| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | System of three linear equations |
| Difficulty | Standard +0.8 This question requires understanding of linear dependence, matrix rank, and geometric interpretation of degenerate systems. While solving 3×3 systems is standard, recognizing non-uniqueness (determinant = 0 or row operations showing dependence) and proving consistency (showing the equations represent three planes meeting in a line) requires deeper conceptual understanding beyond routine calculation, placing it moderately above average difficulty. |
| Spec | 4.03l Singular/non-singular matrices4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Shows determinant is zero or uses row operations to obtain e.g. \(x+2y+3z=1\), \(y+2z=1\) | M1 A1 | Shows that determinant is zero or row operations to obtain e.g. \(x+2y+3z=1\), \(y+2z=1\) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Uses *all three* equations to reduce to one equation with two unknowns | M1 A1 | Reducing to two equations scores M1 A0 |
| The three planes form a sheaf | B1 | Accept clear sketch or the three planes intersect along a common line |
| [3] |
## Question 1:
### Part (a)
$$\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix} = (5\times9-8\times6)-2(4\times9-7\times6)+3(4\times8-7\times5)$$
$$= -3+12-9=0$$
| Answer | Marks | Guidance |
|--------|-------|----------|
| Shows determinant is zero or uses row operations to obtain e.g. $x+2y+3z=1$, $y+2z=1$ | **M1 A1** | Shows that determinant is zero or row operations to obtain e.g. $x+2y+3z=1$, $y+2z=1$ |
| | **[2]** | |
---
### Part (b)
$$x+2y+3z=1,$$
$$4x+5y+6z=1, \Rightarrow z=\tfrac{1}{3}(1-x-2y)\Rightarrow y=-2x-1$$
$$7x+8y+9z=1,$$
The three planes form a sheaf.
| Answer | Marks | Guidance |
|--------|-------|----------|
| Uses *all three* equations to reduce to one equation with two unknowns | **M1 A1** | Reducing to two equations scores M1 A0 |
| The three planes form a sheaf | **B1** | Accept clear sketch or the three planes intersect along a common line |
| | **[3]** | |1
\begin{enumerate}[label=(\alph*)]
\item Show that the system of equations
$$\begin{array} { r }
x + 2 y + 3 z = 1 \\
4 x + 5 y + 6 z = 1 \\
7 x + 8 y + 9 z = 1
\end{array}$$
does not have a unique solution.
\item Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q1 [5]}}