The maximum value of the expression $${1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}$$ is

The number of all possible values of $$\theta $$ where $$0 < \theta < \pi ,$$ for which the system of equations $$$\left( {y + z} \right)\cos {\mkern 1mu} 3\theta = \left( {xyz} \right){\mkern 1mu} \sin 3\theta $$$ $$$x\sin 3\theta = {{2\cos 3\theta } \over y} + {{2\sin 3\theta } \over z}$$$ $$$\left( {xyz} \right){\mkern 1mu} \sin 3\theta = \left( {y + 2z} \right){\mkern 1mu} \cos 3\theta + y{\mkern 1mu} sin3\theta $$$

have a solution $$\left( {{x_0},{y_0},{z_0}} \right)$$ with $${y_0}{z_0}{\mkern 1mu} \ne {\mkern 1mu} 0,$$ is

Two parallel chords of a circle of radius 2 are at a distance $$\sqrt 3 + 1$$ apart. If the chords subtend at the center , angles of $${\pi \over k}$$ and $${{2\pi } \over k},$$ where$$k > 0,$$ then the value of $$\left[ k \right]$$ is

[Note :[k] denotes the largest integer less than or equal to k ]