If $q_1$ , $q_2$ , $q_3$ are roots of the equation $x^3 + 64$ = $0$ , then the value of $\left| {\begin{array}{*{20}{c}}
  {{q_1}}&{{q_2}}&{{q_3}} \\ 
  {{q_2}}&{{q_3}}&{{q_1}} \\ 
  {{q_3}}&{{q_1}}&{{q_2}} 
\end{array}} \right|$ is

If the system of linear equations $2 \mathrm{x}+2 \mathrm{ay}+\mathrm{az}=0$ ; $2 x+3 b y+b z=0$ ; $2 \mathrm{x}+4 \mathrm{cy}+\mathrm{cz}=0$ ; where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then 

The values of $x $ in the following determinant equation, $\left| {\,\begin{array}{*{20}{c}}{a + x}&{a – x}&{a – x}\\{a – x}&{a + x}&{a – x}\\{a – x}&{a – x}&{a + x}\end{array}\,} \right| = 0$ are

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then det. $[( I+ B)^{50} -50B]$ is equal to

Let $\omega $ be a complex number such that  $2\omega + 1 = z$ where $z = \sqrt { – 3} $ . If $\left| {\begin{array}{*{20}{c}}1&1&1\\1&{ – {\omega ^2} – 1}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^7}}\end{array}} \right| = 3k$ then $k$ is equal to :