JEE Main 2025 · 23 January · evening · Single correct · +4 / −1
Let X=R × R. Define a relation R on X as :
(a₁, b₁) R(a₂, b₂) ⇔ b₁=b₂
Statement I: R is an equivalence relation.
Statement II : For some (a, b) ∈ X, the set S={(x, y) ∈ X:(x, y) R(a, b)} represents a line parallel to y=x.
In the light of the above statements, choose the correct answer from the options given below :
- (a) Both Statement I and Statement II are true
- (b) Statement I is true but Statement II is false
- (c) Both Statement I and Statement II are false
- (d) Statement I is false but Statement II is true
Answer and solution
Answer (b) Statement I is true but Statement II is false
Solution
Maths sir · 5 steps
The relation only looks at the second coordinate, so its classes are horizontal lines, never slope 1.
- Check reflexive. For any (a, b), we have b = b, so (a, b) R (a, b). The relation is reflexive.
- Symmetric and transitive. If b₁ = b₂ then b₂ = b₁, so it is symmetric. If b₁ = b₂ and b₂ = b₃ then b₁ = b₃, so it is transitive. Hence R is an equivalence relation and Statement I is true.
- Find the set S. (x, y) R (a, b) means y = b, with x free to be any real number. So S is the line y = b, a horizontal line with slope 0.
- Compare slopes. A line parallel to y = x has slope 1. No choice of (a, b) turns y = b into a slope-1 line, so Statement II is false.
- So. Statement I is true and Statement II is false, which is option (b).
The idea to keep. When a relation says two points are related if one coordinate matches, it is always an equivalence relation, and each class is the line where that coordinate stays fixed.
Answer from the NTA final answer key. Solution by PrepWiser.