JEE MAIN - Mathematics (2024 - 29th January Morning Shift - No. 9)

Let $$R$$ be a relation on $$Z \times Z$$ defined by $$(a, b) R(c, d)$$ if and only if $$a d-b c$$ is divisible by 5. Then $$R$$ is

Reflexive and transitive but not symmetric

Reflexive and symmetric but not transitive

Reflexive but neither symmetric nor transitive

Reflexive, symmetric and transitive

$$(a, b) R(a, b)$$ as $$a b-a b=0$$

Therefore reflexive

Let $$(a, b) R(c, d) \Rightarrow a d-b c$$ is divisible by 5

$$\Rightarrow \mathrm{bc}-\mathrm{ad}$$ is divisible by $$5 \Rightarrow(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b})$$

Therefore symmetric

Relation not transitive as $$(3,1) \mathrm{R}(10,5)$$ and $$(10,5) \mathrm{R}(1,1)$$ but $$(3,1)$$ is not related to $$(1,1)$$