Limits, Continuity & Differentiability · JEE Main previous year questions

2021 25 questions

1. Q12021 · 1 Sept 2021 · Shift 2

   Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. Then ${\lim }_{x\to \frac{\pi }{4}}\frac{\frac{\pi }{4}{\int }_2^{{\sec }^{2}x}f(x)dx}{x^2-\frac{{\pi }^2}{16}}$ is equal to :

   1. A.$f(2)$
   2. B.$2f(2)$
   3. C.$2f\left(\sqrt{2}\right)$
   4. D.$4f(2)$
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   Answer: B

2. Q222021 · 1 Sept 2021 · Shift 2· Numerical answer

   Let $f(x)=x^6+2x^4+x^3+2x+3$, $x\in \mathbb{R}$. Then the natural number n for which ${\lim }_{x\to 1}\frac{x^{n}f(1)-f(x)}{x-1}=44$ is ___________.

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   Answer: 7

3. Q282021 · 1 Sept 2021 · Shift 2· Numerical answer

   Let $[t]$ denote the greatest integer $\le t$. The number of points where the function $f(x)=[x]|x^2-1|+\sin \left(\frac{\pi }{[x]+3}\right)-[x+1]$, $x\in (-2,2)$ is not continuous is ___________.

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   Answer: 2

4. Q162021 · 26 Aug 2021 · Shift 2

   ${\lim }_{x\to 2}\left({\sum }_{n=1}^9\frac{x}{n(n+1)x^2+2(2n+1)x+4}\right)$ is equal to :

   1. A.$\frac{9}{44}$
   2. B.$\frac{5}{24}$
   3. C.$\frac{1}{5}$
   4. D.$\frac{7}{36}$
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   Answer: A

5. Q92021 · 27 Jul 2021 · Shift 2

   The value of $\underset{x\to 0}{\lim }\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$ is equal to :

   1. A.0
   2. B.4
   3. C.$-4$
   4. D.$-1$
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   Answer: C

6. Q42021 · 25 Jul 2021 · Shift 1

   Let f : R $\to$ R be defined as $$f(x)=\{\begin{array}{ll}\frac{\lambda \left|x^2-5x+6\right|}{\mu (5x-x^2-6)},& x<2\\ e^{\frac{\tan (x-2)}{x-[x]}},& x>2\\ \mu ,& x=2\end{array}$$ where [x] is the greatest integer is than or equal to x. If f is continuous at x = 2, then $\lambda +\mu$ is equal to :

   1. A.$e(-e+1)$
   2. B.$e(e-2)$
   3. C.1
   4. D.$2e-1$
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   Answer: A

7. Q212021 · 25 Jul 2021 · Shift 2· Numerical answer

   Consider the function $$f(x)=\{\begin{array}{ll}\frac{P(x)}{\sin (x-2)},& x\ne 2\\ 7,& x=2\end{array}$$ where P(x) is a polynomial such that P''(x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equal to ____.

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   Answer: 39

8. Q132021 · 22 Jul 2021 · Shift 2

   Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $$f(x)=\{\begin{array}{ll}\frac{x^3}{(1-\cos 2x{)}^2}{\log }_e\left(\frac{1+2x{e}^{-2x}}{(1-x{e}^{-x}{)}^2}\right),& x\ne 0\\ \alpha ,& x=0\end{array}$$ If f is continuous at $x=0$, then $\alpha$ is equal to :

   1. A.$1$
   2. B.$3$
   3. C.$0$
   4. D.$2$
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   Answer: A

9. Q272021 · 22 Jul 2021 · Shift 2· Numerical answer

   Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined as $$f(x)=\{\begin{array}{ll}3\left(1-\frac{|x|}{2}\right)& \text{if }|x|\le 2\\ 0& \text{if }|x|>2\end{array}$$ Let $g:\mathbb{R}\to \mathbb{R}$ be given by $g(x)=f(x+2)-f(x-2)$. If n and m denote the number of points in $\mathbb{R}$ where g is not continuous and not differentiable, respectively, then $n+m$ is equal to ________.

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   Answer: 4

10. Q132021 · 20 Jul 2021 · Shift 1

    Let a function $f:\mathrm{R}\to \mathrm{R}$ be defined as $f(x)=\{\begin{array}{llc}\sin x-e^x& if& x\le 0\\ a+[-x]& if& 0<x<1\\ 2x-b& if& x\ge 1\end{array}$ where [ x ] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:

    1. A.4
    2. B.3
    3. C.2
    4. D.5
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    Answer: B

11. Q282021 · 20 Jul 2021 · Shift 1· Numerical answer

    If the value of ${\lim }_{x\to 0}(2-\cos x\sqrt{\cos 2x}{)}^{\left(\frac{x+2}{x^2}\right)}$ is equal to $e^a$, then a is equal to ________.

