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 :

A
$λ^{2} - λ - 6 = 0$
B
$λ^{2} - 3 λ - 4 = 0$
C
$λ^{2} + λ - 6 = 0$
D
$λ^{2} + 3 λ - 4 = 0$

Solution:

For infinitely many solutions

$\begin{array}{l} |\begin{array}{ccc} 1 & 1 & 1 \\ 4 & λ & - λ \\ 3 & 2 & - 4 \end{array}| = 0 \\ \Rightarrow 1 (- 4 λ + 2 λ) - 1 (- 16 + 3 λ) + 1 (8 - 3 λ) = 0 \\ \Rightarrow - 2 λ + 16 - 3 λ + 8 - 3 λ = 0 \Rightarrow - 8 λ + 24 = 0 \Rightarrow λ = 3 \end{array}$

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

$λ^{2} - λ - 6 = 9 - 3 - 6 = 0$

$∴ λ = 3$ is a root of the quadratic equation

$λ^{2} - λ - 6 = 0$ .

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

Solution:

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