| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Parameter values for unique solution |
| Difficulty | Standard +0.3 This is a standard Further Maths question on matrix singularity and solution existence. Part (i) requires computing a 3×3 determinant and solving a quadratic—routine FP1 technique. Part (ii) applies the theory directly: checking if det≠0 for uniqueness, then using row reduction for the singular cases. While it's Further Maths content, it follows a predictable template with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.03j Determinant 3x3: calculation4.03l Singular/non-singular matrices4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| M1 | Attempt to find det \(\mathbf{D}\); Or Cramer's rule or similar | |
| M1 | Show correct process for a \(3\times3\), condone sign errors | |
| M1 | Show correct processes for a \(2\times2\) | |
| \(a^2 - 6a + 5\) | A1 | Obtain correct answer |
| M1 | Attempt to solve det \(\mathbf{D} = 0\) | |
| \(a = 5\) or \(1\) | A1 | Obtain correct answers |
| [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| B1 | State unique solution | |
| B1 | State non unique solutions | |
| M1 | Attempt to solve equations with \(a=1\) | |
| A1 | Explain inconsistency with correct working | |
| [4] | S.C. Answer to (i) wrong, allow correct unique/non-unique B1ft, B1ft only |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| | M1 | Attempt to find det $\mathbf{D}$; Or Cramer's rule or similar |
| | M1 | Show correct process for a $3\times3$, condone sign errors |
| | M1 | Show correct processes for a $2\times2$ |
| $a^2 - 6a + 5$ | A1 | Obtain correct answer |
| | M1 | Attempt to solve det $\mathbf{D} = 0$ |
| $a = 5$ or $1$ | A1 | Obtain correct answers |
| **[6]** | | |
---
## Question 9(ii)(a)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1 | State unique solution |
| | B1 | State non unique solutions |
| | M1 | Attempt to solve equations with $a=1$ |
| | A1 | Explain inconsistency with correct working |
| **[4]** | **S.C. Answer to (i) wrong, allow correct unique/non-unique B1ft, B1ft only** | |
---9 The matrix $\mathbf { D }$ is given by $\mathbf { D } = \left( \begin{array} { l l l } 1 & 3 & 4 \\ 2 & a & 3 \\ 0 & 1 & a \end{array} \right)$.\\
(i) Find the values of $a$ for which $\mathbf { D }$ is singular.\\
(ii) Three simultaneous equations are shown below.
$$\begin{array} { r }
x + 3 y + 4 z = 3 \\
2 x + a y + 3 z = 2 \\
y + a z = 0
\end{array}$$
For each of the following values of $a$, determine whether or not there is a unique solution. If a unique solution does not exist, determine whether the equations are consistent or inconsistent.
\begin{enumerate}[label=(\alph*)]
\item $a = 3$
\item $a = 1$
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2015 Q9 [10]}}