Question 1: Palette 1/139

JEE Mains 2026 MCQ

Given below are two statements :

Statement I : The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \frac{x}{1 + |x|}$ is one-one.

Statement II : The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \frac{x^2 + 4x - 30}{x^2 - 8x + 18}$ is many-one.

In the light of the above statements, choose the correct answer from the options given below :

A

Statement I is true but Statement II is false

B

Both Statement I and Statement II are false

C

Both Statement I and Statement II are true

D

Statement I is false but Statement II is true

Question 2: Palette 2/139

JEE Mains 2026 MCQ

The sum of all the elements in the range of $f(x) = \text{Sgn}(\sin x) + \text{Sgn}(\cos x) + \text{Sgn}(\tan x) + \text{Sgn}(\cot x)$, $x \neq \frac{n\pi}{2}, n \in \mathbb{Z}$, where

$\text{Sgn}(t) = \begin{cases} 1, & \text{if } t > 0 \\ -1, & \text{if } t < 0 \end{cases}$

is :

A

4

B

0

C

2

D

-2

Question 3: Palette 3/139

JEE Mains 2026 MCQ

If $g(x)=3 x^2+2 x-3, f(0)=-3$ and $4 g(f(x))=3 x^2-32 x+72$, then $f(g(2))$ is equal to:

A

$\frac{7}{2}$

B

$-\frac{25}{6}$

C

$\frac{25}{6}$

D

$-\frac{7}{2}$

Question 4: Palette 4/139

JEE Mains 2026 MCQ

Let $f$ be a function such that $3 f(x)+2 f\left(\frac{m}{19 x}\right)=5 x, x \neq 0$, where $m=\sum\limits_{i=1}^9(i)^2$. Then $f(5)-f(2)$ is equal to

A

36

B

9

C

-9

D

18

Question 5: Palette 5/139

JEE Mains 2026 MCQ

Let $f(x)=[x]^2-[x+3]-3, x \in \mathbf{R}$, where [.] is the greatest integer funtion. Then

A

$f(x)=0$ for finitely many values of $x$

B

$f(x)<0$ only for $x \in[-1,3)$

C

$\int\limits_0^2 f(x) \mathrm{d} x=-6$

D

$f(x)>0$ only for $x \in[4, \infty)$

Question 6: Palette 6/139

JEE Mains 2026 MCQ

Let the domain of the function $f(x)=\log _3 \log _5\left(7-\log _2\left(x^2-10 x+85\right)\right)+\sin ^{-1}\left(\left|\frac{3 x-7}{17-x}\right|\right)$ be $(\alpha, \beta]$. Then $\alpha+\beta$ is equal to :

A

12

B

8

C

10

D

9

Question 7: Palette 7/139

JEE Mains 2026 MCQ

Let $f$ and $g$ be functions satisfying $f(x+y)=f(x) f(y), f(1)=7$ and $g(x+y)=g(x y), g(1)=1$, for all $x, y \in \mathbf{N}$. If $\sum\limits_{x=1}^{\mathrm{n}}\left(\frac{f(x)}{\mathrm{g}(x)}\right)=19607$, then n is equal to :

A

6

B

7

C

4

D

5

Question 8: Palette 8/139

JEE Mains 2026 MCQ

If the domain of the function $f(x)=\sin ^{-1}\left(\frac{5-x}{3+2 x}\right)+\frac{1}{\log _e(10-x)}$ is $(-\infty, \alpha] \cup[\beta, \gamma)-\{\delta\}$, then $6(\alpha+\beta+\gamma+\delta)$ is equal to

A

66

B

68

C

70

D

67

Question 9: Palette 9/139

JEE Mains 2026 MCQ

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=\frac{2 x^2-3 x+2}{3 x^2+x+3}$. Then $f$ is :

A

both one-one and onto

B

one-one but not onto

C

onto but not one-one

D

neither one-one nor onto

Question 10: Palette 10/139

JEE Mains 2026 MCQ

Let [ • ] denote the greatest integer function. If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x+[x]}{3}\right)$ is $[\alpha, \beta)$, then $\alpha^2+\beta^2$ is equal to:

A

2

B

5

C

10

D

13

Question 11: Palette 11/139

JEE Mains 2026 MCQ

For the function $f:[1, \infty) \rightarrow[1, \infty)$ defined by $f(x)=(x-1)^4+1$, among the two statements:

(I) The set $\mathrm{S}=\left\{x \in[1, \infty): f(x)=f^{-1}(x)\right\}$ contains exactly two elements, and

(II) The set $\mathrm{S}=\left\{x \in[1, \infty): f(x)=f^{-1}(x+1)\right\}$ is an empty set,

A

only (I) is TRUE

B

only (II) is TRUE

C

both (I) and (II) are TRUE

D

neither (I) nor (II) is TRUE

Question 12: Palette 12/139

JEE Mains 2026 MCQ

Let for some $\alpha \in \mathbb{R}, f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying $f(x+y)=f(x)+2 y^2+y+\alpha x y$ for all $x, y \in \mathbb{R}$. If $f(0)=-1$ and $f(1)=2$, then the value of $\sum\limits_{n=1}^5(\alpha+f(n))$ is :

A

110

B

140

C

150

D

170

Question 13: Palette 13/139

JEE Mains 2026 MCQ

Let [•] denote the greatest integer function. If the domain of the function

$f(x)=\cos ^{-1}\left(\frac{4 x+2[x]}{3}\right)$ is $[\alpha, \beta]$, then $12(\alpha+\beta)$ is equal to :

A

6

B

8

C

9

D

4

Question 14: Palette 14/139

JEE Mains 2026 MCQ

The number of functions $f:\{1,2,3,4\} \rightarrow\{a, b, c\}$, which are not onto, is :

A

48

B

45

C

51

D

35

Question 15: Palette 15/139

JEE Mains 2025 MCQ

If the range of the function $ f(x) = \frac{5-x}{x^2 - 3x + 2} , \ x \neq 1, 2, $ is $ (-\infty , \alpha] \cup [\beta, \infty) $, then $ \alpha^2 + \beta^2 $ is equal to :

A

188

B

192

C

190

D

194

Question 16: Palette 16/139

JEE Mains 2025 MCQ

Let the domains of the functions $f(x)=\log _4 \log _3 \log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)$ and $\mathrm{g}(x)=\sin ^{-1}\left(\frac{7 x+10}{x-2}\right)$ be $(\alpha, \beta)$ and $[\gamma, \delta]$, respectively. Then $\alpha^2+\beta^2+\gamma^2+\delta^2$ is equal to :

A

15

B

13

C

16

D

14

Question 17: Palette 17/139

JEE Mains 2025 MCQ

Let $f, g:(1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x)=\frac{2 x+3}{5 x+2}$ and $g(x)=\frac{2-3 x}{1-x}$. If the range of the function fog: $[2,4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta-\alpha}$ is equal to

A

56

B

2

C

29

D

68

Question 18: Palette 18/139

JEE Mains 2025 MCQ

Let $f$ be a function such that $f(x)+3 f\left(\frac{24}{x}\right)=4 x, x \neq 0$. Then $f(3)+f(8)$ is equal to

