For $\alpha \in \mathrm{N}$, consider a relation R on N given by $\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): 3 \mathrm{x}+\alpha \mathrm{y}$ is a multiple of 7\}. The relation R is an equivalence relation if and only if :

(D) 4 is the remainder when $\alpha$ is divided by 7

(B) $\alpha$ is a multiple of 4

(C) 4 is the remainder when $\alpha$ is divided by 10

JEE Main · 2022 · 28 Jul 2022 · Shift 1 · Sets Relations and Functions

For

α \in N

, consider a relation R on N given by

R = {(x, y) : 3 x + α y

is a multiple of 7}. The relation R is an equivalence relation if and only if :

Answer: (D)

For R to be reflexive:

x R x

means

3 x + α x = (3 + α) x

is a multiple of 7, requiring

3 + α \equiv 0 (mod 7)

, so

α \equiv 4 (mod 7)

. For symmetry: if

x R y

then

y R x

, meaning

3 x + α y \equiv 0 (mod 7)

and

3 y + α x \equiv 0 (mod 7)

. Adding these:

(3 + α) (x + y) \equiv 0 (mod 7)

, which holds when

3 + α \equiv 0 (mod 7)

. For transitivity: if

x R y

and

y R z

then

x R z

. This also requires

3 + α \equiv 0 (mod 7)

. Therefore, R is an equivalence relation if and only if

α \equiv 4 (mod 7)

, i.e., 4 is the remainder when

α

is divided by 7.

JEE Main · 2022 · 28 Jul 2022 · Shift 1 · Single correct option