Let R be a relation defined on the set \(\{1,2,3,4\} \times\{1,2,3,4\}\) by \(\mathrm{R}=\{((\mathrm{a}, \mathrm{b}),(\mathrm{c}, \mathrm{d})): 2 \mathrm{a}+3 \mathrm{~b}=3 \mathrm{c}+4 \mathrm{~d}\}\). Then the number of elements in R is

JEE Main · 2026 · 24 Jan 2026 · Shift 1 · Sets Relations and Functions

Let R be a relation defined on the set

{1, 2, 3, 4} \times {1, 2, 3, 4}

R = {((a, b), (c, d)) : 2 a + 3 b = 3 c + 4 d}

.
Then the number of elements in R is

Answer: (C)

Counting pairs

((a, b), (c, d))

with

a, b, c, d \in {1, 2, 3, 4}

and

2 a + 3 b = 3 c + 4 d

amounts to matching the values of

2 a + 3 b

with those of

3 c + 4 d

. Listing

2 a + 3 b

over the sixteen

(a, b)

and

3 c + 4 d

over the sixteen

(c, d)

and pairing equal values yields exactly the twelve solutions

((1, 3), (1, 2)), ((1, 4), (2, 2)), ((2, 1), (1, 1)), ((2, 2), (2, 1)), ((2, 3), (3, 1)), ((2, 4), (4, 1)), ((3, 3), (1, 3)), ((3, 4), (2, 3)), ((4, 1), (1, 2)), ((4, 2), (2, 2)), ((4, 3), (3, 2)), ((4, 4), (4, 2))

. The number of elements in

R

12

JEE Main · 2026 · 24 Jan 2026 · Shift 1 · Single correct option