Relations and Functions

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

[JEE Main 2025, 22 Jan (Shift 2)]

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 \(\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}\). If both the functions are onto and \(\mathrm{S}=\{x \in \mathbf{Z}: x \in \mathrm{~A}\) or \(x \in B\}\), then \(\mathrm{n}(\mathrm{S})\) is equal to :

[JEE Main 2025, 28 Jan (Shift 2)]

Let \(A=\{-3,-2,-1,0,1,2,3\}\). Let \(R\) be a relation on \(A\) defined by \(x R y\) if and only if \(0 \leq x^2+2 y \leq 4\). Let \(l\) be the number of elements in \(R\) and \(m\) be the minimum number of elements required to be added in \(R\) to make it a reflexive relation, then \(l+m\) is equal to

[JEE Main 2025, 3 Apr (Shift 1)]

Let \(A=\{1,2,3,4\}\) and \(B=\{1,4,9,16\}\).
If \(f: A \rightarrow B\), then number of many-one functions from \(A\) to \(B\) are

[JEE Main 2025]

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

[JEE Main 2024, 6 Apr (Shift 1)]

Define a relation \(R\) on the interval \(\left[0, \frac{\pi}{2}\right)\) by \(x R y\) if and only if \(\sec ^2 x-\tan ^2 y=1\). Then \(R\) is:

[JEE Main 2025, 29 Jan (Shift 1)]

Consider the sets \(\mathrm{A}=\left\{(\mathrm{x},\mathrm{y})\in \mathrm{ℝ}\times \mathrm{ℝ}:{\mathrm{x}}^{2}+{\mathrm{y}}^{2}=25\right\}\), \(\mathrm{B}=\left\{(\mathrm{x},\mathrm{y})\in \mathrm{ℝ}\times \mathrm{ℝ}:{\mathrm{x}}^{2}+9{\mathrm{y}}^{2}=144\right\},\)\(\mathrm{C}={(\mathrm{x},\mathrm{y})\)\(\left.\in \mathrm{ℤ}\times \mathrm{ℤ}:{x}^{2}+{y}^{2}\leq 4\right\}\), and \(\mathrm{D}=\mathrm{A}\cap \mathrm{B}\). The total number of one-one functions from the set D to the set C is:

[JEE Main 2025, 4 Apr (Shift 1)]

Let \(\mathrm{f}:\mathrm{ℝ}-{0}\to \mathrm{ℝ}\) be a function such that \(\mathrm{f}(\mathrm{x})-6\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=\frac{35}{3\mathrm{x}}-\frac{5}{2}.\) If the \(\lim _{x\to 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ;\) \(\alpha ,\beta \in \mathrm{ℝ}\) then \(\alpha +2\beta\) is equal to

[JEE Main 2025, 24 Jan (Shift 1)]

Let \(\mathrm{R}={(1,2),(2,3),(3,3)}\) be a relation defined on the set \({1,2,3,4}.\) Then the minimum number of elements, needed to be added in R so the R becomes an equivalence relation, is :

[JEE Main 2025, 23 Jan (Shift 1)]

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

[JEE Main 2025, 28 Jan (Shift 2)]

If the domain of the function

\(f\left(x\right)=\frac{1}{\sqrt{10+3\mathrm{x}-{\mathrm{x}}^{2}}}+\frac{1}{\sqrt{\mathrm{x}+|\mathrm{x}|}}\) is \((a,b)\),

then \((1+\mathrm{a}{)}^{2}+{\mathrm{b}}^{2}\) is equal to:

[JEE Main 2025, 2 Apr (Shift 2)]

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

[JEE Main 2025, 7 Apr (Shift 2)]

If \(f(x)=2{x}^{3}-15{x}^{2}+36x+7:[0,3]\to A\) \(g(x)=\frac{{x}^{2025}}{1+{x}^{2025}}:[0,\infty )\to B\) \(f(x)\) and g(x) are onto functions. \(S={x∣x\in Z,x\in A\)or \(x\in B}\). Find n(S).

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 :

[JEE Main 2025, 7 Apr (Shift 2)]

Consider two sets \(A=\left\{x\in ℤ:\left|\left(\left|x−3\right|−3\right)\right|\leq 1\right\}\) and \(B=\left\{x\in ℝ−\left\{1,2\right\}:\frac{\left(x−2\right)\left(x−4\right)}{x−1}{\text{log}}_{e}\left(\left|x−2\right|\right)=0\right\}.\)

Then the number of onto functions \(f: A \rightarrow B\) is equal to

[JEE Main 2026, 23 Jan (Shift 2)]

If R be a relation defined on \((0,\pi /2)\) such that \(xRy\Rightarrow {\sec }^{2}x-{\tan }^{2}y=1\), then the relation.

[JEE Main 2025]

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

[JEE Main 2025, 24 Jan (Shift 2)]

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

[JEE Main 2025, 22 Jan (Shift 2)]

\(\text{ If }f(x)=\frac{{2}^{x}}{{2}^{x}+\sqrt{2}},x\in R\text{, then }\sum _{k=1}^{81}f\left(\frac{k}{82}\right)\text{ is equal to }\) (28 Jan, Shift I, Memory Based)

A relation defined on set \(A=\{1,2,3,4\}\), then how many minimum ordered pairs are added to \(R=\{(1,2),(2,3),(3,3)\}\) so that it becomes equivalence relation?

[JEE Main 2025]

The number of solutions, of the equation \({e}^{\sin x}-2{e}^{-\sin x}=2\), is :

[JEE Main 2024, 31 Jan (Shift 2)]

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 :

[JEE Main 2024, 1 Feb (Shift 2)]

Let \(R\) be the relation "is congruent to" on the set of all triangles in a plane. Is R:

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

[JEE Main 2025, 23 Jan (Shift 1)]

The number of functions \(f:\left\{1,2,3,4\right\}\to \left\{a,b,c\right\}\) which are not onto, is:

[JEE Main 2026, 4 Apr (Shift 1)]

Let \(R\) denote the set of all real numbers. Let \(f: R \rightarrow R\) and \(g: R \rightarrow(0,4)\) be functions defined by \(f\left(x\right)={\log }_{e}\left({x}^{2}+2x+4\right),\text{ and }g\left(x\right)=\frac{4}{1+{e}^{-2x}}\) Define the composite function \(f \circ g^{-1}\) by \(\left(f \circ g^{-1}\right)(x)=f\left(g^{-1}(x)\right)\), where \(g^{-1}\) is the inverse of the function \(g\). Then the value of the derivative of the composite function \(f \circ g^{-1}\) at \(x=2\) is________.

