| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| 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 FP1 question on determinants and systems of equations. Part (i) requires routine calculation of a 3×3 determinant with a parameter. Part (ii) applies the theory (unique solution when det≠0, then checking consistency when det=0) to three specific values. While it involves multiple steps and understanding of the theory, it follows a predictable template with no novel insight required, making it slightly easier than average. |
| Spec | 4.03j Determinant 3x3: calculation4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show correct expansion process for \(3\times3\) | M1 | |
| Correct evaluation of any \(2\times2\) | M1 | |
| \(a^3 - 4a\) | A1 | Obtain correct answer |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\det \mathbf{D} = 15\) so unique solution or solve to find correct solution \(\left(-\frac{2}{5}, 1, \frac{4}{5}\right)\) | B1 | SC B1 once if unique solution following their incorrect \(\det \mathbf{D}\) non zero |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Their \(\det \mathbf{D} = 0\), so non-unique solutions | B1 | |
| Attempt to solve equations with \(a = 2\) | M1 | |
| Explain inconsistency with correct working | A1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Their \(\det \mathbf{D} = 0\), so non-unique solutions | B1 | |
| Attempt to solve equations with \(a = 0\) | M1 | |
| Explain consistency with correct working | A1 | |
| [3] |
## Question 10:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show correct expansion process for $3\times3$ | M1 | |
| Correct evaluation of any $2\times2$ | M1 | |
| $a^3 - 4a$ | A1 | Obtain correct answer |
| **[3]** | | |
### Part (ii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\det \mathbf{D} = 15$ so unique solution or solve to find correct solution $\left(-\frac{2}{5}, 1, \frac{4}{5}\right)$ | B1 | **SC** B1 once if unique solution following their incorrect $\det \mathbf{D}$ non zero |
| **[1]** | | |
### Part (ii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Their $\det \mathbf{D} = 0$, so non-unique solutions | B1 | |
| Attempt to solve equations with $a = 2$ | M1 | |
| Explain inconsistency with correct working | A1 | |
| **[3]** | | |
### Part (ii)(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Their $\det \mathbf{D} = 0$, so non-unique solutions | B1 | |
| Attempt to solve equations with $a = 0$ | M1 | |
| Explain consistency with correct working | A1 | |
| **[3]** | | |10 The matrix $\mathbf { D }$ is given by $\mathbf { D } = \left( \begin{array} { r r r } a & 2 & - 1 \\ 2 & a & 1 \\ 1 & 1 & a \end{array} \right)$.\\
(i) Find the determinant of $\mathbf { D }$ in terms of $a$.\\
(ii) Three simultaneous equations are shown below.
$$\begin{array} { r }
a x + 2 y - z = 0 \\
2 x + a y + z = a \\
x + y + a z = a
\end{array}$$
For each of the following values of $a$, determine whether or not there is a unique solution. If the solution is not unique, determine whether the equations are consistent or inconsistent.
\begin{enumerate}[label=(\alph*)]
\item $\quad a = 3$
\item $a = 2$
\item $\quad a = 0$
\section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2012 Q10 [10]}}