JEE 2023 — Mathematics PYQ

1. Reflexivity

Ek relation reflexive hota hai agar har $a\in \mathbb{R}$ ke liye $(a,a)\in R$ ho.

Check karte hain:

$3a-3a+\sqrt{7}=0+\sqrt{7}=\sqrt{7}$

Chunki $\sqrt{7}$ ek irrational number hai, isliye $(a,a)\in R$ hamesha sahi hoga.

Atah, $R$ reflexive hai.

2. Symmetry

Ek relation symmetric hota hai agar $(a,b)\in R⟹(b,a)\in R$.

Maan lijiye $a=\frac{\sqrt{7}}{3}$ aur $b=0$.

$(a,b)\in R$ check karein: $3(\frac{\sqrt{7}}{3})-3(0)+\sqrt{7}=\sqrt{7}+\sqrt{7}=2\sqrt{7}$ (Irrational) $⟹$ Sahi hai.

Ab $(b,a)\in R$ check karein: $3(0)-3(\frac{\sqrt{7}}{3})+\sqrt{7}=-\sqrt{7}+\sqrt{7}=0$.

Chunki $0$ ek rational number hai, $(b,a)\notin R$.

Atah, $R$ symmetric nahi hai.

3. Transitivity

Ek relation transitive hota hai agar $(a,b)\in R$ aur $(b,c)\in R⟹(a,c)\in R$.

Maan lijiye:

$a=\frac{\sqrt{7}}{3}$, $b=0$, aur $c=\frac{2\sqrt{7}}{3}$.

• $(a,b)\in R$: $3(\frac{\sqrt{7}}{3})-3(0)+\sqrt{7}=2\sqrt{7}$ (Irrational) $\to$ Sahi hai.

• $(b,c)\in R$: $3(0)-3(\frac{2\sqrt{7}}{3})+\sqrt{7}=-2\sqrt{7}+\sqrt{7}=-\sqrt{7}$ (Irrational) $\to$ Sahi hai.

• $(a,c)\in R$: $3(\frac{\sqrt{7}}{3})-3(\frac{2\sqrt{7}}{3})+\sqrt{7}=\sqrt{7}-2\sqrt{7}+\sqrt{7}=0$.

  Chunki $0$ rational hai, $(a,c)\notin R$.

  Atah, $R$ transitive nahi hai.

Conclusion

Relation reflexive hai, lekin na toh symmetric hai aur na hi transitive.