Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of $$\Delta$$ABC, then (R + r) is equal to :

Let f : R $$-$$ {3} $$ \to $$ R $$-$$ {1} be defined by f(x) = $${{x - 2} \over {x - 3}}$$.

Let g : R $$ \to $$ R be given as g(x) = 2x $$-$$ 3. Then, the sum of all the values of x for which f^$$-$$1(x) + g^$$-$$1(x) = $${{13} \over 2}$$ is equal to :

Let S₁ be the sum of first 2n terms of an arithmetic progression. Let S₂ be the sum of first 4n terms of the same arithmetic progression. If (S₂ $$-$$ S₁) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :

If $$\sum\limits_{r = 1}^{10} {r!({r^3} + 6{r^2} + 2r + 5) = \alpha (11!)} $$, then the value of $$\alpha$$ is equal to ___________.