JEE Mathematics 2025 Question with Solution

Given: $X = \mathbb{R} \times \mathbb{R}$ and $(a_1,b_1) \, R \, (a_2,b_2) \iff b_1=b_2$.

Find: The truth values of Statement-I and Statement-II.

For Statement-I, we check the three properties of an equivalence relation.

• Reflexive: For any $(a,b) \in X$, we have $b=b$, so $(a,b) \, R \, (a,b)$.
• Symmetric: If $(a_1,b_1) \, R \, (a_2,b_2)$, then $b_1=b_2$. Hence $b_2=b_1$, so $(a_2,b_2) \, R \, (a_1,b_1)$.
• Transitive: If $(a_1,b_1) \, R \, (a_2,b_2)$ and $(a_2,b_2) \, R \, (a_3,b_3)$, then $b_1=b_2$ and $b_2=b_3$. Therefore $b_1=b_3$, so $(a_1,b_1) \, R \, (a_3,b_3)$.

Thus, $R$ is reflexive, symmetric, and transitive. So Statement-I is true.

For Statement-II,

Using the definition of $R$, this means

Hence,

This is a horizontal line, parallel to the $x$-axis, not a line parallel to $y=x$. So Statement-II is false.

Therefore, the correct option is B.

Given: The relation depends only on equality of the second coordinates.

Find: Whether the equivalence relation claim and geometric claim are correct.

If two ordered pairs are related whenever their second coordinates are equal, then all points with the same $y$-coordinate belong to the same equivalence class. For a fixed point $(a,b)$, every point related to it must satisfy $y=b$.

So the equivalence class of $(a,b)$ is the set of all points on the line

This is a horizontal line. A line parallel to $y=x$ must have slope $1$, whereas the line $y=b$ has slope $0$. Therefore the geometric statement is false, while the equivalence relation statement is true.

Hence, Statement-I is true but Statement-II is false, so the answer is B.