Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N=N M$. Further, if $M \neq N^2$ and $M^2=N^4$, then

$(A)$ determinant of $\left( M ^2+ MN ^2\right)$ is $0$

$(B)$ there is a $3 \times 3$ non-zero matrix $U$ such that $\left( M ^2+ MN ^2\right) U$ is the zero matrix

$(C)$ determinant of $\left( M ^2+ MN ^2\right) \geq 1$

$(D)$ for a $3 \times 3$ matrix $U$, if $\left( M ^2+ MN ^2\right) U$ equals the zero matrix then $U$ is the zero matrix

The system of equations  $-k x+3 y-14 z=25$  $-15 x+4 y-k z=3$  $-4 x+y+3 z=4$  is consistent for all $k$ in the set

If system of equations $kx + 2y – z = 2,$$\left( {k – 1} \right)x + ky + z = 1,x + \left( {k – 1} \right)y + kz = 3$ has only one solution, then number of possible real value$(s)$ of $k$ is –
 

The number of positive integral solutions $\left| {\,\,\begin{array}{*{20}{c}}{1 – \lambda }&2&1\\{ – 3}&\lambda &{ – 2}\\2&{ – 2}&{1 + \lambda }\end{array}\,\,} \right|$ $= 0$ is

Let the system of linear equations $x +2 y + z =2$, $\alpha x +3 y – z =\alpha,-\alpha x + y +2 z =-\alpha$ be inconsistent. Then $\alpha$ is equal to