If ${a^2} + {b^2} + {c^2} + ab + bc + ca \leq 0\,\forall a,\,b,\,c\, \in \,R$ , then the value of determinant $\left| {\begin{array}{*{20}{c}}
  {{{(a + b + c)}^2}}&{{a^2} + {b^2}}&1 \\ 
  1&{{{(b + c + 2)}^2}}&{{b^2} + {c^2}} \\ 
  {{c^2} + {a^2}}&1&{{{(c + a + 2)}^2}} 
\end{array}} \right|$ 

If ${\left| {\,\begin{array}{*{20}{c}}4&1\\2&1\end{array}\,} \right|^2} = \left| {\,\begin{array}{*{20}{c}}3&2\\1&x\end{array}\,} \right| - \left| {\,\begin{array}{*{20}{c}}x&3\\{ - 2}&1\end{array}\,} \right|$, then $ x =$

If the system of linear equation $x - 4y + 7z = g,\,3y - 5z = h, \,-\,2x + 5y - 9z = k$ is
consistent, then

Let $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha\end{array}\right]$ and $|2 A|^3=2^{21}$ where $\alpha, \beta \in Z$, Then a value of $\alpha $ is

The number of real values $\lambda$, such that the system of linear equations $2 x-3 y+5 z=9$  ;  $x+3 y-z=-18$    ; $3 x-y+\left(\lambda^{2}-1 \lambda \mid\right) z=16$ has no solution, is :-

The value of the determinant $\left| {\,\begin{array}{*{20}{c}}1&a&{b + c}\\1&b&{c + a}\\1&c&{a + b}\end{array}\,} \right|$is