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JEE Main 2023 (Online) 29th January Evening Shift
Subjects
Mathematics
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Competency Based
Knowledge Based
2023
JEE Main 2023 (Online) 29th January Evening Shift
Full Paper Analysis & Solutions
Mathematics
Multiple Choice Questions
1
COMPETENCY
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MCQ_SINGLE
4 Mark(s)
2023
JEE Main 2023 (Online) 29th January Evening Shift
#1038
Let R be a relation defined on $N$ as $aRb$ if $2a + 3b$ is a multiple of $5$, $a, b \in N$. Then R is
(A)
an equivalence relation
(B)
non reflexive
(C)
symmetric but not transitive
(D)
transitive but not symmetric
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AI Explanation
HARD
Correct Answer:
A
Explanation
Given that $aRb$ if $2a + 3b = 5m$, $m \in I$. (1) $(a, a) \in R$ as $2a + 3a = 5a$, $a \in N$. Hence, R is reflexive. (2) If $(a, b) \in R$ then $2a + 3b = 5m$. Now, $5(a + b) = 5n$. $3a + 2b + 2a + 3b = 5n$ $\therefore 3a + 2b = 5(n - m)$. $\therefore (b, a) \in R$. $\therefore R$ is symmetric. (3) If $(a, b) \in R$ and $(b, c) \in R$ then $2a + 3b = 5m$, $2b + 3c = 5n$. $\Rightarrow 2a + 5b + 3c = 5(m + n)$. $\Rightarrow 2a + 3c = 5(m + n - b)$. $\therefore (a, c) \in R$. $\therefore R$ is transitive. Hence, R is an equivalence relation.
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