Three digit numbers $x17, 3y6$ and $12z$ where $x, y, z$ are integers from $0$ to $9$, are divisible by a fixed constant $k$. Then the determinant $\left| {\,\begin{array}{*{20}{c}}x&3&1\\7&6&z\\1&y&2\end{array}\,} \right|$ + $48$ must be divisible by

If the system of linear equations $x+y+3 z=0$

$x+3 y+k^{2} z=0$

$3 x+y+3 z=0$

has a non-zero solution $(x, y, z)$ for some $k \in R ,$ then $x +\left(\frac{ y }{ z }\right)$ is equal to

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

If the sum of squares of all real values of $\alpha$, for which the lines $2 x-y+3=0,6 x+3 y+1=0$ and $\alpha x+2 y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to $\mathrm{p}$ is $………$

If $a, b, c$ are non-zero real numbers and if the system of equations $(a – 1 )x = y + z,$  $(b – 1 )y = z + x ,$ $(c – 1 )z= x + y,$ has a non-trivial solution, then $ab + bc + ca$ equals