Limits, Continuity & Differentiability · JEE Main previous year questions

Let $f(x)=\left|2x^2+5|x|-3\right|,x\in \mathbf{R}$. If $\mathrm{m}$ and $\mathrm{n}$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $\mathrm{m}+\mathrm{n}$ is equal to :

1. A.5
2. B.3
3. C.2
4. D.0

Let $\alpha$ be a non-zero real number. Suppose $f:\mathbf{R}\to \mathbf{R}$ is a differentiable function such that $f(0)=2$ and ${\lim }_{x\to -\infty }f(x)=1$. If $f^{\prime }(x)=\alpha f(x)+3$, for all $x\in \mathbf{R}$, then $f(-{\log }_{e}2)$ is equal to :

1. A.7
2. B.9
3. C.3
4. D.5

${\lim }_{x\to 0}\frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}$

1. A.is equal to 1
2. B.does not exist
3. C.is equal to $-1$
4. D.is equal to 2

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

1. A.${\log }_e\left(\frac{4}{9}\right)-1$
2. B.$\frac{1}{3}{\log }_e\left(\frac{4}{9e^{1/3}}\right)$
3. C.${\log }_e\left(\frac{4}{9e^{1/3}}\right)$
4. D.$\frac{1}{3}{\log }_e\left(\frac{4}{9}\right)+1$

Let $a$ be the sum of all coefficients in the expansion of ${\left(1-2x+2x^2\right)}^{2023}{\left(3-4x^2+2x^3\right)}^{2024}$ and $b=\underset{x\to 0}{\lim }\left(\frac{\underset{0}{\overset{x}{\int }}\frac{\log (1+t)}{t^{2024}+1}\mathrm{d}t}{x^2}\right)$. If the equation $c{x}^2+dx+e=0$ and $2b{x}^2+ax+4=0$ have a common root, where $c,d,e\in \mathbb{R}$, then $\mathrm{d}:\mathrm{c}:\mathrm{e}$ equals

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

Let $f:\mathbb{R}\to (0,\infty )$ be a strictly increasing function such that ${\lim }_{x\to \infty }\frac{f(7x)}{f(x)}=1$. Then, the value of ${\lim }_{x\to \infty }\left[\frac{f(5x)}{f(x)}-1\right]$ is equal to

1. A.0
2. B.4
3. C.1
4. D.7/5

The temperature $T(t)$ of a body at time $t=0$ is $160^{\circ }\mathrm{F}$ and it decreases continuously as per the differential equation $\frac{dT}{dt}=-K(T-80)$, where $K$ is a positive constant. If $T(15)=120^{\circ }\mathrm{F}$, then $T(45)$ is equal to

1. A.$90^{\circ }$ F
2. B.$85^{\circ }$ F
3. C.$80^{\circ }$ F
4. D.$95^{\circ }$ F

Consider the function $f:(0,\infty )\to \mathbb{R}$ defined by $f(x)=e^{-|{\log }_{e}x|}$. 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

1. A.0
2. B.1
3. C.2
4. D.3

If ${\lim }_{x\to 0}\frac{a{x}^2{e}^x-b{\log }_e(1+x)+cx{e}^{-x}}{x^2\sin x}=1$, then $16\left(a^2+b^2+c^2\right)$ is equal to ____.

The value of ${\lim }_{n\to \infty }{\sum }_{k=1}^n\frac{n^3}{(n^2+k^2)(n^2+3k^2)}$ is :

1. A.$\frac{\pi }{8(2\sqrt{3}+3)}$
2. B.$\frac{(2\sqrt{3}+3)\pi }{24}$
3. C.$\frac{13\pi }{8(4\sqrt{3}+3)}$
4. D.$\frac{13(2\sqrt{3}-3)\pi }{8}$

Let $f:\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\to \mathbf{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If the ${\lim }_{x\to 0}\frac{x{\int }_0^{x}f(t)dt}{e^{x^2}-1}=\alpha$, then $8{\alpha }^2$ is equal to :

1. A.4
2. B.2
3. C.1
4. D.16

${\lim }_{x\to \frac{\pi }{2}}\left(\frac{1}{{\left(x-\frac{\pi }{2}\right)}^2}{\int }_{x^3}^{{\left(\frac{\pi }{2}\right)}^3}\cos \left(t^{\frac{1}{3}}\right)\mathrm{d}t\right)$ is equal to

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

Let the slope of the line $45x+5y+3=0$ be $27r_1+\frac{9r_2}{2}$ for some $r_1,r_2\in \mathbb{R}$. Then ${\lim }_{x\to 3}\left({\int }_3^x\frac{8t^2}{\frac{3r_{2}x}{2}-r_2{x}^2-r_1{x}^3-3x}dt\right)$ is equal to _________.

