The domain of the real valued function $f(x)=\log _{\sqrt{2}}\left(\sqrt{x^2+x}+\sqrt{x^2-x}\right)$ is

$t_1, t_2, t_3, \ldots, t_n$ are positive integers, $S_n=t_1+t_2+t_3+\ldots+t_n$, $S_1=1^2, S_2=3^2, S_3=6^2, S_4=10^2, S_5=15^2$ and similarly other terms are there. Following this pattern, if $S_{10}=k^2$ then $k=$

If $x=\alpha, y=\beta, z=\gamma$ is the solution of the system of equations $2 x+3 y+z=-1,3 x+y+z=4$, $x-3 y-2 z=1$, then the value of $\beta$ is

The positive value of ' $a$ ' for which the system of linear homogeneous equations $x+a y+z=0, a x+2 y-z=0$, $2 x+3 y+z=0$ has non-trivial solution is