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

Consider the system of linear equation $x+y+z=$ $4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15$, where $\lambda, \mu \in R$. Which one of the following statements is $NOT$ correct?

The value of $\left| {\,\begin{array}{*{20}{c}}1&{\cos (\beta – \alpha )}&{\cos (\gamma – \alpha )}\\{\cos (\alpha – \beta )}&1&{\cos (\gamma – \beta )}\\{\cos (\alpha – \gamma )}&{\cos (\beta – \gamma )}&1\end{array}} \right|$ is

The value of $k \in R$, for which the following system of linear equations

$3 x-y+4 z=3$

$x+2 y-3 x=-2$

$6 x+5 y+k z=-3$

has infinitely many solutions, is:

If the system of linear equations $x + ky + 3z = 0;3x + ky – 2z = 0$ ; $2x + 4y – 3z = 0$  has a non-zero solution $\left( {x,y,z} \right)$ then $\frac{{xz}}{{{y^2}}} = $. . . . .