If the system of equations

$2 x+y-z=5$

$2 x-5 y+\lambda z=\mu$

$x+2 y-5 z=7$

has infinitely many solutions, then $(\lambda+\mu)^2+(\lambda-\mu)^2$ is equal to

Value of $\left| {\begin{array}{*{20}{c}}
  0&{x - y}&{x - z} \\ 
  {y - x}&0&{y - z} \\ 
  {z - x}&{z - y}&0 
\end{array}} \right|$ is

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

The number of positive integral solutions $\left| {\,\,\begin{array}{*{20}{c}}{1 - \lambda }&2&1\\{ - 3}&\lambda &{ - 2}\\2&{ - 2}&{1 + \lambda }\end{array}\,\,} \right|$ $= 0$ is

If $f(\theta ) =\left| {\begin{array}{*{20}{c}}
1&{\cos {\mkern 1mu} \theta }&1\\
{ - \sin {\mkern 1mu} \theta }&1&{ - \cos {\mkern 1mu} \theta }\\
{ - 1}&{\sin {\mkern 1mu} \theta }&1
\end{array}} \right|$ and $A$ and $B$ are respectively the maximum and the minimum values of $f(\theta )$, then $(A , B)$ is equal to

The system of linear equation $x + y + z = 2, 2x + 3y + 2z = 5$, $2x + 3y + (a^2 -1)\,z = a + 1$ then