Limits, Continuity & Differentiability · JEE Main previous year questions

Let the function $f(x)=(x^2-1)|x^2-ax+2|+\cos |x|$ be not differentiable at the two points $x=\alpha =2$ and $x=\beta$. Then the distance of the point $(\alpha ,\beta )$ from the line $12x+5y+10=0$ is equal to :

1. A.$5$
2. B.$2$
3. C.$4$
4. D.$3$

If $\underset{t\to 0}{\lim }{\left(\underset{0}{\overset{1}{\int }}(3x+5{)}^t\mathrm{dx}\right)}^{\frac{1}{t}}=\frac{\alpha }{5\mathrm{e}}{\left(\frac{8}{5}\right)}^{\frac{2}{3}}$, then $\alpha$ is equal to ____.

Let $f(x)=\{\begin{array}{ll}3x,& x<0\\ \min \{1+x+[x],x+2[x]\},& 0\le x\le 2\\ 5,& x>2\end{array}$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where $f$ is not continuous and is not differentiable, respectively, then $\alpha +\beta$ equals ____ .

${\lim }_{x\to 0}\mathrm{cosec}x\left(\sqrt{2{\cos }^{2}x+3\cos x}-\sqrt{{\cos }^{2}x+\sin x+4}\right)$ is:

1. A.$\frac{1}{\sqrt{15}}$
2. B.$\frac{1}{2\sqrt{5}}$
3. C.$0$
4. D.$-\frac{1}{2\sqrt{5}}$

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

1. A.$6$
2. B.$5$
3. C.$3$
4. D.$4$

For some $a,b$, let $f(x)=\left|\begin{array}{ccc}a+\frac{\sin x}{x}& 1& b\\ a& 1+\frac{\sin x}{x}& b\\ a& 1& b+\frac{\sin x}{x}\end{array}\right|$, $x\ne 0$, ${\lim }_{x\to 0}f(x)=\lambda +\mu a+\nu b$. Then $(\lambda +\mu +\nu {)}^2$ is equal to :

1. A.25
2. B.16
3. C.9
4. D.36

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x)=[x]+|x-2|,-2<x<3$, is not continuous and not differentiable. Then $\mathrm{m}+\mathrm{n}$ is equal to :

1. A.6
2. B.9
3. C.8
4. D.7

${\lim }_{x\to \infty }\frac{(2x^2-3x+5)(3x-1{)}^{\frac{x}{2}}}{(3x^2+5x+4)\sqrt{(3x+2{)}^x}}$ is equal to :

1. A.$\frac{2\mathrm{e}}{3}$
2. B.$\frac{2}{3\sqrt{\mathrm{e}}}$
3. C.$\frac{2\mathrm{e}}{\sqrt{3}}$
4. D.$\frac{2}{\sqrt{3\mathrm{e}}}$

Let ${\lim }_{n\to \infty }\left(\frac{n}{\sqrt{n^4+1}}-\frac{2n}{(n^2+1)\sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8n}{(n^2+4)\sqrt{n^4+16}}+\cdots +\frac{n}{\sqrt{n^4+n^4}}-\frac{2n\cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}}\right)$ be $\frac{\pi }{k}$, using only the principal values of the inverse trigonometric functions. Then $k^2$ is equal to ______.

Let $f:(0,\pi )\to \mathbf{R}$ be a function given by $$f(x)=\{\begin{array}{ll}{\left(\frac{8}{7}\right)}^{\frac{\tan 8x}{\tan 7x}},& 0<x<\frac{\pi }{2}\\ \mathrm{a}-8,& x=\frac{\pi }{2}\\ (1+|\cot x|{)}^{\frac{\mathrm{b}}{\mathrm{a}}|\tan x|},& \frac{\pi }{2}<x<\pi \end{array}$$ where $\mathrm{a},\mathrm{b}\in \mathbf{Z}$. If $f$ is continuous at $x=\frac{\pi }{2}$, then ${\mathrm{a}}^2+{\mathrm{b}}^2$ is equal to ______.

$\underset{x\to \frac{\pi }{2}}{\lim }\left(\frac{{\int }_{x^3}^{(\pi /2{)}^3}(\sin (2t^{1/3})+\cos (t^{1/3}))dt}{{\left(x-\frac{\pi }{2}\right)}^2}\right)$ is equal to

1. A.$\frac{3{\pi }^2}{2}$
2. B.$\frac{9{\pi }^2}{8}$
3. C.$\frac{5{\pi }^2}{9}$
4. D.$\frac{11{\pi }^2}{10}$

$\underset{x\to 0}{\lim }\frac{e-(1+2x{)}^{\frac{1}{2x}}}{x}$ is equal to

1. A.$\frac{-2}{e}$
2. B.$e-e^2$
3. C.0
4. D.$e$

For a differentiable function $f:\mathbb{R}\to \mathbb{R}$, suppose $f^{\prime }(x)=3f(x)+\alpha$, where $\alpha \in \mathbb{R}$, $f(0)=1$ and $\underset{x\to -\infty }{\lim }f(x)=7$. Then $9f(-{\log }_{e}3)$ is equal to ____.

