JEE MAIN - Mathematics (2021 - 24th February Morning Shift - No. 18)

Let  A = {n $$ \in $$ N: n is a 3-digit number}

       B = {9k + 2: k $$ \in $$ N}

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

If the sum of all the elements of the set A $$ \cap $$ (B $$ \cup $$ C) is 274 $$ \times $$ 400, then $$l$$ is equal to ________.
Answer
5

Explanation

In this problem, we're dealing with 3-digit numbers in set $A$, and subsets $B$ and $C$ which represent numbers of specific forms.

1. First, we consider the numbers of the form $9k + 2$ (Set $B$) within the 3-digit range, which starts at 101 and ends at 992.

2. We calculate the sum of these numbers, denoted as $s_1$. To calculate $s_1$, you use the formula for the sum of an arithmetic series :

$$(n/2) \times (\text{{first term}} + \text{{last term}})$$

Here, $n$ is the total count of such numbers. These are 3-digit numbers of the form $9k + 2$, and we can find the total count by subtracting the smallest such number (101) from the largest (992), dividing the result by 9 (because we're considering numbers with a difference of 9), and then adding 1.

The sum $s_1$ is calculated as follows :

$$(100/2) \times (101 + 992) = 54650$$

3. According to the problem, the sum of all elements of the set $A \cap (B \cup C)$ is $274 \times 400 = 109600$.

Since the set $A \cap (B \cup C)$ is the union of two disjoint sets (the set of all three-digit numbers of form $9k + 2$ and the set of all three-digit numbers of form $9k + l$), we can write this sum as :

$$s_1 $$(for numbers of the form 9k + 2) + $$s_2$$ (for numbers of the form 9k + l) = 109600

4. Solving this equation for $s_2$ (the sum of numbers of the form $9k + l$), we get :

$$s_2 = 109600 - s_1 = 109600 - 54650 = 54950$$

5. The sum $s_2$ can be expressed as $(n/2) \times (\text{{first term}} + \text{{last term}})$, where $n$ is the count of numbers of the form $9k + l$. The first term here is the smallest 3-digit number of this form, which is $99 + l$, and the last term is the largest such number, which is $990 + l$.

We equate this to $s_2$ to solve for $l$ :

$$54950 = (100/2)[(99 + l) + (990 + l)]$$

6. Simplifying this equation, we get :

$$2l + 1089 = 1099$$

Solving for $l$, we find :

$$l = 5$$

So, the correct answer is 5.

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