The solutions of the equation $\left|\begin{array}{ccc}1+\sin ^{2} x & \sin ^{2} x & \sin ^{2} x \\ \cos ^{2} x & 1+\cos ^{2} x & \cos ^{2} x \\ 4 \sin 2 x & 4 \sin 2 x & 1+4 \sin 2 x\end{array}\right|=0$ for $(0 < x < \pi)$ are:

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 $A$ is a skew-symmetric matrix of order $3$ and $X$ is another matrix of the same order,then $|XA + AX^T|$ is (where $|P|$ denotes the determinant of matrix $P$).

Matrix $A$ satisfies $A^2 = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,$A^n$ is equal to $(n \in N)$:

If $A = \begin{bmatrix} 0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0 \end{bmatrix}$ and $I$ is the identity matrix of order $2$,show that $I+A = (I-A) \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$.

If $\omega$ is a root of the equation $x+\frac{1}{x}+1=0$,then the value of the determinant $\left|\begin{array}{ccc}1 & 1+\omega & 1+\omega+\omega^2 \\ 3 & 4+3 \omega & 5+4 \omega+3 \omega^2 \\ 6 & 9+6 \omega & 11+9 \omega+6 \omega^2\end{array}\right|$ is equal to