Q JEE Main 2024

Consider the function $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $...

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Q JEE-Main 2024

Let $\mathrm{f}: \mathrm{R}-\left\{\frac{-1}{2}\right\} \rightarrow \mathrm{R}$ and $\mathrm{g}: \mathrm{R}-\left\{\frac{-5}{2}\right\} \rightarrow \mathrm{R}$ be defined as $f(x)=\frac{2 x+3}{2 x+1}$ and $g(x)=\frac{|x|+1}{2 x+5}$. Then...

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Q JEE MAIN 2026

Given below are two statementsi Statement I: The function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=\frac{x}{1+|x|}$ is one-one. Statement II:...

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Q JEE MAIN 2026

The sum of all the elements in the range of $$ \begin{aligned} & f(x)=\operatorname{Sgn}(\sin x)+\operatorname{Sgn}(\cos x)+\operatorname{Sgn}(\tan x)+\operatorname{Sgn}(\cot x), x \neq...

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Q JEE MAIN 2026

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:

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Q JEE MAIN 2026

Consider two sets $\mathrm{A}=\{x \in \mathbb{Z}:(|x-3|-3) \mid \leq 1\}$ and $\mathrm{B}=\left\{x \in \mathbb{R}-\{1,2\}: \frac{(x-2)(x-4)}{x-1} \log _e(|x-2|)=0\right\}$. Then the number of...

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Q JEE MAIN 2026

If the domain of the function $$ f(x)=\log _{\left(10 x^2-17 x+7\right)}\left(18 x^2-11 x+1\right) $$ is $...

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Q JEE MAIN 2026

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...

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Q JEE MAIN_2026_

Let the domain of the function $f(x)=\log _3 \log _5 \log _7\left(9 x-x^2-13\right)$ be the interval $(\mathrm{m}, \mathrm{n})$. Let the...

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Q JEE MAIN_2026_

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

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Q JEE MAIN 2026

The number of strictly increasing functions $f$ from the set $\{1,2,3,4,5,6\}$ to the set $\{1,2,3, \ldots, 9\}$ such that $...

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Q JEE MAIN_2019

Which one of the following Boolean expressions is a tautology ?

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Q JEE MAIN 2019

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

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Q JEE MAIN 2019

Let $\sum_{k=1}^{10} f(a+k)=16\left(2^{10}-1\right)$ , where the function f satisfies f(x + y) = f(x) f(y) for all natural numbers x,...

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Q JEE MAIN 2019

For $x \in R$, let $[x]$ denote the greatest integer $\leq x$, then the sum of the series $...

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Q JEE MAIN 2019

If [x] denotes the greatest integer ≤ x, then the system of linear equations [sinθ]x + [– cosθ]y = 0...

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Q JEE Main 2020

The domain of the function $f(x)=\sin ^{-1}\left(\frac{|x|+5}{x^2+1}\right)$ is $(-\infty,-a] \cup[a, \infty)$. Then a is equal to:

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Q JEE MAIN 2020

The number of distinct solutions of the equation, $\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|$ in the interval $[0,2 \pi]$, is $\_\_\_\_$...

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Q JEE MAIN 2020

For a suitably chosen real constant a , let a function, $\mathrm{f}: \mathrm{R}-\{-\mathrm{a}\} \rightarrow \mathrm{R}$ be defined by $f(x)=\frac{a-x}{a+x}$. Further...

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Q JEE MAIN 2020

Let $A=\{a, b, c\}$ and $B=\{1,2,3,4\}$. Then the number of elements in the set $...

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Q JEE MAIN 2020

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

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Q JEE MAIN 2020

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

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Q JEE MAIN 2020

Let $a-2 b+c=1$. $f(x)=\left|\begin{array}{lll}x+a & x+2 & x+1 \\ x+b & x+3 & x+2\end{array}\right|$, then

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Q JEE MAIN 2020

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

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Q JEE MAIN 2019

Let N be the set of natural numbers and two functions f and g be defined as $...

