In $\triangle ABC$ with usual notations, if $1 + \tan\left(\dfrac{A}{2}\right)\tan\left(\dfrac{B}{2}\right) = \dfrac{k}{s}$ (where $s$ is the semi-perimeter), then the value of $k$ is...

Let $A = \begin{bmatrix} 2k-1 & 1 & 1 \\ 0 & 2k-1 & 1 \\ 0 & 0 & 2k-1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2k-1 & 1 \\ 1-2k & 0 & k \\ -1 & -k & 0 \end{bmatrix}$ where k is a real number. If $\det(\text{adj} A) + \det(\text{adj} B) = 11^6$, then the value of $k - 5$ is equal to...

If $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$, then the matrix $A^{-3}$ when $\theta = \dfrac{\pi}{6}$ is equal to...

If $\tan^{-1}\left[\dfrac{\sqrt{5-2\sqrt{6}}}{1+\sqrt{6}}\right] = \dfrac{\pi}{3} - \tan^{-1}(k)$, then $\sec^{-1}(k) = ...$