A relation $R$ is defined on $\mathbb{Z}$ as $aRb$ if and only if $a^2 - 7ab + 6b^2 = 0$. Then, $R$ is

reflexive but not symmetric

The correct answer is Option (4) → reflexive but not symmetric ##

Given, $aRb, a, b \in \mathbb{Z}$

Reflexive: For $a \in \mathbb{Z}$, we have

$a^2 - 7a \cdot a + 6a^2 = a^2 - 7a^2 + 6a^2 = 0 \Rightarrow (a, a) \in R$

$∴$ Relation is reflexive.

Symmetric: Since, $(6, 1) \in R$

As $6^2 - 7 \times 6 \times 1 + 6 \times 1^2 = 36 - 42 + 6 = 0$

But $(1, 6) \notin R$

$∴$ Relation is not symmetric.