JEE Mains JEE Advance BITSAT
JEE Mains | PYQ

Year

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Let $ f : (0,1) \to IR $ be a function defined by $ f(x) = \dfrac{1}{1 - e^{-x}} $, and $ g(x) = (f(-x) - f(x)) $. Consider two statements.
($ I $) $ g $ is an increasing function in $ (0, 1) $
($ II $) $ g $ is one-one in $ (0, 1) $ Then :

2.

Let $ R $ be a relation on $ \mathbb{N} \times \mathbb{N} $ defined by $ (a,b) R (c,d) $ if and only if $ ad\,(b-c) = bc\,(a-d) $. Then $ R $ is:

3.

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

4.

Let $ S=\{1,2,3,4,5,6,7,8,9,10\} $. Define $ f:S\to S $ as

Then $ g(10)\big(g(1)+g(2)+g(3)+g(4)+g(5)\big) $ is equal to _______.

5.

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

6.

Let $ R=\{a,b,c,d,e\} $ and $ S=\{1,2,3,4\} $. The total number of onto functions $ f:R\to S $ such that $ f(a)\neq 1 $, is equal to ______.

7.

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

8.

Let $ f $ be a polynomial function such that $ \log_{2}(f(x))=\big[\log_{2}\!\big(2+\frac{2}{3}+\frac{2}{9}+\ldots \infty\big)\big]\cdot \log_{3}\!\left(1+\frac{f(x)}{f(1/x)}\right) $, $ x>0 $ and $ f(6)=37 $. Then $ \displaystyle\sum_{n=1}^{10} f(n) $ is equal to ______.

9.

Let $ f : \mathbb{R} \to \mathbb{R} $ be a continuous function satisfying $ f(0) = 1 $ and $ f(2x) - f(x) = x $ for all $ x \in \mathbb{R} $. If $ \displaystyle \lim_{n \to \infty} \Big[ f(x) - f\!\left(\dfrac{x}{2^n}\right) \Big] = G(x) $, then $ \displaystyle \sum_{r=1}^{10} G(r^2) $ is equal to:

10.

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

11.

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

12.

Let $ f(x) = x^5 + 2x^3 + 3x + 1, \ x \in \mathbb{R} $, and $ g(x) $ be a function such that $ g(f(x)) = x $ for all $ x \in \mathbb{R} $. Then $ \dfrac{g(7)}{g'(7)} $ is equal to:

13.

$ f(\alpha)=\displaystyle\int\limits_{1}^{\alpha}\frac{\log_{10} t}{1+t}\,dt, \ \alpha>0 $ If , then $ f(e^{3})+f(e^{-3}) $ is equal to:

14.

If the domain of the function $\displaystyle f(x) = \frac{\sqrt{x^2 - 25}}{(4 - x^2)} + \log_{10}\!\left( x^2 + 2x - 15 \right) $ is $ (-\infty, \alpha) \cup (\beta, \infty) $, then $ \alpha^2 + \beta^3 $ is equal to:

15.

The domain of the function $ cosec^{-1}\!\left(\frac{1+x}{x}\right) $ is:

16.

Consider the relation $ R $ on the set $ \{ -2, -1, 0, 1, 2 \} $ defined by $ (a, b) \in R $ if and only if $ 1 + ab > 0 $. Then, among the statements:
I. The number of elements in $ R $ is $ 17 $.
II. $ R $ is an equivalence relation.

17.

Let $ A = \{1, 2, 3, 5, 8, 9\} $. Then the number of possible functions $ f : A \to A $ such that $ f(m \cdot n) = f(m) \cdot f(n) $ for every $ m, n \in A $ with $ m \cdot n \in A $ is equal to _________.

18.

Let $ x=2 $ be a local minima of the function $ f(x)=2x^{4}-18x^{2}+8x+12,\, x\in(-4,4) $. If $ M $ is local maximum value of the function $ f $ in $ (-4,4) $, then $ M= $

19.

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

20.

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

21.

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

22.

If $ f(x)=\sin\!\left(\cos^{-1}\!\left(\dfrac{1-2^{2x}}{1+2^{2x}}\right)\right) $ and its first derivative with respect to $ x $ is $ -\dfrac{b}{a}\log_e 2 $ when $ x=1 $, where $ a $ and $ b $ are integers, then the minimum value of $ \left|a^{2}-b^{2}\right| $ is __________.

23.

