The value of $$\sum\limits_{k = 1}^6 {(\sin {{2\pi k} \over 7}} - i\,\cos \,{{2\pi k} \over 7})$$ is

The number of all possible triplets $$\left( {{a_1},\,{a_2},\,{a_3}} \right)$$ such that $${a_1} + {a_2}\,\,\cos \left( {2x} \right) + {a_3}{\sin ^2}\left( x \right) = 0\,$$ for all $$x$$ is

If $$a,\,b,\,c,\,d$$ and p are distinct real numbers such that $$$\left( {{a^2} + {b^2} + {c^2}} \right){p^2} - 2\left( {ab + bc + cd} \right)p + \left( {{b^2} + {c^2} + {d^2}} \right) \le 0$$$
then $$a,\,b,\,c,\,d$$

Find the set of all $$x$$ for which $${{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, > \,{1 \over {\left( {x + 1} \right)}}$$