Q38 / 90 · JEE Main 2021 (16 Mar Shift 2) · Mathematics

Let $A = \{2, 3, 4, 5, \dots ., 30\}$ and $' ≃'$ be an equivalence relation on $A \times A,$ defined by $(a, b) ≃ (c, d),$ 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 :

Solution

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

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

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

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

$\frac{c}{d} = \frac{4}{3} & c, d \in \{2, 3, \dots \dots, 30\}$

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

No. of ordered pair $= 7$

Hence, the correct answer is option: 'D'.