The value of $k \in R$,for which the system of linear equations
$3x - y + 4z = 3$
$x + 2y - 3z = -2$
$6x + 5y + kz = -3$
has infinitely many solutions,is:

For the system of linear equations:
$2x - y + 3z = 5$
$3x + 2y - z = 7$
$4x + 5y + \alpha z = \beta$
Which of the following is $NOT$ correct?

The sum of three numbers is $6$. Thrice the third number when added to the first number gives $7$. On adding three times the first number to the sum of the second and third numbers,we get $12$. The product of these numbers is:

Let $\lambda \in R$. The system of linear equations
$2x_{1} - 4x_{2} + \lambda x_{3} = 1$
$x_{1} - 6x_{2} + x_{3} = 2$
$\lambda x_{1} - 10x_{2} + 4x_{3} = 3$
is inconsistent for:

Investigate the values of $\lambda$ and $\mu$ for the system $x+2y+3z=6, x+3y+5z=9, 2x+5y+\lambda z=\mu$ and match the values in List-$I$ with the items in List-$II$.

List-$I$	List-$II$
$(A)$ $\lambda=8, \mu \neq 15$	$1$. Infinitely many solutions
$(B)$ $\lambda \neq 8, \mu \in R$	$2$. No solution
$(C)$ $\lambda=8, \mu=15$	$3$. Unique solution

The existence of the unique solution of the system $x + y + z = \lambda$,$5x - y + \mu z = 10$,and $2x + 3y - z = 6$ depends on