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

The value of r for which ²⁰C_r ²⁰C₀ + ²⁰C_r$$-$$1 ²⁰C₁ + ²⁰C_r$$-$$2 ²⁰C₂ + . . . . .+ ²⁰C₀ ²⁰C_r  is maximum, is

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 :

The positive value of $$\lambda $$ for which the co-efficient of x² in the expression x² $${\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}}$$ is 720, is -