If $[x]$ denotes the greatest integer  $ \leq x$, then the system of linear equations
$[sin \,\theta ] x + [-cos\,\theta ] y = 0$

$[cot \,\theta ] x + y = 0$

If $\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2\end{array}\right|=\frac{9}{8}(103 x+81)$, then $\lambda$, $\frac{\lambda}{3}$ are the roots of the equation

If $p + q + r = 0 = a + b + c$, then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}{pa}&{qb}&{rc}\\{qc}&{ra}&{pb}\\{rb}&{pc}&{qa}\end{array}\,} \right|$ is

The system of equations $\begin{array}{l}\alpha x + y + z = \alpha – 1\\x + \alpha y + z = \alpha – 1\\x + y + \alpha z = \alpha – 1\end{array}$ has no solution, if $\alpha $ 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