Let A ={2,3,4,5, ldots ., 30} and ^{prime} ∼eq ^{prime} be an equivalence relation on A × A

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1.Relation and Function

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Let $A =\{2,3,4,5, \ldots ., 30\}$ and $^{\prime} \simeq ^{\prime}$ be an equivalence relation on $A \times A ,$ defined by $(a, b) \simeq (c, d),$ if and only if $a d=b c .$ Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :

A

$5$

B

$6$

C

$8$

D

$7$

(JEE MAIN-2021)

Solution

$A =\{2,3,4,5, \ldots ., 30\}$

$(a, b)=(c, d) \quad \Rightarrow \quad a d=b c$

$(4,3)=( c , d ) \quad \Rightarrow \quad 4 d =3 c$

$\Rightarrow \frac{4}{3}=\frac{c}{d}$

$\frac{c}{d}=\frac{4}{3}$ and $c,d\, \in \, \{2,3, \ldots \ldots, 30\}$

$( c , d )=\{(4,3),(8,6),(12,9),(16,12),(20,15), (24,18),(28,21)\}$

No. of ordered pair $=7$

Standard 12

Mathematics

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