JEE Mathematics 2023 Question with Solution

Given: $R$ is defined on $\mathbb{N} \times \mathbb{N}$ by

Find: Which properties among reflexive, symmetric and transitive are satisfied.

From the solution:

For symmetry, it checks

which is the same condition written after interchanging the ordered pairs. Hence the relation is symmetric.

For reflexivity, the solution tests

and concludes that the relation is not reflexive.

For transitivity, the solution gives a counterexample:

but

Therefore the relation is not transitive.

So, based on the extracted solution working, $R$ is symmetric but neither reflexive nor transitive. The source solution states The Correct Option is B, but this conflicts with the listed options. Among the given options, the statement matches option A.

Given: $ad(b-c)=bc(a-d)$. Find: The correct classification of the relation.

1. Symmetry If

then

After swapping the two ordered pairs, the checked form in the solution is

which is equivalent to the same relation. So the relation is symmetric.

2. Reflexivity For reflexivity, we must have

for every $(a,b) \in \mathbb{N} \times \mathbb{N}$. The solution evaluates this as

and therefore concludes not reflexive.

3. Transitivity To test transitivity, the solution uses specific pairs:

but

does not hold. Hence the relation is not transitive.

Therefore the correct description is: symmetric but neither reflexive nor transitive. Hence the defensible answer from the given options is A.