The set of all values of $a$ for which $\displaystyle \lim_{x \to a} ([x - 5] - [2x + 2]) = 0$, where $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$ is equal to:

24.

Let $ f:\mathbb{R}\to\mathbb{R} $ and $ g:\mathbb{R}\to\mathbb{R} $ be defined as where $ a, b $ are non-negative real numbers. If $ (g\circ f)(x) $ is continuous for all $ x\in\mathbb{R} $, then $ a+b $ is equal to ____________.

25.

Let $[.]$ denote the greatest integer function. If the domain of the function $ f(x) = \cos^{-1}\!\left(\frac{4x + 2[x]}{3}\right) $ is $[\alpha,\beta]$, then $ 12(\alpha + \beta) $ is equal to:

26.

Let $ A $ be the set of all functions $ f : Z \to Z $ and $ R $ be a relation on $ A $ such that $ R=\{(f,g): f(0)=g(1) \text{ and } f(1)=g(0)\} $. Then $ R $ is:

27.

The number of distinct real roots of the equation $ x^{7} - 7x - 2 = 0 $ is:

28.

The relation $ R=\{(a,b):\gcd(a,b)=1,\, 2a\neq b,\, a,b\in \mathbb{Z}\} $ is:

29.

The number of real solution(s) of the equation $ x^2 + 3x + 2 = m\ \, in\{\,|x - 3|, \, |x + 2|\,\} $ is :

30.

If $ S = \{ a \in \mathbb{R} \ :|2a - 1| = 3\ [a] + 2\{a\} \} $, where [t] denotes the greatest integer less than or equal to $ t $ and $ \{t\} $ represents the fractional part of $ t $, then $ 72 \sum\limits_{a \in S} a $ is equal to $ \_\_\_\_\_\_\_\_\_\_\_ $.

31.

Let $ A=\{1,2,3,4,5,6\} $. The number of one-one functions $ f:A\to A $ such that $ f(1)\ge 3 $, $ f(3)\le 4 $ and $ f(2)+f(3)=5 $, is ______.

32.

Consider the function $ f:\mathbb{R}\to\mathbb{R} $ defined by $ f(x)=\dfrac{2x}{\sqrt{1+9x^{2}}} $. If the composition of $ (\underbrace{f \circ f \circ f \circ \dots \circ f}_{10\text{ times}})(x) = \frac{2^{10}x}{\sqrt{1 + 9\alpha x^2}} $, then the value of $ \sqrt{3\alpha+1} $ is equal to __________.

33.

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

34.

For the function $ f : [1,\infty) \to [1,\infty) $ defined $ f(x) = (x-1)^4 + 1 $, among the two statements:
(I) The set $ S=\{x \in [1,\infty) : f(x)=f^{-1}(x)\} $ contains exactly two elements, and
(II) The set $ S=\{x \in [1,\infty) : f(x)=f^{-1}(x+1)\} $ is an empty set,

35.

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:

36.

The domain of the function $ f(x) := \dfrac{\cos^{-1}\!\left( \dfrac{x^{2} - 5x + 6}{x^{2} - 9} \right)}{\log_{e}(x^{2} - 3x + 2)} $ is:

37.

Let $ [ x ] $ denote the greatest integer less than or equal to $ x $. Then the domain of $ f(x)=\sec^{-1}(2[ x ] + 1) $ is:

38.

Let $ f(x)=2x^{2}-x-1 $ and $ S=\{\,n\in \mathbb{Z}:\; |f(n)|\le 800\,\} $. Then, the value of $ \sum\limits_{n\in S} f(n) $ is equal to ______.

39.

Let $ A=\{x\in \mathbb{R}:[x+3]+[x+4]\le 3\} $, $ B=\left\{x\in \mathbb{R}:3^{x}\left(\displaystyle\sum_{r=1}^{\infty}\dfrac{3}{10^{r}}\right)^{x-3}<3^{-3x}\right\} $, where $[t]$ denotes greatest integer function. Then:

40.

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

41.

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

42.

Let $ S = \{1, 2, 3, 4, 5, 6\} $. Then the number of one-one functions $ f : S \to P(S) $, where $ P(S) $ denotes the power set of $ S $, such that $ f(n) \subset f(m) $ where $ n < m $ is ________.

43.

