Limits, Continuity & Differentiability · JEE Main previous year questions

2020 16 questions

1. Q122020 · 6 Sept 2020 · Shift 1

   $$\underset{x\to 1}{\lim }\left(\frac{{\int }_0^{(x-1{)}^2}t\cos (t^2)dt}{(x-1)\sin (x-1)}\right)$$

   1. A.is equal to 0
   2. B.is equal to $\frac{1}{2}$
   3. C.does not exist
   4. D.is equal to $-\frac{1}{2}$
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   Answer: A

2. Q172020 · 3 Sept 2020 · Shift 1

   Let $[t]$ denote the greatest integer $\le t$. If for some $\lambda \in \mathbb{R}-\{1,0\}$, ${\lim }_{x\to 0}\left|\frac{1-x+|x|}{\lambda -x+[x]}\right|=\mathrm{L}$, then L is equal to :

   1. A.1
   2. B.2
   3. C.0
   4. D.$\frac{1}{2}$
   Show answer

   Answer: B

3. Q222020 · 3 Sept 2020 · Shift 1· Numerical answer

   If ${\lim }_{x\to 0}\left\{\frac{1}{x^8}\left(1-\cos \frac{x^2}{2}-\cos \frac{x^2}{4}+\cos \frac{x^2}{2}\cos \frac{x^2}{4}\right)\right\}=2^{-k}$ then the value of k is _______.

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   Answer: 8

4. Q142020 · 3 Sept 2020 · Shift 2

   ${\lim }_{x\to a}\frac{(a+2x{)}^{\frac{1}{3}}-(3x{)}^{\frac{1}{3}}}{(3a+x{)}^{\frac{1}{3}}-(4x{)}^{\frac{1}{3}}}(a\ne 0)$ is equal to :

   1. A.$\left(\frac{2}{9}\right){\left(\frac{2}{3}\right)}^{\frac{1}{3}}$
   2. B.$\left(\frac{2}{3}\right){\left(\frac{2}{9}\right)}^{\frac{1}{3}}$
   3. C.${\left(\frac{2}{3}\right)}^{\frac{4}{3}}$
   4. D.${\left(\frac{2}{9}\right)}^{\frac{4}{3}}$
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   Answer: B

5. Q22020 · 2 Sept 2020 · Shift 1

   If a function f(x) defined by $$f(x)=\{\begin{array}{ll}a{e}^x+b{e}^{-x},& -1\le x<1\\ c{x}^2,& 1\le x\le 3\\ a{x}^2+2cx,& 3<x\le 4\end{array}$$ be continuous for some $a,b,c\in \mathbb{R}$ and $f^{\prime }(0)+f^{\prime }(2)=e$, then the value of $a$ is :

   1. A.$\frac{e}{e^2-3e-13}$
   2. B.$\frac{1}{e^2-3e+13}$
   3. C.$\frac{e}{e^2-3e+13}$
   4. D.$\frac{e}{e^2+3e+13}$
   Show answer

   Answer: C

6. Q92020 · 2 Sept 2020 · Shift 2

   ${\lim }_{x\to 0}{\left(\tan \left(\frac{\pi }{4}+x\right)\right)}^{\frac{1}{x}}$ is equal to :

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

7. Q102020 · 9 Jan 2020 · Shift 1

   Let $f$ be any function continuous on $[a,b]$ and twice differentiable on $(a,b)$. If for all $x\in (a,b)$, $f^{\prime }(x)>0$ and $f^{\prime \prime }(x)<0$, then for any $c\in (a,b)$, $\frac{f(c)-f(a)}{f(b)-f(c)}$ is greater than :

   1. A.1
   2. B.$\frac{b-c}{c-a}$
   3. C.$\frac{b+a}{b-a}$
   4. D.$\frac{c-a}{b-c}$
   Show answer

   Answer: D

8. Q142020 · 9 Jan 2020 · Shift 1

   If $$f(x)=\{\begin{array}{llc}\frac{\sin (a+2)x+\sin x}{x}& ;& x<0\\ b& ;& x=0\\ \frac{(x+3x^2{)}^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}}& ;& x>0\end{array}$$ is continuous at $x=0$, then $a+2b$ is equal to :

   1. A.0
   2. B.-1
   3. C.-2
   4. D.1
   Show answer

   Answer: A

9. Q12020 · 9 Jan 2020 · Shift 2

   Let $[t]$ denote the greatest integer $\le t$ and ${\lim }_{x\to 0}x\left[\frac{4}{x}\right]=A$. Then the function, $f(x)=[x^2]\sin (\pi x)$ is discontinuous, when x is equal to :

   1. A.$\sqrt{A+1}$
   2. B.$\sqrt{A+5}$
   3. C.$\sqrt{A+21}$
   4. D.$\sqrt{A}$
   Show answer

   Answer: A

10. Q82020 · 9 Jan 2020 · Shift 2

    Let a function $f:[0,5]\to \mathbb{R}$ be continuous, $f(1)=3$ and F be defined as : $$F(x)={\int }_1^x{t}^{2}g(t)dt\text{ , where }g(t)={\int }_1^{t}f(u)du$$ Then for the function F, the point $x=1$ is :

    1. A.a point of inflection.
    2. B.a point of local maxima.
    3. C.a point of local minima.
    4. D.not a critical point.
    Show answer

    Answer: C

11. Q42020 · 8 Jan 2020 · Shift 1

    ${\lim }_{x\to 0}{\left(\frac{3x^2+2}{7x^2+2}\right)}^{\frac{1}{x^2}}$ is equal to

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

    Answer: C

12. Q42020 · 8 Jan 2020 · Shift 2

    Let S be the set of all functions $f:[0,1]\to \mathrm{R}$, which are continuous on [0, 1] and differentiable on (0, 1). Then for every f in S, there exists a $c\in (0,1)$, depending on f, such that

    1. A.$|f(c)-f(1)|<|f^{\prime }(c)|$
    2. B.$|f(c)+f(1)|<(1+c)|f^{\prime }(c)|$
    3. C.$|f(c)-f(1)|<(1-c)|f^{\prime }(c)|$
    4. D.None
    Show answer

    Answer: D

13. Q92020 · 8 Jan 2020 · Shift 2

    ${\lim }_{x\to 0}\frac{{\int }_0^{x}t\sin (10t)dt}{x}$ is equal to

    1. A.$-\frac{1}{5}$
    2. B.$-\frac{1}{10}$
    3. C.0
    4. D.$\frac{1}{10}$
    Show answer

    Answer: C

14. Q182020 · 7 Jan 2020 · Shift 1

    Let the function, $f:[-7,0]\to \mathbb{R}$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }(x)\le 2$, for all $x\in (-7,0)$, then for all such functions f, $f(-1)+f(0)$ lies in the interval:

    1. A.$[-6,20]$
    2. B.$(-\infty ,20]$
    3. C.$[-3,11]$
    4. D.$(-\infty ,11]$
    Show answer

    Answer: B

15. Q242020 · 7 Jan 2020 · Shift 1· Numerical answer

    ${\lim }_{x\to 2}\frac{3^x+3^{3-x}-12}{3^{-x/2}-3^{1-x}}$ is equal to_____.

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    Answer: 36

16. Q252020 · 7 Jan 2020 · Shift 2· Numerical answer

    If the function f defined on $\left(-\frac{1}{3},\frac{1}{3}\right)$ by $$f(x)=\{\begin{array}{ll}\frac{1}{x}{\log }_e\left(\frac{1+3x}{1-2x}\right),& \text{when }x\ne 0\\ k,& \text{when }x=0\end{array}$$ is continuous, then k is equal to_______.

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    Answer: 5