A) 1
B) \[-\,\,1\]
C) 0
D) \[\frac{1}{3}\]
Correct Answer: A
Solution :
| \[\frac{1}{1+p+{{q}^{-1}}}+\frac{1}{1+q+{{r}^{-1}}}+\frac{1}{1+r+{{p}^{-1}}}\] |
| \[=\frac{1}{1+p+\frac{1}{q}}+\frac{1}{1+q+\frac{1}{r}}+\frac{1}{1+r+\frac{1}{p}}\] |
| \[=\frac{q}{1+pq+q}+\frac{r}{r+rq+1}+\frac{p}{p+rp+1}\] |
| \[=\frac{q}{1+pq+q}+\frac{r}{\frac{1}{pq}+\frac{1}{p}+1}+\frac{p}{p+\frac{1}{q}+1}\] |
| \[(\because pqr=1)\] |
| \[=\frac{q}{1+pq+q}+\frac{rpq}{1+q+pq}+\frac{pq}{pq+1+q}\] |
| \[=\frac{q+rpq+pq}{1+pq+q}\] |
| \[=\frac{q+1+pq}{1+pq+q}=1\] |
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