JEE Mathematics 2025 Question with Solution

Given: $A = \{-3,-2,-1,0,1,2,3\}$ and $R$ is defined by $0 \le x^2 + 2y \le 4$.

Find: $l$ = number of ordered pairs in $R$, $m$ = minimum number of ordered pairs to be added to make $R$ reflexive, and then compute $l+m$.

For $(x,y) \in R$,

so

and hence

Now check each value of $x \in A$:

• For $x=-3$ or $x=3$, $x^2=9$, so $-4.5 \le y \le -2.5$. Hence only $y=-3$ is possible. This gives $2$ ordered pairs.
• For $x=-2$ or $x=2$, $x^2=4$, so $-2 \le y \le 0$. Hence $y=-2,-1,0$. This gives $6$ ordered pairs.
• For $x=-1$ or $x=1$, $x^2=1$, so $-0.5 \le y \le 1.5$. Hence $y=0,1$. This gives $4$ ordered pairs.
• For $x=0$, $x^2=0$, so $0 \le y \le 2$. Hence $y=0,1,2$. This gives $3$ ordered pairs.

Therefore,

To make $R$ reflexive, every $(a,a)$ for $a \in A$ must belong to $R$. So test

for each $a \in A$.

Checking diagonal pairs:

• $a=-3$: $(-3)^2+2(-3)=3$, so $(-3,-3) \in R$
• $a=-2$: $(-2)^2+2(-2)=0$, so $(-2,-2) \in R$
• $a=-1$: $(-1)^2+2(-1)=-1$, so $(-1,-1) \notin R$
• $a=0$: $0^2+2(0)=0$, so $(0,0) \in R$
• $a=1$: $1^2+2(1)=3$, so $(1,1) \in R$
• $a=2$: $2^2+2(2)=8$, so $(2,2) \notin R$
• $a=3$: $3^2+2(3)=15$, so $(3,3) \notin R$

Thus the missing self-pairs are $(-1,-1)$, $(2,2)$, and $(3,3)$. Hence

Finally,

Therefore, the correct option is D.

Given: $0 \le x^2 + 2y \le 4$ with $x,y \in A = \{-3,-2,-1,0,1,2,3\}$.

Find: $l+m$.

Rewrite the condition as

Now fix each value of $y$ and count the possible values of $x$.

• For $y=-3$: $6 \le x^2 \le 10$, so $x=\pm 3$
• For $y=-2$: $4 \le x^2 \le 8$, so $x=\pm 2$
• For $y=-1$: $2 \le x^2 \le 6$, so $x=\pm 2$
• For $y=0$: $0 \le x^2 \le 4$, so $x=-2,-1,0,1,2$
• For $y=1$: $-2 \le x^2 \le 2$, so $x=-1,0,1$
• For $y=2$: $-4 \le x^2 \le 0$, so $x=0$
• For $y=3$: $-6 \le x^2 \le -2$, so no such $x$

Hence the total number of ordered pairs is

For reflexivity, all diagonal pairs $(a,a)$ must be present. From the condition $0 \le a^2+2a \le 4$, the missing pairs are $(-1,-1)$, $(2,2)$, and $(3,3)$. Therefore

So,

Thus the correct option is D.