JEE Main 2024 (Online) 31st January Evening ShiftQuestion 1
Let $A=\{1,2,3,\ldots,100\}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $2x=3y$. Let $R_1$ be a symmetric relation on $A$ such that $R\subset R_1$ and the number of elements in $R_1$ is $n$. Then, the minimum value of $n$ is $\_\_\_\_\_\_$ .
Sets, Relations and Function
Answer$66$
Solution
$2x=3y\Rightarrow x=3k,\,y=2k$, $3k\le100\Rightarrow k\le33$: $|R|=33$. No pair has $x=y$, and reverse pairs $(2k,3k)$ are not in $R$. Symmetric closure adds 33 reverses: $n=66$.