The value of ${\lim }_{x\to 0}2\left(\frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}\dots \sqrt[10]{\cos 10x}}{x^2}\right)$ is ________.

For $a,b>0$, let $f(x)=\{\begin{array}{ll}\frac{\tan ((a+1)x)+b\tan x}{x},& x<0\\ 3,& x=0\\ \frac{\sqrt{ax+b^2{x}^2}-\sqrt{ax}}{b\sqrt{ax}\sqrt{x}},& x>0\end{array}$ be a continuous function at $x=0$. Then $\frac{b}{a}$ is equal to :

1. A.4
2. B.5
3. C.8
4. D.6

If $\alpha =\underset{x\to 0^{+}}{\lim }\left(\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$ and $\beta =\underset{x\to 0}{\lim }(1+\sin x{)}^{\frac{1}{2}\cot x}$ are the roots of the quadratic equation $a{x}^2+bx-\sqrt{e}=0$, then $12{\log }_e(a+b)$ is equal to ____.

${\lim }_{n\to \infty }\frac{(1^2-1)(n-1)+(2^2-2)(n-2)+\cdots +((n-1{)}^2-(n-1))\cdot 1}{(1^3+2^3+\cdots +n^3)-(1^2+2^2+\cdots +n^2)}$ is equal to :

1. A.$\frac{2}{3}$
2. B.$\frac{1}{2}$
3. C.$\frac{3}{4}$
4. D.$\frac{1}{3}$

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

1. A.5
2. B.6
3. C.4
4. D.3

Let $a>0$ be a root of the equation $2x^2+x-2=0$. If $\underset{x\to \frac{1}{a}}{\lim }\frac{16\left(1-\cos (2+x-2x^2)\right)}{(1-ax{)}^2}=\alpha +\beta \sqrt{17}$, where $\alpha ,\beta \in \mathbb{Z}$, then $\alpha +\beta$ is equal to _________.

Let $f:\mathbf{R}\to \mathbf{R}$ be a function given by $$f(x)=\{\begin{array}{ll}\frac{1-\cos 2x}{x^2},& x<0\\ \alpha ,& x=0,\\ \frac{\beta \sqrt{1-\cos x}}{x},& x>0\end{array}$$ where $\alpha ,\beta \in \mathbf{R}$. If $f$ is continuous at $x=0$, then ${\alpha }^2+{\beta }^2$ is equal to :

1. A.48
2. B.6
3. C.3
4. D.12

If $\underset{x\to 1}{\lim }\frac{(5x+1{)}^{1/3}-(x+5{)}^{1/3}}{(2x+3{)}^{1/2}-(x+4{)}^{1/2}}=\frac{m\sqrt{5}}{n(2n{)}^{2/3}}$, where $\gcd (\mathrm{m},\mathrm{n})=1$, then $8\mathrm{m}+12\mathrm{n}$ is equal to ____.

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

1. A.968
2. B.1250
3. C.1152
4. D.746

Let $f(x)={\int }_0^x\left(t+\sin \left(1-e^t\right)\right)dt,x\in \mathbb{R}$. Then, ${\lim }_{x\to 0}\frac{f(x)}{x^3}$ is equal to

1. A.$\frac{1}{6}$
2. B.$-\frac{1}{6}$
3. C.$\frac{2}{3}$
4. D.$-\frac{2}{3}$

Let $f:\mathbf{R}\to \mathbf{R}$ be defined as : $$f(x)=\{\begin{array}{ll}\frac{a-b\cos 2x}{x^2};& x<0\\ x^2+cx+2;& 0\le x\le 1\\ 2x+1;& x>1\end{array}$$ If $f$ is continuous everywhere in $\mathbf{R}$ and $m$ is the number of points where $f$ is NOT differentiable then $\mathrm{m}+\mathrm{a}+\mathrm{b}+\mathrm{c}$ equals :

1. A.1
2. B.4
3. C.3
4. D.2

Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\frac{{\cos }^{-1}\left(1-\{x{\}}^2\right){\sin }^{-1}(1-\{x\})}{\{x\}-\{x{\}}^3},x\ne 0$. If $\mathrm{L}$ and $\mathrm{R}$ respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x=0$, then $\frac{32}{{\pi }^2}\left({\mathrm{L}}^2+{\mathrm{R}}^2\right)$ is equal to ____.