Question

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?

• A. Both I and II are true
• B. I is true, II is false
• C. I is false, II is true
• D. Both I and II are false

Answer 1

Answer: (C)

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.