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    Answer: 3

12. Q62021 · 18 Mar 2021 · Shift 1

    If ${\lim }_{x\to 0}\frac{{\sin }^{-1}x-{\tan }^{-1}x}{3x^3}$ is equal to $L$, then the value of $(6L+1)$ is

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

13. Q122021 · 18 Mar 2021 · Shift 1

    If $f(x)=\{\begin{array}{ll}\frac{1}{|x|}& ;|x|\ge 1\\ a{x}^2+b& ;|x|<1\end{array}$ is differentiable at every point of the domain, then the values of a and b are respectively :

    1. A.$\frac{1}{2},\frac{1}{2}$
    2. B.$\frac{1}{2},-\frac{3}{2}$
    3. C.$\frac{5}{2},-\frac{3}{2}$
    4. D.$-\frac{1}{2},\frac{3}{2}$
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    Answer: D

14. Q72021 · 18 Mar 2021 · Shift 2

    Let $f:\mathrm{R}\to \mathrm{R}$ be a function defined as $$f(x)=\{\begin{array}{ll}\frac{\sin (a+1)x+\sin 2x}{2x},& \text{if }x<0\\ b,& \text{if }x=0\\ \frac{\sqrt{x+b{x}^3}-\sqrt{x}}{b{x}^{5/2}},& \text{if }x>0\end{array}$$ If f is continuous at x = 0, then the value of a + b is equal to :

    1. A.$-3$
    2. B.$-2$
    3. C.$-\frac{5}{2}$
    4. D.$-\frac{3}{2}$
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    Answer: D

15. Q82021 · 18 Mar 2021 · Shift 2

    Let $g(x)={\int }_0^{x}f(t)dt$, where f is continuous function in [ 0, 3 ] such that $\frac{1}{3}\le f(t)\le 1$ for all $t\in [0,1]$ and $0\le f(t)\le \frac{1}{2}$ for all $t\in (1,3]$. The largest possible interval in which g(3) lies is :

    1. A.$[-1,-\frac{1}{2}]$
    2. B.$[-\frac{3}{2},-1]$
    3. C.$[1,3]$
    4. D.$[\frac{1}{3},2]$
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    Answer: D

16. Q262021 · 18 Mar 2021 · Shift 2· Numerical answer

    Let $f:\mathrm{R}\to \mathrm{R}$ satisfy the equation $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in \mathrm{R}$ and $f(x)\ne 0$ for any $x\in \mathrm{R}$. If the function f is differentiable at $x=0$ and $f^{\prime }(0)=3$, then ${\lim }_{h\to 0}\frac{1}{h}(f(h)-1)$ is equal to ____.

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    Answer: 3

17. Q22021 · 16 Mar 2021 · Shift 1

    Let $S_k={\sum }_{r=1}^k{\tan }^{-1}\left(\frac{6^r}{2^{2r+1}+3^{2r+1}}\right)$. Then ${\lim }_{k\to \infty }{S}_k$ is equal to :

    1. A.${\cot }^{-1}\left(\frac{3}{2}\right)$
    2. B.$\frac{\pi }{2}$
    3. C.${\tan }^{-1}(3)$
    4. D.${\tan }^{-1}\left(\frac{3}{2}\right)$
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    Answer: A

18. Q222021 · 16 Mar 2021 · Shift 1· Numerical answer

    Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f(x)+f(x+1)=2$, for all $x\in \mathbb{R}$. If $I_1={\int }_0^{8}f(x)dx$ and $I_2={\int }_{-1}^{3}f(x)dx$, then the value of $I_1+2I_2$ is equal to _________.

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    Answer: 16

19. Q252021 · 16 Mar 2021 · Shift 1· Numerical answer

    If ${\lim }_{x\to 0}\frac{a{e}^x-b\cos x+c{e}^{-x}}{x\sin x}=2$, then $a+b+c$ is equal to _________.

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    Answer: 4

20. Q292021 · 16 Mar 2021 · Shift 1· Numerical answer

    Let $f:(0,2)\to \mathbb{R}$ be defined as $f(x)={\log }_2\left(1+\tan \left(\frac{\pi x}{4}\right)\right)$. Then, ${\lim }_{n\to \infty }\frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots +f(1)\right)$ is equal to _________.

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    Answer: 1

21. Q22021 · 26 Feb 2021 · Shift 1

    The value of ${\lim }_{h\to 0}2\left\{\frac{\sqrt{3}\sin \left(\frac{\pi }{6}+h\right)-\cos \left(\frac{\pi }{6}+h\right)}{\sqrt{3}h\left(\sqrt{3}\cos h-\sin h\right)}\right\}$ is :

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

22. Q12021 · 26 Feb 2021 · Shift 2

    Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then ${\lim }_{x\to a}\frac{xf(a)-af(x)}{x-a}$ equals :

    1. A.$4-2a$
    2. B.$2a+4$
    3. C.$a+4$
    4. D.$2a-4$
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    Answer: A

23. Q112021 · 26 Feb 2021 · Shift 2

    Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $f(x)=\{\begin{array}{ll}2\sin \left(-\frac{\pi x}{2}\right),& \text{if }x<-1\\ |a{x}^2+x+b|,& \text{if }-1\le x\le 1\\ \sin (\pi x),& \text{if }x>1\end{array}$ If f(x) is continuous on $\mathbb{R}$, then $a+b$ equals :

    1. A.$-3$
    2. B.3
    3. C.$-1$
    4. D.1
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    Answer: C

24. Q202021 · 25 Feb 2021 · Shift 2

    $$\underset{n\to \infty }{\lim }\left[\frac{1}{n}+\frac{n}{(n+1{)}^2}+\frac{n}{(n+2{)}^2}+\cdots +\frac{n}{(2n-1{)}^2}\right]\text{ is equal to :}$$

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

25. Q272021 · 25 Feb 2021 · Shift 2· Numerical answer

    If ${\lim }_{x\to 0}\frac{ax-(e^{4x}-1)}{ax(e^{4x}-1)}$ exists and is equal to b, then the value of $a-2b$ is ______ .

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    Answer: 5