If the coefficient of x¹⁰ in the binomial expansion of $${\left( {{{\sqrt x } \over {{5^{{1 \over 4}}}}} + {{\sqrt 5 } \over {{x^{{1 \over 3}}}}}} \right)^{60}}$$ is $${5^k}\,.\,l$$, where l, k $$\in$$ N and l is co-prime to 5, then k is equal to _____________.

$${A_1} = \left\{ {(x,y):|x| \le {y^2},|x| + 2y \le 8} \right\}$$ and

$${A_2} = \left\{ {(x,y):|x| + |y| \le k} \right\}$$. If 27 (Area A₁) = 5 (Area A₂), then k is equal to :

If the sum of the first ten terms of the series

$${1 \over 5} + {2 \over {65}} + {3 \over {325}} + {4 \over {1025}} + {5 \over {2501}} + \,\,....$$

is $${m \over n}$$, where m and n are co-prime numbers, then m + n is equal to ______________.

A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x $$-$$ y + 4 = 0, then the area of R is ____________.