Q. If $\mathcal{Q}$ denotes the set of all rational numbers and $f\left( \frac{p}{q} \right) = \sqrt{p^{2} - q^{2}}$ for any $\frac{p}{q}\mathcal{\in Q}$, then observe the following statements. I. $f\left( \frac{p}{q} \right)$ is real for each $\frac{p}{q}\mathcal{\in Q}$. II. $f\left( \frac{p}{q} \right)$ is a complex number for each $\frac{p}{q}\mathcal{\in Q}$. 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
Explanation
(c)
Given, $f\left( \frac{p}{q} \right) = \sqrt{p^{2} - q^{2}}$, for $\frac{p}{q}\mathcal{= Q}$
If $p < q$, then $f\left( \frac{p}{q} \right)$ is not real.
Hence, statement I is false while statement II is true.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.