Class JEE Mathematics Sets, Relations, and Functions Q #1036
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4 Marks 2023 JEE Main 2023 (Online) 31st January Morning Shift MCQ SINGLE
Let $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d)$ if and only if $ad(b - c) = bc(a - d)$. Then $R$ is
(A) symmetric and transitive but not reflexive
(B) reflexive and symmetric but not transitive
(C) transitive but neither reflexive nor symmetric
(D) symmetric but neither reflexive nor transitive
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Correct Answer: D
Explanation
Given $(a, b) R (c, d) \implies ad(b - c) = bc(a - d)$.

Symmetric:
$(c, d) R (a, b) \implies cb(d - a) = da(c - b)$. This is symmetric.

Reflexive:
$(a, b) R (a, b) \implies ab(b - a) \neq ba(a - b)$. Thus, not reflexive.

Transitive:
$(2, 3) R (3, 2)$ and $(3, 2) R (5, 30)$ but $((2, 3), (5, 30)) \notin R$. Thus, not transitive.

Therefore, R is symmetric but neither reflexive nor transitive.

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