Let $f(x)=\sqrt{{\lim }_{r\to x}\left\{\frac{2r^2[(f(r){)}^2-f(x)f(r)]}{r^2-x^2}-r^3{e}^{\frac{f(r)}{r}}\right\}}$ be differentiable in $(-\infty ,0)\cup (0,\infty )$ and $f(1)=1$. Then the value of $ea$, such that $f(a)=0$, is equal to _________.

If $\mathrm{a}=\underset{x\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $\mathrm{b}=\underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $a{b}^3$ is :

1. A.36
2. B.25
3. C.32
4. D.30

If ${\lim }_{x\to 0}\frac{3+\alpha \sin x+\beta \cos x+{\log }_e(1-x)}{3{\tan }^{2}x}=\frac{1}{3}$, then $2\alpha -\beta$ is equal to :

1. A.$2$
2. B.$1$
3. C.$5$
4. D.$7$

Among (S1): $\underset{n\to \infty }{\lim }\frac{1}{n^2}(2+4+6+\cdots +2n)=1$ (S2) : $\underset{n\to \infty }{\lim }\frac{1}{n^{16}}(1^{15}+2^{15}+3^{15}+\cdots +n^{15})=\frac{1}{16}$

1. A.Only (S1) is true
2. B.Both (S1) and (S2) are true
3. C.Both (S1) and (S2) are false
4. D.Only (S2) is true

If $\underset{x\to 0}{\lim }\frac{e^{ax}-\cos (bx)-\frac{cx{e}^{-cx}}{2}}{1-\cos (2x)}=17$, then $5a^2+b^2$ is equal to

1. A.$64$
2. B.$68$
3. C.$72$
4. D.$76$

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

If $f:\mathbb{R}\to \mathbb{R}$ be a continuous function satisfying ${\int }_0^{\frac{\pi }{2}}f(\sin 2x)\sin xdx+\alpha {\int }_0^{\frac{\pi }{4}}f(\cos 2x)\cos xdx=0$, then the value of $\alpha$ is :

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

Let $f:(-2,2)\to \mathbb{R}$ be defined by $$f(x)=\{\begin{array}{ll}x[x],& -2<x<0\\ (x-1)[x],& 0\le x<2\end{array}$$ where $[x]$ denotes the greatest integer function. If m and n respectively are the number of points in $(-2,2)$ at which $y=|f(x)|$ is not continuous and not differentiable, then $m+n$ is equal to _________.

Let $f$ be a continuous function satisfying ${\int }_0^{t^2}(f(x)+x^2)dx=\frac{4}{3}{t}^3,\forall t>0$. Then $f\left(\frac{{\pi }^2}{4}\right)$ is equal to :

1. A.$-\pi \left(1+\frac{{\pi }^3}{16}\right)$
2. B.$\pi \left(1-\frac{{\pi }^3}{16}\right)$
3. C.$-{\pi }^2\left(1+\frac{{\pi }^2}{16}\right)$
4. D.${\pi }^2\left(1-\frac{{\pi }^2}{16}\right)$

${\lim }_{x\to 0}\left(\left(\frac{1-{\cos }^2(3x)}{{\cos }^3(4x)}\right)\left(\frac{{\sin }^3(4x)}{({\log }_e(2x+1){)}^5}\right)\right)$ is equal to ___________.

1. A.$15$
2. B.$18$
3. C.$9$
4. D.$24$

Let $I(x)=\int \frac{(x+1)}{x(1+x{e}^x{)}^2}dx,x>0$. If ${\lim }_{x\to \infty }I(x)=0$, then $I(1)$ is equal to :

1. A.$\frac{e+1}{e+2}-{\log }_e(e+1)$
2. B.$\frac{e+1}{e+2}+{\log }_e(e+1)$
3. C.$\frac{e+2}{e+1}-{\log }_e(e+1)$
4. D.$\frac{e+2}{e+1}+{\log }_e(e+1)$

If $\alpha >\beta >0$ are the roots of the equation $a{x}^2+bx+1=0$, and ${\lim }_{x\to \frac{1}{\alpha }}{\left(\frac{1-\cos (x^2+bx+a)}{2(1-ax{)}^2}\right)}^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta }-\frac{1}{\alpha }\right)$, then $k$ is equal to :

1. A.$2\beta$
2. B.$\beta$
3. C.$\alpha$
4. D.$2\alpha$