Value of $$c$$ satisfying the conditions and conclusions of Rolle's theorem for the function $$\mathrm{f}(x)=x \sqrt{x+6}, x \in[-6,0]$$ is

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with these three vertices is equilateral, equals ___________.

If the direction cosines $$l, \mathrm{~m}, \mathrm{n}$$ of two lines are connected by relations $$l-5 \mathrm{~m}+3 \mathrm{n}=0$$ and $$7 l^2+5 \mathrm{~m}^2-3 \mathrm{n}^2=0$$, then value of $$l+\mathrm{m}+\mathrm{n}$$ is

Let $$\mathrm{f}(x)=\log (\sin x), 0 < x < \pi$$ and $$\mathrm{g}(x)=\sin ^{-1}\left(\mathrm{e}^{-x}\right), x \geq 0$$. If $$\alpha$$ is a positive real number such that $$\mathrm{a}=(\mathrm{fog})^{\prime}(\alpha)$$ and $$\mathrm{b}=(\mathrm{fog})(\alpha)$$, then