Class JEE Mathematics Sets, Relations, and Functions Q #1015
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4 Marks 2024 JEE Main 2024 (Online) 8th April Evening Shift MCQ SINGLE
Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$. Let $R$ be a relation on $A \times B$ defined by $(a, b)R(c, d)$ if and only if $3ad - 7bc$ is an even integer. Then the relation $R$ is
(A) reflexive but not symmetric.
(B) an equivalence relation.
(C) reflexive and symmetric but not transitive.
(D) transitive but not symmetric.
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Correct Answer: C
Explanation
Given that $(a, b)R(c, d) \Rightarrow 3ad - 7bc \in \text{even}$.

For reflexive, we need to check if $(a, b)R(a, b)$.
$(a, b)R(a, b) \Rightarrow 3ab - 7ba = -4ab \in \text{even}$. This is true since the product $ab$ will be an integer, and $-4$ times an integer is even.
Thus, $R$ is reflexive.

For symmetric, if $(a, b)R(c, d)$, then $(c, d)R(a, b)$ must also be true.
$(a, b)R(c, d) \Rightarrow 3ad - 7bc = 2m$ for some integer $m$.
$(c, d)R(a, b) \Rightarrow 3cb - 7ad$. We want to show this is even.
$3cb - 7ad = -7ad + 3cb = -(7ad - 3cb) = -(7ad - 3cb + 3ad - 3ad - 7bc + 7bc) = -(10ad - 10bc - (3ad - 7bc)) = -10(ad - bc) + (3ad-7bc)$

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