A

13

B

11

C

10

D

12

Question 19: Palette 19/139

JEE Mains 2025 MCQ

If the domain of the function $f(x)=\log _7\left(1-\log _4\left(x^2-9 x+18\right)\right)$ is $(\alpha, \beta) \cup(\gamma, o)$, then $\alpha+\beta+\gamma+\hat{o}$ is equal to

A

17

B

15

C

16

D

18

Question 20: Palette 20/139

JEE Mains 2025 MCQ

$ \text { If the domain of the function } f(x)=\log _e\left(\frac{2 x-3}{5+4 x}\right)+\sin ^{-1}\left(\frac{4+3 x}{2-x}\right) \text { is }[\alpha, \beta) \text {, then } \alpha^2+4 \beta \text { is equal to } $

A

4

B

3

C

7

D

5

Question 21: Palette 21/139

JEE Mains 2025 MCQ

If the domain of the function $f(x)=\frac{1}{\sqrt{10+3 x-x^2}}+\frac{1}{\sqrt{x+|x|}}$ is $(a, b)$, then $(1+a)^2+b^2$ is equal to :

A

29

B

30

C

25

D

26

Question 22: Palette 22/139

JEE Mains 2025 MCQ

If the domain of the function $ \log_5(18x - x^2 - 77) $ is $ (\alpha, \beta) $ and the domain of the function $ \log_{(x-1)} \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) $ is $(\gamma, \delta)$, then $ \alpha^2 + \beta^2 + \gamma^2 $ is equal to:

A

186

B

179

C

195

D

174

Question 23: Palette 23/139

JEE Mains 2025 MCQ

Let $f:[0,3] \rightarrow$ A be defined by $f(x)=2 x^3-15 x^2+36 x+7$ and $g:[0, \infty) \rightarrow B$ be defined by $g(x)=\frac{x^{2025}}{x^{2025}+1}$, If both the functions are onto and $S=\{ x \in Z ; x \in A$ or $x \in B \}$, then $n(S)$ is equal to :

A

29

B

31

C

30

D

36

Question 24: Palette 24/139

JEE Mains 2025 MCQ

If $f(x)=\frac{2^x}{2^x+\sqrt{2}}, \mathrm{x} \in \mathbb{R}$, then $\sum_\limits{\mathrm{k}=1}^{81} f\left(\frac{\mathrm{k}}{82}\right)$ is equal to

A

$82$

B

$81 \sqrt{2}$

C

$41$

D

$\frac{81}{2}$

Question 25: Palette 25/139

JEE Mains 2025 MCQ

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=(2+3 a) x^2+\left(\frac{a+2}{a-1}\right) x+b, a \neq 1$. If $f(x+y)=f(x)+f(\mathrm{y})+1-\frac{2}{7} x \mathrm{y}$, then the value of $28 \sum\limits_{i=1}^5|f(i)|$ is

A

735

B

675

C

715

D

545

Question 26: Palette 26/139

JEE Mains 2025 MCQ

The function $f:(-\infty, \infty) \rightarrow(-\infty, 1)$, defined by $f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}}$ 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

Question 27: Palette 27/139

JEE Mains 2025 MCQ

Let $f(x)=\frac{2^{x+2}+16}{2^{2 x+1}+2^{x+4}+32}$. Then the value of $8\left(f\left(\frac{1}{15}\right)+f\left(\frac{2}{15}\right)+\ldots+f\left(\frac{59}{15}\right)\right)$ is equal to

A

108

B

92

C

118

D

102

Question 28: Palette 28/139

JEE Mains 2025 MCQ

Let $f(x)=\log _{\mathrm{e}} x$ and $g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}$. Then the domain of $f \circ g$ is

A

$(0, \infty)$

B

$[1, \infty)$

C

$\mathbb{R}$

D

$[0, \infty)$

Question 29: Palette 29/139

JEE Mains 2025 MCQ

Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{1,4,9,16\}$. Then the number of many-one functions $f: \mathrm{A} \rightarrow \mathrm{B}$ such that $1 \in f(\mathrm{~A})$ is equal to :

A

151

B

139

C

163

D

127

Question 30: Palette 30/139

JEE Mains 2024 MCQ

Let the range of the function $f(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \mathbb{R}$ be $[a, b]$. If $\alpha$ and $\beta$ ar respectively the A.M. and the G.M. of $a$ and $b$, then $\frac{\alpha}{\beta}$ is equal to

A

$\pi$

B

$\sqrt{\pi}$

C

$\sqrt{2}$

D

2

Question 31: Palette 31/139

JEE Mains 2024 MCQ

If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$ is $\mathbf{R}-(\alpha, \beta)$, then $12 \alpha \beta$ is equal to :

A

40

B

36

C

24

D

32

Question 32: Palette 32/139

JEE Mains 2024 MCQ

Let $f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0< x \leq \mathrm{a}\end{array}\right.$ where $\mathrm{a}> 0$ and $\mathrm{g}(x)=(f(|x|)-|f(x)|) / 2$. Then the function $g:[-a, a] \rightarrow[-a, a]$ is

A

neither one-one nor onto.

B

both one-one and onto.

C

one-one.

D

onto

Question 33: Palette 33/139

JEE Mains 2024 MCQ

If the function $f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0$ attains the maximum value at $x=\frac{1}{\mathrm{e}}$ then :

A

$\mathrm{e}^\pi<\pi^{\mathrm{e}}$

B

$\mathrm{e}^{2 \pi}<(2 \pi)^{\mathrm{e}}$

C

$(2 e)^\pi>\pi^{(2 e)}$

D

$\mathrm{e}^\pi>\pi^{\mathrm{e}}$

Question 34: Palette 34/139

JEE Mains 2024 MCQ

Let $f(x)=\frac{1}{7-\sin 5 x}$ be a function defined on $\mathbf{R}$. Then the range of the function $f(x)$ is equal to :

A

$\left[\frac{1}{8}, \frac{1}{5}\right]$

B

$\left[\frac{1}{7}, \frac{1}{6}\right]$

C

$\left[\frac{1}{7}, \frac{1}{5}\right]$

D

$\left[\frac{1}{8}, \frac{1}{6}\right]$

Question 35: Palette 35/139

JEE Mains 2024 MCQ

The function $f(x)=\frac{x^2+2 x-15}{x^2-4 x+9}, x \in \mathbb{R}$ is

A

both one-one and onto.

B

onto but not one-one.

C

neither one-one nor onto.

D

one-one but not onto.

Question 36: Palette 36/139

JEE Mains 2024 MCQ

Let $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as :

$f(x)=|x-1| \text { and } g(x)= \begin{cases}\mathrm{e}^x, & x \geq 0 \\ x+1, & x \leq 0 .\end{cases}$

Then the function $f(g(x))$ is

A

neither one-one nor onto.

B

one-one but not onto.

C

both one-one and onto.

D

onto but not one-one.