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Q JEE Main 2021

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

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Q JEE MAIN 2019

If $f(x)=\log _e\left(\frac{1-x}{1+x}\right),|x|

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Q JEE MAIN 2021

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

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Q JEE Main 2019

For $x \in R-\{0,1\}$, let $f_1(x)=\frac{1}{x}, f_2(x)=1-x$ and $f_3(x)=\frac{1}{1-x}$ be three given functions. If a function, $J(x)$ satisfies $\left(f_2 \operatorname{OJof}_1\right)(x)=f_3(x)$...

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Q JEE MAIN 2021

Let $f(x)$ be a polynomial of degree 3 such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5$. Then the value of $52-10 f(10)$ is...

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Q JEE MAIN 2021

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 :

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Q _JEE MAIN_2021

Let $\mathrm{g}: \mathrm{N} \rightarrow \mathrm{N}$ be defined as $$ \begin{aligned} & g(3 n+1)=3 n+2 \\ & g(3 n+2)=3 n+3 \\...

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Q JEE MAIN 2021 S2

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

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Q JEE MAIN 2019

Let $f(x)=a^x(a>0)$ be written as $f(x)=f_1(x)+f_2(x)$, where $f_1(x)$ is an even function and $f_2(x)$ is an odd function. Then $f_1(x+y)+f_1(x-y)$...

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Q JEE MAIN 2021

Let S={1,2,3,4,5,6,7}. Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for...

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Q JEE MAIN_2021_

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...

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Q JEE MAIN 2021

Consider function $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}(\mathrm{A}, \mathrm{B}, \mathrm{C} \subseteq \mathrm{R})$ such that (gof) ${ }^{-1}$...

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Q _JEE MAIN_2021_

Let $[x]$ denote the greatest integer less than or equal to $x$. Then, the values of $x \in R$ satisfying...

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Q JEE MAIN 2021

Let $[x]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function...

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Q JEE MAIN 2021

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

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Q JEE MAIN 2021

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...

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Q JEE MAIN 2021

Lef $f: R-\left\{\frac{\alpha}{6}\right\} \rightarrow R$ be defined by $f(x)=\frac{5 x+3}{6 x-\alpha}$. Then the value of $\alpha$ for which (fof) $(x)=x$,...

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Q JEE MAIN 2021

Let $f ; \mathrm{R} \rightarrow \mathrm{R}$ be defined as $f(\mathrm{x})=2 \mathrm{x}-1$ and $\mathrm{g}: \mathrm{R}-\{1\} \rightarrow \mathrm{R}$ be defined as $\mathrm{g}(\mathrm{x})=\frac{\mathrm{x}-\frac{1}{2}}{\mathrm{x}-1}$...

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Q JEE MAIN 2021

The real valued function $f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{x-[x]}}$, where $[\mathrm{x}]$ denotes the greatest integer less than or equal to x , is...

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Q JEE MAIN 2021

Let $f, g: N \rightarrow N$ such that $f(n+1)=f(n)+f(1) \forall n \in N$ and $g$ be any arbitrary function. Which...

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. Define $$ f: S \rightarrow S \text { as } f(n)=\left\{\begin{array}{cc} 2 n, & \text { if }...

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Q JEE MAIN 2022

Let $f(x)$ and $g(x)$ be two real polynomials of degree 2 and 1 respectively. If $f(g(x))=8 x^2-2 x$, and $...

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Q JEE MAIN 2022

Let $S=\{1,2,3,4\}$. Then the number of elements in the set $\{f: S \times S \rightarrow S: f$ is onto and...

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

Let $f: R \rightarrow R$ be a function defined $f(x)=\frac{2 e^{2 x}}{e^{2 x}+e}$. 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

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

Let $f(x)=2 \cos ^{-1} x+4 \cot ^{-1} x-3 x^2-2 x+10, x \in[-1,1]$. If $[a, b]$ is the range of the...

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Q JEE MAIN 2022

Let $f: R \rightarrow R$ 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 $g(x)=f(f(f(x)))+f(f(x))$, the greatest integer less than...

