Question 57

If $$pqr = 1$$ then
$$\left(\left(\frac{1}{(1 + p + q^{-1})}\right) + \left(\frac{1}{(1 + q + r^{-1})}\right) + \left(\frac{1}{(1 + r + p^{-1})}\right)\right)$$ is equal to

Solution

Given : $$pqr=1$$

Expression : $$\left(\left(\frac{1}{(1 + p + q^{-1})}\right) + \left(\frac{1}{(1 + q + r^{-1})}\right) + \left(\frac{1}{(1 + r + p^{-1})}\right)\right)$$

= $$\frac{1}{(1+p+\frac{1}{q})}+\frac{1}{(1+q+\frac{1}{r})}+\frac{1}{(1+r+\frac{1}{p})}$$

= $$\frac{q}{(1+q+pq)}+\frac{1}{(1+q+pq)}+\frac{1}{(1+\frac{1}{pq}+\frac{1}{p})}$$

= $$\frac{q}{(1+q+pq)}+\frac{1}{(1+q+pq)}+\frac{pq}{(1+q+pq)}$$

= $$\frac{1+q+pq}{1+q+pq}=1$$

=> Ans - (A)

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