JEE Mathematics 2026 Question with Solution

Given: $A = \{2, 3, 5, 7, 9\}$ and $R$ is defined by $xRy \iff 2x \le 3y$.

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

A relation is symmetric if whenever $(x,y) \in R$, then $(y,x) \in R$ must also hold.

We first count all ordered pairs in $R$.

For $x=2$:

$$2(2) = 4 \le 3y \implies y \ge \frac{4}{3}$$

Possible $y$ are $\{2,3,5,7,9\}$, so there are $5$ pairs.

For $x=3$:

$$2(3) = 6 \le 3y \implies y \ge 2$$

Possible $y$ are $\{2,3,5,7,9\}$, so there are $5$ pairs.

For $x=5$:

$$2(5) = 10 \le 3y \implies y \ge \frac{10}{3}$$

Possible $y$ are $\{5,7,9\}$, so there are $3$ pairs.

For $x=7$:

$$2(7) = 14 \le 3y \implies y \ge \frac{14}{3}$$

Possible $y$ are $\{5,7,9\}$, so there are $3$ pairs.

For $x=9$:

$$2(9) = 18 \le 3y \implies y \ge 6$$

Possible $y$ are $\{7,9\}$, so there are $2$ pairs.

Therefore,

$$l = 5 + 5 + 3 + 3 + 2 = 18$$

Now check which pairs need their reverse ordered pair to make the relation symmetric.

Pairs with $$x

$$$$

Given: $R = \{(x,y) \in A \times A : 2x \le 3y\}$ on $A = \{2,3,5,7,9\}$.

Find: First compute $l = |R|$, then compute the minimum additional pairs $m$ needed for symmetry.

To count $l$, fix each value of $x$ and determine all allowed values of $y$.

1. For $x=2$, the inequality becomes $4 \le 3y$, so every element of $A$ works.
2. For $x=3$, the inequality becomes $6 \le 3y$, so every element of $A$ works.
3. For $x=5$, the inequality becomes $10 \le 3y$, so only $5,7,9$ work.
4. For $x=7$, the inequality becomes $14 \le 3y$, so only $5,7,9$ work.
5. For $x=9$, the inequality becomes $18 \le 3y$, so only $7,9$ work.

Thus,

$$l = 5 + 5 + 3 + 3 + 2 = 18$$

For symmetry, diagonal pairs such as $(2,2), (3,3), (5,5), (7,7), (9,9)$ do not create any issue because their reverse is the same pair.

So we only inspect non-diagonal pairs. Every time $(x,y) \in R$ but $(y,x) \notin R$, one new pair must be added.

The missing reverse pairs are exactly:

• from $(2,5)$, add $(5,2)$
• from $(2,7)$, add $(7,2)$
• from $(2,9)$, add $(9,2)$
• from $(3,5)$, add $(5,3)$
• from $(3,7)$, add $(7,3)$
• from $(3,9)$, add $(9,3)$
• from $(5,9)$, add $(9,5)$

Therefore,

$$m = 7$$

And hence,

$$l+m = 18+7 = 25$$

Therefore, the value of $l+m$ is $25$.