The sum $$\sum\limits_{k = 1}^{20} {k{1 \over {{2^k}}}} $$ is equal to

If three distinct numbers a, b, c are in G.P. and the equations ax² + 2bx + c = 0 and dx² + 2ex + ƒ = 0 have a common root, then which one of the following statements is correct?

The sum of all natural numbers 'n' such that 100 < n < 200 and H.C.F. (91, n) > 1 is :

If sin⁴$$\alpha $$ + 4 cos⁴$$\beta $$ + 2 = 4$$\sqrt 2 $$ sin $$\alpha $$ cos $$\beta $$; $$\alpha $$, $$\beta $$ $$ \in $$ [0, $$\pi $$],
then cos($$\alpha $$ + $$\beta $$) $$-$$ cos($$\alpha $$ $$-$$ $$\beta $$) is equal to :