Kerala KEAM 5 June 2024 Solved Paper

We are given a relation $R$ on the set of natural numbers $\mathbb{N}$ defined by $aRb$ if and only if $a + 2b = 11$. We need to determine the properties of this relation: whether it is reflexive, symmetric, or both.Step 1: Check if the relation is reflexive.A relation $R$ is reflexive if for all $a \in \mathbb{N}$, we have $aRa$, i.e., $a + 2a = 11$. Simplifying this: $$a + 2a = 3a = 11.$$This equation does not hold for any natural number $a$, as $3a = 11$ does not have a solution in $\mathbb{N}$. Therefore, the relation is not reflexive.Step 2: Check if the relation is symmetric.A relation $R$ is symmetric if for all $a, b \in \mathbb{N}$, whenever $aRb$ (i.e., $a + 2b = 11$), we also have $bRa$ (i.e., $b + 2a = 11$).Let’s assume $a + 2b = 11$. For the relation to be symmetric, we need $b + 2a = 11$. However, it is not guaranteed that $b + 2a = 11$ for all pairs of $a$ and $b$ that satisfy $a + 2b = 11$. For example, for $a = 7$ and $b = 2$, we have: $$a + 2b = 7 + 2(2) = 11,$$but: $$b + 2a = 2 + 2(7) = 16 \neq 11.$$Thus, the relation is not symmetric.Step 3: Conclusion.Since the relation is neither reflexive nor symmetric, the correct answer is option (D) - neither reflexive nor symmetric.Thus, the correct answer is option (D). Quick Tip: To check if a relation is reflexive, verify if $aRa$ holds for all elements in the set. To check for symmetry, verify if $aRb$ implies $bRa$ for all pairs $a, b$.