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) 26th July Evening Shift
+4
-1

If the maximum value of $$a$$, for which the function $$f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$$ is non-decreasing in $$\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$$, is $$\bar{a}$$, then $$f_{\bar{a}}\left(\frac{\pi}{8}\right)$$ is equal to

A
$$8-\frac{9 \pi}{4\left(9+\pi^{2}\right)}$$
B
$$8-\frac{4 \pi}{9\left(4+\pi^{2}\right)}$$
C
$$8\left(\frac{1+\pi^{2}}{9+\pi^{2}}\right)$$
D
$$8-\frac{\pi}{4}$$
3
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1

Let f : R $$\to$$ R be a continuous function such that $$f(3x) - f(x) = x$$. If $$f(8) = 7$$, then $$f(14)$$ is equal to :

A
4
B
10
C
11
D
16
4
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
$$\frac{50!}{2}$$
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