Let $${a_1},{a_2},.......,{a_{30}}$$ be an A.P.,

$$S = \sum\limits_{i = 1}^{30} {{a_i}} $$ and $$T = \sum\limits_{i = 1}^{15} {{a_{\left( {2i - 1} \right)}}} $$.

If $$a_5$$ = 27 and S - 2T = 75, then $$a_{10}$$ is equal to :

For $$x \in R - \left\{ {0,1} \right\}$$, Let f₁(x) = $$1\over x$$, f₂ (x) = 1 – x

and f₃ (x) = $$1 \over {1 - x}$$ be three given

functions. If a function, J(x) satisfies

(f₂ o J o f₁) (x) = f₃ (x) then J(x) is equal to :

$$\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}$$

For any $$\theta \in \left( {{\pi \over 4},{\pi \over 2}} \right)$$, the expression

$$3{(\cos \theta - \sin \theta )^4}$$$$ + 6{(\sin \theta + \cos \theta )^2} + 4{\sin ^6}\theta $$

equals :