If $Q$ denotes the set of all rational numbers and $f\left(\frac{p}{q}\right)=\sqrt{p^2-q^2}$ for any…
If $Q$ denotes the set of all rational numbers and $f\left(\frac{p}{q}\right)=\sqrt{p^2-q^2}$ for any $\frac{p}{q} \in Q$, then observe the following statements.
I. $f\left(\frac{p}{q}\right)$ is real for each $\frac{p}{q} \in Q$
II. $f\left(\frac{p}{q}\right)$ is a complex number for each $\frac{p}{q} \in Q$.
Which of the following is correct?
Both I and II are true
I is true, II is false
I is false, II is true
Both I and II are false
Solution
Given, $f\left(\frac{p}{q}\right)=\sqrt{p^2-q^2}$, for $\frac{p}{q} \in Q$ If $p < q$, then $f\left(\frac{p}{q}\right)$ is not real.