Let $ f^1(x) = \frac{3x + 2}{2x + 3}, \; x \in \mathbb{R} - \left\{ \frac{-3}{2} \right\} $. For $ n \ge 2 $, define $ f^n(x) = f^1 \circ f^{n-1}(x) $. If $ f^5(x) = \frac{ax + b}{bx + a} $, $ \gcd(a,b) = 1 $, then $ a + b $ is equal to _______.

44.

Let $ R_1 = \{(a,b) \in N \times N : |a-b| \le 13\} $ and $ R_2 = \{(a,b) \in N \times N : |a-b| \ne 13\} $. Then on $ N $:

45.

Let the function $ f:[0,2]\to \mathbb{R} $ be defined as where $ [t] $ denotes the greatest integer less then or equal to $ t $. Then the value of the integral $\displaystyle \int_{0}^{2} x f(x)\,dx $ is:

46.

Let $ f : R \to R $ and $ g : R \to R $ be defined as . Then, $ gof : R \to R $ is:

47.

fg

48.

Let the domain of the function $ f(x)=\log_{3}\log_{5}\!\big(7-\log_{2}(x^{2}-10x+85)\big)+\sin^{-1}\!\left(\left|\dfrac{3x-7}{17-x}\right|\right) $ be $ (\alpha,\beta] $. Then $ \alpha+\beta $ is equal to:

49.

Let the domain of the function $ f(x) = \log_{3}\!\big(\log_{5}\!\big(\log_{7}(9x - x^{2} - 13)\big)\big) $ be the interval $ (m, n) $. Let the hyperbola $ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 $ have eccentricity $ \frac{n}{3} $ and the length of the latus rectum $ \frac{8m}{3} $. Then $ b^{2} - a^{2} $ is equal to:

50.

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

51.

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

52.

The number of functions $ f $, from the set $ A=\left\{x\in \mathbb{N}:x^{2}-10x+9\le 0\right\} $ to the set $ B=\left\{n^{2}:n\in \mathbb{N}\right\} $ such that $ f(x)\le (x-3)^{2}+1 $, for every $ x\in A $, is ________.

53.

The number of distinct real roots of the equation $ |x|\,|x+2| - 5|x+1| - 1 = 0 $ is $ \_\_\_\_\_\_\_\_\_\_\_ $.

54.

The number of relations, on the set $ \{1,2,3\} $ containing $ (1,2) $ and $ (2,3) $, which are reflexive and transitive but not symmetric, is_____.

55.

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\left( 2\left( 1-\frac{x^{25}}{2} \right) \left( 2+x^{25} \right) \right)^{\frac{1}{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, then the greatest integer less than or equal to $g(1)$ is ______.

56.

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

57.

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:

58.

Let $ D $ be the domain of the function $ f(x) = \sin^{-1}\!\left( \log_{3x} \left( \dfrac{6 + 2\log_3{x}}{-5x} \right) \right) $. If the range of the function $ g : D \to \mathbb{R} $ defined by $ g(x) = x - [x ] $, ($ [ x ] $ is the greatest integer function), is $ (\alpha, \beta) $, then $ \alpha^{2} + \dfrac{5}{\beta} $ is equal to:

59.

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

60.

The equation $ e^{4x} + 8e^{3x} + 13e^{2x} - 8e^{x} + 1 = 0, \, x \in \mathbb{R} $ has:

61.

If the domain of the function $ f(x) = \cos^{-1}\!\left(\frac{2x - 5}{11 - 3x}\right) + \sin^{-1}\!\left(2x^2 - 3x + 1\right) $ is the interval $ [\alpha, \beta] $, then $ \alpha + 2\beta $ is equal to:

62.

Let $ R = \{(x,y) \in \mathbb{N} \times \mathbb{N} : \log_{a}(x+y) \le 2\} $. Then the minimum number of elements, required to be added in $ R $ to make it a transitive relation, is __________.

63.

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

64.

Let $ A = \{(x,y): 2x + 3y = 23, \, x,y \in \mathbb{N}\} $ and $ B = \{ x : (x,y) \in A \} $. Then the number of one-one functions from $ A $ to $ B $ is equal to ________.

65.

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

66.

Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function that satisfies the relation $ f(x+y) = f(x) + f(y) - 1 $, $ \forall x, y \in \mathbb{R} $. If $ f'(0) = 2 $, then $ | f(-2) | $ is equal to __________.

67.

