QJEEM23Ma4a8a: JEE Main 2023 Mathematics - Class 11 JEEM23Ma4a8a Solution: Let R be a relation on R , given by R = { ( a , b ) : 3 a − 3 b …

Question

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Let $R$ be a relation on $R$ , given by $R = {(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number $}$ . Then $R$ is

Accepted Answer

For reflexive :

3 a-3 a+\sqrt{7}

is an irrational number

\forall a \in R R

is reflexive

For symmetric :

Let

3 a-3 b+\sqrt{7}

is an irrational number

\Rightarrow 3 b-3 a+\sqrt{7}

is an irrational number

For example, Let

3 a-3 b=\sqrt{7}

\sqrt{7}+\sqrt{7}

is irrational but

-\sqrt{7}+\sqrt{7}

is not.

\therefore R

is not symmetric

For transitive :

Let

3 a-3 b+\sqrt{7}

is irrational and

3 b-3 c+\sqrt{7}

is irrational.

\Rightarrow 3 a-3 c+\sqrt{7}

is irrational.

For example, take

a=0, b=-\sqrt{7}, c=\frac{\sqrt{7}}{3}

R

is not transitive.

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