The foot of the perpendicular from the point $$(1,2,3)$$ on the line $$\mathbf{r}=(6 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})$$ has the coordinates

Water is running in a hemispherical bowl of radius $$180 \mathrm{~cm}$$ at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is $$120 \mathrm{~cm}$$ ? (1 decimeter $$=10 \mathrm{~m}$$)

The solution of $$\sin x+\sin 5 x=\sin 3 x$$ in $$(0, \pi / 2)$$ are

If $$(1+\sqrt{1+x}) \tan x=1+\sqrt{1-x}$$, then $$\sin 4 x$$ is