Algebra

Relations & Functions

Easy

If $R$ is a relation on $Z$ (set of all integers) defined by $xRy$ , iff $|x - y| \leq 1$ , then

(a) $R$ is reflexive

(b) $R$ is symmetric

(c) $R$ is transitive

(d) $R$ is not symmetric

(e) $R$ is not transitive

Choose the most appropriate answer from the options given below

(a) and (d) only

(a), (b) and (c) only

(b) and (c) only

(a), (b) and (e) only

✅ Correct Option: 4

Reflexive: $|x-x|=0\le 1$ , true.

Symmetric: $|x-y|=|y-x|$ , so symmetric.

Transitive: take $x=1,y=2,z=3$ . $|1-2|=1\le1$ and $|2-3|=1\le1$ , but $|1-3|=2 > 1$ . So not transitive.

Hence (a), (b) and (e) hold.

2022: 17 Aug Shift 2

Let $f : [2, \infty) \to \mathbf{R}$ be a function defined by $f(x) = x^2 - 4x + 5$ . The range of $f$ is :

2022: 25 May Shift 1

The range of the function $f(x) = \frac{1}{3 - \sin 4x}$ is:

2025: 22 May Shift 2

The graph given below represents which of the following function?

2022: 16 July Shift 2

A relation R = {(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(2,3)} on A = {1,2,3} will be an equivalence relation, if we delete:

Choose the correct answer from the options given below:

2026: 29 May Shift 1

Let $R_1, R_2, R_3, R_4$ be the relations defined on the set $A=\{1,2,3\}$ . then

Match List-I with List-II

List-I (Relation)	List-II (Type)
(A) $R_1=\{(1,1),(2,2),(3,3)\}$	(I) Only transitive
(B) $R_2=\{(1,2),(2,3),(1,3)\}$	(II) Equivalence relation
(C) $R_3=\{(1,2),(2,1),(1,1),(2,2)\}$	(III) Reflexive and symmetric but not transitive
(D) $R_4=\{(1,1),(2,2),(3,3),(2,3),(3,2),(1,2),(2,1)\}$	(IV) Symmetric and transitive but not reflexive

Choose the correct answer from the options given below:

2025: 3 June Shift 1

The range of function $f(x) = 4x^2 + 12x + 7, x \in \mathbb{R}$ is

Solution