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

Let the area of the triangle with vertices $A (1, \alpha)$, $B (\alpha, 0)$ and $C (0, \alpha)$ be $4\, sq.$ units. If the point $(\alpha,-\alpha),(-\alpha, \alpha)$ and $\left(\alpha^{2}, \beta\right)$ are collinear, then $\beta$ is equal to

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

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 $n$  be the number of values of $x$ for which
matrix $\Delta (x) =\left[ {\begin{array}{*{20}{c}}
{ – x}&x&2\\
2&x&{ – x}\\
x&{ – 2}&{ – x}
\end{array}} \right]$ will be singular, then $det(\Delta\,(n))$ is

$($ where $det(B)$ denotes determinant of Matrix $B) -$