Let $ f : R \to R $ be defined as $ f(x) = x - 1 $ and $ g : R - \{ 1, -1 \} \to R $ be defined as $ g(x) = \frac{x^2}{x^2 - 1} $. Then the function $ fog $ is:

68.

If domain of the function $ \log_{e}\!\left(\dfrac{6x^{2}+5x+1}{2x-1}\right)+\cos^{-1}\!\left(\dfrac{2x^{2}-3x+4}{3x-5}\right) $ is $ (\alpha,\beta)\cup(\gamma,\delta] $, then $ 18(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}) $ is equal to ________.

69.

If the domain of the function $ f(x)=\sin^{-1}\!\left(\dfrac{1}{x^{2}-2x-2}\right) $ is $ (-\infty,\alpha] \cup [\beta,\gamma] \cup (\delta,\infty) $, then $ \alpha+\beta+\gamma+\delta $ is equal to:

70.

If $ y=y(x) $ is the solution of the differential equation $ \dfrac{dy}{dx}+\dfrac{4x}{(x^{2}-1)}y=\dfrac{x+2}{(x^{2}-1)^{5/2}},\, x>1 $ such that $ y(2)=\dfrac{2}{9}\log_{e}(2+\sqrt{3}) $ and $ y(\sqrt{2})=\alpha \log_{e}(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma},\; \alpha,\beta,\gamma \in \mathbb{N} $, then $ \alpha\beta\gamma $ is equal to __________.

71.

Let $ S=\{4,6,9\} $ and $ T=\{9,10,11,\ldots,1000\} $. If $ A=\{ a_1+a_2+\ldots+a_k : k\in \mathbb{N}, a_1,a_2,a_3,\ldots,a_k\in S \} $, then the sum of all the elements in the set $ T-A $ is equal to__________.

72.

The number of functions $ f : \{1,2,3,4\} \to \{ a \in Z \, \, |a| \le 8 \} $ satisfying $ f(n) + \frac{1}{n} f(n+1) = 1 , \forall n \in \{1,2,3\} $ is:

73.

For an integer $ n \ge 2 $, if the arithmetic mean of all coefficients in the binomial expansion of $ (x + y)^{2n - 3} $ is $ 16 $, then the distance of point $ P(2n - 1, n^2 - 4n) $ from the line $ x + y = 8 $ is:

74.

Let the domain of the function $ f(x)=\cos^{-1}\!\Big(\dfrac{4x+5}{3x-7}\Big) $ be $ [\alpha,\beta] $ and the domain of $ g(x)=\log_{2}\!\big(2-6\log_{27}(2x+5)\big) $ be $ (\gamma,\delta) $. Then $ \big|7(\alpha+\beta)+4(\gamma+\delta)\big| $ is equal to ______.

75.

If the function $ f:(-\infty,-1)\to (a,b] $ defined by $ f(x) = e^{x^3-3x+1} $ is one – one and onto, then the distance of the point $ P(2b+4,a+2) $ from the line $ x+e^{-3}y=4 $ is:

76.

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

77.

Consider two sets $ A=\{x\in\mathbb{Z}:\,|(|x-3|-3)|\le 1\} $ and $ B=\left\{x\in\mathbb{R}-\{1,2\}:\,\dfrac{(x-2)(x-4)}{x-1}\,\log_e(|x-2|)=0\right\} $. Then the number of onto functions $ f\!:\!A\to B $ is equal to:

78.

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

79.

Let $ f , g : \mathbb{R} \to \mathbb{R} $ be defined as:

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

80.

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

81.

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

82.

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

83.

If the domain of the function $ f(x) = \log_{9}\!\left( \frac{2x-3}{5+4x} \right) + \sin^{-1}\!\left( \frac{4+3x}{2-x} \right) $ is $ (\alpha, \beta] $, then $ \alpha^{2} + 4\beta $ is equal to:

84.

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

85.

Let $ R_1 $ and $ R_2 $ be two relations defined on $ \mathbb{R} $ by $ a \, R_1 \, b \Leftrightarrow ab \ge 0 $ and $ a \, R_2 \, b \Leftrightarrow a \ge b $. Then:

86.

Suppose $f : \mathbb{R} \rightarrow (0, \infty)$ be a differentiable function such that $5 f(x+y) = f(x)\cdot f(y), \; \forall \, x, y \in \mathbb{R}$. If $f(3) = 320$, then $\displaystyle \sum_{n=0}^{5} f(n)$ is equal to:

87.

