Question

Suppose f is a function satisfying $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{y}\right)$ for  all $\mathrm{x},\mathrm{y}\in \mathrm{\mathbb{N}}$ and $\mathrm{f}\left(1\right)=\frac{1}{5}$. If $\sum _{n=1}^m\frac{f\left(n\right)}{n(n+1)(n+2)}=\frac{1}{12}$ then $\mathrm{m}$ is equal to ______.

JEE Main 2023 (29 Jan Shift 1)

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Question

Let $f\left(x\right)=\left[2x^2+1\right]$ and $g\left(x\right)=\left\{\begin{array}{ll}2x-3,& x<0\\ 2x+3,& x\ge 0\end{array}\right.$, where $\left[t\right]$ is the greatest integer $\le t$. Then, in the open interval $\left(-1,1\right)$, the number of points where fog is discontinuous is equal to ______.

JEE Main 2022 (25 Jun Shift 2)

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Question

If the function $f(x)= \begin{cases}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{cases}$ is continuous at $x=0$, then the value of $a^2$ is equal to

JEE Main 2024 (04 Apr Shift 2)

Options

• A: 968
• B: 1152
• C: 746
• D: 1250

Question

Let $f:R-\left\{0\right\}\to R$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f\left(x\right)}{f\left(y\right)}$ for all $x,y,f\left(y\right)\ne 0$. If $f^{\text{'}}\left(1\right)=2024$, then

JEE Main 2024 (30 Jan Shift 2)

Options

• A: $x{f}^{\text{'}}\left(x\right)-2024f\left(x\right)=0$
• B: $x{f}^{\text{'}}\left(x\right)+2024f\left(x\right)=0$
• C: $x^{\text{'}}\left(x\right)+f\left(x\right)=2024$
• D: $x{f}^{\text{'}}\left(x\right)-2023f\left(x\right)=0$

Question

Let $g\left(x\right)$ be a linear function and $f\left(x\right)=\left\{\begin{array}{cc}g\left(x\right),& x\le 0\\ {\left(\frac{1+x}{2+x}\right)}^{\frac{1}{x}},& x>0\end{array}\right.$, is continuous at $x=0$. If $f^{\text{'}}\left(1\right)=f\left(-1\right)$, then the value of $g\left(3\right)$ is

JEE Main 2024 (31 Jan Shift 1)

Options

• A: $\frac{1}{3}{\log }_e\left(\frac{4}{9e^{{\displaystyle \frac{1}{3}}}}\right)$
• B: $\frac{1}{3}{\log }_e\left(\frac{4}{9}\right)+1$
• C: ${\log }_e\left(\frac{4}{9}\right)-1$
• D: ${\log }_e\left(\frac{4}{9e^{{\displaystyle \frac{1}{3}}}}\right)$

Question

Let $f\left(x\right)=\left\{\begin{array}{cc}\left|4x^2-8x+5\right|,& \mathrm{if} 8x^2-6x+1\ge 0\\ \left[4x^2-8x+5\right],& \mathrm{if} 8x^2-6x+1<0\end{array}\right.$, where $\left[\alpha \right]$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $R$ where $f$ is not differentiable is _____ .

JEE Main 2022 (25 Jul Shift 1)

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Question

Let $f:R\to R$ be a function defined by : $f\left(x\right)=\left\{\begin{array}{cc}\underset{t\le x}{\max }\left\{t^3-3t\right\};& x\le 2\\ x^2+2x-6;& 2<x<3\\ \left[x-3\right]+9;& 3\le x\le 5\\ 2x+1; & x>5\end{array}\right.$
Where $\left[t\right]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I={\int }_{-2}^{2}f\left(x\right)dx$. Then the ordered pair $\left(m,I\right)$ is equal to

JEE Main 2022 (29 Jun Shift 1)

Options

• A: $\left(3,\frac{27}{4}\right)$
• B: $\left(3,\frac{23}{4}\right)$
• C: $\left(4,\frac{27}{4}\right)$
• D: $\left(4,\frac{23}{4}\right)$

Question

Let $f\left(x\right)=\min \left\{1,1+x\sin x\right\},0\le x\le 2\pi$. If $m$ is the number of points, where $f$ is not differentiable and $n$ is the number of points, where $f$ is not continuous, then the ordered pair $\left(m,n\right)$ is equal to

JEE Main 2022 (26 Jun Shift 2)

Options

• A: $\left(2,0\right)$
• B: $\left(1,0\right)$
• C: $\left(1,1\right)$
• D: $\left(2,1\right)$

