1
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

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

A
$$\alpha=14$$
B
$$\alpha$$ is a multiple of 4
C
4 is the remainder when $$\alpha$$ is divided by 10
D
4 is the remainder when $$\alpha$$ is divided by 7
2
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1

Let $$R_{1}$$ and $$R_{2}$$ be two relations defined on $$\mathbb{R}$$ by

$$a \,R_{1} \,b \Leftrightarrow a b \geq 0$$ and $$a \,R_{2} \,b \Leftrightarrow a \geq b$$

Then,

A
$$R_{1}$$ is an equivalence relation but not $$R_{2}$$
B
$$R_{2}$$ is an equivalence relation but not $$R_{1}$$
C
both $$R_{1}$$ and $$R_{2}$$ are equivalence relations
D
neither $$R_{1}$$ nor $$R_{2}$$ is an equivalence relation
3
JEE Main 2022 (Online) 29th June Morning Shift
+4
-1

Let a set A = A1 $$\cup$$ A2 $$\cup$$ ..... $$\cup$$ Ak, where Ai $$\cap$$ Aj = $$\phi$$ for i $$\ne$$ j, 1 $$\le$$ j, j $$\le$$ k. Define the relation R from A to A by R = {(x, y) : y $$\in$$ Ai if and only if x $$\in$$ Ai, 1 $$\le$$ i $$\le$$ k}. Then, R is :

A
reflexive, symmetric but not transitive.
B
reflexive, transitive but not symmetric.
C
reflexive but not symmetric and transitive.
D
an equivalence relation.
4
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1

Let R1 = {(a, b) $$\in$$ N $$\times$$ N : |a $$-$$ b| $$\le$$ 13} and

R2 = {(a, b) $$\in$$ N $$\times$$ N : |a $$-$$ b| $$\ne$$ 13}. Then on N :

A
Both R1 and R2 are equivalence relations
B
Neither R1 nor R2 is an equivalence relation
C
R1 is an equivalence relation but R2 is not
D
R2 is an equivalence relation but R1 is not
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