A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of $${\left( {{2^{1/3}} + {1 \over {2{{\left( 3 \right)}^{1/3}}}}} \right)^{10}}$$ is :

Let S_n = 1 + q + q² + . . . . . + qⁿ and T_n = 1 + $$\left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \right)^2}$$ + . . . . . .+ $${\left( {{{q + 1} \over 2}} \right)^n}$$ where q is a real number and q $$ \ne $$ 1. If   ¹⁰¹C₁ + ¹⁰¹C₂ . S₁ + .... + ¹⁰¹C₁₀₁ . S₁₀₀ = $$\alpha $$T₁₀₀ then $$\alpha $$ is equal to

Let (x + 10)⁵⁰ + (x $$-$$ 10)⁵⁰ = a₀ + a₁x + a₂x² + . . . . + a₅₀x⁵⁰, for all x $$ \in $$ R; then $${{{a_2}} \over {{a_0}}}$$ is equal to

The sum of the real values of x for which the middle term in the binomial expansion of $${\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8}$$ equals 5670 is :