$\left| {\,\begin{array}{*{20}{c}}{bc}&{bc' + b'c}&{b'c'}\\{ca}&{ca' + c'a}&{c'a'}\\{ab}&{ab' + a'b}&{a'b'}\end{array}\,} \right|$ is equal to

If $1,\omega ,{\omega ^2}$ are the cube roots of unity, then $\Delta = \left| {\,\begin{array}{*{20}{c}}1&{{\omega ^n}}&{{\omega ^{2n}}}\\{{\omega ^n}}&{{\omega ^{2n}}}&1\\{{\omega ^{2n}}}&1&{{\omega ^n}}\end{array}\,} \right|$ is equal to

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-

If $A = \left[ {\begin{array}{*{20}{c}}\alpha &2\\2&\alpha \end{array}} \right]$ and $|{A^3}|$=125, then $\alpha = $

If ${a_1},{a_2},{a_3}.....{a_n}....$ are in $G.P.$ then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}{\log {a_n}}&{\log {a_{n + 1}}}&{\log {a_{n + 2}}}\\{\log {a_{n + 3}}}&{\log {a_{n + 4}}}&{\log {a_{n + 5}}}\\{\log {a_{n + 6}}}&{\log {a_{n + 7}}}&{\log {a_{n + 8}}}\end{array}\,} \right|$ is

Find area of the triangle with vertices at the point given in each of the following: $(1,0),(6,0),(4,3)$