Let $$\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]$$ be such that
$$2\cos \theta \left( {1 - \sin \,\varphi } \right) = {\sin ^2}\theta \,\,\left( {\tan {\theta \over 2} + \cot {\theta \over 2}} \right)\cos \varphi - 1,\,\tan \left( {2\pi - \theta } \right) > 0$$ and $$ - 1 < \sin \theta \, < - {{\sqrt 3 } \over 2},$$

then $$\varphi $$ cannot satisfy

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle $${x^2}\, + \,{y^2} = 9$$ is

The ellipse $${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ is inscribed in a rectangle $$R$$ whose sides are parallel to the coordinate axes. Another ellipse $${E_2}$$ passing through the point $$(0, 4)$$ circumscribes the rectangle $$R$$. The eccentricity of the ellipse $${E_2}$$ is