1
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1

Let $$f, g: \mathbb{N}-\{1\} \rightarrow \mathbb{N}$$ be functions defined by $$f(a)=\alpha$$, where $$\alpha$$ is the maximum of the powers of those primes $$p$$ such that $$p^{\alpha}$$ divides $$a$$, and $$g(a)=a+1$$, for all $$a \in \mathbb{N}-\{1\}$$. Then, the function $$f+g$$ is

A
one-one but not onto
B
onto but not one-one
C
both one-one and onto
D
neither one-one nor onto
2
JEE Main 2022 (Online) 25th July Evening Shift
+4
-1

The number of bijective functions $$f:\{1,3,5,7, \ldots, 99\} \rightarrow\{2,4,6,8, \ldots .100\}$$, such that $$f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots . . f(99)$$, is ____________.

A
$${ }^{50} P_{17}$$
B
$${ }^{50} P_{33}$$
C
$$33 ! \times 17$$!
D
$$\frac{50!}{2}$$
3
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1

The total number of functions,

$$f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\}$$ such that $$f(1)+f(2)=f(3)$$, is equal to :

A
60
B
90
C
108
D
126
4
JEE Main 2022 (Online) 28th June Morning Shift
+4
-1

Let a function f : N $$\to$$ N be defined by

$$f(n) = \left[ {\matrix{ {2n,} & {n = 2,4,6,8,......} \cr {n - 1,} & {n = 3,7,11,15,......} \cr {{{n + 1} \over 2},} & {n = 1,5,9,13,......} \cr } } \right.$$

then, f is

A
one-one but not onto
B
onto but not one-one
C
neither one-one nor onto
D
one-one and onto
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