Q JEE MAIN 2026

Let $f(x)=\lim _{\boldsymbol{\theta} \rightarrow 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\boldsymbol{\theta}}\right)} \sin (x-1)}{1+x^{\left(\frac{2}{\boldsymbol{\theta}}\right)}(x-1)}\right), x \in \mathbf{R}$. Consider the following two statements : (I) $f(x)$...

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Q JEE MAIN 2026

If $f(x)=\left\{\begin{array}{cc}\frac{a|x|+x^2-2(\sin |x|)(\cos |x|)}{x} & , x \neq 0 \\ b & , x=0\end{array}\right.$ is continuous at $x=0$, then $a+b$...

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Q JEE MAIN 2026

Let $\alpha, \beta \in \mathbb{R}$ be such that the function $f(x)=\left\{\begin{array}{ll}2 \alpha\left(x^2-2\right)+2 \beta x & , x

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Q JEE MAIN 2026

If the function $f(x)=\frac{e^x\left(e^{\tan x-x}-1\right)+\log _e(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal...

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Q JEE MAIN 2026

Let $[\bullet]$ denote the greatest integer function, and let $f(x)=\min \left\{\sqrt{2} x, x^2\right\}$. Let $\mathrm{S}=\left\{x \in(-2,2)\right.$ : the function $\mathrm{g}(x)=|x|\left[x^2\right]$...

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Q JEE MAIN_2026_

Let $f(x)=\left\{\begin{array}{ll}\frac{\mathrm{a} x^2+2 \mathrm{ax}+3}{4 x^2+4 x-3} & , x \neq-\frac{3}{2}, \frac{1}{2} \\ \mathrm{~b} & , x=-\frac{3}{2}, \frac{1}{2}\end{array}\right.$ be continuous at...

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Q JEE MAIN 2026

Let $f$ and $g$ be functions satisfying $f(x+y)=f(x) f(y), f(1)=7$ and $g(x+y)=g(x y), g(1)=1$, for all $x, y \in \mathbf{N}$....

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Q JEE MAIN 2026

Let $f: \mathbf{R} \rightarrow(0, \infty)$ be a twice differentiable function such that $f(3)=18, f^{\prime}(3)=0$ and $f^{\prime \prime}(3)=4$. Then $...

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Q JEE MAIN_2019_

Let $f \sim R \rightarrow R$ be differentiable at $c \in R$ and $f(c)=$ 0. If $g(x)=|f(x)|$, then at $...

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Q _ JEE MAIN_2019_

$$ \text { If } f(x)=\left\{\begin{array}{cc} \frac{\sin (p+1) x+\sin x}{x} & , x0 \end{array}\right. $$ is continuous at $x=0$, then...

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Q JEE MAIN 2019

Let f(x) = 15 – |x – 10|; x $\in$ R. Then the set of all values of x, at...

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Q JEE MAIN 2019

If the function f defined on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ by $...

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Q JEE MAIN 2019

If the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the plane $2 x+3 y-z+13=0$ at a point $P$ and the plane $3 x+y+4 z=16$...

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Q JEE MAIN 2020

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function which satisfies $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x})+f(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in \mathrm{R}$. If $f(1)=2$ and $...

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Q JEE-MAIN 2019

Let $S$ be the set of all points in $(-\pi, \pi)$ at which the function, $f(x)=\min \{\sin x, \cos x\}$...

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Q JEE Main 2020

If a function f(x) defined by be continuous for some $a, b, c \in R$ and $f^{\prime}(0)+f^{\prime}(2)=e$, then the value...

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Q JEE MAIN 2020

If $f(x)= \begin{cases}\frac{\sin (a+2) x+\sin x}{x} & ; x0\end{cases}$ is continuous at $x=0$, then $a+2 b$ is equal to

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Q JEE MAIN 2020

Let $f(x)=x \cdot\left[\frac{x}{2}\right]$ for $-10

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Q JEE MAIN 2020

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Q JEE MAIN 2020

If the function $f(x)\left\{\begin{array}{ll}k_1(x-\pi)^2-1, & x \leq \pi \\ k_2 \cos x, & x>\pi\end{array}\right.$ is twice differentiable, then the ordered...

