JEE Main 2025 — Sets and Relations Question with Solution
JEE Main 2025 (29 Jan Shift 1)
Question
Define a relation R on the interval by if and only if . Then R is :
Choose an option
Show full solutionCorrect option: B
Correct answer
Ban equivalence relation
Step-by-step explanation
$\begin{aligned}
& \sec ^2 x-\tan ^2 x=1 \quad(\text { on replacing } y \text { with } x) \\
& \Rightarrow \text { Reflexive } \\
& \sec ^2 x-\tan ^2 y=1 \\
& \Rightarrow 1+\tan ^2 x+1-\sec ^2 y=1 \\
& \Rightarrow \sec ^2 y-\tan ^2 x=1 \\
& \Rightarrow \text { symmetric } \\
& \sec ^2 x-\tan ^2 y=1 \\
& \sec ^2 y-\tan ^2 z=1
\end{aligned}$
Adding both
$\begin{aligned}
& \Rightarrow \sec ^2 x-\tan ^2 y+\sec ^2 y-\tan ^2 z=1+1 \\
& \sec ^2 x+1-\tan ^2 z=2 \\
& \sec ^2 x-\tan ^2 z=1 \\
& \Rightarrow \text { Transitive }
\end{aligned}$
hence equivalence releation
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