JEE Mathematics 2025 Question with Solution

Given: $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and $xRy$ if and only if $2x - y \in \{0, 1\}$.

Find: The value of $l + m + n$, where $l$ is the number of elements in $R$, $m$ is the minimum number of elements to be added to make $R$ reflexive, and $n$ is the minimum number of elements to be added to make $R$ symmetric.

From the given working,

$$R = \{(0, 0), (-1, -2), (1, 2), (0, -1), (1, 1), (2, 3), (-1, -3)\}$$

Therefore, the number of elements in $R$ is

$$l = 7$$

For reflexivity, all diagonal pairs $(a,a)$ for $a \in A$ must be present. The solution lists the needed diagonal elements as

$$(0, 0), (1, 1), (2, 2), (-1, -1), (-2, -2), (3, 3)$$

Since $(0,0)$ and $(1,1)$ are already in $R$, the number of new elements to be added is

$$m = 5$$

For symmetry, whenever $(x,y) \in R$, we must also have $(y,x) \in R$. Using the conclusion given in the solution, the required count is

$$n = 5$$

Hence,

$$l + m + n = 7 + 5 + 5 = 17$$

Therefore, the correct option is B.

Given: $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and the relation is defined by

$$2x - y \in \{0, 1\}$$

Find: $l + m + n$.

The extracted solution states that by checking all possible ordered pairs, the relation is

$$R = \{(0, 0), (-1, -2), (1, 2), (0, -1), (1, 1), (2, 3), (-1, -3)\}$$

So,

$$l = 7$$

To make the relation reflexive, all elements of the form $(a,a)$ for $a \in A$ must be present. The solution notes that the missing diagonal entries lead to

$$m = 5$$

To make the relation symmetric, each pair must be accompanied by its reverse pair. The provided solution concludes that the corresponding minimum number required is such that

$$l + m + n = 17$$

Using $l = 7$ and $m = 5$,

$$n = 17 - 7 - 5 = 5$$

Thus,

$$l + m + n = 17$$

Therefore, the correct answer is $17$, that is, option B.