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1.Relation and Function
hard
Among the relations $S =\left\{( a , b ): a , b \in R -\{0\}, 2+\frac{ a }{ b } > 0\right\}$ And $T =\left\{( a , b ): a , b \in R , a ^2- b ^2 \in Z \right\}$,
A
$S$ is transitive but $T$ is not
B
$T$ is symmetric but $S$ is not
C
Neither $S$ nor $T$ is transitive
D
Both $S$ and $T$ are symmetric
(JEE MAIN-2023)
Solution
For relation $T=a^2-b^2=-I$
Then,$(b, a)$ on relation $R$
$\Rightarrow b ^2- a ^2=- I$
$\therefore T \text { is symmetric }$
$S =\left\{( a , b ): a , b \in R -\{0\}, 2+\frac{ a }{ b } > 0\right\}$
$2+\frac{ a }{ b } > 0 \Rightarrow \frac{ a }{ b } > -2, \Rightarrow \frac{ b }{ a } < \frac{-1}{2}$
If $(b, a) \in S$ then
$2+\frac{b}{a}$ not necessarily positive
$\therefore S$ is not symmetric
Standard 12
Mathematics
