The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $${{27} \over {19}}$$.Then the common ratio of this series is :

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x² – c² = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

The value of r for which ²⁰C_r ²⁰C₀ + ²⁰C_r$$-$$1 ²⁰C₁ + ²⁰C_r$$-$$2 ²⁰C₂ + . . . . .+ ²⁰C₀ ²⁰C_r  is maximum, is

The value of the integral $$\int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx$$ (where [x] denotes the greatest integer less than or equal to x) is