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

The least value of the product $xyz$ for which the determinant $\left| \begin{array}{ccc} x & 1 & 1 \\ 1 & y & 1 \\ 1 & 1 & z \end{array} \right|$ is non-negative,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 value of $\det A$, where $A = \begin{bmatrix} 1 & \cos \theta & 0 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}$, lies

The number of real values of $t$ such that the system of homogeneous equations
$\begin{aligned}
t x+(t+1) y+(t-1) z &=0 \\
(t+1) x+t y+(t+2) z &=0 \\
(t-1) x+(t+2) y+t z &=0
\end{aligned}$
has non-trivial solutions is

The constant term of the polynomial $\left|\begin{array}{ccc}x+3 & x & x+2 \\ x & x+1 & x-1 \\ x+2 & 2x & 3x+1\end{array}\right|$ is