An ordered pair $(\alpha , \beta )$ for which the system of linear equations

$\left( {1 + \alpha } \right)x + \beta y + z = 2$ ; $\alpha x + \left( {1 + \beta } \right)y + z = 3$ ; $\alpha x  + \beta y + 2z = 2$ has a unique solution, is

Evaluate the determinants

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

Let $\lambda \in R .$ The system of linear equations

$2 x_{1}-4 x_{2}+\lambda x_{3}=1$

$x_{1}-6 x_{2}+x_{3}=2$

$\lambda x_{1}-10 x_{2}+4 x_{3}=3$  is inconsistent for 

Let $\alpha, \beta, \gamma$ be the real roots of the equation, $x ^{3}+ ax ^{2}+ bx + c =0,( a , b , c \in R$ and $a , b \neq 0)$ If the system of equations (in, $u,v,w$) given by $\alpha u+\beta v+\gamma w=0, \beta u+\gamma v+\alpha w=0$ $\gamma u +\alpha v +\beta w =0$ has non-trivial solution, then the value of $\frac{a^{2}}{b}$ is

Let the system of equations $x+2 y+3 z=5$, $2 x+3 y+z=9,4 x+3 y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2 \mu$ is equal to :

If $\left| {\begin{array}{*{20}{c}}{x - 4}&{2x}&{2x}\\{2x}&{x - 4}&{2x}\\{2x}&{2x}&{x - 4}\end{array}} \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2},$ then the ordered pair $\left( {A,B} \right) = $. . . . .