If abc = 1, then $${\frac{1}{{1 + a + {b^{ - 1}}}} + }$$ $${\frac{1}{{1 + b + {c^{ - 1}}}} + }$$ $${\frac{1}{{1 + c + {a^{ - 1}}}}}$$ = ?
A. 0
B. 1
C. $$\frac{1}{{{\text{ab}}}}$$
D. ab
Answer: Option B
Solution(By Examveda Team)
Given expression,$${\frac{1}{{1 + a + {b^{ - 1}}}} + }$$ $${\frac{1}{{1 + b + {c^{ - 1}}}} + }$$ $${\frac{1}{{1 + c + {a^{ - 1}}}}}$$
$$ = \frac{1}{{1 + a + {b^{ - 1}}}} + $$ $$\frac{b^{ - 1}}{{{b^{ - 1}} + 1 + {b^{ - 1}}{c^{ - 1}}}} + $$ $$\frac{1}{{a + ac + 1}}$$
$$ = \frac{1}{{1 + a + {b^{ - 1}}}} + $$ $$\frac{{{b^{ - 1}}}}{{1 + {b^{ - 1}} + a}} + $$ $$\frac{a}{{a + {b^{ - 1}} + 1}}$$
$$\eqalign{ & = \frac{{1 + a + {b^{ - 1}}}}{{1 + a + {b^{ - 1}}}} \cr & = 1 \cr} $$
$$\left[ {\because abc = 1 \Rightarrow {{\left( {bc} \right)}^{ - 1}} = a \Rightarrow {b^{ - 1}}{c^{ - 1}} = a,{\text{and }}ac = {b^{ - 1}}} \right]$$
This Question Belongs to Arithmetic Ability >> Surds And Indices
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