Question 37: Palette 37/139

JEE Mains 2024 MCQ

Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f: A \rightarrow B$, such that $f(1)+f(3)=14$, is :

A

120

B

180

C

240

D

480

Question 38: Palette 38/139

JEE Mains 2024 MCQ

If the domain of the function

$f(x)=\frac{\sqrt{x^2-25}}{\left(4-x^2\right)}+\log _{10}\left(x^2+2 x-15\right)$ is $(-\infty, \alpha) \cup[\beta, \infty)$, then $\alpha^2+\beta^3$ is equal to :

A

140

B

175

C

125

D

150

Question 39: Palette 39/139

JEE Mains 2024 MCQ

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as

$f(x)=\left\{\begin{array}{ll}\log _{\mathrm{e}} x, & x>0 \\ \mathrm{e}^{-x}, & x \leq 0\end{array}\right.$ and

$g(x)=\left\{\begin{array}{ll}x, & x \geqslant 0 \\ \mathrm{e}^x, & x<0\end{array}\right.$. Then, gof : $\mathbf{R} \rightarrow \mathbf{R}$ is :

A

one-one but not onto

B

neither one-one nor onto

C

onto but not one-one

D

both one-one and onto

Question 40: Palette 40/139

JEE Mains 2024 MCQ

If $f(x)=\frac{4 x+3}{6 x-4}, x \neq \frac{2}{3}$ and $(f \circ f)(x)=g(x)$, where $g: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\}$, then $(g ogog)(4)$ is equal to

A

$-4$

B

$\frac{19}{20}$

C

$-\frac{19}{20}$

D

4

Question 41: Palette 41/139

JEE Mains 2024 MCQ

If the domain of the function $f(x)=\log _e\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5 \beta-4 \alpha$ is equal to

A

9

B

12

C

11

D

10

Question 42: Palette 42/139

JEE Mains 2024 MCQ

If the domain of the function $f(x)=\cos ^{-1}\left(\frac{2-|x|}{4}\right)+\left\{\log _e(3-x)\right\}^{-1}$ is $[-\alpha, \beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is equal to :

A

11

B

12

C

9

D

8

Question 43: Palette 43/139

JEE Mains 2024 MCQ

If $f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\begin{array}{cc}-x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1\end{array}\right.\right.$, then range of $(f o g)(x)$ is

A

$[0,1)$

B

$[0,3)$

C

$(0,1]$

D

$[0,1]$

Question 44: Palette 44/139

JEE Mains 2024 MCQ

Let $f: \mathbf{R}-\left\{\frac{-1}{2}\right\} \rightarrow \mathbf{R}$ and $g: \mathbf{R}-\left\{\frac{-5}{2}\right\} \rightarrow \mathbf{R}$ be defined as $f(x)=\frac{2 x+3}{2 x+1}$ and $g(x)=\frac{|x|+1}{2 x+5}$. Then, the domain of the function fog is :

A

$\mathbf{R}-\left\{-\frac{7}{4}\right\}$

B

$\mathbf{R}$

C

$\mathbf{R}-\left\{-\frac{5}{2},-\frac{7}{4}\right\}$

D

$\mathbf{R}-\left\{-\frac{5}{2}\right\}$

Question 45: Palette 45/139

JEE Mains 2024 MCQ

The function $f: \mathbf{N}-\{1\} \rightarrow \mathbf{N}$; defined by $f(\mathrm{n})=$ the highest prime factor of $\mathrm{n}$, is :

A

one-one only

B

neither one-one nor onto

C

onto only

D

both one-one and onto

Question 46: Palette 46/139

JEE Mains 2023 MCQ

The range of $f(x)=4 \sin ^{-1}\left(\frac{x^{2}}{x^{2}+1}\right)$ is

A

$[0,2 \pi]$

B

$[0,2 \pi)$

C

$[0, \pi)$

D

$[0, \pi]$

Question 47: Palette 47/139

JEE Mains 2023 MCQ

For $x \in \mathbb{R}$, two real valued functions $f(x)$ and $g(x)$ are such that, $g(x)=\sqrt{x}+1$ and $f \circ g(x)=x+3-\sqrt{x}$. Then $f(0)$ is equal to

A

5

B

0

C

$-$3

D

1

Question 48: Palette 48/139

JEE Mains 2023 MCQ

Let $\mathrm{D}$ be the domain of the function $f(x)=\sin ^{-1}\left(\log _{3 x}\left(\frac{6+2 \log _{3} x}{-5 x}\right)\right)$. If the range of the function $\mathrm{g}: \mathrm{D} \rightarrow \mathbb{R}$ defined by $\mathrm{g}(x)=x-[x],([x]$ is the greatest integer function), is $(\alpha, \beta)$, then $\alpha^{2}+\frac{5}{\beta}$ is equal to

A

45

B

136

C

46

D

nearly 135

Question 49: Palette 49/139

JEE Mains 2023 MCQ

The domain of the function $f(x)=\frac{1}{\sqrt{[x]^{2}-3[x]-10}}$ is : ( where $[\mathrm{x}]$ denotes the greatest integer less than or equal to $x$ )

A

$(-\infty,-2) \cup[6, \infty)$

B

$(-\infty,-3] \cup[6, \infty)$

C

$(-\infty,-2) \cup(5, \infty)$

D

$(-\infty,-3] \cup(5, \infty)$

Question 50: Palette 50/139

JEE Mains 2023 MCQ

If $f(x) = {{(\tan 1^\circ )x + {{\log }_e}(123)} \over {x{{\log }_e}(1234) - (\tan 1^\circ )}},x > 0$, then the least value of $f(f(x)) + f\left( {f\left( {{4 \over x}} \right)} \right)$ is :

A

2

B

4

C

0

D

8

Question 51: Palette 51/139

JEE Mains 2023 MCQ

Let the sets A and B denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$, where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Then among the statements

(S1) : $A \cap B=(1, \infty)-\mathbb{N}$ and

(S2) : $A \cup B=(1, \infty)$

A

only $(\mathrm{S} 2)$ is true

B

only (S1) is true

C

neither (S1) nor (S2) is true

D

both (S1) and (S2) are true

Question 52: Palette 52/139

JEE Mains 2023 MCQ

Let $f:\mathbb{R}-{0,1}\to \mathbb{R}$ be a function such that $f(x)+f\left(\frac{1}{1-x}\right)=1+x$. Then $f(2)$ is equal to

A

$\frac{9}{4}$

B

$\frac{7}{4}$

C

$\frac{7}{3}$

D

$\frac{9}{2}$

Question 53: Palette 53/139

JEE Mains 2023 MCQ

Let $f(x) = \left| {\matrix{ {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \sin 2x} \cr } } \right|,\,x \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

A

${\alpha ^2} - {\beta ^2} = 4\sqrt 3 $

B

${\beta ^2} - 2\sqrt \alpha = {{19} \over 4}$

C

${\beta ^2} + 2\sqrt \alpha = {{19} \over 4}$

D

${\alpha ^2} + {\beta ^2} = {9 \over 2}$

Question 54: Palette 54/139

JEE Mains 2023 MCQ

Let $f: \mathbb{R}-\{2,6\} \rightarrow \mathbb{R}$ be real valued function

defined as $f(x)=\frac{x^2+2 x+1}{x^2-8 x+12}$.