Question

Consider the function $\mathrm{f}:(0,2)\to \mathrm{R}$ defined by $\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}}{2}+\frac{2}{\mathrm{x}}$ and the function $\mathrm{g}\left(\mathrm{x}\right)$ defined by $\mathrm{g}\left(\mathrm{x}\right)=\left\{\begin{array}{cc}\text{min}\left\{\mathrm{f}\right(\mathrm{t}\left)\right\},& 0<\mathrm{t}\le \mathrm{x}\text{ and }0<\mathrm{x}\le 1\\ \frac{3}{2}+\mathrm{x},& 1<\mathrm{x}<2\end{array}\right.$. Then

JEE Main 2024 (27 Jan Shift 2)

Options

• A: $\mathrm{g}$ is continuous but not differentiable at $\mathrm{x}=1$
• B: $\mathrm{g}$ is not continuous for all $\mathrm{x}\in (0,2)$
• C: $\mathrm{g}$ is neither continuous nor differentiable at $\mathrm{x}=1$
• D: $\mathrm{g}$ is continuous and differentiable for all $\mathrm{x}\in (0,2)$

Question

Let $f:R\to R$ be a function defined as $f\left(x\right)=\left\{\begin{array}{ccc}3\left(1-\frac{\left|x\right|}{2}\right)& \text{if}& \left|x\right|\le 2\\ 0& \text{if}& \left|x\right|>2\end{array}\right..$

Let $g:R\to R$ be given by $g\left(x\right)=f(x+2)-f(x-2).$ If $n$ and $m$ denote the number of points in $R$ where $g$ is not continuous and not differentiable, respectively, then $n+m$ is equal to ________.

JEE Main 2021 (22 Jul Shift 1)

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Question

The number of points, where the function $f:R\to R, f\left(x\right)=\left|x-1\right|\cos \left|x-2\right|\sin \left|x-1\right|+\left(x-3\right)\left|x^2-5x+4\right|$, is NOT differentiable, is

JEE Main 2022 (29 Jul Shift 1)

Options

• A: $1$
• B: $2$
• C: $3$
• D: $4$

Question

Let $f$ and $g$ be twice differentiable functions on $R$ such that

$f^{"}\left(x\right)=g^{"}\left(x\right)+6x$

$f^{\text{'}}\left(1\right)=4g^{\text{'}}\left(1\right)-3=9$

$f\left(2\right)=3 g\left(2\right)=12$

Then which of the following is NOT true ?

JEE Main 2023 (29 Jan Shift 2)

Options

• A: $g\left(-2\right)-f\left(-2\right)=20$
• B: If $-1<x<2$, then $\left|f\left(x\right)-g\left(x\right)\right|<8$
• C: $\left|f^{\text{'}}\left(x\right)-g^{\text{'}}\left(x\right)\right|<6\Rightarrow -1<x<1$
• D: There exists $x_0\in \left(1,\frac{3}{2}\right)$ such that $f\left(x_0\right)=g\left(x_0\right)$

Question

Consider the function $f:\left(0,\infty \right)\to R$ defined by $f\left(x\right)=e^{-\left|{\log }_{e}x\right|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m+n$ is

JEE Main 2024 (31 Jan Shift 2)

Options

• A: $0$
• B: $3$
• C: $1$
• D: $2$

Question

Let $f\left(x\right)=x·\left[\frac{x}{2}\right],$ for $-10<x<10,$ where $\left[t\right]$ denotes the greatest integer function. Then the number of points of discontinuity of $f\left(x\right)$ is equal to

JEE Main 2020 (05 Sep Shift 1)

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Question

Let $f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin \left(x-\left[x\right]\right)}{x-\left[x\right]},& x\in \left(-2,-1\right)\\ \max \left(2x,3\left[\left|x\right|\right]\right),& \left|x\right|<1\\ 1,& \mathrm{otherwise}\end{array}\right.$
where $\left[t\right]$ denotes greatest integer $\le t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, the ordered pair $\left(m,n\right)$ is:

JEE Main 2022 (24 Jun Shift 2)

Options

• A: $\left(3,3\right)$
• B: $\left(2,4\right)$
• C: $\left(2,3\right)$
• D: $\left(3,4\right)$

Question

If $f\left(x\right)=\left\{\begin{array}{ll}x+a,& x\le 0\\ \left|x-4\right|,& x>0\end{array}\right.$ and $g\left(x\right)=\left\{\begin{array}{ll}x+1,& x<0\\ {\left(x-4\right)}^2+b,& x\ge 0\end{array}\right.$ are continuous on $R$, then $\left(gof\right)\left(2\right)+\left(fog\right)\left(-2\right)$ is equal to:

