The roots of the determinant equation (in $x$) $\left| \begin{array}{ccc} a & a & x \\ m & m & m \\ b & x & b \end{array} \right| = 0$ are:

Evaluate the determinant: $\left|\begin{array}{rrr}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$

$\begin{aligned} & \text { If }\left|\begin{array}{ccc}n^2 & (n+1)^2 & (n+2)^2 \\ (n+1)^2 & (n+2)^2 & (n+3)^2 \\ (n+2)^2 & (n+3)^2 & (n+4)^2\end{array}\right|=\Delta \text { and } \\ & \left|\begin{array}{ccc}1 & -4 & 7 \\ -2 & 3 & -5 \\ 3 & x & -3\end{array}\right|=2 \Delta+1, \text { then } x=\end{aligned}$

The greatest value of $c \in R$ for which the system of linear equations $x - cy - cz = 0$,$cx - y + cz = 0$,$cx + cy - z = 0$ has a non-trivial solution,is

If $C = 2 \cos \theta$,then the value of the determinant $\Delta = \begin{vmatrix} C & 1 & 0 \\ 1 & C & 1 \\ 6 & 1 & C \end{vmatrix}$ is

If $a, b, c$ are in $A.P.$,then the value of $\left| \begin{array}{ccc} x+2 & x+3 & x+a \\ x+4 & x+5 & x+b \\ x+6 & x+7 & x+c \end{array} \right|$ is