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Q JEE MAIN 2022

The domain of the function $f(x)=\frac{\cos ^{-1}\left(\frac{x^2-5 x+6}{x^2-9}\right)}{\log _e\left(x^2-3 x+2\right)}$ is

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

Let $f: N \rightarrow R$ be a function such that $f(x+y)=2 f(x) f(y)$ for natural numbers $x$ and $y$. If...

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Q JEE MAIN 2023

Let $R=\{a, b, c, d, e\}$ and $S=\{1,2,3,4\}$. Total number of onto function $f \mid: R \rightarrow S$ such that...

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Q JEE MAIN 2023

If domain of the function $\log _e\left(\frac{6 x^2+5 x+1}{2 x-1}\right)+\cos ^{-1}\left(\frac{2 x^2-3 x+4}{3 x-5}\right)$ is $(\alpha, \beta) \cup(\gamma, \delta]$, then...

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Q JEE MAIN 2023

If the domain of the function $f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$ is $[\alpha, \beta] \cup[\gamma, \delta]$, then $|3 \alpha+10(\beta+\gamma)+21 \delta|$ is...

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Q JEE MAIN_2025

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

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Q JEE MAIN 2025

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 $...

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

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.

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Q JEE MAIN 2025

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:

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Q JEE MAIN 2023

If the domain of the function Then $36|\alpha+\beta|$ is equal to-

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Q JEE MAIN 2023

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

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Q JEE MAIN 2023

https://competishun.com/let-d-be-the-domain-of-the-function-fxsin-1leftlog-_3-xleftfrac62-log-_3-x-5-xrightright-if-the-range-of-the-function-g_i-d-rightarrow-r-defined-by-gxx-xleftxright-is-the-greatest-integer-function-is-alpha-beta-then-alpha2frac5beta-is-equal-to/

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Q JEE MAIN 2023

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

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Q JEE MAIN

Let $f(x)=x^2+9, g(x)=\frac{x}{x-9}$ and $a=\operatorname{fog}(10), b=\operatorname{gof}(3)$. If e and 1 denote the eccentricity and the length of the latus rectum...

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Q JEE MAIN

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

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Q JEE MAIN 2024

If $S=\{a \in R:|2 a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents...

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Q JEE MAIN 2024

If the range of $f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$,...

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Q JEE MAIN 2024

Let $[t]$ be the greatest integer less than or equal to t. Let $A$ be the set of al prime...

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Q JEE-Main 2024

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

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Q JEE MAIN 2024

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 :

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Q JEE MAIN 2024

Let A={1,2,3,….7} and let P(1) denote the power set of A. If the number of functions f:A→P(A) such that a∈f(a),∀a∈A...

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Q JEE-Main 2024

Let f:R→R and g:R→R be defined as $$f(x)=\left\{\begin{array}{lll} \log _e x & , & x>0 \\ e^{-x} & , &...

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Q JEE MAIN 2024

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

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Q JEE MAIN 2025

If the domain of the function $\log _{5}\left(18 x-x^{2}-77\right)$ is ( $\alpha, \beta$ ) and the domain of the function...

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Q JEE MAIN 2024

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

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Q JEE MAIN 2024

Consider the matrix $...

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Q JEE MAIN 2025

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 $\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}$....

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Q JEE MAIN 2025

Consider the sets $A=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+y^2=25\right\}, B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+9 y^2=144\right\}$, $...

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Q JEE MAIN 2025

Let $f,g:(1,\infty ) \to $ $\mathbb{R}$ be defined as $f(x) = \frac{{2x + 3}}{{5x + 2}}$ and $...

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Q JEE MAIN 2025

If the domain of the function $...

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Q 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 :

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

If $f(x) = \frac{{{2^x}}}{{{2^x} + \sqrt 2 }},{\rm{x}} \in $, then $\sum\limits_{{\rm{k}} = 1}^{81} f \left( {\frac{{\rm{k}}}{{82}}} \right)$ is equal...

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Q JEE MAIN 2025

Let $f(x) = \frac{{{2^{x + 2}} + 16}}{{{2^{2x + 1}} + {2^{x + 4}} + 32}}$. Then the value of...

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

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 $1 \in f(\mathrm{~A})$ is...

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