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Question:

Let λ be a real number for which the system of linear equations x+y+z=6, 4x+λy-λz=λ-2, 3x+2y-4z=-5 has infinitely many solutions. Then λ is a root of the quadratic equation

λ²−3λ−4=0

λ²−λ−6=0

λ²+λ−6=0

λ²+3λ−4=0

Solution:

Correct option is B. λ²−λ−6=0
D=0|1 1 1| = 0
|4 λ -λ| => λ = 3
|3 2 -4|
Now,(3)²−3(3)−4=−4≠0
(3)²+3(3)−4=14≠0
(3)²+3−6=6≠0
(3)²−3−6=0
Therefore, Option B is correct; i.e λ=3 is the root of λ²−λ−6=0