If $f(x) = \left| {\begin{array}{*{20}{c}}1&x&{x + 1}\\{2x}&{x(x – 1)}&{(x + 1)x}\\{3x(x – 1)}&{x(x – 1)(x – 2)}&{(x + 1)x(x – 1)}\end{array}} \right|$ then $f(100)$ is equal to

If the minimum and the maximum values of the function $f :\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow R ,$ defined by : 

$f (\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 1 \\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 1 \\ 12 & 10 & -2\end{array}\right|$ are $m$ and $M$ respectively, then the ordered pair $( m , M )$ is equal to

If ${a_1},{a_2},{a_3},……..,{a_n},……$ are in G.P. and ${a_i} > 0$  for each $i$, then the value of the determinant $\Delta = \left| {\,\begin{array}{*{20}{c}}{\log {a_n}}&{\log {a_{n + 2}}}&{\log {a_{n + 4}}}\\{\log {a_{n + 6}}}&{\log {a_{n + 8}}}&{\log {a_{n + 10}}}\\{\log {a_{n + 12}}}&{\log {a_{n + 14}}}&{\log {a_{n + 16}}}\end{array}} \right|$ is equal to

The value of $\left| {\,\begin{array}{*{20}{c}}{265}&{240}&{219}\\{240}&{225}&{198}\\{219}&{198}&{181}\end{array}\,} \right|$ is equal to

Let the numbers $2, b, c$ be in an $A.P$ and $A = \left[ {\begin{array}{*{20}{c}}
  1&1&1 \\ 
  2&b&c \\ 
  4&{{b^2}}&{{c^2}} 
\end{array}} \right]$. If $det(A) \in [2,16]$ then $c$ lies in the interval