If $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = k(a + b + c)({a^2} + {b^2} + {c^2}$ $ – bc – ca – ab)$, then $k =$

The system of equations  $-k x+3 y-14 z=25$  $-15 x+4 y-k z=3$  $-4 x+y+3 z=4$  is consistent for all $k$ in the set

If ${2^{{a_1}}},{2^{{a_2}}},{2^{{a_3}}},{……2^{{a_n}}}$ are in $G.P.$ then $\left| {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\ 
  {{a_{n + 1}}}&{{a_{n + 2}}}&{{a_{n + 3}}} \\ 
  {{a_{2n + 1}}}&{{a_{2n + 2}}}&{{a_{2n + 3}}} 
\end{array}} \right|$ is equal to

If $A = \left[ {\begin{array}{*{20}{c}}
1&{\sin \,\theta }&1\\
{ – \,\sin \,\theta }&1&{\sin \,\theta }\\
{ – 1}&{ – \,\sin \,\theta }&1
\end{array}} \right];$ then for all $\theta \, \in \,\left( {\frac{{3\pi }}{4},\frac{{5\pi }}{4}} \right),$ det $(A)$ lies in the interval

$\left| {\,\begin{array}{*{20}{c}}5&3&{ – 1}\\{ – 7}&x&{ – 3}\\9&6&{ – 2}\end{array}\,} \right| = 0$, then $ x$ is equal to