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Q JEE MAIN 2020

Let $f: R \rightarrow R$ be a function defined by $f(x)=\max \left\{x, x^2\right\}$. Let $S$ denote the set of all...

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Q JEE MAIN 2020

If $f(x+y)=f(x) f(y)$ and $\sum_{x=1}^{\infty} f(x)=2 x, y \in N$, where $N$ is the set of all natural numbers, then...

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Q JEE MAIN 2019

Let $f(x)=\left\{\begin{array}{cc}-1, & -2 \leq x

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Q JEE MAIN 2020

Let $[t]$ denote the greatest integer $\leq t$ and $\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=A$. Then the function, $...

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Q JEE Main 2019

Let $f(x)=\left\{\begin{array}{ll}\max \left\{|x|, x^2\right\}, & |x| \leq 2 \\ 8-2|x|, & 2

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Q JEE MAIN 2020

Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y)=f(x)+f(y)+x y^2+x^2 y$, for all real $x$ and $...

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Q JEE MAIN 2020

Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8, f^{\prime}(2)=5, f^{\prime}(x) \geq 1$ and $f^{\prime \prime}(x) \geq 4$,...

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Q JEE MAIN 2020

The function $f(x)=\left\{\begin{array}{l}\frac{\pi}{4}+\tan ^{-1} x,|x| \leq 1 \\ \frac{1}{2}(|x|-1),|x|>1\end{array}\right.$ is :

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Q JEE MAIN 2020

Let $f:(0, \infty) \rightarrow(0, \infty)$ be a dif ferentiable function such that $f(1)=e$ and $...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

Let $K$ be the set of all real values of $x$ where the function $f(x)=\sin |x|-|x|+2(x-\pi)$ is not differentiable. Then...

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Q JEE MAIN 2020

Let $S$ be the set of points where the function, $f(x)=|2-|x-3| \|, x \in R$, is not differentiable. Then $...

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Q JEE MAIN 2019

If the function $f(x)=\left\{\begin{array}{ll}a|\pi-x|+1, & x \leq 5 \\ b|x-\pi|+3, & x>5\end{array}\right.$ is continuous at $x=5$, then the value of...

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Q JEE MAIN 2021

If the function $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}\frac{1}{x} \log _e\left(\frac{1+\frac{a}{x}}{1-\frac{x}{b}}\right) & , x0\end{array}\right.$ Is continuous at $\mathrm{x}=0$, then $\frac{1}{a}+\frac{1}{b}+\frac{4}{k}$ is equal to :

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Q JEE MAIN 2019

Let $f:(-1,1) \rightarrow R$ be a function defined by $f(x)=\max \left\{-|x|,-\sqrt{1-x^2}\right\}$. If $K$ be the set of all points at...

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Q JEE MAIN 2021

The function $f(x)=\left|x^2-2 x-3\right| . e^{\left|9 x^2-12 x+4\right|}$ is not differentiable at exactly:

JEE Main Mathematics Easy

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Q JEE Main 2021

Let $a, b \in R, b \neq 0$, Define a function $...

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Q JEE Main 2019

Let $f: R \rightarrow R$ be a function defined as $$ f(x)=\left\{\begin{array}{rrr} 5, & \text { if } & x...

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Q JEE MAIN 2021

Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f(x)=x-[x], g(x)=1-x+[x]$, and $...

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Q JEE MAIN 2021

Let $[t]$ denote the greatest integer $\leq t$. The number of points where the function $f(x)=[x]\left|x^2-1\right|+\sin \left(\frac{\pi}{[x]+3}\right)-[x+1], x \in(-2,2)$ is...

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Q JEE MAIN 2021 S2

Let $f:[0, \infty) \rightarrow[0,3]$ be a function defined by $...

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Q JEE MAIN_2021_

Let $f: R \rightarrow R$ be defined as $...