Then range of $f$ is

A

$ \left(-\infty,-\frac{21}{4}\right] \cup[1, \infty) $

B

$\left(-\infty,-\frac{21}{4}\right) \cup(0, \infty) $

C

$\left(-\infty,-\frac{21}{4}\right] \cup[0, \infty) $

D

$\left(-\infty,-\frac{21}{4}\right] \cup\left[\frac{21}{4}, \infty\right)$

Question 55: Palette 55/139

JEE Mains 2023 MCQ

The absolute minimum value, of the function

$f(x)=\left|x^{2}-x+1\right|+\left[x^{2}-x+1\right]$,

where $[t]$ denotes the greatest integer function, in the interval $[-1,2]$, is :

A

$\frac{3}{4}$

B

$\frac{3}{2}$

C

$\frac{1}{4}$

D

$\frac{5}{4}$

Question 56: Palette 56/139

JEE Mains 2023 MCQ

If the domain of the function $f(x)=\frac{[x]}{1+x^{2}}$, where $[x]$ is greatest integer $\leq x$, is $[2,6)$, then its range is

A

$\left(\frac{5}{37}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$

B

$\left(\frac{5}{37}, \frac{2}{5}\right]$

C

$\left(\frac{5}{26}, \frac{2}{5}\right]$

D

$\left(\frac{5}{26}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$

Question 57: Palette 57/139

JEE Mains 2023 MCQ

The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is :

A

$[2 \sqrt{2}, \sqrt{11}]$

B

$[\sqrt{5}, \sqrt{13}]$

C

$[\sqrt{2}, \sqrt{7}]$

D

$[\sqrt{5}, \sqrt{10}]$

Question 58: Palette 58/139

JEE Mains 2023 MCQ

Consider a function $f:\mathbb{N}\to\mathbb{R}$, satisfying $f(1)+2f(2)+3f(3)+....+xf(x)=x(x+1)f(x);x\ge2$ with $f(1)=1$. Then $\frac{1}{f(2022)}+\frac{1}{f(2028)}$ is equal to

A

8000

B

8400

C

8100

D

8200

Question 59: Palette 59/139

JEE Mains 2023 MCQ

The domain of $f(x) = {{{{\log }_{(x + 1)}}(x - 2)} \over {{e^{2{{\log }_e}x}} - (2x + 3)}},x \in \mathbb{R}$ is

A

$( - 1,\infty ) - \{ 3\} $

B

$\mathbb{R} - \{ - 1,3)$

C

$(2,\infty ) - \{ 3\} $

D

$\mathbb{R} - \{ 3\} $

Question 60: Palette 60/139

JEE Mains 2023 MCQ

Let $f:R \to R$ be a function such that $f(x) = {{{x^2} + 2x + 1} \over {{x^2} + 1}}$. Then

A

$f(x)$ is many-one in $( - \infty , - 1)$

B

$f(x)$ is one-one in $( - \infty ,\infty )$

C

$f(x)$ is one-one in $[1,\infty )$ but not in $( - \infty ,\infty )$

D

$f(x)$ is many-one in $(1,\infty )$

Question 61: Palette 61/139

JEE Mains 2023 MCQ

The number of functions

$f:\{ 1,2,3,4\} \to \{ a \in Z|a| \le 8\} $

satisfying $f(n) + {1 \over n}f(n + 1) = 1,\forall n \in \{ 1,2,3\} $ is

A

2

B

3

C

1

D

4

Question 62: Palette 62/139

JEE Mains 2023 MCQ

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x) = {\log _{\sqrt m }}\{ \sqrt 2 (\sin x - \cos x) + m - 2\} $, for some $m$, such that the range of $f$ is [0, 2]. Then the value of $m$ is _________

A

4

B

3

C

5

D

2

Question 63: Palette 63/139

JEE Mains 2023 MCQ

Let $f(x) = 2{x^n} + \lambda ,\lambda \in R,n \in N$, and $f(4) = 133,f(5) = 255$. Then the sum of all the positive integer divisors of $(f(3) - f(2))$ is

A

60

B

58

C

61

D

59

Question 64: Palette 64/139

JEE Mains 2023 MCQ

Let $f(x)$ be a function such that $f(x+y)=f(x).f(y)$ for all $x,y\in \mathbb{N}$. If $f(1)=3$ and $\sum\limits_{k = 1}^n {f(k) = 3279} $, then the value of n is

A

9

B

7

C

6

D

8

Question 65: Palette 65/139

JEE Mains 2023 MCQ

If $f(x) = {{{2^{2x}}} \over {{2^{2x}} + 2}},x \in \mathbb{R}$, then $f\left( {{1 \over {2023}}} \right) + f\left( {{2 \over {2023}}} \right)\, + \,...\, + \,f\left( {{{2022} \over {2023}}} \right)$ is equal to

A

2011

B

2010

C

1010

D

1011

Question 66: Palette 66/139

JEE Mains 2022 MCQ

$ \text { Let } f(x)=a x^{2}+b x+c \text { be such that } f(1)=3, f(-2)=\lambda \text { and } $ $f(3)=4$. If $f(0)+f(1)+f(-2)+f(3)=14$, then $\lambda$ is equal to :

A

$-$4

B

$\frac{13}{2}$

C

$\frac{23}{2}$

D

4

Question 67: Palette 67/139

JEE Mains 2022 MCQ

Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha x^{5}+\beta x^{3}+\gamma x, x \in \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be such that $g(f(x))=x$ for all $x \in \mathbf{R}$. If $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots, \mathrm{a}_{\mathrm{n}}$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\frac{1}{\mathrm{n}} \sum\limits_{i=1}^{\mathrm{n}} f\left(\mathrm{a}_{i}\right)\right)\right)$ is equal to :

A

0

B

3

C

9

D

27

Question 68: Palette 68/139

JEE Mains 2022 MCQ

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

Question 69: Palette 69/139

JEE Mains 2022 MCQ

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

$\frac{50!}{2}$

Question 70: Palette 70/139

JEE Mains 2022 MCQ

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

Question 71: Palette 71/139

JEE Mains 2022 MCQ

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

Question 72: Palette 72/139

JEE Mains 2022 MCQ

Let f : R $\to$ R be defined as f (x) = x $-$ 1 and g : R $-$ {1, $-$1} $\to$ R be defined as $g(x) = {{{x^2}} \over {{x^2} - 1}}$.

