Standard +0.8 This question requires recognizing that the first and third equations represent parallel planes (third is -3/2 times the first), understanding the geometric interpretation of no unique solution, and analyzing how the parameter k affects the system. It goes beyond routine Gaussian elimination to require conceptual understanding of linear dependence and geometric interpretation of solution sets.
1 Show that the system of equations
$$\begin{aligned}
14 x - 4 y + 6 z & = 5 \\
x + y + k z & = 3 \\
- 21 x + 6 y - 9 z & = 14
\end{aligned}$$
where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically.
Evaluates determinant. Can expand along any row e.g. \(-(36-36)+(-126+126)+k(84-84)\). If using row operations, they must show an inconsistent system for M1. All their row operations must be correct for A1.
Two parallel planes, not identical.
B1
Other plane not parallel.
B1
Total: 4
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{vmatrix} 14 & -4 & 6 \\ 1 & 1 & k \\ -21 & 6 & -9 \end{vmatrix} = 14\begin{vmatrix} 1 & k \\ 6 & -9 \end{vmatrix} + 4\begin{vmatrix} 1 & k \\ -21 & -9 \end{vmatrix} + 6\begin{vmatrix} 1 & 1 \\ -21 & 6 \end{vmatrix}$ $= 14(-9-6k)+4(-9+21k)+6(6+21)=0$ | M1 A1 | Evaluates determinant. Can expand along any row e.g. $-(36-36)+(-126+126)+k(84-84)$. If using row operations, they must show an inconsistent system for M1. All their row operations must be correct for A1. |
| Two parallel planes, not identical. | B1 | |
| Other plane not parallel. | B1 | |
| **Total: 4** | | |
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1 Show that the system of equations
$$\begin{aligned}
14 x - 4 y + 6 z & = 5 \\
x + y + k z & = 3 \\
- 21 x + 6 y - 9 z & = 14
\end{aligned}$$
where $k$ is a constant, does not have a unique solution and interpret this situation geometrically.\\
\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q1 [4]}}