Binomial Theorem

If the fourth term in the binomial expansion of $${\left( {{2 \over x} + {x^{{{\log }_8}x}}} \right)^6}$$ (x > 0) is 20 × 8⁷, then a value of x is :

If the fourth term in the binomial expansion of $${\left( {\sqrt {{x^{\left( {{1 \over {1 + {{\log }_{10}}x}}} \right)}}} + {x^{{1 \over {12}}}}} \right)^6}$$ is equal to 200, and x > 1, then the value of x is :

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 :

If n is the number of irrational terms in the expansion of $${\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}$$, then (n $$-$$ 1) is divisible by :

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 -

If the third term in the binomial expansion of $${\left( {1 + {x^{{{\log }_2}x}}} \right)^5}$$ equals 2560, then a possible value of x is -

If the constant term in the binomial expansion of $${\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}$$ is 405, then |k| equals :

If the term independent of x in the expansion of $${\left( {{3 \over 2}{x^2} - {1 \over {3x}}} \right)^9}$$ is k, then 18 k is equal to :

If the number of integral terms in the expansion of (3^1/2 + 5^1/8)^n is exactly 33, then the least value of n is :

Let $$\alpha$$ > 0, $$\beta$$ > 0 be such that $$\alpha$$³ + $$\beta$$² = 4. If the maximum value of the term independent of x in the binomial expansion of $${\left( {\alpha {x^{{1 \over 9}}} + \beta {x^{ - {1 \over 6}}}} \right)^{10}}$$ is 10k, then k is equal to :