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

The value of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6,3 x+5 y+5 z=26, x+2 y+\lambda z=\mu$ has no solution, are :

The number of integers $x$ satisfying $-3 x^4+\operatorname{det}\left[\begin{array}{ccc}1 & x & x^2 \\ 1 & x^2 & x^4 \\ 1 & x^3 & x^6\end{array}\right]=0$ is equal to

The roots of the equation $\left| {\,\begin{array}{*{20}{c}}1&4&{20}\\1&{ - 2}&5\\1&{2x}&{5{x^2}}\end{array}\,} \right| = 0$ are

Evaluate $\Delta=\left|\begin{array}{ccc}0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$

If the system of linear equations  $2 x + y - z =7$ ; $x-3 y+2 z=1$  ; $x +4 y +\delta z = k$, where $\delta, k \in R$  has infinitely many solutions, then $\delta+ k$ is equal to