Let $X = R \times R$. Define a relation R on X as: $(a_1, b_1) R (a_2, b_2) \Leftrightarrow b_1 = b_2$ Statement I: $R$ is an equivalence relation. Statement II: For some $(a, b) \in X$, the set $S = \{(x, y) \in X : (x, y)R(a, b)\}$ 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
MEDIUM
Correct Answer:B
Explanation
Statement I: Reflexive: $(a_1, b_1) R (a_1, b_1) \Rightarrow b_1 = b_1$ (True). Symmetric: $(a_1, b_1) R (a_2, b_2) \Rightarrow b_1 = b_2$ and $(a_2, b_2) R (a_1, b_1) \Rightarrow b_2 = b_1$ (True). Transitive: $(a_1, b_1) R (a_2, b_2) \Rightarrow b_1 = b_2$ and $(a_2, b_2) R (a_3, b_3) \Rightarrow b_2 = b_3$. Thus, $b_1 = b_3$, so $(a_1, b_1) R (a_3, b_3)$ (True). Hence, relation $R$ is an equivalence relation, and Statement I is true. For Statement II: $(x, y) R (a, b) \Rightarrow y = b$. This represents a line $y=b$, which is parallel to the x-axis. It is not parallel to the line $y=x$. Therefore, Statement II is false.
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4 Mark(s)
2025
JEE Main 2025 (Online) 23rd January Evening Shift
Let $A = {(x, y) ∈ R × R : |x + y| ⩾ 3}$ and $B = {(x, y) ∈ R × R : |x| + |y| ≤ 3}$. If $C = {(x, y) ∈ A ∩ B : x = 0$ or $y = 0}$, then $\sum_{(x, y) ∈ C} |x + y|$ is :
(A)
18
(B)
24
(C)
15
(D)
12
MEDIUM
Correct Answer:D
Explanation
From the image, we can determine that the points in set C are $(3,0)$, $(-3,0)$, $(0,3)$ and $(0,-3)$.
Thus, $C = {(3, 0), (-3, 0), (0, 3), (0, -3)}$.
$\sum |x + y| = |3 + 0| + |-3 + 0| + |0 + 3| + |0 + (-3)| = 3 + 3 + 3 + 3 = 12$.