Let $\alpha, \beta \in R$ be such that the system of linear equations $x+2y+z=5, 2x+y+\alpha z=5, 8x+4y+\beta z=18$ has no solution. Then $\frac{\beta}{\alpha}$ is equal to:

If a system of three linear equations in three unknowns,which is in the matrix equation form of $AX = D$,is inconsistent,then $\frac{\text{rank of } A}{\text{rank of } AD}$ is

If the system of linear equations $2x + 3y - z = -2$; $x + y + z = 4$; $x - y + |\lambda|z = 4\lambda - 4$ (where $\lambda \in R$) has no solution,then:

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x + y + z = 1$,$x + \lambda y + z = 1$,and $x + y + \lambda z = 1$ is inconsistent. Then,$\sum_{\lambda \in S} (|\lambda|^2 + |\lambda|)$ is equal to

Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations
$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma$
is consistent. Let $|M|$ represent the determinant of the matrix
$M=\begin{bmatrix} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$
Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent,and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.
$(1)$ The value of $|M|$ is
$(2)$ The value of $D$ is

Consider the system of linear equations $x+y+z=5$,$x+2y+\lambda^2 z=9$,and $x+3y+\lambda z=\mu$,where $\lambda, \mu \in R$. Then,which of the following statements is $NOT$ correct?