| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Geometric interpretation of systems |
| Difficulty | Standard +0.8 This is a Further Maths linear algebra question requiring systematic analysis of a 3×3 system using row reduction or matrix methods to determine consistency and solution type, followed by geometric interpretation (likely a unique point of intersection). While methodical, it requires solid understanding of rank, solution spaces, and 3D geometry—more demanding than standard A-level but routine for Further Maths students. |
| Spec | 4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix}1 & 2 & 0 & 3\\2 & -5 & 3 & 8\\0 & 6 & -2 & 0\end{pmatrix}\) | M1 | Attempt at row reduction |
| \(\begin{pmatrix}1 & 2 & 0 & 3\\0 & -9 & 3 & 2\\0 & 6 & -2 & 0\end{pmatrix}\) | A1 | 1 row a multiple of another row |
| \(\begin{pmatrix}1 & 2 & 0 & 3\\0 & -9 & 3 & \frac{2}{4}\\0 & 0 & 0 & \frac{4}{3}\end{pmatrix}\) | A1 | oe |
| Valid statement. E.g. As \(0x + 0y + 0z \neq \frac{4}{3}\) there are no solutions | E1 | If M0, SC1 det A = 0; SC1 No unique solutions |
| Answer | Marks | Guidance |
|---|---|---|
| A correct statement involving 3 planes with no incorrect statements | B1 | FT their (a) |
| e.g. 3 planes do not meet at a single point |
**Part a)**
$\begin{pmatrix}1 & 2 & 0 & 3\\2 & -5 & 3 & 8\\0 & 6 & -2 & 0\end{pmatrix}$ | M1 | Attempt at row reduction
$\begin{pmatrix}1 & 2 & 0 & 3\\0 & -9 & 3 & 2\\0 & 6 & -2 & 0\end{pmatrix}$ | A1 | 1 row a multiple of another row
$\begin{pmatrix}1 & 2 & 0 & 3\\0 & -9 & 3 & \frac{2}{4}\\0 & 0 & 0 & \frac{4}{3}\end{pmatrix}$ | A1 | oe
Valid statement. E.g. As $0x + 0y + 0z \neq \frac{4}{3}$ there are no solutions | E1 | If M0, SC1 det A = 0; SC1 No unique solutions
**Part b)**
A correct statement involving 3 planes with no incorrect statements | B1 | FT their (a)
e.g. 3 planes do not meet at a single point | |
---\begin{enumerate}[label=(\alph*)]
\item Determine the number of solutions of the equations
\begin{align}
x + 2y &= 3,\\
2x - 5y + 3z &= 8,\\
6y - 2z &= 0.
\end{align} [4]
\item Give a geometric interpretation of your answer in part (a). [1]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 2022 Q5 [5]}}