Then the function fog 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

Question 73: Palette 73/139

JEE Mains 2022 MCQ

Let $f(x) = {{x - 1} \over {x + 1}},\,x \in R - \{ 0, - 1,1\} $. If ${f^{n + 1}}(x) = f({f^n}(x))$ for all n $\in$ N, then ${f^6}(6) + {f^7}(7)$ is equal to :

A

${7 \over 6}$

B

$ - {3 \over 2}$

C

${7 \over {12}}$

D

$ - {{11} \over {12}}$

Question 74: Palette 74/139

JEE Mains 2022 MCQ

Let f : N $\to$ R be a function such that $f(x + y) = 2f(x)f(y)$ for natural numbers x and y. If f(1) = 2, then the value of $\alpha$ for which

$\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)} $

holds, is :

A

2

B

3

C

4

D

6

Question 75: Palette 75/139

JEE Mains 2022 MCQ

Let $f:R \to R$ and $g:R \to R$ be two functions defined by $f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$ and $g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}$. Then, for which of the following range of $\alpha$, the inequality $f\left( {g\left( {{{{{(\alpha - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha -{5 \over 3}} \right)} \right)$ holds ?

A

(2, 3)

B

($-$2, $-$1)

C

(1, 2)

D

($-$1, 1)

Question 76: Palette 76/139

JEE Mains 2021 MCQ

The range of the function,

$f(x) = {\log _{\sqrt 5 }}\left( {3 + \cos \left( {{{3\pi } \over 4} + x} \right) + \cos \left( {{\pi \over 4} + x} \right) + \cos \left( {{\pi \over 4} - x} \right) - \cos \left( {{{3\pi } \over 4} - x} \right)} \right)$ is :

A

$\left( {0,\sqrt 5 } \right)$

B

[$-$2, 2]

C

$\left[ {{1 \over {\sqrt 5 }},\sqrt 5 } \right]$

D

[0, 2]

Question 77: Palette 77/139

JEE Mains 2021 MCQ

Let f : N $\to$ N be a function such that f(m + n) = f(m) + f(n) for every m, n$\in$N. If f(6) = 18, then f(2) . f(3) is equal to :

A

6

B

54

C

18

D

36

Question 78: Palette 78/139

JEE Mains 2021 MCQ

Let f : R $\to$ R be defined as $f(x + y) + f(x - y) = 2f(x)f(y),f\left( {{1 \over 2}} \right) = - 1$. Then, the value of $\sum\limits_{k = 1}^{20} {{1 \over {\sin (k)\sin (k + f(k))}}} $ is equal to :

A

cosec²(21) cos(20) cos(2)

B

sec²(1) sec(21) cos(20)

C

cosec²(1) cosec(21) sin(20)

D

sec²(21) sin(20) sin(2)

Question 79: Palette 79/139

JEE Mains 2021 MCQ

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

Question 80: Palette 80/139

JEE Mains 2021 MCQ

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

Question 81: Palette 81/139

JEE Mains 2021 MCQ

Let $f:R - \left\{ {{\alpha \over 6}} \right\} \to R$ be defined by $f(x) = {{5x + 3} \over {6x - \alpha }}$. Then the value of $\alpha$ for which (fof)(x) = x, for all $x \in R - \left\{ {{\alpha \over 6}} \right\}$, is :

A

No such $\alpha$ exists

B

5

C

8

D

6

Question 82: Palette 82/139

JEE Mains 2021 MCQ

Let [ x ] denote the greatest integer $\le$ x, where x $\in$ R. If the domain of the real valued function $f(x) = \sqrt {{{\left| {[x]} \right| - 2} \over {\left| {[x]} \right| - 3}}} $ is ($-$ $\infty$, a) $]\cup$ [b, c) $\cup$ [4, $\infty$), a < b < c, then the value of a + b + c is :

A

8

B

1

C

$-$2

D

$-$3

Question 83: Palette 83/139

JEE Mains 2021 MCQ

Let f : R $-$ {3} $ \to $ R $-$ {1} be defined by f(x) = ${{x - 2} \over {x - 3}}$.

Let g : R $ \to $ R be given as g(x) = 2x $-$ 3. Then, the sum of all the values of x for which f^$-$1(x) + g^$-$1(x) = ${{13} \over 2}$ is equal to :

A

3

B

5

C

2

D

7

Question 84: Palette 84/139

JEE Mains 2021 MCQ

The real valued function
$f(x) = {{\cos e{c^{ - 1}}x} \over {\sqrt {x - [x]} }}$, where [x] denotes the greatest integer less than or equal to x, is defined for all x belonging to :

A

all real except integers

B

all non-integers except the interval [ $-$1, 1 ]

C

all integers except 0, $-$1, 1

D

all real except the interval [ $-$1, 1 ]

Question 85: Palette 85/139

JEE Mains 2021 MCQ

If the functions are defined as $f(x) = \sqrt x $ and $g(x) = \sqrt {1 - x} $, then what is the common domain of the following functions :

f + g, f $-$ g, f/g, g/f, g $-$ f where $(f \pm g)(x) = f(x) \pm g(x),(f/g)x = {{f(x)} \over {g(x)}}$

A

$0 \le x \le 1$

B

$0 \le x < 1$

C

$0 < x < 1$

D

$0 < x \le 1$

Question 86: Palette 86/139

JEE Mains 2021 MCQ

The inverse of $y = {5^{\log x}}$ is :

A

$x = {5^{\log y}}$

B

$x = {y^{{1 \over {\log 5}}}}$

C

$x = {5^{{1 \over {\log y}}}}$

D

$x = {y^{\log 5}}$

Question 87: Palette 87/139

JEE Mains 2021 MCQ

The range of a$\in$R for which the

function f(x) = (4a $-$ 3)(x + log_e 5) + 2(a $-$ 7) cot$\left( {{x \over 2}} \right)$ sin²$\left( {{x \over 2}} \right)$, x $\ne$ 2n$\pi$, n$\in$N has critical points, is :

A

[1, $\infty $)

B

($-$3, 1)

C

$\left[ { - {4 \over 3},2} \right]$

D

($-$$\infty $, $-$1]

Question 88: Palette 88/139

JEE Mains 2021 MCQ

Let $A = \{ 1,2,3,....,10\} $ and $f:A \to A$ be defined as

$f(k) = \left\{ {\matrix{ {k + 1} & {if\,k\,is\,odd} \cr k & {if\,k\,is\,even} \cr } } \right.$

Then the number of possible functions $g:A \to A$ such that $gof = f$ is :

A

5⁵

B

10⁵

C

5!

D

¹⁰C₅

Question 89: Palette 89/139

JEE Mains 2021 MCQ

A function f(x) is given by $f(x) = {{{5^x}} \over {{5^x} + 5}}$, then the sum of the series $f\left( {{1 \over {20}}} \right) + f\left( {{2 \over {20}}} \right) + f\left( {{3 \over {20}}} \right) + ....... + f\left( {{{39} \over {20}}} \right)$ is equal to :

A

${{{39} \over 2}}$

B

${{{19} \over 2}}$

C

${{{49} \over 2}}$

D

${{{29} \over 2}}$

Question 90: Palette 90/139

JEE Mains 2021 MCQ

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set A $\times$ B. Then :

A

2y = 273x

B

y = 91x

C

2y = 91x

D

y = 273x

Question 91: Palette 91/139

JEE Mains 2021 MCQ

Let f, g : N $ \to $ N such that f(n + 1) = f(n) + f(1) $\forall $ n$\in$N and g be any arbitrary function. Which of the following statements is NOT true?

