If $A = \left[ {\begin{array}{*{20}{c}}
1&1\\
1&1
\end{array}} \right]$ and $\det ({A^n} – I) = 1 – {\lambda ^n}\,,\,n \in N$ then $\lambda $ is

Given the system of equation $a(x + y + z)=x,b(x + y + z) = y, c(x + y + z) = z$ where $a,b,c$  are non-zero real numbers. If the real numbers $x,y,z$ are such that $xyz \neq 0,$ then  $(a + b + c)$ is equal to-

The value of $\left| {\,\begin{array}{*{20}{c}}1&{\cos (\beta – \alpha )}&{\cos (\gamma – \alpha )}\\{\cos (\alpha – \beta )}&1&{\cos (\gamma – \beta )}\\{\cos (\alpha – \gamma )}&{\cos (\beta – \gamma )}&1\end{array}} \right|$ is

Let $[.]$ , $ \{.\} $ and $sgn$$(.)$ denotes greatest integer function, fractional part function and signum function respectively, then value of determinant

$\left| {\begin{array}{*{20}{c}}
  {\left[ \pi  \right]}&{amp(1 + i\sqrt 3 )}&1 \\ 
  1&0&2 \\ 
  {\operatorname{sgn} ({{\cot }^{ – 1}}x)}&1&{\{ \pi \} } 
\end{array}} \right|$ is-

The number of real values of $\lambda $ for which the system of linear equations $2x + 4y – \lambda  z = 0$ ;$4x + \lambda y + 2z = 0$ ; $\lambda x + 2y+ 2z = 0$ has infinitely many solutions, is