JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Sets Relations and Functions

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Let A and B be two finite sets with m and n elements respectively. The total number of subsets of the set A is 56 more than the total number of subsets of B . Then the distance of the point

P (m, n)

from the point

Q (- 2, - 3)

(A)10✓ Correct
(B)6
(C)4
(D)8

Answer: (A)

Solution

A set with

k

elements has

2^{k}

subsets. With

∣ A ∣ = m

∣ B ∣ = n

2^{m} - 2^{n} = 56 = 2^{3} \times 7 \Rightarrow 2^{n} (2^{m - n} - 1) = 2^{3} \times 7.

Matching factors:

2^{n} = 2^{3} \Rightarrow n = 3

, and

2^{m - n} - 1 = 7 \Rightarrow 2^{m - n} = 8 \Rightarrow m - n = 3 \Rightarrow m = 6

.

Distance of

P (6, 3)

from

Q (- 2, - 3)

P Q = (6 + 2)^{2} + (3 + 3)^{2} = 64 + 36 = 100 = 10.

JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Single correct option

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