If $x$ is a positive integer,then $\Delta = \left| \begin{array}{ccc} x! & (x+1)! & (x+2)! \\ (x+1)! & (x+2)! & (x+3)! \\ (x+2)! & (x+3)! & (x+4)! \end{array} \right|$ is equal to

If $\left| \begin{array}{ccc} x+1 & 1 & 1 \\ 2 & x+2 & 2 \\ 3 & 3 & x+3 \end{array} \right| = 0$,then $x$ is

Let $\sigma_1, \sigma_2, \sigma_3$ be planes passing through the origin. Assume that $\sigma_1$ is perpendicular to the vector $(1, 1, 1)$,$\sigma_2$ is perpendicular to a vector $(a, b, c)$,and $\sigma_3$ is perpendicular to the vector $(a^2, b^2, c^2)$. What are all the positive values of $a, b$,and $c$ so that $\sigma_1 \cap \sigma_2 \cap \sigma_3$ is a single point?

If the area of the triangle with vertices $(x, 0), (1, 1)$ and $(0, 2)$ is $4$ square units,then a value of $x$ is

Find the equation of the line joining $A(1, 3)$ and $B(0, 0)$ using determinants and find $k$ if $D(k, 0)$ is a point such that the area of triangle $ABD$ is $3 \, \text{sq units}$.

$\left| {\begin{array}{*{20}{c}}1&1&1\\1&{1 + x}&1\\1&1&{1 + y}\end{array}} \right| = $