$\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{m{a_1}}&{{b_1}}\\{{a_2}}&{m{a_2}}&{{b_2}}\\{{a_3}}&{m{a_3}}&{{b_3}}\end{array}\,} \right| = $

If the system of linear equations $2 x-3 y=\gamma+5$ ; $\alpha x+5 y=\beta+1$, where $\alpha, \beta, \gamma \in R$ has infinitely many solutions, then the value of $|9 \alpha+3 \beta+5 \gamma|$ is equal to

Let the system of linear equations $x +2 y + z =2$, $\alpha x +3 y - z =\alpha,-\alpha x + y +2 z =-\alpha$ be inconsistent. Then $\alpha$ is equal to

If $\alpha ,\beta \ne 0$ and $f\left( n \right) = {\alpha ^n} + {\beta ^n}$ and $\left| {\begin{array}{*{20}{c}}3&{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}\\{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}\\{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}&{1 + f\left( 4 \right)}\end{array}} \right|\; = K{\left( {1 - \alpha } \right)^2}$ ${\left( {1 - \beta } \right)^2}{\left( {\alpha - \beta } \right)^2}$ ,then $K=$ . . . . . .

The ordered pair $(a, b)$, for which the system of linear equations  $3 x-2 y+z=b$  ;  $5 x-8 y+9 z=3$  ;  $2 x+y+a z=-1$ has no solution, is

Consider system of equations  $ x + y -az = 1$  ;  $2x + ay + z = 1$   ; $ax + y -z = 2$