If $\left|\begin{array}{ccc}a^{2} & b c & c^{2}+a c \\ a^{2}+a b & b^{2} & c a \\ a b & b^{2}+b c & c^{2}\end{array}\right|=k a^{2} b^{2} c^{2}$, then $k=$

If $\Delta = \begin{vmatrix} x+y+z^2 & x^2+y+z & x+y^2+z \\ z^2 & x^2 & y^2 \\ x+y & y+z & x+z \end{vmatrix}$,(where $x \neq y \neq z$ and $x, y, z \in \mathbb{R} - \{0\}$),then $\Delta = $ . . . . . . .

The determinant $\left| {\begin{array}{*{20}{c}}{{a^2} + {x^2}}&{ab}&{ca}\\{ab}&{{b^2} + {x^2}}&{bc}\\{ca}&{bc}&{{c^2} + {x^2}}\end{array}} \right|$ is a divisor of:

Show that $\left|\begin{array}{ccc}a & b & c \\ a+2x & b+2y & c+2z \\ x & y & z\end{array}\right|=0$.

$f(x) = \left| \begin{array}{ccc} 1 & x & x+1 \\ 2x & x(x-1) & (x+1)x \\ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1) \end{array} \right|$, then $f(100)$ is equal to:

If $a - 2b + c = 1$,then the value of $\left| \begin{array}{ccc} x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c \end{array} \right|$ is