A

If g is onto, then fog is one-one

B

f is one-one

C

If f is onto, then f(n) = n $\forall $n$\in$N

D

If fog is one-one, then g is one-one

Question 92: Palette 92/139

JEE Mains 2021 MCQ

Let f : R → R be defined as f (x) = 2x – 1 and g : R - {1} → R be defined as g(x) = ${{x - {1 \over 2}} \over {x - 1}}$. Then the composition function f(g(x)) 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

Question 93: Palette 93/139

JEE Mains 2020 MCQ

For a suitably chosen real constant a, let a

function, $f:R - \left\{ { - a} \right\} \to R$ be defined by

$f(x) = {{a - x} \over {a + x}}$. Further suppose that for any real number $x \ne - a$ and $f(x) \ne - a$,

(fof)(x) = x. Then $f\left( { - {1 \over 2}} \right)$ is equal to :

A

$ {1 \over 3}$

B

–3

C

$ - {1 \over 3}$

D

3

Question 94: Palette 94/139

JEE Mains 2020 MCQ

If f(x + y) = f(x)f(y) and $\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$ , x, y $ \in $ N, where N is the set of all natural number, then the value of ${{f\left( 4 \right)} \over {f\left( 2 \right)}}$ is :

A

${2 \over 3}$

B

${1 \over 9}$

C

${1 \over 3}$

D

${4 \over 9}$

Question 95: Palette 95/139

JEE Mains 2020 MCQ

Let f : R $ \to $ R be a function which satisfies
f(x + y) = f(x) + f(y) $\forall $ x, y $ \in $ R. If f(1) = 2 and
g(n) = $\sum\limits_{k = 1}^{\left( {n - 1} \right)} {f\left( k \right)} $, n $ \in $ N then the value of n, for which g(n) = 20, is :

A

20

B

9

C

5

D

4

Question 96: Palette 96/139

JEE Mains 2020 MCQ

Let a – 2b + c = 1.

If $f(x)=\left| {\matrix{ {x + a} & {x + 2} & {x + 1} \cr {x + b} & {x + 3} & {x + 2} \cr {x + c} & {x + 4} & {x + 3} \cr } } \right|$, then:

A

ƒ(50) = 1

B

ƒ(–50) = –1

C

ƒ(50) = –501

D

ƒ(–50) = 501

Question 97: Palette 97/139

JEE Mains 2020 MCQ

Let ƒ : (1, 3) $ \to $ R be a function defined by
$f(x) = {{x\left[ x \right]} \over {1 + {x^2}}}$ , where [x] denotes the greatest integer $ \le $ x. Then the range of ƒ is

A

$\left( {{2 \over 5},{1 \over 2}} \right) \cup \left( {{3 \over 4},{4 \over 5}} \right]$

B

$\left( {{3 \over 5},{4 \over 5}} \right)$

C

$\left( {{2 \over 5},{4 \over 5}} \right]$

D

$\left( {{2 \over 5},{3 \over 5}} \right] \cup \left( {{3 \over 4},{4 \over 5}} \right)$

Question 98: Palette 98/139

JEE Mains 2020 MCQ

The inverse function of

f(x) = ${{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}$, x $ \in $ (-1, 1), is :

A

${1 \over 4}{\log _e}\left( {{{1 - x} \over {1 + x}}} \right)$

B

${1 \over 4}\left( {{{\log }_8}e} \right){\log _e}\left( {{{1 - x} \over {1 + x}}} \right)$

C

${1 \over 4}\left( {{{\log }_8}e} \right){\log _e}\left( {{{1 + x} \over {1 - x}}} \right)$

D

${1 \over 4}{\log _e}\left( {{{1 + x} \over {1 - x}}} \right)$

Question 99: Palette 99/139

JEE Mains 2020 MCQ

If g(x) = x² + x - 1 and
(goƒ) (x) = 4x² - 10x + 5, then ƒ$\left( {{5 \over 4}} \right)$ is equal to:

A

${1 \over 2}$

B

${3 \over 2}$

C

-${1 \over 2}$

D

-${3 \over 2}$

Question 100: Palette 100/139

JEE Mains 2019 MCQ

For x $ \in $ (0, 3/2), let f(x) = $\sqrt x $ , g(x) = tan x and h(x) = ${{1 - {x^2}} \over {1 + {x^2}}}$. If $\phi $ (x) = ((hof)og)(x), then $\phi \left( {{\pi \over 3}} \right)$ is equal to :

A

$\tan {{7\pi } \over {12}}$

B

$\tan {{11\pi } \over {12}}$

C

$\tan {\pi \over {12}}$

D

$\tan {{5\pi } \over {12}}$

Question 101: Palette 101/139

JEE Mains 2019 MCQ

Let f(x) = e^x – x and g(x) = x² – x, $\forall $ x $ \in $ R. Then the set of all x $ \in $ R, where the function h(x) = (fog) (x) is increasing, is :

A

[0, $\infty $)

B

$\left[ { - 1, - {1 \over 2}} \right] \cup \left[ {{1 \over 2},\infty } \right)$

C

$\left[ { - {1 \over 2},0} \right] \cup \left[ {1,\infty } \right)$

D

$\left[ {0,{1 \over 2}} \right] \cup \left[ {1,\infty } \right)$

Question 102: Palette 102/139

JEE Mains 2019 MCQ

Let f(x) = x² , x $ \in $ R. For any A $ \subseteq $ R, define g (A) = { x $ \in $ R : f(x) $ \in $ A}. If S = [0,4], then which one of the following statements is not true ?

A

g(f(S)) $ \ne $ S

B

f(g(S)) = S

C

f(g(S)) $ \ne $ f(S)

D

g(f(S)) = g(S)

Question 103: Palette 103/139

JEE Mains 2019 MCQ

The domain of the definition of the function

$f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)$ is

A

(-1, 0) $ \cup $ (1, 2) $ \cup $ (2, $\infty $)

B

(-2, -1) $ \cup $ (-1,0) $ \cup $ (2, $\infty $)

C

(1, 2) $ \cup $ (2, $\infty $)

D

(-1, 0) $ \cup $ (1,2) $ \cup $ (3, $\infty $)

Question 104: Palette 104/139

JEE Mains 2019 MCQ

Let $\sum\limits_{k = 1}^{10} {f(a + k) = 16\left( {{2^{10}} - 1} \right)} $ where the function ƒ satisfies
ƒ(x + y) = ƒ(x)ƒ(y) for all natural numbers x, y and ƒ(1) = 2. then the natural number 'a' is

A

2

B

16

C

4

D

3

Question 105: Palette 105/139

JEE Mains 2019 MCQ

If the function ƒ : R – {1, –1} $ \to $ A defined by
ƒ(x) = ${{{x^2}} \over {1 - {x^2}}}$ , is surjective, then A is equal to

A

R – (–1, 0)

B

R – {–1}

C

R – [–1, 0)

D

[0, $\infty $)

