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1
JEE Main 2021 (Online) 27th July Evening Shift
+4
-1
Let N be the set of natural numbers and a relation R on N be defined by $$R = \{ (x,y) \in N \times N:{x^3} - 3{x^2}y - x{y^2} + 3{y^3} = 0\}$$. Then the relation R is :
A
symmetric but neither reflexive nor transitive
B
reflexive but neither symmetric nor transitive
C
reflexive and symmetric, but not transitive
D
an equivalence relation
2
JEE Main 2021 (Online) 25th July Evening Shift
+4
-1
If [x] be the greatest integer less than or equal to x,

then $$\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]}$$ is equal to :
A
0
B
4
C
$$-$$2
D
2
3
JEE Main 2021 (Online) 25th July Evening Shift
+4
-1
Consider function f : A $$\to$$ B and g : B $$\to$$ C (A, B, C $$\subseteq$$ R) such that (gof)$$-$$1 exists, then :
A
f and g both are one-one
B
f and g both are onto
C
f is one-one and g is onto
D
f is onto and g is one-one
4
JEE Main 2021 (Online) 25th July Morning Shift
+4
-1
Let g : N $$\to$$ N be defined as

g(3n + 1) = 3n + 2,

g(3n + 2) = 3n + 3,

g(3n + 3) = 3n + 1, for all n $$\ge$$ 0.

Then which of the following statements is true?
A
There exists an onto function f : N $$\to$$ N such that fog = f
B
There exists a one-one function f : N $$\to$$ N such that fog = f
C
gogog = g
D
There exists a function : f : N $$\to$$ N such that gof = f
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