JEE Main · 2020 · 08 Jan 2020 · Shift 1 · Matrices

The coefficient determinant $\left|\begin{array}{ccc}1& 2& 3\\ 3& 4& 5\\ 4& 4& 4\end{array}\right|$ works out to $0$, so the system has either no solution or infinitely many. The third equation's coefficients are a combination of the first two, since $-2(1,2,3)+2(3,4,5)=(4,4,4)$, so consistency demands the same combination of the right-hand sides, $\delta =-2(1)+2\mu =2\mu -2$. The system is inconsistent whenever $\delta \ne 2\mu -2$, and the pair $(\mu ,\delta )=(4,3)$ gives $2\mu -2=6\ne 3$, making the system inconsistent.