The sum of the series

$${1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}$$ when x = 2 is :

If $${{}^{20}{C_r}}$$ is the co-efficient of x^r in the expansion of (1 + x)²⁰, then the value of $$\sum\limits_{r = 0}^{20} {{r^2}.{}^{20}{C_r}} $$ is equal to :

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

The equation $$\arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4}$$ represents a circle with :