An ordered pair $(\alpha, \beta)$ for which the system of linear equations $(1 + \alpha)x + \beta y + z = 2$; $\alpha x + (1 + \beta)y + z = 3$; $\alpha x + \beta y + 2z = 2$ has a unique solution,is

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

Let $S$ be the set of values of $\lambda$,for which the system of equations
$6 \lambda x - 3 y + 3 z = 4 \lambda^2$
$2 x + 6 \lambda y + 4 z = 1$
$3 x + 2 y + 3 \lambda z = \lambda$
has no solution. Then $12 \sum_{\lambda \in S} |\lambda|$ is equal to $...........$.

If the system of equations $\begin{bmatrix} \alpha & -1 & -1 \\ 1 & -\alpha & -1 \\ 1 & -1 & -\alpha \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \alpha-1 \\ \alpha-1 \\ \alpha-1 \end{bmatrix}$ is inconsistent, then $\alpha=$

Given the system of linear equations: $2x + 3y + 4z = 9$,$4x + 9y + 3z = 10$,and $5x + 10y + 5z = 11$. The value of $x$ is given by:

If $x=a, y=b, z=c$ is the solution of the system of simultaneous linear equations $x+y+z=4$, $x-y+z=2$, and $x+2y+2z=1$, then $ab+bc+ca=$