| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| 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 Pure 1 question on matrix determinants and singularity. Part (i) requires routine calculation of a 3×3 determinant, part (ii) is direct application (setting det=0), and part (iii) tests understanding that singular matrices mean no unique solution. While it's Further Maths content, it follows a predictable template with no novel problem-solving required, making it slightly easier than average overall. |
| Spec | 4.03j Determinant 3x3: calculation4.03l Singular/non-singular matrices4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) Expansion of \(a \begin{bmatrix} a & 0 \\ 2 & 1 \end{bmatrix} - 4 \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} + 2 \begin{bmatrix} 1 & a \\ 1 & 2 \end{bmatrix}\) | M1 | Correct expansion process shown |
| A1 | Obtain correct unsimplified expression | |
| A1, 3 | Obtain correct answer: \(a^2 - 2a\) | |
| Part (ii) Set \(\det M = 0\) | M1 | Solve their \(\det M = 0\) |
| A1, A1, 1ft, 3 | Obtain correct answers: \(a = 0\) or \(a = 2\) | |
| Part (iii) (a) | B1, B1 | Solution, as inverse matrix exists or \(M\) non-singular or \(\det M \neq 0\) |
| Part (iii) (b) | B1, B1, 4 | Solutions, eqn. 1 is multiple of eqn 3 |
**Part (i)** Expansion of $a \begin{bmatrix} a & 0 \\ 2 & 1 \end{bmatrix} - 4 \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} + 2 \begin{bmatrix} 1 & a \\ 1 & 2 \end{bmatrix}$ | M1 | Correct expansion process shown
| A1 | Obtain correct unsimplified expression
| A1, 3 | Obtain correct answer: $a^2 - 2a$
**Part (ii)** Set $\det M = 0$ | M1 | Solve their $\det M = 0$
| A1, A1, 1ft, 3 | Obtain correct answers: $a = 0$ or $a = 2$
**Part (iii) (a)** | B1, B1 | Solution, as inverse matrix exists or $M$ non-singular or $\det M \neq 0$
**Part (iii) (b)** | B1, B1, 4 | Solutions, eqn. 1 is multiple of eqn 38 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 2 \\ 1 & a & 0 \\ 1 & 2 & 1 \end{array} \right)$.\\
(i) Find, in terms of $a$, the determinant of $\mathbf { M }$.\\
(ii) Hence find the values of $a$ for which $\mathbf { M }$ is singular.\\
(iii) State, giving a brief reason in each case, whether the simultaneous equations
$$\begin{aligned}
a x + 4 y + 2 z & = 3 a \\
x + a y & = 1 \\
x + 2 y + z & = 3
\end{aligned}$$
have any solutions when
\begin{enumerate}[label=(\alph*)]
\item $a = 3$,
\item $a = 2$.
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2006 Q8 [10]}}