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
• Correct answer
  C

  I is false, II is true
• D

  Both I and II are false

Hence, the correct answer is option: 'C'.