The system of equations $4x + y – 2z = 0\ ,\ x – 2y + z = 0$ ; $x + y – z =0 $ has

The determinant $\left| {\,\begin{array}{*{20}{c}}a&b&{a – b}\\b&c&{b – c}\\2&1&0\end{array}\,} \right|$ is equal to zero if $a,b,c$ are in

The set of all values of $\lambda $ for which the system of linear equations $x – 2y – 2z = \lambda x$ ; $x + 2y + z = \lambda y$ ; $-x – y = \lambda z$ has non zero solutions.

Evaluate the determinants : $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$

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