NIMCET 2012 Question Paper with solutions

Let coefficient matrix
A = $\left[\begin{array}{ccc}1& {\mathit{\omega }}^2& \mathit{\omega }\\ \mathit{\omega }& 1& {\mathit{\omega }}^2\\ {\mathit{\omega }}^2& \mathit{\omega }& 1\end{array}\right]$
Since ${\mathit{\omega }}^3$= 1 , 1 + ω + ${\mathit{\omega }}^2$ = 0
Use operation,
${\mathit{R}}_2$ → ${\mathit{R}}_2$ - ω ${\mathit{R}}_1$ , ${\mathit{R}}_3$ → ${\mathit{R}}_3$ - ${\mathit{\omega }}^2{\mathit{R}}_1$
A ~ $\left[\begin{array}{ccc}1& {\mathit{\omega }}^2& \mathit{\omega }\\ 0& 1-{\mathit{\omega }}^3& {\mathit{\omega }}^2-{\mathit{\omega }}^2\\ 0& \mathit{\omega }-{\mathit{\omega }}^4& 1-{\mathit{\omega }}^3\end{array}\right]$ ~ $\left[\begin{array}{ccc}1& {\mathit{\omega }}^2& \mathit{\omega }\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
So f (A) = r = 1
and number of unknowns , n = 3
Since, r < n,so the system of equations is consistent andhas more than one solution