Jee mains pyq maths

Question

Let R be a relation defined on N  as a R b is 2a + 3b is a multiple of 5, a, b ∈ N  . Then R is  (1) not reflexive  (2) transitive but not symmetric  (3) symmetric but not transitive  (4) an equivalence relation

Answer 1

Correct option is (4) an equivalence relation

Given,

\(R = a, b:2a + 3b\) is divisible by 5 for \(a, b\in N\)

Reflexive:

\(2a + 3a = 5a\) is always divisible by 5.

So, \(a, a\in R\)

Symmetric:

Let \(a, b\in R\) i.e., \(2a + 3b = 5\lambda\), where \(\lambda \in Z^+\).

Now,

\(5a + 5b\) = multiple of 5

⇒ \(2a + 3b + 2b + 3a\) = multiple of 5

⇒ \(5\lambda + 2b + 3a\) = multiple of 5

⇒ \(2b + 3a\) = multiple of \(5 -5 \lambda\)

So, \(2b + 3a\) is divisible by 5.

Transitive:

Let \(a, b, b , c \in R\), then

\(2a+ 3b = 5\lambda_1;\lambda_1 \in Z^+\)

\(2b+ 3c= 5\lambda_2;\lambda_2 \in Z^+\)

So,

\(2a + 5b + 3c = 5(\lambda_1 + \lambda_2)\)

⇒ \(2a + 3c = 5(\lambda_1 + \lambda_2 - b)\)

Hence, \(2a+ 3c\) is divisible by 5, so \(a, c \in R\)

Hence, R is an equivalence relation.