If the sum of the first $$2n$$ terms of the A.P.$$2,5,8,......,$$ is equal to the sum of the first $$n$$ terms of the A.P.$$57,59,61,.....,$$ then $$n$$ equals

The maximum value of $$\left( {\cos {\alpha _1}} \right).\left( {\cos {\alpha _2}} \right).....\left( {\cos {\alpha _n}} \right),$$ under the restrictions
$$0 \le {\alpha _1},{\alpha _2},....,{\alpha _n} \le {\pi \over 2}$$ vand $$\left( {\cot {\alpha _1}} \right).\left( {\cot {\alpha _2}} \right)....\left( {\cot {\alpha _n}} \right) = 1$$ is

If $$\alpha + \beta = \pi /2$$ and $$\beta + \gamma = \alpha ,$$ then $$\tan \,\alpha \,$$ equals

The complex numbers $${z_1},\,{z_2}$$ and $${z_3}$$ satisfying $${{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt 3 } \over 2}\,$$ are the vertices of a triangle which is