The equation $\left| {\begin{array}{*{20}{c}}{{{(1 + x)}^2}}&{{{(1 - x)}^2}}&{ - \,(2 + {x^2})}\\{2x + 1}&{3x}&{1 - 5x}\\{x + 1}&{2x}&{2 - 3x}\end{array}} \right|$ $+$ $\left| {\begin{array}{*{20}{c}}{{{(1 + x)}^2}}&{2x + 1}&{x + 1}\\{{{(1 - x)}^2}}&{3x}&{2x}\\{1 - 2x}&{3x - 2}&{2x - 3}\end{array}} \right|$ $= 0$

The system of equations $kx + 2y\,-z = 1$  ;  $(k\,-\,1)y\,-2z = 2$  ;  $(k + 2)z = 3$ has unique solution, if $k$ is equal to

Find area of the triangle with vertices at the point given in each of the following: $(-2,-3),(3,2),(-1,-8)$

If $\left| {\,\begin{array}{*{20}{c}}a&b&c\\m&n&p\\x&y&z\end{array}\,} \right| = k$, then $\left| {\,\begin{array}{*{20}{c}}{6a}&{2b}&{2c}\\{3m}&n&p\\{3x}&y&z\end{array}\,} \right| = $

Let $\omega $ be a complex number such that  $2\omega + 1 = z$ where $z = \sqrt { - 3} $ . If $\left| {\begin{array}{*{20}{c}}1&1&1\\1&{ - {\omega ^2} - 1}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^7}}\end{array}} \right| = 3k$ then $k$ is equal to :

The system of linear equations  $3 x-2 y-k z=10$; $2 x-4 y-2 z=6$ ; $x+2 y-z=5\, m$ is inconsistent if