JEE Mathematics 2024 Question with Solution

Given: A relation $R$ on $\mathbb{Z} \times \mathbb{Z}$ is defined by $(a,b)R(c,d)$ if and only if $ad-bc$ is divisible by $5$.

Find: Whether $R$ is reflexive, symmetric, and transitive.

For reflexivity, check whether $(a,b)R(a,b)$ holds for every $(a,b)$.

Since $0$ is divisible by $5$, the relation is reflexive.

For symmetry, assume $(a,b)R(c,d)$. Then

is divisible by $5$. Hence

is also divisible by $5$. Therefore $(c,d)R(a,b)$, so the relation is symmetric.

For transitivity, suppose

and

for some integers $k_1$ and $k_2$. Then

and

Adding,

that is,

This does not guarantee that $af - be$ is divisible by $5$ for all integers, so transitivity does not follow.

The extracted the solution contains an internal inconsistency because one heading says option B, but both the detailed conclusion and the worked classification state reflexive and symmetric but not transitive. That classification matches option A in the given options.

Therefore, the relation is reflexive and symmetric but not transitive, so the correct option is A.

1. Reflexive:

So $R$ is reflexive.

2. Symmetric: If $(a,b)R(c,d)$, then $ad-bc$ is divisible by $5$. Hence $-(ad-bc)=bc-ad$ is also divisible by $5$, so $(c,d)R(a,b)$. Thus $R$ is symmetric.

3. Transitive: From

it is not forced that

for every choice of integers. Hence $R$ is not transitive.

So the relation is reflexive and symmetric but not transitive.