If the solution of the equation $$\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1, x \in\left(0, \frac{\pi}{2}\right)$$, is $$\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$$, where $$\alpha$$, $$\beta$$ are integers, then $$\alpha+\beta$$ is equal to :

Let $$A=\left(\begin{array}{cc}\mathrm{m} & \mathrm{n} \\ \mathrm{p} & \mathrm{q}\end{array}\right), \mathrm{d}=|\mathrm{A}| \neq 0$$ and $$\mathrm{|A-d(A d j A)|=0}$$. Then

If $$\tan 15^\circ + {1 \over {\tan 75^\circ }} + {1 \over {\tan 105^\circ }} + \tan 195^\circ = 2a$$, then the value of $$\left( {a + {1 \over a}} \right)$$ is :

The line $$l_1$$ passes through the point (2, 6, 2) and is perpendicular to the plane $$2x+y-2z=10$$. Then the shortest distance between the line $$l_1$$ and the line $$\frac{x+1}{2}=\frac{y+4}{-3}=\frac{z}{2}$$ is :