NIMCET 2026 — Mathematics PYQ

To determine the properties of the relation $a\sim b⟺a-2b=3k$ (where $k$ is an integer), we check for reflexivity, symmetry, and transitivity.

1. Reflexivity:

A relation is reflexive if $a\sim a$ for all $a$.

$$a-2a=-a$$

$-a$ is divisible by $3$ only if $a$ is a multiple of $3$. Since it must hold for all $a\in \{1,2,\dots ,10\}$ (e.g., $a=1$, $1-2=-1$, which is not divisible by $3$), the relation is not reflexive.

2. Symmetry:

A relation is symmetric if $a\sim b⟹b\sim a$.

If $a\sim b$, then $a-2b=3k$.

For $b\sim a$, we need $b-2a$ to be divisible by $3$.

Taking $a=1,b=2$: $1-2(2)=-3$ (divisible by $3$, so $1\sim 2$ holds).

Check $2\sim 1$: $2-2(1)=0$ (divisible by $3$, so $2\sim 1$ holds).

Wait, check $a=1,b=4$: $1-2(4)=-7$ (not divisible by $3$).

Check $a=2,b=1$: $2-2(1)=0$ (divisible by $3$).

Since $1\sim 2$ is true and $2\sim 1$ is true, but $1\sim 4$ is false, symmetry depends on the pair. However, generally, if $a-2b=3k$, then $b-2a=b-2(2b+3k)=-3b-6k$, which is divisible by $3$. Let's re-verify: $a\sim b⟹a\equiv 2b(\mathrm{mod}3)$. Then $b\equiv 2^{-1}a\equiv 2a(\mathrm{mod}3)$ because $2\times 2=4\equiv 1(\mathrm{mod}3)$. So $b\sim a$ is true. The relation is symmetric.

3. Transitivity:

A relation is transitive if $a\sim b$ and $b\sim c⟹a\sim c$.

$a\sim b⟹a\equiv 2b(\mathrm{mod}3)$

$b\sim c⟹b\equiv 2c(\mathrm{mod}3)$

Then $a\equiv 2(2c)=4c\equiv c(\mathrm{mod}3)$.

But we need $a\sim c⟹a\equiv 2c(\mathrm{mod}3)$.

Since $c\not\equiv 2c(\mathrm{mod}3)$ in general, the relation is not transitive.

Conclusion:

The relation is symmetric but neither reflexive nor transitive.

Correct Option: (c) $\sim$ is symmetric but neither reflexive nor transitive