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JEE Mathematics 2024 Question with Solution

Let A and B be two finite sets with mm and nn elements, respectively. If the total subsets of set A are 5656 more than B's subsets, then the distance of the point P(m,n)P(m,n) from Q(−2,−3)Q(-2,-3) is:

  • A

    1010

  • B

    66

  • C

    44

  • D

    88

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Set A has mm elements and set B has nn elements.

Find: The distance between P(m,n)P(m,n) and Q(−2,−3)Q(-2,-3).

The number of subsets of a set with kk elements is 2k2^k. Therefore, for sets A and B, the numbers of subsets are 2m2^m and 2n2^n respectively.

According to the question,

2m=2n+562^m = 2^n + 56

so,

2m−2n=562^m - 2^n = 56

From the working,

2n(2m−n−1)=562^n(2^{m-n} - 1) = 56

Since

56=23×756 = 2^3 \times 7

we get

2n=8  ⟹  n=32^n = 8 \implies n = 3

and

2m−n−1=7  ⟹  2m−n=8  ⟹  m−n=32^{m-n} - 1 = 7 \implies 2^{m-n} = 8 \implies m-n = 3

Hence,

m=6,n=3m = 6, \quad n = 3

Now the points are P(6,3)P(6,3) and Q(−2,−3)Q(-2,-3). Using the distance formula,

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Substituting the coordinates,

d=((−2)−6)2+((−3)−3)2d = \sqrt{((-2)-6)^2 + ((-3)-3)^2} d=(−8)2+(−6)2d = \sqrt{(-8)^2 + (-6)^2} d=64+36d = \sqrt{64 + 36} d=100=10d = \sqrt{100} = 10

Therefore, the distance is 1010 and the correct option is A.

Factorisation Method

Given: The total number of subsets of A is 5656 more than that of B.

Find: The distance of P(m,n)P(m,n) from Q(−2,−3)Q(-2,-3).

Using the subset formula,

2m=2n+562^m = 2^n + 56

Rearranging,

2m−2n=562^m - 2^n = 56

Factorising,

2n(2m−n−1)=562^n(2^{m-n} - 1) = 56

Since

56=23×756 = 2^3 \times 7

match powers of 22 and the odd factor:

2n=8  ⟹  n=32^n = 8 \implies n=3 2m−n−1=7  ⟹  2m−n=82^{m-n} - 1 = 7 \implies 2^{m-n} = 8 m−n=3  ⟹  m=6m-n=3 \implies m=6

Hence P(m,n)=P(6,3)P(m,n) = P(6,3). Now,

Distance=(6−(−2))2+(3−(−3))2\text{Distance} = \sqrt{(6-(-2))^2 + (3-(-3))^2} =82+62= \sqrt{8^2 + 6^2} =64+36=100=10= \sqrt{64+36} = \sqrt{100} = 10

Therefore, the required distance is 1010, so the correct option is A.

Common mistakes

  • Using the number of elements itself instead of the number of subsets is incorrect. A set with mm elements has 2m2^m subsets, not mm subsets. First convert the condition into an equation involving powers of 22.

  • Writing the condition as 2m+2n=562^m + 2^n = 56 is wrong because the question says the subsets of A are 5656 more than those of B. The correct relation is 2m=2n+562^m = 2^n + 56.

  • Applying the distance formula directly to mm and nn without first finding their values is a conceptual error. Determine m=6m=6 and n=3n=3 first, then substitute into the coordinate formula.

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