| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | System of three linear equations |
| Difficulty | Standard +0.8 This is a Further Maths question on linear systems requiring understanding of consistency, uniqueness, and geometric interpretation. While the algebraic manipulation is straightforward (subtracting equations to show dependence), it demands conceptual maturity about solution existence/uniqueness and 3D geometric visualization of planes—topics beyond standard A-level. The multi-part structure with proof elements and geometric interpretation elevates it above routine exercises. |
| Spec | 4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{vmatrix} 1 & -1 & 2 \\ 1 & -1 & -3 \\ 1 & -1 & 7 \end{vmatrix} = -7 - 3 + 10 - 2(0) = 0\) | M1 A1 | Shows that determinant is zero. |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x - y + 2z = 4\), \(\quad x - y - 3z = -5\), \(\quad x - y + 7z = 13\) \(\Rightarrow z = \frac{9}{5},\ x - y = \frac{2}{5}\) | M1 A1 | M1 for row operations or eliminating one variable. A1 for deriving one correct equation with two unknowns. |
| The three planes form a sheaf. | B1 | All three planes have the same common line. |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x - y + 2z = 4\), \(x - y - 3z = a\), \(\Rightarrow z = \frac{9}{5}\), \(3z + a = 4 - 2z \Rightarrow a = -5\) | B1 | Derives contradiction |
| The three planes form a triangular prism | B1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{vmatrix} 1 & -1 & 2 \\ 1 & -1 & -3 \\ 1 & -1 & 7 \end{vmatrix} = -7 - 3 + 10 - 2(0) = 0$ | **M1 A1** | Shows that determinant is zero. |
| **Total: 2** | | |
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## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x - y + 2z = 4$, $\quad x - y - 3z = -5$, $\quad x - y + 7z = 13$ $\Rightarrow z = \frac{9}{5},\ x - y = \frac{2}{5}$ | **M1 A1** | M1 for row operations or eliminating one variable. A1 for deriving one correct equation with two unknowns. |
| The three planes form a sheaf. | **B1** | All three planes have the same common line. |
| **Total: 3** | | |
## Question 2(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x - y + 2z = 4$, $x - y - 3z = a$, $\Rightarrow z = \frac{9}{5}$, $3z + a = 4 - 2z \Rightarrow a = -5$ | B1 | Derives contradiction |
| The three planes form a triangular prism | B1 | |
---2
\begin{enumerate}[label=(\alph*)]
\item Show that the system of equations
$$\begin{aligned}
& x - y + 2 z = 4 \\
& x - y - 3 z = a \\
& x - y + 7 z = 13
\end{aligned}$$
where $a$ is a constant, does not have a unique solution.
\item Given that $a = - 5$, show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
\item Given instead that $a \neq - 5$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2022 Q2 [7]}}