Examine the consistency of the system of equations. $2 x-y=5$ ; $x+y=4$

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 the system of the following equations  $\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4$ ; $\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1$  ; $\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$

Let $S$ be the set of values of $\lambda$, for which the system of equations

$6 \lambda x-3 y+3 z=4 \lambda^2$

$2 x+6 \lambda y+4 z=1$

$3 x+2 y+3 \lambda z=\lambda \text { has no solution. Then } 12 \sum_{\lambda \in S}|\lambda|$ is equal to $………..$.

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2+x-1=0$. Consider the set $T=\{1, \alpha, \beta\}$. For a $3 \times 3$ matrix $M=\left(a_{\ell}\right) 3 \times 3_3$, define $R_l=a_{l 1}+a_{l 2}+a_\beta$ and $C_j=a_{1 j}+a_{2 l}+a_{3 j}$ for $i=1,2,3$ and $j=1,2,3$

Match each entry in $List-I$ to the correct entry in $List-II$.

The correct option is