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

88.

The range of the function $ f(x) = \log_{\sqrt{5}}\!\left( 3 + \cos\!\left(\frac{3\pi}{4} + x\right) + \cos\!\left(\frac{\pi}{4} + x\right) + \cos\!\left(\frac{\pi}{4} - x\right) - \cos\!\left(\frac{3\pi}{4} - x\right) \right) $ is:

89.

Let $ A = \{1,2,3,4,5\} $ and $ B = \{1,2,3,4,5,6\} $. Then the number of functions $ f : A \to B $ satisfying $ f(1) + f(2) = f(4) - 1 $ is equal to _______.

90.

The domain of the function $ f(x) = \sin^{-1}\!\left(\dfrac{x^{2}-3x+2}{x^{2}+2x+7}\right) $ is :

91.

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

92.

Let $ f $ be a real valued continuous function defined on the positive real axis such that $\displaystyle g(x) = \int_{0}^{x} t\, f(t)\, dt $. If $ g(x^{3}) = x^{6} + x^{7} $, then value of $\displaystyle \sum_{r=1}^{15} f(r^{3}) $ is:

93.

Let $f:\mathbb{R}\to \mathbb{R}$ be defined as:

Let $A=\{x\in \mathbb{R}:f \text{ is increasing}\}$. Then $A$ is equal to:

94.

Let $ f(x) = \log_e x $ and $ g(x) = \dfrac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} $. Then the domain of $ fog $ is:

95.

Let $ f:\mathbb{R}-\{2,6\}\to\mathbb{R} $ be real valued function defined as $ f(x)=\frac{x^2+2x+1}{x^2-8x+12} $. Then range of $ f $ is

96.

Let $ f:\mathbb{R}\to\mathbb{R} $ be a function defined by $ f(x)=\dfrac{2e^{2x}}{e^{2x}+e} $. Then $ f\!\left(\frac{1}{100}\right)+f\!\left(\frac{2}{100}\right)+f\!\left(\frac{3}{100}\right)+\ldots+f\!\left(\frac{99}{100}\right) $ is equal to $ \_\_\_ $.

97.

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:

98.

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

99.

Let $ f $ and $ g $ be two functions defined by and . Then $ (g \circ f)(x) $ is:

100.

Let $ f,g : \mathbb{N} - \{1\} \to \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} $ divided $ a $, and $ g(a) = a + 1 $, for all $ a \in \mathbb{N} - \{1\} $. Then, the function $ f + g $ is:

101.

Let $ R_1 $ and $ R_2 $ be relations on the set $ \{ 1, 2, \ldots, 50 \} $ such that
$ R_1 = \{ (p, p^n) : p \text{ is a prime and } n \ge 0 \text{ is an integer} \} $ and
$ R_2 = \{ (p, p^n) : p \text{ is a prime and } n = 0 \text{ or } 1 \} $.
Then, the number of elements in $ R_1 - R_2 $ is ________.

102.

Let $ f:\mathbb{R}-\{0\}\rightarrow(-\infty,1) $ be a polynomial of degree $ 2 $, satisfying $\displaystyle f(x)f\!\left(\frac{1}{x}\right)=f(x)+f\!\left(\frac{1}{x}\right) $. If $ f(k)=-2k $, then the sum of squares of all possible values of $ k $ is:

103.

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

104.

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

105.

Let $ g : \mathbb{N} \to \mathbb{N} $ be defined as
$ g(3n+1) = 3n + 2, $
$ g(3n+2) = 3n + 3, $
$ g(3n+3) = 3n + 1, \\ \text{for all } n \ge 0. $
Then which of the following statements is true?

106.

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:

107.

If $ a+\alpha=1, \; b+\beta=2 $ and $ a f(x)+\alpha f\!\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}, \; x\neq 0 $, then the value of the expression
$ \displaystyle \frac{f(x) + f\left(\frac{1}{x}\right)}{x + \frac{1}{x}}$ is __________.

108.

If the domain of the function $ f(x) = \log_{7} \left( 1 - \log_{4} (x^{2} - 9x + 18) \right) $ is $ (\alpha, \beta) \cup (\gamma, \delta) $, then $ \alpha + \beta + \gamma + \delta $ is equal to:

109.

The number of points in the interval $ [2,4] $, at which the function $ f(x)=\left[x^{2}-x-\frac{1}{2}\right] $, where $ [.] $ denotes the greatest integer function, is _________.

