The area of a triangle with vertices $A(k, 1)$,$B(2, 4)$,and $C(1, 1)$ is $6$ sq. units. Find the value of $k$.

If $A = \begin{bmatrix} \alpha & 2 \\ 2 & \alpha \end{bmatrix}$ and $|A^3| = 125$,then $\alpha = $ . . . . . .

If $a, b, c$ are distinct and rational numbers,then the value of the determinant $\left| \begin{array}{ccc} (a^2 + b^2 + c^2) & (ab + bc + ca) & (ab + bc + ca) \\ (ab + bc + ca) & (a^2 + b^2 + c^2) & (ab + bc + ca) \\ (ab + bc + ca) & (ab + bc + ca) & (a^2 + b^2 + c^2) \end{array} \right|$ is always:

Prove that the determinant $\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$ is independent of $\theta$.

$\left| {\begin{array}{*{20}{c}}{{{\log }_3}512}&{{{\log }_4}3}\\{{{\log }_3}8}&{{{\log }_4}9}\end{array}} \right| \times \left| {\begin{array}{*{20}{c}}{{{\log }_2}3}&{{{\log }_8}3}\\{{{\log }_3}4}&{{{\log }_3}4}\end{array}} \right| = $

If $A = \begin{bmatrix} 3 & 2 & 4 \\ 1 & 2 & 1 \\ 3 & 2 & 6 \end{bmatrix}$ and $A_{ij}$ are cofactors of the elements $a_{ij}$ of $A$,then $a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13}$ is equal to