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Q JEE MAIN 2021

Let $f:[0,3] \rightarrow R$ be defined by $f(x)=\min \{x-[x], 1+[x]-x\}$ where $[x]$ is the greatest integer less than or equal...

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Q JEE MAIN 2021

Let $f(x)$ be a differentiable function at $x=a$ with $f^{\prime}(a)=2$ and $f(a)=4$. Then $\lim _{x \rightarrow 0} \frac{x f(a)-a f(x)}{x-a}$...

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Q JEE MAIN 2021

Let $f:\left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{cc}(1+|\sin x|)^{\frac{3 a}{|\sin x|}}, & -\frac{\pi}{4}

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Q JEE MAIN_2021_

Let $f: R \rightarrow R$ be defined as $...

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Q JEE MAIN 2021

A function $f$ is defined on $[-3,3]$ as $$ f(x)=\left\{\begin{array}{cc} \min \left\{|x|, 2-x^2\right\}, & -2 \leq x \leq 2 \\...

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ satisfy the equation $f(x+y)=f(x)$. $f(y)$ for all $x, y \in R$ and $f(x) \neq 0$...

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Q JEE MAIN 2021

Let a function $\mathrm{g}:[0,4] \rightarrow R$ be defined as $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ be a function defined as $...

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Q JEE MAIN_2021

If the function $f(x)=\frac{\cos (\sin x)-\cos x}{x^4}$ is continuous at each point in its domain and $f(0)=\frac{1}{k}$, then $k$ is...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ and $g \mid: R \rightarrow R$ be defined as $$ \begin{aligned} & f(x)=\left\{\begin{array}{ll} x+a, &...

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Q JEE MAIN 2021

Let $f: S \rightarrow$ Swhere $S=(0, \infty)$ be a twice differentiable function such that $\mathrm{f}(\mathrm{x}+1)= \mathrm{xf}(\mathrm{x})$. If $...

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Q JEE MAIN 2021

Let $\alpha \in \mathrm{R}$ be such that the function $$ f(x)= \begin{cases}\frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, & x \neq 0 \\...

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Q JEE MAIN 2021

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 expression $\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ is $\_\_\_\_$ .

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Q JEE MAIN 2021

If $\quad f(x)=\left\{\begin{array}{cl}\frac{1}{|x|} ; & |x| \geq 1 \\ a x^2+b ; & |x|

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be defined as

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let f be any function defined on R and let it satisfy the condition : $$ |f(x)-f(y)| \leq\left|(x-y)^2\right|, \forall(x, y)...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ be defined as $$ f(x)=\left\{\begin{array}{cc} 2 \sin \left(-\frac{\pi x}{2}\right), & \text { if } x1...

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Q JEE MAIN 2021

If $f: \mathrm{R} \rightarrow \mathrm{R}$ is a function defined by $f(\mathrm{x})=[\mathrm{x}-1] \cos \left(\frac{2 \mathrm{x}-1}{2}\right) \pi$, where $[$.$...

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Q JEE MAIN 2021

The number of points, at which of function $f(x)=|2 x+1|-3|x+2|+\left|x^2+x-2\right|, x \in R$ is not differentiable, is $\_\_\_\_$

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Q JEE MAIN_2022

If $f(x)=\left\{\begin{array}{cc}x+a, & x \leq 0 \\ |x-4|, & x>0\end{array}\right.$ and $g(x)=\left\{\begin{array}{cc}x+1, & x

JEE Main Mathematics Medium

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Q JEE MAIN_2022

If the function $f(x)=\left\{\frac{\log e\left(1-x+x^2\right)+\log e\left(1+x+x^2\right)}{\sec x-\cos x}, x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\{0\}\right.$ is continuous at $x=0$, then $k$ is equal to:...

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Q JEE MAIN 2022

Let $...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

Given $\pi a^2-\pi a b=30 \pi$ and $\pi a b-\pi b^2=18 \pi$ on subtracting, we get $(a-b)^2=a^2-2 a b+b^2=12$ Let...

