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1.Relation and Function
hard
Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R=\left\{(x, y) \in N \times N: x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0\right\} .$ Then the relation $R$ is:
A
reflexive and symmetric, but not transitive
B
reflexive but neither symmetric nor transitive
C
an equivalence relation
D
symmetric but neither reflexive nor transitive
(JEE MAIN-2021)
Solution
$x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0$
$\Rightarrow x(x-y)(x+y)-3 y(x-y)(x+y)=0$
$\Rightarrow(x-3 y)(x-y)(x+y)=0$
Now, $x=y \forall(x, y) \in N \times N$ so reflexive
But not symmetric \& transitive
See, $(3,1)$ satisfies but $(1,3)$ does not.
Also $(3,1) \,\&\,(1,-1)$ satisfies but $(3,-1)$ does not
Standard 12
Mathematics