[JEE Advanced 2025]

If \(f: \mathbb{R} \rightarrow \mathbb{R}, g: \mathbb{R} \rightarrow \mathbb{R}\) are defined by \(f(x)=5 x-3, g(x)=x^2+3\), then \(g \circ f^{-1}(3)\) is equal to

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

[JEE Main 2025, 28 Jan (Shift 1)]

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

[JEE Main 2025, 28 Jan (Shift 1)]

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

[JEE Main 2025, 3 Apr (Shift 2)]

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

[JEE Main 2025, 2 Apr (Shift 2)]

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

[JEE Main 2025, 3 Apr (Shift 1)]

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 \(\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}\). If both the functions are onto and \(\mathrm{S}=\{x \in \mathbf{Z}: x \in \mathrm{~A}\) or \(x \in B\}\), then \(\mathrm{n}(\mathrm{S})\) is equal to :

[JEE Main 2025, 28 Jan (Shift 2)]

If \(f: \mathbb{R} \rightarrow \mathbb{R}, g: \mathbb{R} \rightarrow \mathbb{R}\) are defined by \(f(x)=5 x-3, g(x)=x^2+3\), then \(g \circ f^{-1}(3)\) is equal to

Define a relation R on the interval \(\left[0,\frac{\pi }{2}\right)\text{ by x R y }\)if and only if \({\sec }^{2}x-{\tan }^{2}y=1\). Then R is :

[JEE Main 2025, 29 Jan (Shift 1)]

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

[JEE Main 2026, 6 Apr (Shift 2)]

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

[JEE Main 2025, 2 Apr (Shift 1)]

Function \(f(x)=\log _e x\) and \(g(x)=\frac{4}{2 x^2-2 x+1}\) Domain of \(y=f(g(x))\)

Let \(R\) be a relation on \(Z \times Z\) defined by \((a, b) R(c, d)\) if and only if \(a d-b c\) is divisible by 5 . Then \(R\) is

[JEE Main 2024, 29 Jan (Shift 1)]

Let \(R\) be a relation on \(Z \times Z\) defined by \((a, b) R(c, d)\) if and only if \(a d-b c\) is divisible by 5 . Then \(R\) is

[JEE Main 2024, 29 Jan (Shift 1)]

\(\text{If f(x) is a 2 degree polynomial satisfying }\\ \text{f(x).f(}\frac{1}{x})=f(x)+f(\frac{1}{x})andf(1)=2;\\ \text{then find the real values of "k" satisfying f(k)=-2k}\)

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

[JEE Main 2025, 3 Apr (Shift 2)]

\(\text{ The relation }R={(x,y)∣x,y\in Z,x+y=even}\text{ then }R\text{ is }\) (28 Jan, Shift I, Memory Based)

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

[JEE Main 2025, 3 Apr (Shift 2)]

If the domain of the function

\(f\left(x\right)={\text{sin}}^{−1}\left(\frac{5−x}{3+2x}\right)+\frac{1}{{\text{log}}_{e}\left(10−x\right)}\) is \(\left(−∞,\alpha \left]\cup \right[\beta ,\gamma \right)−\left\{\delta \right\}\), then \(6\left(\alpha +\beta +\gamma +\delta \right)\) is equal to

[JEE Main 2026, 22 Jan (Shift 1)]

If \(R\) is the smallest equivalence relation on the set \(\{1,2,3,4\}\) such that \(\{(1,2),(1,3)\} \subset R\), then the number of elements in \(R\) is ____

[JEE Main 2024, 29 Jan (Shift 2)]

If \(R\) is the smallest equivalence relation on the set \(\{1,2,3,4\}\) such that \(\{(1,2),(1,3)\} \subset R\), then the number of elements in \(R\) is ____

[JEE Main 2024, 29 Jan (Shift 2)]

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

[JEE Main 2025, 28 Jan (Shift 1)]

The relation \(R={(x,y):x,y\in Z\text{ and }x+y\text{ is even }}\) is:

[JEE Main 2025, 28 Jan (Shift 1)]

The number of elements in the set \(S=\left\{\left(r,k\right):k\in Z\text{and}{}^{36}C_{r+1}=\frac{6\left({}^{35}C_{r}\right)}{{k}^{2}-3}\right\}\)

[JEE Main 2026, 2 Apr (Shift 1)]

The relation \(R={(x,y):x,y\in Z\text{ and }x+y\text{ is even }}\) is:

[JEE Main 2025, 28 Jan (Shift 1)]

Let \(f,g:R\to R\) be defined as:

\(f\left(x\right)=\left|x-1\right|\) and \(g\left(x\right)=\left\{\begin{matrix}{e}^{x} & x⩾0 \\ x+1 & x⩽0\end{matrix}\right.\)

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

[JEE Main 2024, 5 Apr (Shift 2)]

If the domain of \({\log }_{x-1}\left(\frac{2{x}^{2}-9x+4}{{x}^{2}-4x+5}\right)\) is \((\alpha ,\infty )\) and \({\log }_{5}\left(18x-{x}^{2}-77\right)\) is \((\beta ,\gamma )\), then the value of \({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2}\) is

Let \(\mathrm{f}(\mathrm{x})+2 \mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=\mathrm{x}^2+5\) and \(2 g(x)-3 g\left(\frac{1}{2}\right)=x, x>0\). If \(\alpha=\int_1^2 f(x) d x\), and \(\beta=\int_1^2 g(x) d x\), then the value of \(9 \alpha+\beta\) is:

[JEE Main 2025, 4 Apr (Shift 2)]

\(\text{ The relation }R={(x,y)∣x,y\in Z,x+y=even}\text{ then }R\text{ is }\) (28 Jan, Shift I, Memory Based)

The number of elements in the relation \(R={(x,y):4{x}^{2}+{y}^{2}<52,x,y\in Z}\) is

If R be a relation defined on \((0,\pi /2)\) such that \(xRy\Rightarrow {\sec }^{2}x-{\tan }^{2}y=1\), then the relation.

[JEE Main 2025]

Let for some \(\alpha \in \mathrm{R},f:\mathrm{R}\to \mathrm{R}\) be a function satisfying \(f(x+y)=f(x)+2{y}^{2}+y+\alpha xy\) for all \(x,y\in R\). If \(f(0)=-1\) and \(f(1)=2\), then the value of \(\sum _{n=1}^{5}(\alpha +f\left(n\right))\) is:

[JEE Main 2026, 4 Apr (Shift 2)]

Let A= \({-3,-2,-1,0,1,2,3}\), . Let R be a relation on A defined by \(xRy\) if and only if \(0\leq {x}^{2}+2y\leq 4\)
Let \(l\) be the number of elements in \(R\)and \(m\)be the minimum number of elements required to be added in R to make it a reflexive relation. then \(l+m\) is equal to

[JEE Main 2025, 3 Apr (Shift 1)]

Consider the sets

\(\mathrm{A}=\left\{(\mathrm{x},\mathrm{y})\in \mathrm{ℝ}\times \mathrm{ℝ}:{\mathrm{x}}^{2}+{\mathrm{y}}^{2}=25\right\}\),

\(\mathrm{B}=\left\{(\mathrm{x},\mathrm{y})\in \mathrm{ℝ}\times \mathrm{ℝ}:{\mathrm{x}}^{2}+9{\mathrm{y}}^{2}=144\right\},\)

\(C=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\}\), and

\(\mathrm{D}=\mathrm{A}\cap \mathrm{B}\).