110.

The number of elements in the set $ S = \left\{ x \in \mathbb{R} : 2\cos\!\left(\dfrac{x^2 + x}{6}\right) = 4^{x} + 4^{-x} \right\} $ is:

111.

Let $ f(x)=2+|x|-|x-1|+|x+1|,\, x\in \mathbb{R} $.
Consider
$(S1):\;\displaystyle f'\!\left(-\frac{3}{2}\right)+f'\!\left(-\frac{1}{2}\right)+f'\!\left(\frac{1}{2}\right)+f'\!\left(\frac{3}{2}\right)=2$
$(S2):\; \displaystyle\int\limits_{-2}^{2} f(x)\,dx = 12$
Then,

112.

If the absolute maximum value of the function $ f(x)=(x^{2}-2x+7)e^{(4x^{3}-12x^{2}-180x+31)} $ in the interval $ [-3,0] $ is $ f(\alpha) $, then :

113.

Let $ c,k \in R $. If $ f(x)=(c+1)x^2+(1-c^2)x+2k $ and $ f(x+y)=f(x)+f(y)-xy $, for all $ x,y \in R $, then the value of $ \left| 2 \left( f(1)+f(2)+f(3)+\ldots+f(20) \right) \right| $ is equal to __________.

114.

Let the domains of the functions $ f(x)=\log_{4}\log_{3}\log_{7}\left(8-\log_{2}\left(x^{2}+4x+5\right)\right) $ and $ g(x)=\sin^{-1}\!\left(\dfrac{7x+10}{x-2}\right) $ be $ (\alpha,\beta) $ and $ [\gamma,\delta] $, respectively. Then $ \alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2} $ is equal to:

115.

If the domain of the function $ f(x)=\sqrt{\log_{0.6}\!\left(\left|\frac{2x-5}{x^{2}-4}\right|\right)} $ is $ (-\infty,a]\cup\{b\}\cup[c,d)\cup(e,\infty) $, then the value of $ a+b+c+d+e $ is _______.

116.

Let $ f:\mathbb{R}\to\mathbb{R} $ satisfies $ f(x+y)=2^{x}f(y)+4^{y}f(x),\ \forall x,y\in\mathbb{R}. $ If $ f(2)=3 $, then $ 14\cdot \frac{f'(4)}{f'(2)} $ is equal to ______.

117.

The number of real roots of the equation $ e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^{x}+1=0 $ is:

118.

Let $ f:\mathbb{N}\to \mathbb{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 $ \displaystyle \sum_{k=1}^{10} f(\alpha+k)=\dfrac{512}{3}\left(2^{20}-1\right) $ holds, is:

119.

For some $ a,b,c \in \mathbb{N} $, let $ f(x)=ax-3 $ and $ g(x)=x^{b}+c,\, x\in \mathbb{R} $. If $ (fog)^{-1}(x)=\left(\frac{x-7}{2}\right)^{\frac{1}{3}} $, then $ (fog)(ac)+(gof)(b) $ is equal to ________.

120.

Let $ [\alpha] $ denote the greatest integer $ \le \alpha $. Then $ [\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \ldots + [\sqrt{120}] $ is equal to _____.

121.

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

122.

Let $ f : \mathbb{R} - \{3\} \to \mathbb{R} - \{1\} $ be defined by $ f(x) = \dfrac{x - 2}{x - 3} $. Let $ g : \mathbb{R} \to \mathbb{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) = \dfrac{13}{2} $ is equal to:

123.

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

124.

If $ f(x) = \dfrac{2^{2x}}{2^{2x} + 2} , \, x \in R $, then $ f\!\left(\dfrac{1}{2023}\right) + f\!\left(\dfrac{2}{2023}\right) + \ldots + f\!\left(\dfrac{2022}{2023}\right) $ is equal to:

125.

Consider functions $ f: A \to B $ and $ g: B \to C $ $ (A,B,C \subseteq R) $ such that $ (g \circ f)^{-1} $ exists, then:

126.

If the domain of the function $ \sin^{-1}\!\left(\dfrac{3x-22}{2x-19}\right) + \log_e\!\left(\dfrac{3x^2-8x+5}{x^2-3x-10}\right) $ is $ (\alpha,\ \beta] $, then $ 3\alpha + 10\beta $ is equal to:

127.

