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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?
Options:
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?
Solution:
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Verified Answer
The correct answer is:
I is false, II is true
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.
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