Question

Let A = {n ∈ N: n is a 3-digit number}

B = {9k + 2: k ∈ N}

and C = {9k + ℓ: k ∈ N} for some ℓ(0 < ℓ < 9)

If the sum of all the elements of the set A ∩ (B ∪ C) is 274 × 400, then ℓ is equal to ______.

पर्याय

• 2

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• 5

Answer 1

Let A = {n ∈ N: n is a 3-digit number}

B = {9k + 2: k ∈ N}

and C = {9k + ℓ: k ∈ N} for some ℓ(0 < ℓ < 9)

If the sum of all the elements of the set A ∩ (B ∪ C) is 274 × 400, then ℓ is equal to 5.

Explanation:

Given: A = {n ∈ N : n is a 3-digit number}

B = {9k + 2: k ∈ N}

C = {9k + ℓ: k ∈ N} for some ℓ(0 < ℓ < 9)

3-digit numbers of the form 9k + 2 are {101, 110, 992}

It forms an A.P. with

First term = a = 101, common difference,

d = 9 and last term = 992

∴ Sum(S₁) = `100/2{101 + 992}`

= 50 × 1093

 S₁ = 54650

Given, 274 × 400 = S₁ + S₂

⇒ 274 × 400 = 50 × 1093 + S₂

S₂ = 109600 – 54650

or S₂ = 54950

Now, 54950 = `100/2[(99 + l) + (990 + l)]`

1099 = 2l + 1089

⇒ l = 5