| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 8 |
| 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 3×3 determinants and matrix singularity. Part (i) requires routine determinant calculation using cofactor expansion. Parts (ii) and (iii) involve straightforward substitution and checking if det(M)=0. While it's Further Maths content, it follows a predictable template with no novel problem-solving required, making it slightly easier than an average A-level question overall. |
| Spec | 4.03j Determinant 3x3: calculation4.03l Singular/non-singular matrices4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(a(a-12) + 32\) | M1 M1 A1 | Show correct expansion process. Show evaluation of a 2 × 2 determinant. Obtain correct answer a.e.f. |
| (ii) \(\det M = 12\) non-singular | M1 A1ft | Substitute \(a = 2\) in their determinant. |
| (iii) EITHER | B1 | |
| M1 A1 | Obtain correct answer and state a consistent conclusion. | |
| OR | ||
| \(\det M = 0\) so non-unique solutions | M1 A1 A1 | \(\det M = 0\) so non-unique solutions. Attempt to solve and obtain 2 inconsistent equations. Deduce that there are no solutions. |
| Substitute \(a = 4\) and attempt to solve. Obtain 2 correct inconsistent equations. Deduce no solutions. |
(i) $a(a-12) + 32$ | M1 M1 A1 | Show correct expansion process. Show evaluation of a 2 × 2 determinant. Obtain correct answer a.e.f. |
(ii) $\det M = 12$ non-singular | M1 A1ft | Substitute $a = 2$ in their determinant. |
(iii) EITHER | B1 | |
| M1 A1 | Obtain correct answer and state a consistent conclusion. |
OR | | |
$\det M = 0$ so non-unique solutions | M1 A1 A1 | $\det M = 0$ so non-unique solutions. Attempt to solve and obtain 2 inconsistent equations. Deduce that there are no solutions. |
| | | Substitute $a = 4$ and attempt to solve. Obtain 2 correct inconsistent equations. Deduce no solutions. |
**Total: 3 + 2 + 3 = 8 marks**7 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 0 \\ 0 & a & 4 \\ 2 & 3 & 1 \end{array} \right)$.\\
(i) Find, in terms of $a$, the determinant of $\mathbf { M }$.\\
(ii) In the case when $a = 2$, state whether $\mathbf { M }$ is singular or non-singular, justifying your answer.\\
(iii) In the case when $a = 4$, determine whether the simultaneous equations
$$\begin{aligned}
a x + 4 y \quad = & 6 \\
a y + 4 z & = 8 \\
2 x + 3 y + z & = 1
\end{aligned}$$
have any solutions.
\hfill \mbox{\textit{OCR FP1 2007 Q7 [8]}}