| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2022 |
| Session | November |
| Marks | 5 |
| 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 Maths linear algebra question requiring determinant calculation to find when a system has a unique solution (det ≠ 0), followed by geometric interpretation of planes. The computation is routine and the concepts are core syllabus material, making it slightly easier than average for Further Maths. |
| Spec | 4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{vmatrix} 1 & 2 & 3 \\ k & 4 & 6 \\ 7 & 8 & 9 \end{vmatrix} = 6k - 12\) | M1 A1 | Evaluates determinant or solves system of equations (eliminate at least one variable for M1): \(x = -\dfrac{2}{k-2}\), \(y = \dfrac{4}{k-2}\), \(z = \dfrac{k-8}{3(k-2)}\) |
| \(k \neq 2\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Two parallel planes. Other plane not parallel. | B1 | SC \(k = 2\), two parallel planes. |
| Parallel planes not identical. | B1 | Accept diagram. |
| Total: 2 |
## Question 1:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{vmatrix} 1 & 2 & 3 \\ k & 4 & 6 \\ 7 & 8 & 9 \end{vmatrix} = 6k - 12$ | M1 A1 | Evaluates determinant or solves system of equations (eliminate at least one variable for M1): $x = -\dfrac{2}{k-2}$, $y = \dfrac{4}{k-2}$, $z = \dfrac{k-8}{3(k-2)}$ |
| $k \neq 2$ | A1 | |
| **Total: 3** | | |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Two parallel planes. Other plane not parallel. | B1 | SC $k = 2$, two parallel planes. |
| Parallel planes not identical. | B1 | Accept diagram. |
| **Total: 2** | | |
---1
\begin{enumerate}[label=(\alph*)]
\item Find the set of values of $k$ for which the system of equations
$$\begin{aligned}
x + 2 y + 3 z & = 1 \\
k x + 4 y + 6 z & = 0 \\
7 x + 8 y + 9 z & = 3
\end{aligned}$$
has a unique solution.
\item Interpret the situation geometrically in the case where the system of equations does not have a unique solution.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2022 Q1 [5]}}