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 :

In the expansion of $${\left( {{x \over {\cos \theta }} + {1 \over {x\sin \theta }}} \right)^{16}}$$, if $${\ell _1}$$ is the least value of the term independent of x when $${\pi \over 8} \le \theta \le {\pi \over 4}$$ and $${\ell _2}$$ is the least value of the term independent of x when $${\pi \over {16}} \le \theta \le {\pi \over 8}$$, then the ratio $${\ell _2}$$ : $${\ell _1}$$ is equal to :

If $$\alpha $$ and $$\beta $$ be the coefficients of x⁴ and x² respectively in the expansion of
$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then