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

Let $\left| \begin{array}{ccc} (a-x)^2 & (a-y)^2 & (a-z)^2 \\ (b-x)^2 & (b-y)^2 & (b-z)^2 \\ (c-x)^2 & (c-y)^2 & (c-z)^2 \end{array} \right| = \frac{-351}{8}$. If $x, y, z$ are the roots of the equation $8t^3 - 62t^2 + 43t - 7 = 0$ and $a, b, c$ are distinct numbers,then the value of $|(a-b)(b-c)(c-a)|$ is:

Let $A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -5\alpha & 0 \\ 0 & 4\alpha & -2\alpha \end{bmatrix} + \text{adj}(A)$. If $\det(B) = 66$, then $\det(\text{adj}(A))$ equals:

Suppose $\det \begin{bmatrix} \sum_{k=0}^n k & \sum_{k=0}^n {^nC_k} k^2 \\ \sum_{k=0}^n {^nC_k} k & \sum_{k=0}^n {^nC_k} 3^k \end{bmatrix} = 0$ holds for some positive integer $n$. Then $\sum_{k=0}^n \frac{{^nC_k}}{k+1}$ equals

Let $A = \begin{bmatrix} 5 & \sin^2 \theta & \cos^2 \theta \\ -\sin^2 \theta & -5 & 1 \\ \cos^2 \theta & 1 & 5 \end{bmatrix}$. Then the maximum value of $\det(A)$ is

If $A$ and $B$ are two matrices such that $AB = B$ and $BA = A,$ then ${A^2} + {B^2} = $