Let $ f:(-2,2)\rightarrow \mathbb{R} $ be defined by

where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2,2)$ at which $y=| f(x)|$ is not continuous and not differentiable then $m+n$ is equal to ____.

128.

Let the function $ f(x)=2x^{3}+(2p-7)x^{2}+3(2p-9)x-6 $ have a maxima for some value of $ x<0 $ and a minima for some value of $ x>0 $. Then, the set of all values of $ p $ is:

129.

Let $ f(x)=3\sqrt{x-2}+\sqrt{4-x} $ be a real valued function. If $ \alpha $ and $ \beta $ are respectively the minimum and the maximum values of $ f $, then $ \alpha^2+2\beta^2 $ is equal to:

130.

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

131.

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

132.

Let $ A = \{1, 2, 3, \ldots, 7\} $ and let $ P(A) $ denote the power set of $ A $. If the number of functions $ f : A \to P(A) $ such that $ a \in f(a),\ \forall\, a \in A $ is $ m^n $, $ m $ and $ n \in \mathbb{N} $ and $ m $ is least, then $ m + n $ is equal to ______.

133.

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

134.

If the domain of the function $ \log_{5}(18x-x^{2}-77) $ is $ (\alpha,\beta) $ and the domain of the function $\displaystyle \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:

135.

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

136.

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

137.

The total number of functions $ f : \{1,2,3,4\} \to \{1,2,3,4,5,6\} $ Such that $ f(1) + f(2) = f(3) $ is equal to :

138.

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

139.

If $ A = \{ x \in \mathbb{R} : |x - 2| > 1 \} $, $ B = \{ x \in \mathbb{R} : \sqrt{x^2 - 3} > 1 \} $, $ C = \{ x \in \mathbb{R} : |x - 4| \ge 2 \} $ and $ Z $ is the set of all integers, then the number of subsets of the set $ (A \cap B \cap C)^{c} \cap Z $ is $ \_\_\_\_\_\_\_\_\_\_ $.

140.

The number of integral solutions $ x $ of $ \log_{\left(x+\dfrac{7}{2}\right)}\left(\dfrac{x-7}{2x-3}\right)^2 \ge 0 $ is:

141.

The domain of the function $ \cos^{-1}\!\left( \dfrac{2 \sin^{-1}\!\left( \dfrac{1}{4x^{2}-1} \right)}{\pi} \right) $ is:

142.

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:

143.

If a function $ f $ satisfies $ f(m+n)=f(m)+f(n) $ for all $ m,n\in \mathbb{N} $ and $ f(1)=1 $, then the largest natural number $ \lambda $ such that $ \displaystyle \sum_{k=1}^{2022} f(\lambda+k)\le (2022)^{2} $ is equal to ________.

144.

Let $ A=\{1,3,4,6,9\} $ and $ B=\{2,4,5,8,10\} $. Let $ R $ be a relation defined on $ A\times B $ such that $ R=\{((a_1,b_1),(a_2,b_2)):\; a_1\le b_2 \;\&\; b_1\le a_2\} $. Then the number of elements in the set $ R $ is:

145.

Let $ [t] $ be the greatest integer less than or equal to $ t $. Let $ A $ be the set of all prime factors of $ 2310 $ and $ f : A \to \mathbb{Z} $ be the function $ f(x) = \left[ \log_{2} \left( x^{3} + \left[ \frac{x^{3}}{5} \right] \right) \right] $. The number of one-to-one functions from $ A $ to the range of $ f $ is:

146.

The domain of the function $ f(x)=\sin^{-1}\!\left[\;2x^{2}-3\;\right]+\log_{2}\!\left(\log_{\dfrac{1}{2}}\!\left(x^{2}-5x+5\right)\right) $, where $[t]$ is
the greatest integer functions, is :

147.

Suppose $ f $ is a function satisfying $ f(x+y)=f(x)+f(y) $ for all $ x,y \in \mathbb{N} $ and $ f(1)=\dfrac{1}{5} $. If $ \displaystyle \sum_{n=1}^{m} \dfrac{f(n)}{n(n+1)(n+2)}=\dfrac{1}{12} $, then $ m $ is equal to _________.

148.

Let $ f(x)=2x+\tan^{-1}x $ and $ g(x)=\log_{e}\!\left(\sqrt{1+x^{2}}+x\right) $, $ x\in[0,3] $.
Then:

149.

then range of $ (f og)(x) $ is:

150.

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

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