If $\left[\begin{array}{rrr}1 & -1 & x \\ 1 & x & 1 \\ x & -1 & 1\end{array}\right]$ has no inverse,then the real value of $x$ is

If $\omega \neq 1$ is a cube root of unity,then the value of the determinant $\left|\begin{array}{ccc}\omega+\omega^2 & \omega^2+\omega^9 & \omega^9+\omega \\ \omega^{27}+\omega^{31} & \omega^{31}+\omega^{17} & \omega^{17}+\omega^{27} \\ \omega^{30}+\omega^{41} & \omega^{41}+\omega^{19} & \omega^{19}+\omega^{30}\end{array}\right|$ is:

If $2\left|\begin{array}{ll}\sin ( A + B ) & \cos ( A + B ) \\ \cos ( A - B ) & \sin ( A - B )\end{array}\right|+\sqrt{3}= 0$,then $A =$ . . . . . . .

The area of $\triangle PQR$ with the vertices $P(k, 1)$,$Q(2, 4)$,and $R(1, 1)$ is $3$ sq. units. Then,$k = $ . . . . . . .

Let $0 \neq a \in \mathbb{Z}$ and $A = \begin{bmatrix} a & a & a-y \\ a & a+x & a \\ a & a & a \end{bmatrix}$ be a matrix. Then,the equation $\det(A) = 16$ represents:

If $\omega$ is an imaginary cube root of unity and $\left|\begin{array}{ccc}x+\omega^2 & \omega & 1 \\ \omega & \omega^2 & 1+x \\ 1 & x+\omega & \omega^2\end{array}\right|=0$,then one of the values of $x$ is