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Mathematics · Sets and Relations

JEE Main 2024 — 29 January, Shift 1 — Question 16

Let R be a relation on Z×Z\mathrm{Z} \times \mathrm{Z} defined by (a,b)R(c,d)(a, b) R(c, d) if and only if ad−bc\mathrm{ad}-\mathrm{bc} is divisible by 5 . Then RR is

  1. Option A:

    Reflexive and symmetric but not transitive

    Correct
  2. Option B:

    Reflexive but neither symmetric not transitive

  3. Option C:

    Reflexive, symmetric and transitive

  4. Option D:

    Reflexive and transitive but not symmetric

Answer: A

Step-by-step solution

(a,b)R(a,b)(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{a}, \mathrm{b}) as ab−ab=0\mathrm{ab}-\mathrm{ab}=0

Therefore reflexive Let (a,b)R(c,d)⇒ad−bc(a, b) R(c, d) \Rightarrow a d-b c

is divisible by 5 ⇒bc−ad\Rightarrow \mathrm{bc}-\mathrm{ad}

is divisible by 5⇒(c,d)R(a,b)5 \Rightarrow(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b})

Therefore symmetric Relation not transitive as (3,1)R(10,5)(3,1) R(10,5) and (10,5)R(1,1)(10,5) R(1,1) but (3,1)(3,1) is not related to (1,1)(1,1)

Answer key and solution verified before publishing.

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Exam
JEE Main 2024
Paper
29 January, Shift 1
Subject
Mathematics
Chapter
Sets and Relations
Topic
Types of Relations
Let R be a relation on Z × Z defined by (a, b) R(c, d) if and only if… | JEE Main 2024 PYQ with Solution · DhiX AI