A value of $\theta  \in  (0, \pi /3)$, for which $\left| {\begin{array}{*{20}{c}}
  {1 + {{\cos }^2}\,\theta }&{{{\sin }^2}\,\theta }&{4\,\cos \,6\theta } \\ 
  {{{\cos }^2}\,\theta }&{1 + {{\sin }^2}\,\theta }&{4\,\cos \,6\theta } \\ 
  {{{\cos }^2}\,\theta }&{{{\sin }^2}\,\theta }&{1 + 4\,\cos \,6\theta } 
\end{array}} \right| = 0$, is

If $a,b,c$ are unequal what is the condition that the value of the following determinant is zero $\Delta = \left| {\,\begin{array}{*{20}{c}}a&{{a^2}}&{{a^3} + 1}\\b&{{b^2}}&{{b^3} + 1}\\c&{{c^2}}&{{c^3} + 1}\end{array}\,} \right|$

At what value of $x,$ will $\left| {\,\begin{array}{*{20}{c}}{x + {\omega ^2}}&\omega &1\\\omega &{{\omega ^2}}&{1 + x}\\1&{x + \omega }&{{\omega ^2}}\end{array}\,} \right| = 0$

By using properties of determinants, show that:

$\left|\begin{array}{ccc}
1 & 1 & 1 \\
a & b & c \\
a^{3} & b^{3} & c^{3}
\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)$

The value of $\theta$ lying between $\theta = 0$ and $\theta = \pi /2$ and satisfying the equation : $\left| {\,\begin{array}{*{20}{c}} {1\,\, + \,\,{{\sin }^2}\,\theta }&{{{\cos }^2}\,\theta }&{4\,\sin \,4\,\theta }\\ {{{\sin }^2}\,\theta }&{1\,\, + \,\,{{\cos }^2}\,\theta }&{4\,\sin \,4\,\theta }\\ {{{\sin }^2}\,\theta }&{{{\cos }^2}\,\theta }&{1\,\, + \,\,4\,\sin \,4\,\theta } \end{array}\,} \right|$ $= 0$ are :