If the system of linear equations,


If the system of linear equations,


$x+2 y+3 z=10$

$3 x+2 y+\lambda z=\mu$

has more than two solutions, then $\mu-\lambda^{2}$ is equal to_________.



$x+2 y+3 z=10$...(ii)

$3 x+2 y+\lambda z=\mu$...(iii)

From (i) and (ii),

If $z=0 \Rightarrow x+y=6$ and $x+2 y=10$

$\Rightarrow y=4, x=2$


If $y=0 \Rightarrow x+z=6$ and $x+3 z=10$

$\Rightarrow \quad z=2$ and $x=4$


So, $3 x+2 y+\lambda z=\mu$, must pass through $(2,4,0)$ and


So, $6+8=\mu \Rightarrow \mu=14$

and $12+2 \lambda=\mu$

$12+2 \lambda=14 \Rightarrow \lambda=1$

So, $\mu-\lambda^{2}=14-1=13$

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