JEE Main 2022 (26 Jul Shift 1)

Options

• A: $-10$
• B: $10$
• C: $8$
• D: $-8$

Question

The function $f:R\to R$ defined by $f\left(x\right)=\underset{n\to \infty }{\lim }\frac{\cos \left(2\pi x\right)-x^{2n}\sin \left(x-1\right)}{1+x^{2n+1}-x^{2n}}$ is continuous for all $x$ in

JEE Main 2022 (28 Jul Shift 2)

Options

• A: $R-\left\{-1\right\}$
• B: $R-\left\{-1,1\right\}$
• C: $R-\left\{1\right\}$
• D: $R-\left\{0\right\}$

Question

Let $a\in \mathrm{\mathbb{Z}}$ and $\left[t\right]$ be the greatest integer $\le t$, then the number of points, where the function $f\left(x\right)=\left[a+13 \sin x\right], x\in \left(0,\pi \right)$ is not differentiable, is $\_\_\_\_\_\_\_\_\_\_\_\_$

JEE Main 2023 (06 Apr Shift 1)

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Question

Let $f:\left[0,\infty \right)\to \left[0,3\right]$ be a function defined by $f\left(x\right)=\left\{\begin{array}{l}\max \left\{\sin t:0\le t\le \pi \right\}, x\in \left[0,\mathrm{\pi }\right]\\ 2+\cos x, x>\pi \end{array}\right.$. Then which of the following is true ?

JEE Main 2021 (27 Jul Shift 2)

Options

• A: $f$ is continuous everywhere but not differentiable exactly at one point in $\left(0,\infty \right)$
• B: $f$ is differentiable everywhere in $\left(0,\infty \right)$
• C: $f$ is not continuous exactly at two points in $\left(0,\infty \right)$
• D: $f$ is continuous everywhere but not differentiable exactly at two points in $\left(0,\infty \right)$

Question

Let $f:[-1,2] \rightarrow \mathbf{R}$ be given by $f(x)=2 x^2+x+\left[x^2\right]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :

JEE Main 2024 (05 Apr Shift 2)

Options

• A: 5
• B: 6
• C: 3
• D: 4

Question

If $f:R\to R$ is a function defined by $f\left(x\right)=\left[x-1\right]\cos \left(\frac{2x-1}{2}\right)\pi ,$ where $\left[·\right]$ denotes the greatest integer function, then $f$ is:

JEE Main 2021 (24 Feb Shift 1)

Options

• A: discontinuous only at $x=1$
• B: discontinuous at all integral values of $x$ except at $x=1$
• C: continuous only at $x=1$
• D: continuous for every real $x$

Question

Let $f:\left[0,3\right]\to R$ be defined by $f\left(x\right)=\min \left\{x-\left[x\right],1+\left[x\right]-x\right\}$ where $\left[x\right]$ is the greatest integer less than or equal to $x.$ Let $P$ denote the set containing all $x\in \left[0,3\right]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x\in \left(0,3\right)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to _____.

JEE Main 2021 (27 Jul Shift 1)

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Question

Let $\left[x\right]$ be the greatest integer $\le x$. Then the number of points in the interval $(–2, 1)$ where the function $f\left(x\right)=\left|\left[x\right]\right|+\sqrt{x-\left[x\right]}$ is discontinuous, is _____.

JEE Main 2023 (12 Apr Shift 1)

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Question

Let $f,g:\mathit{R}\to \mathit{R}$ be functions defined by
$f\left(x\right)=\left\{\begin{array}{ll}\left[x\right]& , x<0\\ \left|1-x\right|& , x\ge 0\end{array}\right.$ and
$g\left(x\right)=\left\{\begin{array}{ll}{e}^x-x,& x<0\\ {\left(x-1\right)}^2-1,& x\ge 0\end{array}\right.$
where $\left[x\right]$ denote the greatest integer less than or equal to $x$. Then, the function fog is discontinuous at exactly

JEE Main 2022 (28 Jun Shift 2)

Options

• A: one point
• B: two points
• C: three points
• D: four points

Question

If $\left[t\right]$ denotes the greatest integer $\le t$, then number of points, at which the function $f\left(x\right)=4\left|2x+3\right|+$ $9\left[x+\frac{1}{2}\right]-12\left[x+20\right]$ is not differentiable in the open interval $\left(-20,20\right)$, is ______.

JEE Main 2022 (29 Jul Shift 2)

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