JEE Main · 2010 · 11 Apr 2010 · Offline · Matrices

Form the coefficient determinant $D=\left|\begin{array}{ccc}1& 2& 1\\ 2& 3& 1\\ 3& 5& 2\end{array}\right|$; since the third row is the sum of the first two, $D=0$, so there is no unique solution. Checking consistency, adding the first two equations gives $3x_1+5x_2+2x_3=6$, whereas the third equation says the same combination equals $1$. These contradict each other, so the system is inconsistent and has no solution.