$\left| {\,\begin{array}{*{20}{c}}1&5&\pi \\{{{\log }_e}e}&5&{\sqrt 5 }\\{{{\log }_{10}}10}&5&e\end{array}\,} \right| = $

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 $B) -$

The following system of linear equations  $7 x+6 y-2 z=0$ ; $3 x+4 y+2 z=0$ ; ${x}-2{y}-6{z}=0,$ has

The existance of the unique solution of the system of equations$2x + y + z = \beta $ , $10x - y + \alpha z = 10$ and $4x+ 3y-z =6$ depends on

Let $a,b,c$ be positive real numbers. The following system of equations in $x, y$  and $ z $ $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1$, $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1, - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$ has

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 :