If the system of equations $3x - 2y + z = 0$,$\lambda x - 14y + 15z = 0$,and $x + 2y + 3z = 0$ has a non-trivial solution,then $\lambda = $

Let $\alpha_1, \alpha_2$ be two values of $\alpha$ for which the system $2\alpha x + y = 5$,$x - 6y = \alpha$,and $x + y = 2$ is consistent. Then $|2(\alpha_1 + \alpha_2)|$ is -

If the system of equations $kx + (k+1)y + (k-1)z = 0$,$(k-1)x + (k+2)y + kz = 0$,and $(k+1)x + ky + (k+2)z = 0$ has a non-trivial solution,then the sum of all possible values of $k$ is:

The values of $\lambda$ and $\mu$ for which the system of linear equations $x+y+z=2$,$x+2y+3z=5$,and $x+3y+\lambda z=\mu$ has infinitely many solutions are,respectively:

Let $a, b, c$ be positive real numbers. The following system of equations in $x, y, z$:
$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$
$\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
$-\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
has:

The system of equations $x+3by+bz=0$, $x+2ay+az=0$ and $x+4cy+cz=0$ has