NIMCET 2014 Question Paper with solutions

We have,
-2x + y + z = I ... (i)
x - 2y + z = m ... (ii)
x + y - 2z = n ... (iii)
Given l + m + n = 0
Coefficient matrix will be
$\left[{\mathit{A}}_{\mathit{k}}\right]$ = $\left[\begin{array}{cccc}-2& 1& 1& \mathit{l}\\ 1& -2& 1& \mathit{m}\\ 1& 1& -2& \mathit{n}\end{array}\right]$
Applying ${\mathit{R}}_1$ → ${\mathit{R}}_1+{\mathit{R}}_2+{\mathit{R}}_3$
= $\left[\begin{array}{cccc}0& 0& 0& \mathit{l}+\mathit{m}+\mathit{n}\\ 1& -2& 1& \mathit{m}\\ 1& 1& -2& \mathit{n}\end{array}\right]$
$\left[{\mathit{A}}_{\mathit{k}}\right]$ = $\left[\begin{array}{cccc}0& 0& 0& 0\\ 1& -2& 1& \mathit{m}\\ 1& 1& -2& \mathit{n}\end{array}\right]$
[l + m + n = 0]
$\mathit{p}\left[{\mathit{A}}_{\mathit{k}}\right]$ = 2
i.e. $\mathit{p}\left[{\mathit{A}}_{\mathit{k}}\right]$ < number of variables.
Hence, it has infinitely many solutions