The total number of one-one functions from the set \(D\) to the set \(C\) is:

[JEE Main 2025, 4 Apr (Shift 1)]

A relation defined on set \(A=\{1,2,3,4\}\), then how many minimum ordered pairs are added to \(R=\{(1,2),(2,3),(3,3)\}\) so that it becomes equivalence relation?

[JEE Main 2025]

Let \(R\) be a relation on the set \(N\) of natural numbers defined by \(nRm\) if \(n\) divides \(m\). Then \(R\) is

Let \(\mathrm{R}={(1,2),(2,3),(3,3)}\) be a relation defined on the set \({1,2,3,4}.\) Then the minimum number of elements, needed to be added in R so the R becomes an equivalence relation, is :

[JEE Main 2025, 23 Jan (Shift 1)]

If \(S\) be the set of 10 distinct primes and let \(A\) be the set of product of two or more elements from the set \(S\). If \(P=\{(x, y): x \in S\) and \(y \in A\) and \(y\) is divided by \(x\}\). Then \(n(P)\) is equal to (24 Jan, Shift I, Memory Based)

The sum of all the elements in the range of \(f\left(x\right)=\text{Sgn}\left(\sin x\right)+\text{ Sgn}\left(\cos x\right)+\)\(\text{Sgn}\left(\tan x\right)+\text{Sgn}\left(\cot x\right)\), \(x\text{ }\neq \text{ }\frac{n\text{ }\pi }{2},\text{  }n\text{ }\in \text{ }Z,\) where \(\text{Sgn}\left(t\right)=\left\{\begin{matrix}1, & \text{if} & t\text{ }>\text{ }0 \\ −1 & \text{if} & t\text{ }<\text{ }0\end{matrix},\right.\) is:

[JEE Main 2026, 28 Jan (Shift 2)]

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 :

[JEE Main 2024, 1 Feb (Shift 2)]

Let \(A=\{0,1,2,3,4,5\}\). Let \(R\) be a relation on \(A\) defined by \((x, y) \in R\) if and only if \(\max \{x, y\} \in\{3,4\}\). Then among the statements
\(\left(S_1\right)\): The number of elements in \(R\) is 18, and
\(\left(S_2\right)\): The relation \(R\) is symmetric but neither reflexive nor transitive

[JEE Main 2025, 8 Apr (Shift 1)]

A function \(f:R\to (-1,1)\) such that \(f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}}\) the \(f(x)\) is

The sum of all the solutions of the equation \((8)^{2 x}-16 \cdot(8)^x+48=0\) is :

[JEE Main 2024, 8 Apr (Shift 1)]

Let \(A={-3,-2,-1,0,1,2,3}\) and R be a relation on A defined by \(xRy\) if and only if \(2\mathrm{x}-\mathrm{y}\in {0,1}\). Let \(l\) be the number of elements in R. Let \(m\) and \(n\) be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then \(l+m+n\) is equal to:-

[JEE Main 2025, 4 Apr (Shift 2)]

Let \(\mathrm{R}={(1,2),(2,3),(3,3)}\) be a relation defined on the set \({1,2,3,4}.\) Then the minimum number of elements, needed to be added in R so the R becomes an equivalence relation, is:

[JEE Main 2025, 23 Jan (Shift 1)]

\(\begin{aligned}&\text { Then find domain of fog (x). }\\&& f(x)=\log _e x \\& g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}\end{aligned}\)

[JEE Main 2025]

Let \(A={-2,-1,0,1,2,3}\). Let \(R\) be a relation on \(A\) defined by \(xRy\) if and only if \(y=\max {x,1}\). Let \(l\) be the number of elements in \(R\). Let \(m\) and \(n\)be the minimum number of elements required to be added in\(R\)to make it reflexive and symmetric relations, respectively. Then \(l+m+n\) is equal to

[JEE Main 2025, 3 Apr (Shift 2)]

Let \(\mathrm{A}={1,2,3,\ldots .,100}\)and R be a relation on A such that \(\mathrm{R}={(\mathrm{a},\mathrm{b}):\mathrm{a}=2\mathrm{b}+1}\). Let \(\left({\mathrm{a}}_{1},{\mathrm{a}}_{2}\right)\), \(\left({\mathrm{a}}_{2},{\mathrm{a}}_{3}\right),\left({\mathrm{a}}_{3},{\mathrm{a}}_{4}\right),\ldots .,\left({\mathrm{a}}_{\mathrm{k}},{\mathrm{a}}_{\mathrm{k}+1}\right)\) be a sequence of k elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k , for which such a sequence exists, is equal to :

If the domain of the function

\(f\left(x\right)={\text{sin}}^{−1}\left(\frac{5−x}{3+2x}\right)+\frac{1}{{\text{log}}_{e}\left(10−x\right)}\) is \(\left(−∞,\alpha \left]\cup \right[\beta ,\gamma \right)−\left\{\delta \right\}\), then \(6\left(\alpha +\beta +\gamma +\delta \right)\) is equal to

[JEE Main 2026, 22 Jan (Shift 1)]

Consider the relations \(R_1\) and \(R_2\) defined as \(a{R}_{1}b\Leftrightarrow {a}^{2}+{b}^{2}=1\) for all \(a, b \in R\) and \((a,b){R}_{2}(c,d)\Leftrightarrow\) \(a+d=b+c\) for all \(( a , b ),( c , d ) \in N \times N\). Then

[JEE Main 2024, 1 Feb (Shift 2)]

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

[JEE Main 2024, 6 Apr (Shift 2)]

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

[JEE Main 2024, 6 Apr (Shift 2)]

Let \(R\) be the relation "is congruent to" on the set of all triangles in a plane. Is R:

If \(f(x)=2{x}^{3}-15{x}^{2}+36x+7:[0,3]\to A\) \(g(x)=\frac{{x}^{2025}}{1+{x}^{2025}}:[0,\infty )\to B\) \(f(x)\) and g(x) are onto functions. \(S={x∣x\in Z,x\in A\)or \(x\in B}\). Find n(S).

