| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| 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 singularity. Part (i) requires routine calculation of a 3×3 determinant with a parameter. Part (ii) is direct application (set det=0). Part (iii) requires understanding that singular matrices may have no solution or infinitely many, but the question only asks students to state and briefly justify, not fully solve the systems. This is slightly easier than average A-level difficulty as it follows a predictable template with straightforward algebraic manipulation. |
| Spec | 4.03j Determinant 3x3: calculation4.03l Singular/non-singular matrices4.03s Consistent/inconsistent: systems of equations4.03t Plane intersection: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(a\begin{vmatrix} a & 1 \\ 1 & 2 \end{vmatrix} - 1\begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} + 1\begin{vmatrix} 1 & a \\ 1 & 1 \end{vmatrix}\) | M1 A1 | |
| \(2a^2 - 2a\) | A1 | 3 |
| (ii) \(a = 0\) or \(1\) | M1 A1 ft A1 ft | 3 |
| (iii)(a) Equations consistent, but non-unique solutions | B1 B1 | |
| (b) Correct equations seen & inconsistent, no solutions | B1 B1 | 4 |
| 10 |
**(i)** $a\begin{vmatrix} a & 1 \\ 1 & 2 \end{vmatrix} - 1\begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} + 1\begin{vmatrix} 1 & a \\ 1 & 1 \end{vmatrix}$ | M1 A1 | | Correct expansion process shown. Obtain correct unsimplified expression
$2a^2 - 2a$ | A1 | 3 | Obtain correct answer
**(ii)** $a = 0$ or $1$ | M1 A1 ft A1 ft | 3 | Equate their det to 0. Obtain correct answers, ft solving a quadratic
**(iii)(a)** Equations consistent, but non-unique solutions | B1 B1 | |
(b) Correct equations seen & inconsistent, no solutions | B1 B1 | 4 |
| | | **10** |9 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l l } a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & 2 \end{array} \right)$.\\
(i) Find, in terms of $a$, the determinant of $\mathbf { A }$.\\
(ii) Hence find the values of $a$ for which $\mathbf { A }$ is singular.\\
(iii) State, giving a brief reason in each case, whether the simultaneous equations
$$\begin{aligned}
a x + y + z & = 2 a \\
x + a y + z & = - 1 \\
x + y + 2 z & = - 1
\end{aligned}$$
have any solutions when
\begin{enumerate}[label=(\alph*)]
\item $a = 0$,
\item $a = 1$.
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2009 Q9 [10]}}