Standard +0.8 This requires recognizing that the coefficient matrix is singular (row 3 is -1.5 times row 1), then checking consistency of the system, and providing geometric interpretation of three planes. It's a standard Further Maths linear algebra question requiring multiple conceptual steps but follows a well-established procedure for analyzing systems without unique solutions.
Show that the system of equations
$$14x - 4y + 6z = 5,$$
$$x + y + kz = 3,$$
$$-21x + 6y - 9z = 14,$$
where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically. [4]
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.
Answer
Marks
Two parallel planes, not identical.
B1
Other plane not parallel.
B1
4
Answer
Marks
Guidance
Question
Answer
Marks
Question 1:
1 | 14 −4 6
1 k 1 k 1 1
1 1 k =14 +4 +6
6 −9 −21 −9 −21 6
−21 6 −9
=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
4
Question | Answer | Marks | Guidance
Show that the system of equations
$$14x - 4y + 6z = 5,$$
$$x + y + kz = 3,$$
$$-21x + 6y - 9z = 14,$$
where $k$ is a constant, does not have a unique solution and interpret this situation geometrically. [4]
\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q1 [4]}}