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function satisfying \(f(0)=1\) and \(f(2 \mathrm{x})-f(\mathrm{x})=\mathrm{x}\) for all \(\mathrm{x} \in \mathbb{R}\). If \(\lim _{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)\), then \(\sum_{r=1}^{10} G\left(r^2\right)\) is equal to

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function satisfying \(f(0)=1\) and \(f(2 \mathrm{x})-f(\mathrm{x})=\mathrm{x}\) for all \(\mathrm{x} \in \mathbb{R}\). If \(\lim _{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)\), then \(\sum_{r=1}^{10} G\left(r^2\right)\) is equal to

[JEE Main 2025, 4 Apr (Shift 1)]

Let the sum of the maximum and the minimum values of the function \(f\left(x\right)=\frac{2{x}^{2}-3x+8}{2{x}^{2}+3x+8}\) be \(\frac{m}{n}\), where \(\gcd (m,n)=1\). Then \(m+n\) is equal to:

Let the relations \({R}_{1}\) and \({R}_{2}\) on the set \(X={1,2,3,\ldots .,20}\) be given by \({R}_{1}={(x,y):2x-3y=2}\) and \({R}_{2}={(x,y):-5x+4y=0}\) . If \(M\) and \(N\) be the minimum number of elements required to be added in \({R}_{1}\) and \({R}_{2}\) , respectively, in order to make the relations symmetric, then \(M+N\) equals

[JEE Main 2024, 6 Apr (Shift 1)]

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

[JEE Main 2025, 24 Jan (Shift 1)]

Let \(\mathrm{X}=\mathrm{R} \times \mathrm{R}\). Define a relation R on X as:

\(\left(a_1, b_1\right) R\left(a_2, b_2\right) \Leftrightarrow b_1=b_2 .\)

Statement-I: R is an equivalence relation.
Statement-II: For some \((\mathrm{a}, \mathrm{b}) \in \mathrm{X}\), the set \(S=\{(x, y) \in X:(x, y) R(a, b)\}\) represents a line parallel to \(\mathrm{y}=\mathrm{x}\).

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

[JEE Main 2025, 23 Jan (Shift 2)]

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

[JEE Main 2024, 5 Apr (Shift 1)]

If \(S\) be the set of 10 distinct primes and let \(A\) be the set of product of two or more elements from the set \(S\). If \(P=\{(x, y): x \in S\) and \(y \in A\) and \(y\) is divided by \(x\}\). Then \(n(P)\) is equal to (24 Jan, Shift I, Memory Based)

Let \(A=\left\{-2,-1,0,1,2,3,4\right\}\). Let \(R\) be a relation on \(A\) defined by \(xRy\) if and only if \(2x+y\leq 2\). Let \(l\) be the number of elements in \(R.\) Let \(m\) and \(n\) be the minimum number of elements required to be added in \(R\) to make it reflexive and symmetric relations respectively. Then \(l+m+n\) is equal to:

[JEE Main 2026, 23 Jan (Shift 2)]

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 \(\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}\). If both the functions are onto and \(\mathrm{S}=\{x \in \mathbf{Z}: x \in \mathrm{~A}\) or \(x \in B\}\), then \(\mathrm{n}(\mathrm{S})\) is equal to :

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

[JEE Main 2024, 27 Jan (Shift 1)]

Let \(f:\mathrm{ℝ}\to \mathrm{ℝ}\)be a continuous function satisfying\(f(0)=1\)and \(f(2\mathrm{x})-f(\mathrm{x})=\mathrm{x}\) for all \(\mathrm{x}\in \mathrm{ℝ}\). If \(\lim _{n\to \infty }\left\{f(x)-f\left(\frac{x}{{2}^{n}}\right)\right\}=G(x)\), then \(\sum _{r=1}^{10}G\left({r}^{2}\right)\) is equal to

[JEE Main 2025, 4 Apr (Shift 1)]

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

[JEE Main 2025, 28 Jan (Shift 1)]

Let \(A={-2,-1,0,1,2,3}\). let R be a relation on A defined by \(xRy\) if and only if \(y=\max {x,l}\). Let \(l\) be the number of elements in \(R\). Let \(m\) and \(n\)be the minimum number of elements required to be added in \(R\)to make it reflexive and symmetric relations, respectively. Then \(l+\mathrm{m}+\mathrm{n}\) is equal to

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

[JEE Main 2024, 30 Jan (Shift 2)]

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

[JEE Main 2024, 30 Jan (Shift 2)]

The number of elements in the relation \(R={(x,y):4{x}^{2}+{y}^{2}<52,x,y\in Z}\) is

[JEE Main 2026, 22 Jan (Shift 2)]

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

[JEE Main 2025, 3 Apr (Shift 2)]

Let \(A=\{1,2,3,4\}\) and \(B=\{1,4,9,16\}\).
If \(f: A \rightarrow B\), then number of many-one functions from \(A\) to \(B\) are

[JEE Main 2025]

Let \(S=\{1,2,3, \ldots, 10\}\). Suppose \(M\) is the set of all the subsets of \(S\), then the relation \(R=\{(A, B): A \cap B \neq \phi ; A, B \in M\}\) is:

[JEE Main 2024, 27 Jan (Shift 1)]

Let \(S=\{1,2,3, \ldots, 10\}\). Suppose \(M\) is the set of all the subsets of \(S\), then the relation \(R=\{(A, B): A \cap B \neq \phi ; A, B \in M\}\) is:

[JEE Main 2024, 27 Jan (Shift 1)]

Let \(X=R\times R.\) Define a relation R on X as:

\(\left({a}_{1},\text{ }{b}_{1}\right)\text{ }R\text{ }\left({a}_{2},\text{ }{b}_{2}\right)\text{ }\Leftrightarrow \text{ }{b}_{1}={b}_{2}.\)

Statement-I: R is an equivalence relation.

Statement-II: For some \(\left(a,b\right)\in X,\) the set

\(S=\left\{\left(x,\text{ }y\right)\text{ }\in \text{ }X\text{ }:\text{ }\left(x,\text{ }y\right)\text{ }R\text{ }\left(a,\text{ }b\right)\right\}\) represents a line parallel to \(y=x.\)

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

Let \(A={1,2,3,\ldots .,100}\) and \(R\) be a relation on \(A\) such that \(R={(a,b):a=2b+1}\). Let \(\left({a}_{1},{a}_{2}\right)\), \(\left({a}_{2},{a}_{3}\right),\left({a}_{3},{a}_{4}\right),\ldots .,\left({a}_{k},{a}_{k+1}\right)\) be a sequence of \(k\) elements of \(R\) such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer \(k\), for which such a sequence exists, is equal to:

Consider the relations \(R_1\) and \(R_2\) defined as \(a{R}_{1}b\Leftrightarrow {a}^{2}+{b}^{2}=1\) for all \(a, b \in R\) and \((a,b){R}_{2}(c,d)\Leftrightarrow\) \(a+d=b+c\) for all \(( a , b ),( c , d ) \in N \times N\). Then

[JEE Main 2024, 1 Feb (Shift 2)]

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

[JEE Main 2024, 31 Jan (Shift 1)]

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

[JEE Main 2024, 31 Jan (Shift 1)]

Function \(f(x)=\log _e x\) and \(g(x)=\frac{4}{2 x^2-2 x+1}\) Domain of \(y=f(g(x))\)

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

[JEE Main 2025, 4 Apr (Shift 1)]

