For some a, b, c $$\in\mathbb{N}$$, let $$f(x) = ax - 3$$ and $$\mathrm{g(x)=x^b+c,x\in\mathbb{R}}$$. If $${(fog)^{ - 1}}(x) = {\left( {{{x - 7} \over 2}} \right)^{1/3}}$$, then $$(fog)(ac) + (gof)(b)$$ is equal to ____________.

Let $$S = \left\{ {\alpha :{{\log }_2}({9^{2\alpha - 4}} + 13) - {{\log }_2}\left( {{5 \over 2}.\,{3^{2\alpha - 4}} + 1} \right) = 2} \right\}$$. Then the maximum value of $$\beta$$ for which the equation $${x^2} - 2{\left( {\sum\limits_{\alpha \in s} \alpha } \right)^2}x + \sum\limits_{\alpha \in s} {{{(\alpha + 1)}^2}\beta = 0} $$ has real roots, is ____________.

Let $$\mathrm{A_1,A_2,A_3}$$ be the three A.P. with the same common difference d and having their first terms as $$\mathrm{A,A+1,A+2}$$, respectively. Let a, b, c be the $$\mathrm{7^{th},9^{th},17^{th}}$$ terms of $$\mathrm{A_1,A_2,A_3}$$, respective such that $$\left| {\matrix{ a & 7 & 1 \cr {2b} & {17} & 1 \cr c & {17} & 1 \cr } } \right| + 70 = 0$$.

If $$a=29$$, then the sum of first 20 terms of an AP whose first term is $$c-a-b$$ and common difference is $$\frac{d}{12}$$, is equal to ___________.

If the area enclosed by the parabolas $$\mathrm{P_1:2y=5x^2}$$ and $$\mathrm{P_2:x^2-y+6=0}$$ is equal to the area enclosed by $$\mathrm{P_1}$$ and $$\mathrm{y=\alpha x,\alpha > 0}$$, then $$\alpha^3$$ is equal to ____________.