Question 106: Palette 106/139

JEE Mains 2019 MCQ

Let ƒ(x) = a^x (a > 0) be written as
ƒ(x) = ƒ₁ (x) + ƒ₂ (x), where ƒ₁ (x) is an even function of ƒ₂ (x) is an odd function.
Then ƒ₁ (x + y) + ƒ₁ (x – y) equals

A

2ƒ₁ (x)ƒ₁ (y)

B

2ƒ₁ (x + y)ƒ₁ (x – y)

C

2ƒ₁ (x)ƒ₂ (y)

D

2ƒ₁ (x + y)ƒ₂ (x – y)

Question 107: Palette 107/139

JEE Mains 2019 MCQ

If $f(x) = {\log _e}\left( {{{1 - x} \over {1 + x}}} \right)$, $\left| x \right| < 1$ then $f\left( {{{2x} \over {1 + {x^2}}}} \right)$ is equal to

A

2f(x²)

B

2f(x)

C

(f(x))²

D

-2f(x)

Question 108: Palette 108/139

JEE Mains 2019 MCQ

Let a function f : (0, $\infty $) $ \to $ (0, $\infty $) be defined by f(x) = $\left| {1 - {1 \over x}} \right|$. Then f is :

A

not injective but it is surjective

B

neiter injective nor surjective

C

injective only

D

both injective as well as surjective

Question 109: Palette 109/139

JEE Mains 2019 MCQ

The number of functions f from {1, 2, 3, ...., 20} onto {1, 2, 3, ...., 20} such that f(k) is a multiple of 3, whenever k is a multiple of 4, is :

A

6⁵ $ \times $ (15)!

B

5⁶ $ \times $ 15

C

(15)! $ \times $ 6!

D

5! $ \times $ 6!

Question 110: Palette 110/139

JEE Mains 2019 MCQ

Let f_k(x) = ${1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$ for k = 1, 2, 3, ... Then for all x $ \in $ R, the value of f₄(x) $-$ f₆(x) is equal to

A

${1 \over 4}$

B

${5 \over {12}}$

C

${{ - 1} \over {12}}$

D

${1 \over {12}}$

Question 111: Palette 111/139

JEE Mains 2019 MCQ

Let f : R $ \to $ R be defined by f(x) = ${x \over {1 + {x^2}}},x \in R$.   Then the range of f is :

A

$\left[ { - {1 \over 2},{1 \over 2}} \right]$

B

$R - \left[ { - {1 \over 2},{1 \over 2}} \right]$

C

($-$ 1, 1) $-$ {0}

D

R $-$ [$-$1, 1]

Question 112: Palette 112/139

JEE Mains 2019 MCQ

Let N be the set of natural numbers and two functions f and g be defined as f, g : N $ \to $ N such that

f(n) = $\left\{ {\matrix{ {{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr {{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr } \,\,} \right.$;

      and g(n) = n $-$($-$ 1)ⁿ.

Then fog is -

A

neither one-one nor onto

B

onto but not one-one

C

both one-one and onto

D

one-one but not onto

Question 113: Palette 113/139

JEE Mains 2019 MCQ

Let A = {x $ \in $ R : x is not a positive integer}.

Define a function $f$ : A $ \to $  R   as  $f(x)$ = ${{2x} \over {x - 1}}$,

then $f$ is :

A

not injective

B

neither injective nor surjective

C

surjective but not injective

D

injective but not surjective

Question 114: Palette 114/139

JEE Mains 2019 MCQ

For $x \in R - \left\{ {0,1} \right\}$, Let f₁(x) = $1\over x$, f₂ (x) = 1 – x

and f₃ (x) = $1 \over {1 - x}$ be three given

functions. If a function, J(x) satisfies

(f₂ o J o f₁) (x) = f₃ (x) then J(x) is equal to :

A

f₁ (x)

B

$1 \over x$ f₃ (x)

C

f₂ (x)

D

f₃ (x)

Question 115: Palette 115/139

JEE Mains 2018 MCQ

Let f : A $ \to $ B be a function defined as f(x) = ${{x - 1} \over {x - 2}},$ Where A = R $-$ {2} and B = R $-$ {1}. Then   f   is :

A

invertible and ${f^{ - 1}}(y) = $ ${{3y - 1} \over {y - 1}}$

B

invertible and ${f^{ - 1}}\left( y \right) = {{2y - 1} \over {y - 1}}$

C

invertible and ${f^{ - 1}}\left( y \right) = {{2y + 1} \over {y - 1}}$

D

not invertible

Question 116: Palette 116/139

JEE Mains 2017 MCQ

The function f : N $ \to $ N defined by f (x) = x $-$ 5 $\left[ {{x \over 5}} \right],$ Where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is :

A

one-one and onto

B

one-one but not onto.

C

onto but not one-one.

D

neither one-one nor onto.

Question 117: Palette 117/139

JEE Mains 2017 MCQ

Let f(x) = 2¹⁰.x + 1 and g(x)=3¹⁰.x $-$ 1. If (fog) (x) = x, then x is equal to :

A

${{{3^{10}} - 1} \over {{3^{10}} - {2^{ - 10}}}}$

B

${{{2^{10}} - 1} \over {{2^{10}} - {3^{ - 10}}}}$

C

${{1 - {3^{ - 10}}} \over {{2^{10}} - {3^{ - 10}}}}$

D

${{1 - {2^{ - 10}}} \over {{3^{10}} - {2^{ - 10}}}}$

Question 118: Palette 118/139

JEE Mains 2017 MCQ

The function $f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right]$ defined as

$f\left( x \right) = {x \over {1 + {x^2}}}$, is

A

invertible

B

injective but not surjective.

C

surjective but not injective

D

neither injective nor surjective.

Question 119: Palette 119/139

JEE Mains 2017 MCQ

Let $a$, b, c $ \in R$. If $f$(x) = ax² + bx + c is such that
$a$ + b + c = 3 and $f$(x + y) = $f$(x) + $f$(y) + xy, $\forall x,y \in R,$

then $\sum\limits_{n = 1}^{10} {f(n)} $ is equal to

A

165

B

190

C

255

D

330

Question 120: Palette 120/139

JEE Mains 2016 MCQ

For x $ \in $ R, x $ \ne $ 0, Let f₀(x) = ${1 \over {1 - x}}$ and
f_n+1 (x) = f₀(f_n(x)), n = 0, 1, 2, . . . .

Then the value of f₁₀₀(3) + f₁$\left( {{2 \over 3}} \right)$ + f₂$\left( {{3 \over 2}} \right)$ is equal to :

A

${8 \over 3}$

B

${5 \over 3}$

C

${4 \over 3}$

D

${1 \over 3}$

Question 121: Palette 121/139

JEE Mains 2016 MCQ

If $f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0$, and $\mathrm{S}=\{x \in \mathbf{R}: f(x)=f(-x)\}$; then $\mathrm{S}:$

A

is an empty set.

B

contains exactly one element.

C

contains exactly two elements.

D

contains more than two elements.

