$\begin{aligned} \frac{1}{1 + x^{(b - a)} + x^{(c - a)}} + \\ \frac{1}{1 + x^{(a - b)} + x^{(c - b)}} + \\ \frac{1}{1 + x^{(b - c)} + x^{(a - c)}} = ? \end{aligned}$ $\eqalign{ & \frac{1}{{1 + {x^{\left( {b - a} \right)}} + {x^{\left( {c - a} \right)}}}} + \cr & \frac{1}{{1 + {x^{\left( {a - b} \right)}} + {x^{\left( {c - b} \right)}}}} + \cr & \frac{1}{{1 + {x^{\left( {b - c} \right)}} + {x^{\left( {a - c} \right)}}}} = ? \cr}$

10
21
3x^{a - b - c}
4None of these

Answer:- 1

Explanation:-

Solution:

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

$\eqalign{ & {\text{Given}}\,{\text{exp}}{\text{.}} = \cr & \frac{1}{{\left( {1 + \frac{{{x^b}}}{{{x^a}}} + \frac{{{x^c}}}{{{x^a}}}} \right)}} + \frac{1}{{\left( {1 + \frac{{{x^a}}}{{{x^b}}} + \frac{{{x^c}}}{{{x^b}}}} \right)}} + \frac{1}{{\left( {1 + \frac{{{x^b}}}{{{x^c}}} + \frac{{{x^a}}}{{{x^c}}}} \right)}} \cr & = \frac{{{x^a}}}{{\left( {{x^a} + {x^b} + {x^c}} \right)}} + \frac{{{x^b}}}{{\left( {{x^a} + {x^b} + {x^c}} \right)}} + \frac{{{x^c}}}{{\left( {{x^a} + {x^b} + {x^c}} \right)}} \cr & = \frac{{\left( {{x^a} + {x^b} + {x^c}} \right)}}{{\left( {{x^a} + {x^b} + {x^c}} \right)}} \cr & = 1 \cr}$