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Q JEE MAIN 2022

Let $f, g: R \rightarrow R$ be functions defined by $f(x)=\left\{\begin{array}{cc}{[x],} & x

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $f(x)=\left[2 x^2+1\right]$ and $g(x)=\left\{\begin{array}{ll}2 x-3, & x

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Q JEE MAIN 2022

Let $c, k \in R$. If $f(x)=(c+1) x^2+\left(1-c^2\right) x+2 k$ and $f(x+y)=f(x)+f(y)-x y$, for all $x, y \in R$, then...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let $f(x)=\min \{1,1+x \sin x\}, 0 \leq x \leq 2 \pi$. If m is the number of points, where f...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let $f(x)=\left\{\begin{array}{ccc}\frac{\sin (x-[x])}{x-[x]}, & x \in(-2,-1) \\ \{\max 2 x, 3[|x|]\}, & |x|

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Q JEE MAIN 2022

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$ f(x)=\left[\begin{array}{ll} {\left[e^x\right],} & x

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $[x]$ denote the greatest integer function and $f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2$. Let $m$ be...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $[x]$ be the greatest integer $\leq x$. Then the number of points in the interval $(-2,1)$, where the function...

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Q JEE MAIN 2023

Let $f(x)=\left[x^2-x\right]+|-x+[x]|$, where $x \in \mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Then, f...

JEE Main Mathematics Medium

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Q JEE-Main 2023

Let $a \in Z$ and [t] be the greatest integer $\leq$ t. Then the number of points, where the function...

JEE Main Mathematics Medium

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Q JEE MAIN

Let f:(0,π)→R be a function given by Where a,b∈Z. If f is continuous at $\mathrm{x}=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If the function $f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3}, x \in R$, is continuous at $x=0$, then $f(0)$...

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Q JEE MAIN 2024

If the function $f(x)=\left\{\begin{array}{cl}\frac{1}{|x|} & , \\ a x^2+2 b, & |x| \geq 2\end{array}\right.$ is differentiable on R , then...

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Q JEE MAIN 2024

Let $g(x)$ be a linear function and $f(x)=\left\{\begin{array}{ll}g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0\end{array}\right.$, is continuous...

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Q JEE-Main 2024

Let f:R→R be defined as $f(x)=\left\{\begin{array}{ccc}\frac{a-b \cos 2 x}{x^2} & ; & x1\end{array}\right.$ If f is continuous everywhere in R...

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Q JEE MAIN 2024

Consider the function $f(x)=\left\{\begin{array}{cl} \frac{a\left(7 x-12-x^2\right)}{b\left|x^2-7 x+12\right|} & , x3 \\ b & , x=3 \end{array}\right.$ Where $[\mathrm{x}]$ denotes the...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the function $f(x)=\left(x^{2}-1\right)\left|x^{2}-\mathrm{a} x+2\right|+\cos |x|$ be not differentiable at the two points $x=\alpha=2$ and $x=\beta$. Then the distance of...

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Q JEE MAIN 2025

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...

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Q JEE MAIN 2025

The number of points of discontinuity of the function $f(x) = \left[ {\frac{{{x^2}}}{2}} \right] - [\sqrt x ],x \in [0,4]$,...

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Q JEE MAIN 2025

Let $m$ and $n$ be the number of points at which the function $...

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Q JEE MAIN 2025

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $f(0) = 1$ and $f(2x) - f(x) = x$ for...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let $[\mathrm{x}]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where...

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Q JEE MAIN 2025

Let $f:R \to R$ be a twice differentiable function such that $...

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Q JEE MAIN 2025

Let $...

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Q JEE MAIN 2025

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

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Q JEE MAIN 2025

If the function $...

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Q JEE MAIN 2025

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x){{f}^{\prime }}(y)+{{f}^{\prime }}(x)f(y)$ for all $x,y\in \mathbf{R}$. Then $...

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Q JEE MAIN 2025

Let the function, $f(x)= \begin{cases}-3 a^2-2, & x1, \mathbf{b} \in \mathbf{R}$. If the area of the region enclosed by $y=f(x)$...

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