If ${\Delta _r} = \left| {\begin{array}{*{20}{c}} r&{2r - 1}&{3r - 2} \\ {\frac{n}{2}}&{n - 1}&a \\ {\frac{1}{2}n\left( {n - 1} \right)}&{{{\left( {n - 1} \right)}^2}}&{\frac{1}{2}\left( {n - 1} \right)\left( {3n - 4} \right)} \end{array}} \right|$,then the value of $\sum\limits_{r = 1}^{n - 1} {{\Delta _r}} $:

Let $A$ be a $3 \times 3$ matrix such that $X^{T}AX = O$ for all nonzero $3 \times 1$ matrices $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$. If $A \left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{l}1 \\ 4 \\ -5\end{array}\right]$,$A \left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]=\left[\begin{array}{l}0 \\ 4 \\ -8\end{array}\right]$,and $\operatorname{det}(\operatorname{adj}(2(A+I)))=2^\alpha 3^\beta 5^\gamma$,where $\alpha, \beta, \gamma \in \mathbb{N}$,then $\alpha^2+\beta^2+\gamma^2$ is

The number of $3 \times 3$ non-singular matrices,with four entries as $1$ and all other entries as $0$,is

If $\begin{bmatrix} -1 & 2 & b \\ a & 5 & 6 \\ 3 & c & 7 \end{bmatrix}$ is a symmetric matrix,then $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} =$

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 $A = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{bmatrix}$ and $\alpha, \beta, \gamma$ are the roots of the characteristic equation $|A - xI| = 0$,then $\alpha^2 + \beta^2 + \gamma^2 = $