inverse |
| $\frac{1}{13}\begin{pmatrix} 5 & 1 \\ -3 & 2 \end{pmatrix}\begin{pmatrix} 1 \\ 21 \end{pmatrix}$ | M1 | Attempt to pre-multiply by their inverse |
| $= \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ | A1ft, A1 | Correct matrix multiplication; c.a.o. |
| $\Rightarrow x = 2,\ y = 3$ | A1ft | At least one correct |
| | **[7]** | |
---
# Question 9(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| There are no unique solutions | B1 | |
| | **[1]** | |
---
# Question 9(iv):
| Answer | Mark | Guidance |
|--------|------|----------|
| (A) Lines intersect | B1 | |
| (B) Lines parallel | B1 | |
| (C) Lines coincident | B1 | |
| | **[3]** | |9 The simultaneous equations
$$\begin{aligned}
& 2 x - y = 1 \\
& 3 x + k y = b
\end{aligned}$$
are represented by the matrix equation $\mathbf { M } \binom { x } { y } = \binom { 1 } { b }$.
\begin{enumerate}[label=(\roman*)]
\item Write down the matrix $\mathbf { M }$.
\item State the value of $k$ for which $\mathbf { M } ^ { - 1 }$ does not exist and find $\mathbf { M } ^ { - 1 }$ in terms of $k$ when $\mathbf { M } ^ { - 1 }$ exists.
Use $\mathbf { M } ^ { - 1 }$ to solve the simultaneous equations when $k = 5$ and $b = 21$.
\item What can you say about the solutions of the equations when $k = - \frac { 3 } { 2 }$ ?
\item The two equations can be interpreted as representing two lines in the $x - y$ plane. Describe the relationship between these two lines\\
(A) when $k = 5$ and $b = 21$,\\
(B) when $k = - \frac { 3 } { 2 }$ and $b = 1$,\\
(C) when $k = - \frac { 3 } { 2 }$ and $b = \frac { 3 } { 2 }$.
RECOGNISING ACHIEVEMENT
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2011 Q9 [13]}}