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 :

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

1. A

${\lambda }^{2}-\lambda -6=0$

2. B

${\lambda }^{2}-3\lambda -4=0$

3. C

${\lambda }^{2}+\lambda -6=0$

4. D

${\lambda }^{2}+3\lambda -4=0$

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### Solution:

For infinitely many solutions

Putting $\lambda =3$ is ${\lambda }^{2}-\lambda -6$, we get

${\lambda }^{2}-\lambda -6=9-3-6=0$

is a root of the quadratic equation

${\lambda }^{2}-\lambda -6=0$.

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