Question

If abc = 1, then find the value of

$\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}$ [4 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

Given,

$\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}$

$=\frac{1}{1+a+b^{-1}}+\frac{b^{-1}}{b^{-1}+b{b}^{-1}+b^{-1}{c}^{-1}}+\frac{a}{a+ac+a{a}^{-1}}$

[Multiplying 2nd term by $\frac{b^{-1}}{b^{-1}}$ and 3rd term by $\frac{a}{a}$]

$=\frac{1}{1+a+b^{-1}}+\frac{b^{-1}}{b^{-1}+1+a}+\frac{a}{a+b^{-1}+1}$

$=\frac{1+b^{-1}+a}{1+a+b^{-1}}$

$=1$