The sum of distinct values of $\lambda$ for which the system of equations
$(\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0$
$(\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0$
$2 x+(3 \lambda+1) y+3(\lambda-1) z=0$
has non-zero solutions,is

Let $[.]$,$\{.\}$ and $\operatorname{sgn}(.)$ denote the greatest integer function,fractional part function,and signum function respectively. Then,the value of the determinant $\left| {\begin{array}{*{20}{c}} {[ \pi ]} & {\operatorname{amp}(1 + i\sqrt 3 )} & 1 \\ 1 & 0 & 2 \\ {\operatorname{sgn} (\cot^{ - 1}x)} & 1 & {\{ \pi \} } \end{array}} \right|$ is:

If the system of equations $x - ky - z = 0$,$kx - y - z = 0$ and $x + y - z = 0$ has a non-zero solution,then the possible values of $k$ are:

If $a, b, c$ are in $A.P.$,then the determinant $\left|\begin{array}{lll}x+2 & x+3 & x+2a \\ x+3 & x+4 & x+2b \\ x+4 & x+5 & x+2c\end{array}\right|$ is

The values of $\theta, \lambda$ for which the following equations $\sin \theta x - \cos \theta y + (\lambda + 1)z = 0$; $\cos \theta x + \sin \theta y - \lambda z = 0$; $\lambda x + (\lambda + 1)y + \cos \theta z = 0$ have a non-trivial solution are:

The area of the triangle with vertices $(a, b)$,$(x_1, y_1)$,and $(x_2, y_2)$,where $a, x_1, x_2$ are in $G.P.$ with common ratio $r$ and $b, y_1, y_2$ are in $G.P.$ with common ratio $s$,is given by