If the domain of the function \(f\left(x\right)={\text{sin}}^{−1}\left(\frac{1}{{x}^{2}−2x−2}\right)\) is \(\left(−∞,\alpha \left]\cup \right[\beta ,\gamma \left]\cup \right[\delta ,∞\right)\), then \(\alpha +\beta +\gamma +\delta\) is equal to

[JEE Main 2026, 24 Jan (Shift 2)]

Let \(A=\{2,3,6,8,9,11\}\) and \(B=\{1,4,5,10,15\}\). Let \(R\) be a relation on \(A \times B\) defined by \((a, b) R(c, d)\) if and only if \(3 a d-7 b c\) is an even integer. Then the relation \(R\) is

[JEE Main 2024, 8 Apr (Shift 2)]

The number of non-empty equivalence relations on the set \(\{1,2,3\}\) is:

[JEE Main 2025, 22 Jan (Shift 1)]

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

[JEE Main 2025, 2 Apr (Shift 1)]

A function \(f:R\to (-1,1)\) such that \(f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}}\) the \(f(x)\) is

\(\begin{aligned}&\text { Then find domain of fog (x). }\\&& f(x)=\log _e x \\& g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}\end{aligned}\)

[JEE Main 2025]

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

Then \(f\left(5\right)-f\left(2\right)\) is equal to

[JEE Main 2026, 24 Jan (Shift 2)]

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

[JEE Main 2025, 24 Jan (Shift 1)]

Let \([t]\) be the greatest integer less than or equal to \(t\). Let \(A\) be the set of all prime factors of \(2310\) and \(f: A \rightarrow Z\) be the function \(f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]\). The number of one-to-one functions from \(A\) to the range of \(f\) is

[JEE Main 2024, 8 Apr (Shift 1)]

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

[JEE Main 2025, 4 Apr (Shift 2)]

Let \(f\left(x\right)={\log }_{e}x\) and

\(g\left(x\right)=\frac{{x}^{4}−2{x}^{3}+3{x}^{2}−2x+2}{2{x}^{2}−2x+1}.\)

Then the domain of \(fog\) is:

Let \(\mathrm{R}={(1,2),(2,3),(3,3)}\) be a relation defined on the set \({1,2,3,4}.\) Then the minimum number of elements, needed to be added in R so the R becomes an equivalence relation, is :

[JEE Main 2025, 23 Jan (Shift 1)]

Let \(A=\left\{0,1,2,\ldots 9\right\}.\) Let \(R\) be a relation on \(A\) defined by \((x,y)\in R\) if and only if \(|x–y|\) is a multiple of \(3.\)

Given below are two statements:

Statement \(\text{I}:n(R)=36.\)

Statement \(\text{II}:R\) is an equivalence relation.

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

[JEE Main 2026, 23 Jan (Shift 2)]

Let a relation R on \(N\times N\) be defined as: \(\left({x}_{1},{y}_{1}\right)R\left({x}_{2},{y}_{2}\right)\) if and only if \({x}_{1}\leq {x}_{2}\) or \({y}_{1}\leq {y}_{2}\). Consider the two statements:
(I) R is reflexive but not symmetric.
(II) R is transitive Then which one of the following is true?

[JEE Main 2024, 4 Apr (Shift 2)]

Let \(R\) be a relation defined on the set \(\left\{1,2,3,4\right\}\times \left\{1,2,3,4\right\}\) by \(R=\left\{\left(\left(a,b\right),\left(c,d\right)\right):2a+3b=3c+4d\right\}\). Then the number of elements in \(R\) is

[JEE Main 2026, 24 Jan (Shift 1)]

If \(g\left(x\right)=3{x}^{2}+2x−3,f\left(0\right)=−3\) and \(4g\left(f\left(x\right)\right)=3{x}^{2}−32x+72\), then \(f\left(g\left(2\right)\right)\) is equal to:

[JEE Main 2026, 28 Jan (Shift 1)]

Given below are two statements:

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

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

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

[JEE Main 2026, 28 Jan (Shift 2)]

Let \( S = \mathbb{N} \cup \{0\} \). Define a relation \( R \) from \( S \) to \( \mathbb{R} \) by: \( R = \{(x, y): \log_e y = x \log_e \left(\frac{2}{5}\right),\ x \in S, y \in \mathbb{R}\} \). Then, the sum of all the elements in the range of \( R \) is equal to:

Let \(D\) 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 \(g: D \rightarrow R\) defined by \(g(x)=x-[x]\), (\([x]\) is the greatest integer function) is \((\alpha, \beta)\), then \(\alpha^2+\frac{5}{\beta}\) is equal to

[JEE Main 2023, 12 Apr (Shift 1)]

\(\begin{aligned}& A=\{1,2,3, \ldots, 10\}, \\& B=\left\{\frac{m}{n}, n>m, m, n \in A, \operatorname{gcd}(m \cdot n)=1\right\}\end{aligned}\)
Then no. of elements in \(\mathrm{B}=\) ?

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

[JEE Main 2025, 28 Jan (Shift 2)]

The sum of all the elements in the range of \(f\left(x\right)=\text{Sgn}\left(\sin x\right)+\text{ Sgn}\left(\cos x\right)+\)\(\text{Sgn}\left(\tan x\right)+\text{Sgn}\left(\cot x\right)\), \(x\text{ }\neq \text{ }\frac{n\text{ }\pi }{2},\text{  }n\text{ }\in \text{ }Z,\) where \(\text{Sgn}\left(t\right)=\left\{\begin{matrix}1, & \text{if} & t\text{ }>\text{ }0 \\ −1 & \text{if} & t\text{ }<\text{ }0\end{matrix},\right.\) is:

[JEE Main 2026, 28 Jan (Shift 2)]

Let \(A={-2,-1,0,1,2,3}\). let R be a relation on A defined by \(xRy\) if and only if \(y=\max {x,1}\). Let \(l\) be the number of elements in \(R\). Let \(m\) and \(n\)be the minimum number of elements required to be added in \(R\)to make it reflexive and symmetric relations, respectively. Then \(l+\mathrm{m}+\mathrm{n}\) is equal to

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

[JEE Main 2025, 24 Jan (Shift 2)]

Let \(\mathrm{A}={0,1,2,3,4,5}\). Let R be a relation on A defined by \((\mathrm{x},\mathrm{y})\in \mathrm{R}\) if and only if max \({\mathrm{x},\mathrm{y}}\in {3,4}\). Then among the statements
\(\left({\mathrm{S}}_{1}\right)\) : The number of elements in R is 18 , and
\(\left({\mathrm{S}}_{2}\right)\): The relation R is symmetric but neither reflexive nor transitive

[JEE Main 2025, 8 Apr (Shift 1)]

Let \(\mathrm{f}: \mathbb{R}-\{0\} \rightarrow \mathbb{R}\) be a function such that \(\mathrm{f}(\mathrm{x})-6 \mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=\frac{35}{3 \mathrm{x}}-\frac{5}{2}\). If the \(\lim _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ; \alpha, \beta \in \mathbb{R}\) then \(\alpha+2 \beta\) is equal to

