Let \(\mathrm{A}=\{-3,-2,-1,0,1,2,3\}\). Let \(\mathrm{R}\) be a relation on A defined by xRy if and only if \(0 \leq x^{2}+2y \leq 4\). Let \(\ell\) be the number of elements in R and \(m\) be the minimum number of elements required to be added in R to make it a reflexive relation. Then \(\ell+m\) is equal to

JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Sets Relations and Functions

Let

A = {- 3, - 2, - 1, 0, 1, 2, 3}

. Let

R

be a relation on A defined by xRy if and only if

0 \leq x^{2} + 2 y \leq 4

. Let

ℓ

be the number of elements in R and

m

be the minimum number of elements required to be added in R to make it a reflexive relation. Then

ℓ + m

is equal to

Answer: (D)

The condition

0 \leq x^{2} + 2 y \leq 4

is tested over

x, y \in {- 3, \dots, 3}

. Counting valid

x

for each

y

y = - 3 \to x \in {- 3, 3}

;

y = - 2 \to {- 2, 2}

;

y = - 1 \to {- 2, 2}

;

y = 0 \to {- 2, - 1, 0, 1, 2}

;

y = 1 \to {- 1, 0, 1}

;

y = 2 \to {0}

;

y = 3 \to

none. This totals

2 + 2 + 2 + 5 + 3 + 1 = 15

, so

ℓ = 15

. For reflexivity,

(a, a)

needs

0 \leq a^{2} + 2 a \leq 4

, which holds only for

a = - 3, - 2, 0, 1

; the pairs

(- 1, - 1), (2, 2), (3, 3)

must be added, so

m = 3

. Therefore

ℓ + m = 15 + 3 = 18

JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Single correct option