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 $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ 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 $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ is inconsistent,is
$(A) 0$ $(B)$ more than $2$ $(C) 2$ $(D) 1$

Let $A$ and $B$ be two $3 \times 3$ real matrices such that $(A^{2}-B^{2})$ is an invertible matrix. If $A^{5}=B^{5}$ and $A^{3} B^{2}=A^{2} B^{3}$,then the value of the determinant of the matrix $A^{3}+B^{3}$ is equal to:

Let $a, b$ and $c$ be three real numbers satisfying $\begin{bmatrix} a & b & c \end{bmatrix} \begin{bmatrix} 1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$ $(E)$.
$1.$ If the point $P(a, b, c)$, with reference to $(E)$, lies on the plane $2x+y+z=1$, then the value of $7a+b+c$ is
$(A) 0$ $(B) 12$ $(C) 7$ $(D) 6$
$2.$ Let $\omega$ be a solution of $x^3-1=0$ with $\operatorname{Im}(\omega)>0$. If $a=2$ with $b$ and $c$ satisfying $(E)$, then the value of $\frac{3}{\omega^a}+\frac{1}{\omega^b}+\frac{3}{\omega^c}$ is equal to
$(A) -2$ $(B) 2$ $(C) 3$ $(D) -3$
$3.$ Let $b=6$, with $a$ and $c$ satisfying $(E)$. If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2+bx+c=0$, then $\sum_{n=0}^{\infty} \left(\frac{1}{\alpha}+\frac{1}{\beta}\right)^n$ is
$(A) 6$ $(B) 7$ $(C) \frac{6}{7}$ $(D) \infty$
Give the answer for questions $1, 2$ and $3$.

If $A = \begin{bmatrix} 2 & 3 \\ 0 & -1 \end{bmatrix}$,then the value of $\det(A^4) + \det(A^{10} - (\operatorname{adj}(2A))^{10})$ is equal to ........

Consider the following linear equations:
$ax+by+cz=0$,$bx+cy+az=0$,$cx+ay+bz=0$
Match the conditions/expressions in Column $I$ with statements in Column $II$:

Column $I$	Column $II$
$(A)$ $a+b+c \neq 0$ and $a^2+b^2+c^2=ab+bc+ca$	$(p)$ The equations represent planes meeting only at a single point.
$(B)$ $a+b+c=0$ and $a^2+b^2+c^2 \neq ab+bc+ca$	$(q)$ The equations represent the line $x=y=z$.
$(C)$ $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq ab+bc+ca$	$(r)$ The equations represent identical planes.
$(D)$ $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$	$(s)$ The equations represent the whole of the three-dimensional space.

Let $R = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ be a non-zero $3 \times 3$ matrix,where $x \sin \theta = y \sin \left(\theta + \frac{2 \pi}{3}\right) = z \sin \left(\theta + \frac{4 \pi}{3}\right) \neq 0$,$\theta \in (0, 2 \pi)$. For a square matrix $M$,let $\text{trace}(M)$ denote the sum of all the diagonal entries of $M$. Then,among the statements:
$(I) \text{ Trace}(R) = 0$
$(II) \text{ If trace}(\text{adj}(\text{adj}(R))) = 0, \text{ then } R \text{ has exactly one non-zero entry.}$