[JEE Main 2025, 24 Jan (Shift 1)]

If \(A=\{1,2,3\}\), find the number of non empty equivalence relation on set \(A\)

[JEE Main 2025]

If \(A=\{1,2,3\}\), find the number of non empty equivalence relation on set \(A\)

[JEE Main 2025]

Let \(\mathrm{f}(\mathrm{x})+2\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)={\mathrm{x}}^{2}+5\) and \(2g(x)-3g\left(\frac{1}{2}\right)=x,x>0\). If \(\alpha ={\int }_{1}^{2}f(x)dx\), and \(\beta ={\int }_{1}^{2}g(x)dx\), then the value of \(9\alpha +\beta\) is :

[JEE Main 2025, 4 Apr (Shift 2)]

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 :

[JEE Main 2024, 27 Jan (Shift 2)]

\(\text{If f(x) is a 2 degree polynomial satisfying }\\ \text{f(x).f(}\frac{1}{x})=f(x)+f(\frac{1}{x})andf(1)=2;\\ \text{then find the real values of "k" satisfying f(k)=-2k}\)

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 \(\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}\). If both the functions are onto and \(\mathrm{S}=\{x \in \mathbf{Z}: x \in \mathrm{~A}\) or \(x \in B\}\), then \(\mathrm{n}(\mathrm{S})\) is equal to:

[JEE Main 2025, 28 Jan (Shift 2)]

\(\text{ If }f(x)=\frac{{2}^{x}}{{2}^{x}+\sqrt{2}},x\in R\text{, then }\sum _{k=1}^{81}f\left(\frac{k}{82}\right)\text{ is equal to }\) (28 Jan, Shift I, Memory Based)

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

[JEE Main 2025, 2 Apr (Shift 1)]

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

[JEE Main 2025, 4 Apr (Shift 1)]

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

[JEE Main 2025, 2 Apr (Shift 1)]

If the domain of \({\log }_{x-1}\left(\frac{2{x}^{2}-9x+4}{{x}^{2}-4x+5}\right)\) is \((\alpha ,\infty )\) and \({\log }_{5}\left(18x-{x}^{2}-77\right)\) is \((\beta ,\gamma )\), then the value of \({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2}\) is

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

[JEE Main 2026, 8 Apr (Shift 2)]

The number of non-empty equivalence relations on the set \(\{1,2,3\}\) is:

[JEE Main 2025, 22 Jan (Shift 1)]

The domain of the function \(f(x)=\frac{1}{\sqrt{x^{12}-x^9+x^4-x+1}} \) is given by

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 :

[JEE Main 2022, 25 Jul (Shift 1)]

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

[JEE Main 2023, 1 Feb (Shift 2)]

Let \( f: S \rightarrow S \) where \( S=(0, \infty) \) be a twice differentiable function such that \( f(x+1)=x f(x) \). If \( g: S \rightarrow R \) be defined as \( g(x)=\log _{e} f(x) \), then the value of \( \left|g^{\prime \prime}(5)-g^{\prime \prime}(1)\right| \) is equal to

If \([x]\) be the greatest integer less than or equal to \(x\), then \(\sum_{n=8}^{100}\left[\frac{(-1)^n n}{2}\right]\) is equal to:

[JEE Main 2021, 25 Jul (Shift 2)]

Consider a function \(\mathrm{f}:\mathrm{IN}\to \mathrm{IR}\), satisfying

\(f(1)+2f(2)+3f(3)+\ldots .+xf(x)=x(x+1)f(x);x\geq 2\)

with \(f(1)=1.\)

Then \(\frac{1}{f(2022)}+\frac{1}{f(2028)}\) is equal to

[JEE Main 2023, 29 Jan (Shift 2)]

Let \( A=\{2,3,4,5, \ldots ., 30\} \) and \( \simeq \) be an equivalence relation on \( A \times A \), defined by \( (a, b) \simeq \) \( (c, d) \), if and only if \( a d=b c \). Then the number of ordered pairs which satisfy this equivalence relation with ordered pair \( (4,3) \) is equal to:

[JEE Main 2021, 16 Mar (Shift 2)]

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

Let \(f: R \rightarrow R\) be defined as \(f(x+y)+f(x-y)\) \(=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1 \quad\) Then, the value of \(\sum_{k=1}^{20} \frac{1}{\sin (k) \sin (k+f(k))}\) is equal to:

[JEE Main 2021, 27 Jul (Shift 2)]

The inverse function of \(f\left(x\right)=\frac{{8}^{2x}-{8}^{-2x}}{{8}^{2x}+{8}^{-2x}},x\in \left(-1,1\right)\), is

[JEE Main 2020, 8 Jan (Shift 1)]

Let \(R\) be a relation on \(N \times N\) defined by \((a, b) R(c, d)\) if and only if \(a d(b-c)=b c(a-d)\). Then \(R\) is

[JEE Main 2023, 31 Jan (Shift 1)]

If the function \(f:R\to R\) is defined by \(f(\mathrm{x})=|x|(x-\sin x)\), then which of the following statements is TRUE?

[JEE Advanced 2020]

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

[JEE Main 2020, 8 Jan (Shift 2)]

Suppose \(f:R\to (0,\infty )\) be a differentiable function such that \(5f(x+y)=f(x).f(y),\forall x,y\in \mathrm{R}\). If \(f(3)=320\), then \(\sum _{n=0}^{5}f\left(n\right)\) is equal to:

[JEE Main 2023, 30 Jan (Shift 1)]

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

Suppose \(f:R\to (0,\infty )\) be a differentiable function such that \(5f(x+y)=f(x).f(y),\forall x,y\in \mathrm{R}\). If \(f(3)=320\), then \(\sum _{n=0}^{5}f(n)\) is equal to:

[JEE Main 2023, 30 Jan (Shift 1)]

Let \([x]\) denote the greatest integer less than or equal to \(x\). Then, the values of \(x \in R\) satisfying the equation \(\left[e^x\right]^2+\left[e^x+1\right]-3=0\) lie in the interval:

[JEE Main 2021, 22 Jul (Shift 2)]

Let \(A={0,1,2,3,4,5,6,7}\). Then the number of bijective functions \(f:A\to A\) such that \(f(1)+f(2)=3-f(3)\) is equal to ____

[JEE Main 2021, 22 Jul (Shift 2)]

Let \(f(x)=\frac{{x}^{2}−6x+5}{{x}^{2}−5x+6}\)

Let \(A=\{1,2,3,4,5,6,7\}\). Then the relation \(R=\{(x, y)\) \(\in A \times A: x+y=7\}\) is

[JEE Main 2023, 8 Apr (Shift 2)]

The number of real roots of the equation \({e}^{4x}+{e}^{3x}-4{e}^{2x}+{e}^{x}+1=0\) is :

