If there exists a $k^{\text{th}}$ order non-singular submatrix in a matrix $P$ of order $m \times n$,then the rank $(\rho)$ of $P$

If the matrix $A=\left[\begin{array}{cccc}1 & 2 & 3 & 0 \\ 2 & 4 & 3 & 2 \\ 3 & 2 & 1 & 3 \\ 6 & 8 & 7 & \alpha\end{array}\right]$ is of rank $3$,then $\alpha$ equals to

The rank of $A = \begin{bmatrix} 1 & x & x+1 \\ 2x & x^2-x & x^2+x \\ 3x(x-1) & x(x^2-3x+2) & x(x^2-1) \end{bmatrix}$ is:

If $f(x) = \left| \begin{array}{ccc} \sin(x + \alpha) & \sin(x + \beta) & \sin(x + \gamma) \\ \cos(x + \alpha) & \cos(x + \beta) & \cos(x + \gamma) \\ \sin(\alpha + \beta) & \sin(\beta + \gamma) & \sin(\gamma + \alpha) \end{array} \right|$ and $f(10) = 10$,then $f(\pi)$ is equal to

If $\alpha, \beta, \text{ and } \gamma$ are real numbers,then $D = \begin{vmatrix} 1 & \cos(\beta - \alpha) & \cos(\gamma - \alpha) \\ \cos(\alpha - \beta) & 1 & \cos(\gamma - \beta) \\ \cos(\alpha - \gamma) & \cos(\beta - \gamma) & 1 \end{vmatrix} = $

If $A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}$ where $a = 7^x$,$b = 7^{7^x}$,$c = 7^{7^{7^x}}$,then $\int |A| \, dx$ (where $|A|$ is the determinant of the matrix $A$) is equal to: