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1. $\frac{1}{1 + x^{(b - a)} + x^{(c - a)}}$ $+ \frac{1}{1 + x^{(a - b)} + x^{(c - b)}}$ $+ \frac{1}{1 + x^{(b - c)} + x^{(a - c)}} = ?$

A. 0

B. 1

C. x^{a - b - c}

D. None of these

Solution:

Given exp. =

= \frac{1}{(1 + \frac{x^{b}}{x^{a}} + \frac{x^{c}}{x^{a}})} +

\frac{1}{(1 + \frac{x^{a}}{x^{b}} + \frac{x^{c}}{x^{b}})} +

\frac{1}{(1 + \frac{x^{b}}{x^{c}} + \frac{x^{a}}{x^{c}})}

= \frac{x^{a}}{(x^{a} + x^{b} + x^{c})} +

\frac{x^{b}}{(x^{a} + x^{b} + x^{c})} +

\frac{x^{c}}{(x^{a} + x^{b} + x^{c})}

\begin{aligned} = \frac{x^{a} + x^{b} + x^{c}}{x^{a} + x^{b} + x^{c}} \\ = 1 \end{aligned}

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