[JEE Main 2020, 9 Jan (Shift 1)]

Let \( A=\{1,2,3 \ldots ., 10\} \) and \( f: A \rightarrow A \) be defined as \( f(k)=\left\{\begin{array}{ll}k+1 & \text { if } k \text { is odd } \\ k, & \text { if } k \text { is even }\end{array}\right. \). Then the number of possible functions \( g: A \rightarrow A \) such that \( g o f=f \) is

[JEE Main 2021, 26 Feb (Shift 2)]

If \(f(x)=\frac{x}{\sqrt{1+x^2}}\), then (fofof) (x) is

The minimum number of elements that must be added to the relation \(R=\{(a, b),(b, c)\}\) on the set \(\{a, b, c\}\) so that it becomes symmetric and transitive is:

[JEE Main 2023, 30 Jan (Shift 1)]

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

[JEE Main 2023, 24 Jan (Shift 2)]

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

[JEE Main 2021, 17 Mar (Shift 1)]

Let \([t]\) denote the greatest integer \(\leq t\). Then the equation in \(x,[x{]}^{2}+2[x+2]-7=0\) has:

If the domain of the function \(f\left(x\right)={\log }_{e}\left(4{x}^{2}+11x+6\right)+{\sin }^{-1}\left(4x+3\right)+{\cos }^{-1}\left(\frac{10x+6}{3}\right)\) is \((\alpha ,\beta ]\), then \(36|\alpha +\beta |\) is equal to:

[JEE Main 2023, 15 Apr (Shift 1)]

Which of the following is not correct for relation \(R\) on the set of real numbers?

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

Let \(R\) be a relation on \(\mathbb{R}\), given by \(R=\{(a, b): 3 a-3 b+\sqrt{7}\) is an irrational number }. Then \(R\) is

[JEE Main 2023, 1 Feb (Shift 1)]

Let \(N\) be the set of natural numbers and a relation \(R\) on \(N\) be defined by \(R=\left\{(x, y) \in N \times N: x^3-3 x^2 y-x y^2+3 y^3=0\right\}\). Then the relation \(R\) is:

[JEE Main 2021, 27 Jul (Shift 2)]

The range of the function
\(f(x)={\log }_{\sqrt{5}}\left(3+\cos \left(\frac{3\pi }{4}+x\right)\right.+\cos \left(\frac{\pi }{4}+x\right)\left.+\cos \left(\frac{\pi }{4}-x\right)-\cos \left(\frac{3\pi }{4}-x\right)\right)\)

[JEE Main 2021, 1 Sep (Shift 2)]

Let the sets \(A\) and \(B\) denote the domain and range respectively of the function \(f(x)=\frac{1}{\sqrt{[x]-x}}\), where \([x]\) denotes the smallest integer greater than or equal to \(x\). Then among the statements \(\left(\mathrm{S}_1\right): A \cap B=(1, \infty)-N\) and \[\left(\mathrm{S}_2\right): A \cup B=(1, \infty)\]

[JEE Main 2023, 6 Apr (Shift 2)]

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

[JEE Main 2023, 1 Feb (Shift 2)]

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

[JEE Main 2023, 29 Jan (Shift 1)]

If \(\mathrm{R}=\left\{(x, y): x, y \in Z, x^2+3 y^2 \leq 8\right\}\) is a relation on the set of integers \(Z\), then the domain of \(\mathrm{R}^{-1}\) is:

[JEE Main 2020, 2 Sep (Shift 1)]

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

[JEE Main 2021, 20 Jul (Shift 2)]

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) 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

[JEE Main 2023, 25 Jan (Shift 2)]

If \( g(x)=x^{2}+x-1 \) and \( (gof)(x)=4 x^{2}-10 x+5 \), then \( \mathrm{f}\left(\frac{5}{4}\right) \) is equal to

[JEE Main 2020, 7 Jan (Shift 1)]

The range of the function
\(f\left(x\right)={\log }_{\sqrt{5}}\left(3+\cos \left(\frac{3\pi }{4}+x\right)\right.+\cos \left(\frac{\pi }{4}+x\right)\left.+\cos \left(\frac{\pi }{4}-x\right)-\cos \left(\frac{3\pi }{4}-x\right)\right)\)

[JEE Main 2021, 1 Sep (Shift 2)]

Let \([x]\) denote the greatest integer less than or equal to \(x\). Then, the values of \(x \in R\) satisfying the equation \(\left[e^x\right]^2+\) \(\left[e^x+1\right]-3=0\) lie in the interval:

[JEE Main 2021, 22 Jul (Shift 2)]

The domain of the function \(f(x)=\sqrt{x-\sqrt{1-x^2}}\) is

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

[JEE Main 2021, 20 Jul (Shift 2)]

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

[JEE Main 2021, 25 Feb (Shift 2)]

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

Let \([x]\) denote the greatest integer \(\leq x\). If \(f(x)= [ x ] \) and \(g(x)=|x|\), then the value of \(\mathrm{f}\left(\mathrm{g}\left(\frac{8}{5}\right)\right)-\mathrm{g}\left(\mathrm{f}\left(-\frac{8}{5}\right)\right)\) is

Let \( R_{1} \) and \( R_{2} \) be two relations defined as follows: \( R_{1}=\left\{(a, b) \in R^{2}: a^{2}+b^{2} \in Q\right\} \) and

\( R_{2}=\left\{(a, b) \in R^{2}: a^{2}+b^{2} \notin Q\right\} \), where \( Q \) is the set of all rational numbers. Then:

[JEE Main 2020, 3 Sep (Shift 2)]

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

[JEE Main 2023, 25 Jan (Shift 2)]

Let the number of elements in sets \(A\) and \(B\) be five and two respectively. Then the number of subsets of \(A \times B\) each having at least \(3\) and at most \(6\) elements is :

[JEE Main 2023, 8 Apr (Shift 1)]

Let \( f: R \rightarrow R \) be defined as \( f(x)=2 x-1 \) and \( g: R-\{1\} \rightarrow R-\{1\} \) be defined a \( g(x)=\frac{x-\frac{1}{2}}{x-1} \). Then the composition function \( f(g(x)) \) is

[JEE Main 2021, 24 Feb (Shift 1)]

Among the relations \(S=\left\{(a, b): a, b \in R -\{0\}, 2+\frac{a}{b}>0\right\}\) and \(T=\left\{(a, b): a, b \in R , a^2-b^2 \in Z\right\}\), which of the following is true?