Question 122: Palette 122/139

JEE Mains 2011 MCQ

The domain of the function f(x) = ${1 \over {\sqrt {\left| x \right| - x} }}$ is

A

$\left( {0,\infty } \right)$

B

$\left( { - \infty ,0} \right)$

C

$\left( { - \infty ,\infty } \right) - \left\{ 0 \right\}$

D

$\left( { - \infty ,\infty } \right)$

Question 123: Palette 123/139

JEE Mains 2009 MCQ

Let $f\left( x \right) = {\left( {x + 1} \right)^2} - 1,x \ge - 1$

Statement - 1 : The set $\left\{ {x:f\left( x \right) = {f^{ - 1}}\left( x \right)} \right\} = \left\{ {0, - 1} \right\}$.

Statement - 2 : $f$ is a bijection.

A

Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1

B

Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1

C

Statement - 1 is true, Statement - 2 is false

D

Statement - 1 is false, Statement - 2 is true

Question 124: Palette 124/139

JEE Mains 2009 MCQ

For real x, let f(x) = x³ + 5x + 1, then

A

f is one-one but not onto R

B

f is onto R but not one-one

C

f is one-one and onto R

D

f is neither one-one nor onto R

Question 125: Palette 125/139

JEE Mains 2008 MCQ

Let $f:N \to Y$ be a function defined as f(x) = 4x + 3 where
Y = { y $ \in $ N, y = 4x + 3 for some x $ \in $ N }.
Show that f is invertible and its inverse is

A

$g\left( y \right) = {{3y + 4} \over 4}$

B

$g\left( y \right) = 4 + {{y + 3} \over 4}$

C

$g\left( y \right) = {{y + 3} \over 4}$

D

$g\left( y \right) = {{y - 3} \over 4}$

Question 126: Palette 126/139

JEE Mains 2007 MCQ

The largest interval lying in $\left( { - {\pi \over 2},{\pi \over 2}} \right)$ for which the function

$f\left( x \right) = {4^{ - {x^2}}} + {\cos ^{ - 1}}\left( {{x \over 2} - 1} \right)$$ + \log \left( {\cos x} \right)$,

is defined, is

A

$\left[ { - {\pi \over 4},{\pi \over 2}} \right)$

B

$\left[ {0,{\pi \over 2}} \right)$

C

$\left[ {0,\pi } \right]$

D

$\left( { - {\pi \over 2},{\pi \over 2}} \right)$

Question 127: Palette 127/139

JEE Mains 2005 MCQ

Let $f:( - 1,1) \to B$, be a function defined by
$f\left( x \right) = {\tan ^{ - 1}}{{2x} \over {1 - {x^2}}}$,
then $f$ is both one-one and onto when B is the interval

A

$\left( {0,{\pi \over 2}} \right)$

B

$\left[ {0,{\pi \over 2}} \right)$

C

$\left[ { - {\pi \over 2},{\pi \over 2}} \right]$

D

$\left( { - {\pi \over 2},{\pi \over 2}} \right)$

Question 128: Palette 128/139

JEE Mains 2005 MCQ

A real valued function f(x) satisfies the functional equation

f(x - y) = f(x)f(y) - f(a - x)f(a + y)

where a is given constant and f(0) = 1, f(2a - x) is equal to

A

- f(x)

B

f(x)

C

f(a) + f(a - x)

D

f(- x)

Question 129: Palette 129/139

JEE Mains 2004 MCQ

The domain of the function
$f\left( x \right) = {{{{\sin }^{ - 1}}\left( {x - 3} \right)} \over {\sqrt {9 - {x^2}} }}$

A

[1, 2]

B

[2, 3)

C

[1, 2)

D

[2, 3]

Question 130: Palette 130/139

JEE Mains 2004 MCQ

If $f:R \to S$, defined by
$f\left( x \right) = \sin x - \sqrt 3 \cos x + 1$,
is onto, then the interval of $S$ is

A

[-1, 3]

B

[-1, 1]

C

[0, 1]

D

[0, 3]

Question 131: Palette 131/139

JEE Mains 2004 MCQ

The range of the function f(x) = ${}^{7 - x}{P_{x - 3}}$ is

A

{1, 2, 3, 4, 5}

B

{1, 2, 3, 4, 5, 6}

C

{1, 2, 3, 4}

D

{1, 2, 3}

Question 132: Palette 132/139

JEE Mains 2004 MCQ

The graph of the function y = f(x) is symmetrical about the line x = 2, then

A

$f\left( x \right) = - f\left( { - x} \right)$

B

$f\left( {2 + x} \right) = f\left( {2 - x} \right)$

C

$f\left( x \right) = f\left( { - x} \right)$

D

$f\left( {x + 2} \right) = f\left( {x - 2} \right)$

Question 133: Palette 133/139

JEE Mains 2003 MCQ

A function $f$ from the set of natural numbers to integers defined by $$f\left( n \right) = \left\{ {\matrix{ {{{n - 1} \over 2},\,when\,n\,is\,odd} \cr { - {n \over 2},\,when\,n\,is\,even} \cr } } \right.$$ is

A

neither one -one nor onto

B

one-one but not onto

C

onto but not one-one

D

one-one and onto both

Question 134: Palette 134/139

JEE Mains 2003 MCQ

The function $f\left( x \right)$ $ = \log \left( {x + \sqrt {{x^2} + 1} } \right)$, is

A

neither an even nor an odd function

B

an even function

C

an odd function

D

a periodic function

Question 135: Palette 135/139

JEE Mains 2003 MCQ

If $f:R \to R$ satisfies $f$(x + y) = $f$(x) + $f$(y), for all x, y $ \in $ R and $f$(1) = 7, then $\sum\limits_{r = 1}^n {f\left( r \right)} $ is

A

${{7n\left( {n + 1} \right)} \over 2}$

B

${{7n} \over 2}$

C

${{7\left( {n + 1} \right)} \over 2}$

D

$7n + \left( {n + 1} \right)$

Question 136: Palette 136/139

JEE Mains 2003 MCQ

Domain of definition of the function f(x) = ${3 \over {4 - {x^2}}}$ + ${\log _{10}}\left( {{x^3} - x} \right)$, is

A

(-1, 0)$ \cup $(1, 2)$ \cup $(2, $\infty $)

B

(1, 2)

C

(-1, 0) $ \cup $ (1, 2)

D

(1, 2)$ \cup $(2, $\infty $)

Question 137: Palette 137/139

JEE Mains 2002 MCQ

The period of ${\sin ^2}\theta $ is

A

${\pi ^2}$

B

$\pi $

C

$2\pi $

D

$\pi /2$

Question 138: Palette 138/139

JEE Mains 2002 MCQ

The domain of ${\sin ^{ - 1}}\left[ {{{\log }_3}\left( {{x \over 3}} \right)} \right]$ is

A

[1, 9]

B

[-1, 9]

C

[9, 1]

D

[-9, -1]

Question 139: Palette 139/139

JEE Mains 2002 MCQ

Which one is not periodic?

A

$\left| {\sin 3x} \right| + {\sin ^2}x$

B

$\cos \sqrt x + {\cos ^2}x$

C

$\cos \,4x + {\tan ^2}x$

D

$cos\,2x + \sin x$