The area of the region enclosed between the parabolas y² = 2x $$-$$ 1 and y² = 4x $$-$$ 3 is

The coefficient of x¹⁰¹ in the expression $${(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}$$, x > 0, is

Water is being filled at the rate of 1 cm³ / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm² / sec) at which the wet conical surface area of the vessel increases is

If $${b_n} = \int_0^{{\pi \over 2}} {{{{{\cos }^2}nx} \over {\sin x}}dx,\,n \in N} $$, then