| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Consistency conditions for systems |
| 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 using cofactor expansion. Part (ii) tests understanding of when systems have unique/no/infinite solutions by examining the determinant, but the analysis is straightforward substitution into the already-computed determinant formula. While it's Further Maths content, it's a textbook application with no novel insight required, making it slightly easier than an average A-level question overall. |
| Spec | 4.03j Determinant 3x3: calculation4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks |
|---|---|
| M1 | Show correct expansion process for 3 × 3 |
| M1 | Correct evaluation of a 2 × 2 |
| A1 3 | Obtain correct answer |
| Answer | Marks |
|---|---|
| M1 | Find a pair of inconsistent equations |
| A1 | State inconsistent or no solutions |
| Answer | Marks |
|---|---|
| M1 | Find a repeated equation |
| A1 | State non unique solutions |
| Answer | Marks |
|---|---|
| B1 | State that \(\det A\) is non-zero or find correct solution |
| B1 6 | State unique solution |
| SC if \(\det A\) incorrect, can score 2 marks for correct deduction of a unique solution, but only once |
## (i)
| M1 | Show correct expansion process for 3 × 3
| M1 | Correct evaluation of a 2 × 2
| A1 3 | Obtain correct answer
$\det A = a^2 - a$
## (ii)
### (a)
| M1 | Find a pair of inconsistent equations
| A1 | State inconsistent or no solutions
### (b)
| M1 | Find a repeated equation
| A1 | State non unique solutions
### (c)
| B1 | State that $\det A$ is non-zero or find correct solution
| B1 6 | State unique solution
| | SC if $\det A$ incorrect, can score 2 marks for correct deduction of a unique solution, but only once
---The matrix $\mathbf{A}$ is given by $\mathbf{A} = \begin{pmatrix} a & a & -1 \\ 0 & a & 2 \\ 1 & 2 & 1 \end{pmatrix}$.
\begin{enumerate}[label=(\roman*)]
\item Find, in terms of $a$, the determinant of $\mathbf{A}$. [3]
\item Three simultaneous equations are shown below.
\begin{align}
ax + ay - z &= -1 \\
ay + 2z &= 2a \\
x + 2y + z &= 1
\end{align}
For each of the following values of $a$, determine whether the equations are consistent or inconsistent. If the equations are consistent, determine whether or not there is a unique solution.
\begin{enumerate}[label=(\alph*)]
\item $a = 0$
\item $a = 1$
\item $a = 2$ [6]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2010 Q9 [9]}}