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

If $\left| {\,\begin{array}{*{20}{c}}{{x^2} + x}&{x + 1}&{x - 2}\\{2{x^2} + 3x - 1}&{3x}&{3x - 3}\\{{x^2} + 2x + 3}&{2x - 1}&{2x - 1}\end{array}\,} \right| = Ax - 12$, then the value of $A $ is

If the system of equations

$2 x+y-z=5$

$2 x-5 y+\lambda z=\mu$

$x+2 y-5 z=7$

has infinitely many solutions, then $(\lambda+\mu)^2+(\lambda-\mu)^2$ is equal to

If $A_1B_1C_1,\, A_2B_2C_2,\, A_3B_3C_3$ are three digit number each of which is divisible by $k$ and $\Delta  = \left| {\begin{array}{*{20}{c}}
  {{A_1}{\kern 1pt} }&{{B_1}}&{{C_1}} \\ 
  {{A_2}}&{{B_2}}&{{C_2}} \\ 
  {{A_3}}&{{B_3}}&{{C_3}} 
\end{array}} \right|$ ; then $\Delta $ is divisible by

The system of equations $kx + y + z =1$ $x + ky + z = k$ and $x + y + zk = k ^{2}$ has no solution if $k$ is equal to

If the lines $x + 2ay + a = 0, x + 3by + b = 0$ and $x + 4cy + c = 0$ are concurrent, then $a, b$  and $c$ are in :-