Statement $-1 :$Determinant of a skew-symmetric matrix of order $3$ is zero

Statement $-2 :$ For any matrix $A,$ $\det \left( {{A^T}} \right) = {\rm{det}}\left( A \right)$ and $\det \left( { - A} \right) = - {\rm{det}}\left( A \right)$ Where $\det \left( A \right) = A$. Then :

$\left| {\,\begin{array}{*{20}{c}}{19}&{17}&{15}\\9&8&7\\1&1&1\end{array}\,} \right| = $

If $\omega $ is cube root of unity, then root of the equation $\left| {\begin{array}{*{20}{c}}
  {x + 2}&\omega &{{\omega ^2}} \\ 
  \omega &{x + 1 + {\omega ^2}}&1 \\ 
  {{\omega ^2}}&1&{x + 1 + \omega } 
\end{array}} \right| = 0$ is 

Consider system of equations in $x$ , $y$ and $z$

$12x + by + cz = 0$ ;   $ax + 24y + cz = 0$  ;   $ax + by + 36z = 0$ .

(where $a$ , $b$ , $c$ are real numbers, $a \ne 12$ , $b \ne 24$ , $c \ne 36$ ).

If system of equation has solution and $z \ne 0$, then value of  $\frac{1}{{a - 12}} + \frac{2}{{b - 24}} + \frac{3}{{c - 36}}$ is

If $\left| {\begin{array}{*{20}{c}}
  {\cos 2x}&{{{\sin }^2}x}&{\cos 4x} \\ 
  {{{\sin }^2}x}&{\cos 2x}&{{{\cos }^2}x} \\ 
  {\cos 4x}&{{{\cos }^2}x}&{\cos 2x} 
\end{array}} \right| = {a_0} + {a_1}\sin x + {a_2}{\sin ^2}x + .....$ then $a_0$ is equal to

If $a,b,c$ be positive and not all equal, then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right|$ is