Let $\Omega$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either $0$ or $1$ . Five of these entries are $1$ and four of them are $0$ .

$1.$ The number of matrices in $\Omega$ is

$(A)$ $12$ $(B)$ $6$ $(C)$ $9$ $(D)$ $3$

$2.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations

$A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has a unique solution, is

$(A)$ less than $4$

$(B)$ at least $4$ but less than $7$

$(C)$ at least $7$ but less than $10$

$(D)$ at least $10$

$3.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations

$A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ is inconsistent, is

$(A)$ $0$ $(B)$ more than $2$ $(C)$ $2$ $(D)$ $1$

 Solve system of linear equations, using matrix method. $x-y+2 z=7$ ;  $3 x+4 y-5 z=-5$ ; $2 x-y+3 z=12$

Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations

$a x+2 y=\lambda$

$3 x-2 y=\mu$Which of the following statement($s$) is(are) correct?

($A$) If $a=-3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$

($B$) If $a \neq-3$, then the system has a unique solution for all values of $\lambda$ and $\mu$

($C$) If $\lambda+\mu=0$, then the system has infinitely many solutions for $a=-3$

($D$) If $\lambda+\mu \neq 0$, then the system has no solution for $a=-3$

If the system of linear equations

$7 x+11 y+\alpha z=13$

$5 x+4 y+7 z=\beta$

$175 x+194 y+57 z=361$

has infinitely many solutions, then $\alpha+\beta+2$ is equal to

The number of values of $k$ for which the system of linear equations, $(k + 2) x + 10y = k,\,\,kx + (k + 3)y = k – 1$  has no solution, is