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 $\left| {\,\begin{array}{*{20}{c}}a&b&{a + b}\\b&c&{b + c}\\{a + b}&{b + c}&0\end{array}\,} \right| = 0$; then $a,b,c$ are in

Let $A=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right),$ where $[t]$ denotes the greatest integer less than or equal to $\mathrm{t}$. If $\operatorname{det}(\mathrm{A})=192$, then the set of values of $\mathrm{x}$ is the interval

The sum of distinct values of $\lambda$ for which the system of equations

$(\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0$

$(\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0$

$2 x+(3 \lambda+1) y+3(\lambda-1) z=0$

has non-zero solutions, is

Consider the system of linear equations

$-x+y+2 z=0$

$3 x-a y+5 z=1$

$2 x-2 y-a z=7$

Let $S_{1}$ be the set of all $\mathrm{a} \in {R}$ for which the system is inconsistent and $S_{2}$ be the set of all $a \in {R}$ for which the system has infinitely many solutions. If $n\left(S_{1}\right)$ and $n\left(S_{2}\right)$ denote the number of elements in $S_{1}$ and $\mathrm{S}_{2}$ respectively, then

If $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$ ,then