The set of all values of $\lambda $ for which the system of linear equations $x - 2y - 2z = \lambda x$ ; $x + 2y + z = \lambda y$ ; $-x - y = \lambda z$ has non zero solutions.

Let $d \in R$, and  $A = \left[ {\begin{array}{*{20}{c}} { - 2}&{4 + d}&{\left( {\sin \,\theta } \right) - 2}\\ 1&{\left( {\sin \,\theta } \right) + 2}&d\\ 5&{\left( {2\sin \,\theta } \right) - d}&{\left( { - \sin \,\theta } \right) + 2 + 2d} \end{array}} \right]$, $\theta  \in \left[ {0,2\pi } \right]$. If the minimum value of det $(A)$ is $8$, then a value of $d$ is

If $\alpha , \beta \, and \, \gamma$ are real numbers , then $D = \left|{\begin{array}{*{20}{c}}1&{\cos \,(\beta \, - \,\alpha )}&{\cos \,(\gamma \, - \,\alpha )}\\{\cos \,(\alpha \, - \,\beta )}&1&{\cos \,(\gamma \, - \,\beta )}\\{\cos \,(\alpha \, - \,\gamma )}&{\cos \,(\beta \, - \,\gamma )}&1 \end{array}} \right|$ =

If the system of equation $3x - 2y + z = 0$, $\lambda x - 14y + 15z = 0$, $x + 2y + 3z = 0$ have a non-trivial solution, then $\lambda = $

if $\left| \begin{gathered}
   - 6\ \ \,\,1\ \ \,\,\lambda \ \  \hfill \\
  \,0\ \ \,\,\,\,3\ \ \,\,7\ \  \hfill \\
   - 1\ \ \,\,0\ \ \,\,5\ \  \hfill \\ 
\end{gathered}  \right| = 5948 $, then $\lambda $  is

The value of $x,$ if $\left| {\,\begin{array}{*{20}{c}}{ - x}&1&0\\1&{ - x}&1\\0&1&{ - x}\end{array}\,} \right| = 0 $ is equal to