The number of $\theta \in (0, 4\pi)$ for which the system of linear equations $3(\sin 3\theta)x - y + z = 2$,$3(\cos 2\theta)x + 4y + 3z = 3$,and $6x + 7y + 7z = 9$ has no solution is:

Examine the consistency of the system of equations: $5x - y + 4z = 5$,$2x + 3y + 5z = 2$,and $5x - 2y + 6z = -1$.

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 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

The system of linear equations $x+y+z=6, x+2y+3z=10$ and $x+2y+az=b$ has no solutions when

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix,where $a_{ij} = \begin{cases} (-1)^{j-i} & \text{if } i < j \\ 2 & \text{if } i = j \\ (-1)^{i+j} & \text{if } i > j \end{cases}$. Then $\det(3 \operatorname{Adj}(2 A^{-1}))$ is equal to: