For two positive real numbers a and b such that $${1 \over {{a^2}}} + {1 \over {{b^3}}} = 4$$, then minimum value of the constant term in the expansion of $${\left( {a{x^{{1 \over 8}}} + b{x^{ - {1 \over {12}}}}} \right)^{10}}$$ is :

Let n $$\ge$$ 5 be an integer. If 9ⁿ $$-$$ 8n $$-$$ 1 = 64$$\alpha$$ and 6ⁿ $$-$$ 5n $$-$$ 1 = 25$$\beta$$, then $$\alpha$$ $$-$$ $$\beta$$ is equal to

If the constant term in the expansion of

$${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}$$ is 2^k.l, where l is an odd integer, then the value of k is equal to:

The term independent of x in the expansion of

$$(1 - {x^2} + 3{x^3}){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}},\,x \ne 0$$ is :