[JEE Main 2023, 31 Jan (Shift 2)]

The range of the function \(f(x)=\sqrt{3 x^2-4 x+5}\) is

If \(f(x)=\frac{{2}^{2x}}{{2}^{2x}+2},x\in R\) then

\(f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\ldots \ldots +f\left(\frac{2022}{2023}\right)\) is equal to

[JEE Main 2023, 24 Jan (Shift 2)]

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 \(R=\left\{\left(\left(a_1, b_1\right)\right.\right.\), \(\left.\left(a_2, b_2\right)\right): a_1 \leq b_2\) and \(\left.b_1 \leq a_2\right\}\). Then the number of elements in the set \(R\) is

[JEE Main 2023, 11 Apr (Shift 2)]

Let \(R\) be a relation on \(N \times N\) defined by \((a, b) R(c, d)\) if and only if \(a d(b-c)=b c(a-d)\). Then \(R\) is

Consider the function \(f\) in \(A=R−\left\{\frac{2}{3}\right\}\) defined as \(f(x)=\frac{4x+3}{6x−4}\), then \({f}^{−1}\) is equal to

Let \([t]\) denote the greatest integer \(\leq t\). Then the equation in \(x,[x]^2+2[x+2]-7=0\) has :

[JEE Main 2020, 4 Sep (Shift 1)]

Let \( f(x)=\sin ^{-1} x \) and \( g(x)=\frac{x^{2}-x-2}{2 x^{2}-x-6} \).

If \( g(2)=\lim _{x \rightarrow 2} g(x) \), then the domain of the function fog is

[JEE Main 2021, 26 Feb (Shift 2)]

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 \(f(0)\) =

[JEE Main 2023, 13 Apr (Shift 1)]

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

[JEE Main 2023, 13 Apr (Shift 1)]

Let \( f(x)=\sin ^{-1} x \) and \( g(x)=\frac{x^{2}-x-2}{2 x^{2}-x-6} \). If \( g(2)=\lim _{x \rightarrow 2} g(x) \), then the domain of the function \(fog\) is

[JEE Main 2021, 26 Feb (Shift 2)]

If domain of the function
\({\log }_{e}\left(\frac{6{x}^{2}+5x+1}{2x-1}\right)+{\cos }^{-1}\left(\frac{2{x}^{2}-3x+4}{3x-5}\right)\text{ is }(\alpha ,\beta )\cup (\gamma ,\delta ]\text{, }\)
then \(18\left({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2}+{\delta }^{2}\right)\) is equal to _____ .

[JEE Main 2023, 8 Apr (Shift 2)]

Among the relations \(S=\left\{(a, b): a, b \in R -\{0\}, 2+\frac{a}{b}>0\right\}\) and \(T=\left\{(a, b): a, b \in R , a^2-b^2 \in Z\right\}\)

[JEE Main 2023, 31 Jan (Shift 2)]

Let \(R\) be a relation on \(R\), given by \(R=\left\{(a,b):3a-3b+\sqrt{7}\right\}\) is an irrational number. Then \(R\) is

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

\(\left({\mathrm{S}}_{1}\right):A\cap B=(1,\infty )-N\) and \(\left({\mathrm{S}}_{2}\right):A\cup B=(1,\infty )\)

[JEE Main 2023, 6 Apr (Shift 2)]

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

[JEE Main 2023, 10 Apr (Shift 1)] $$

Let \(M\) and \(m\) respectively be the maximum and minimum values of the function \(f(x)=\tan ^{-1}(\sin x+\cos x)\) in \(\left[0, \frac{\pi}{2}\right]\). Then the value of \(\tan (M-m)\) is equal to

[JEE Main 2021, 27 Aug (Shift 2)]

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

Let \( R=\{P, Q) \mid P \) and \( Q \) are at the same distance from the origin \( \} \) be a relation, then the equivalence class of \( (1,-1) \) is the set

[JEE Main 2021, 26 Feb (Shift 1)]

The domain of the function \(f\left(x\right)=\sqrt{x−1}+\sqrt{6−x}\) is

Let \(N\) be the set of natural numbers and a relation R on N be defined by \(R=\left\{(x,y)\in N\times N:{x}^{3}-3{x}^{2}y-x{y}^{2}+3{y}^{3}=0\right\}\). Then the relation R is :

[JEE Main 2021, 27 Jul (Shift 2)]

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

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

[JEE Main 2023, 24 Jan (Shift 1)]

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

[JEE Main 2021, 31 Aug (Shift 2)]

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:

[JEE Main 2023, 30 Jan (Shift 2)]

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

[JEE Main 2023, 11 Apr (Shift 2)]

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

[JEE Main 2021, 25 Jul (Shift 2)]

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

[JEE Main 2021, 22 Jul (Shift 2)]

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

[JEE Main 2021, 20 Jul (Shift 2)]

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

[JEE Main 2023, 10 Apr (Shift 1)]

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

[JEE Main 2023, 1 Feb (Shift 2)]

The domain of \(f\left(x\right)=\frac{{\log }_{(x+1)}(x-2)}{{e}^{2{\log }_{e}x}-(2x+3)},x\in R\) is

[JEE Main 2023, 29 Jan (Shift 1)]

Let \( f(x)=2 x^{\mathrm{n}}+\lambda , \lambda \in \mathrm{R}, n \in N \) and \( \mathrm{f}(4)=133 \),

\( f(5)=255 \). Then the sum of all positive integer

divisors of \( f(3)-f(2) \) is

Let \([x]\) denote the greatest integer \(\leq x\), where \(x\in R\). If the domain of the real valued function \(f(x)=\sqrt{\frac{|[x]|-2}{|[x]|-3}}\) is \((-\infty ,a)\cup [b,c)\cup [4,\infty ),a

[JEE Main 2021, 20 Jul (Shift 1)]

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

[JEE Main 2021, 18 Mar (Shift 2)]

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

[JEE Main 2021, 18 Mar (Shift 2)]

If \( f(x+y)=f(x) f(y) \) and \( \sum_{x=1}^{\infty} f(x)=2, x, y \in N \) Where \( \mathrm{N} \) is the set of all natural numbers, then the value of \( \frac{f(4)}{f(2)} \) is :

[JEE Main 2020, 6 Sep (Shift 1)]

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

[JEE Main 2023, 30 Jan (Shift 2)]

Let \(R\) be a relation defined on \(N\) as \( a R b \) if \(2 a+3 b\) is a multiple of \(5, a, b \in N\). Then \(R\) is

[JEE Main 2023, 29 Jan (Shift 2)]

Define a relation \(R\) over a class of \(n \times n\) real matrices \(A\) and \(B\) as \(ARB\) iff there exists a non-singular matrix \(P\) such that \(P A P^{-1}=B\). Then which of the following is true?

[JEE Main 2021, 18 Mar (Shift 2)]

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

[JEE Main 2021, 31 Aug (Shift 2)]

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

[JEE Main 2023, 30 Jan (Shift 1)]

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

Let \(R\) be a relation on \(R\), given by \(R=\{(a, b): 3 a-3 b+\sqrt{7}\) is an irrational number}. Then \(R\) is

[JEE Main 2023, 1 Feb (Shift 1)]

The domain of the function \(f(x)=\sqrt{x−\sqrt{1−{x}^{2}}}\) is

If \(f(x)=x^3-x^2 f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R\), then