Limits, Continuity & Differentiability · JEE Main previous year questions

2019 19 questions

1. Q42019 · 12 Apr 2019 · Shift 1

   If $\alpha$ and $\beta$ are the roots of the equation $375x^2-25x-2=0$, then $\underset{n\to \infty }{\lim }\sum _{r=1}^n{\alpha }^r+\underset{n\to \infty }{\lim }\sum _{r=1}^n{\beta }^r$ is equal to :

   1. A.$\frac{7}{116}$
   2. B.$\frac{29}{358}$
   3. C.$\frac{1}{12}$
   4. D.$\frac{21}{346}$
   Show answer

   Answer: C

2. Q202019 · 12 Apr 2019 · Shift 1

   Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable function such that $f(2)=6$ and $f^{\prime }(2)=\frac{1}{48}$. If ${\int }_6^{f(x)}4t^{3}dt=(x-2)g(x)$, then ${\lim }_{x\to 2}g(x)$ is equal to :

   1. A.18
   2. B.36
   3. C.12
   4. D.24
   Show answer

   Answer: A

3. Q122019 · 12 Apr 2019 · Shift 2

   ${\lim }_{x\to 0}\frac{x+2\sin x}{\sqrt{x^2+2\sin x+1}-\sqrt{{\sin }^{2}x-x+1}}$ is :

   1. A.6
   2. B.1
   3. C.3
   4. D.2
   Show answer

   Answer: D

4. Q152019 · 12 Apr 2019 · Shift 2

   Let $f(x)=5-|x-2|$ and $g(x)=|x+1|$, $x\in \mathbb{R}$. If $f(x)$ attains maximum value at $\alpha$ and $g(x)$ attains minimum value at $\beta$, then ${\lim }_{x\to -\alpha \beta }\frac{(x-1)(x^2-5x+6)}{x^2-6x+8}$ is equal to :

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

   Answer: A

5. Q122019 · 10 Apr 2019 · Shift 1

   If $f(x)=\{\begin{array}{ll}\frac{\sin (p+1)x+\sin x}{x},& x<0\\ q,& x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}},& x>0\end{array}$ is continuous at x = 0, then the ordered pair (p, q) is equal to

   1. A.$\left(-\frac{3}{2},-\frac{1}{2}\right)$
   2. B.$\left(-\frac{1}{2},\frac{3}{2}\right)$
   3. C.$\left(-\frac{3}{2},\frac{1}{2}\right)$
   4. D.$\left(\frac{5}{2},\frac{1}{2}\right)$
   Show answer

   Answer: C

6. Q132019 · 10 Apr 2019 · Shift 1

   Let $f:\mathrm{R}\to \mathrm{R}$ be differentiable at $c\in \mathrm{R}$ and f(c) = 0. If g(x) = |f(x)|, then at x = c, g is :

   1. A.differentiable if $f^{\prime }(c)=0$
   2. B.differentiable if $f^{\prime }(c)\ne 0$
   3. C.not differentiable
   4. D.not differentiable if $f^{\prime }(c)=0$
   Show answer

   Answer: A

7. Q292019 · 10 Apr 2019 · Shift 1

   ${\lim }_{n\to \infty }\left(\frac{(n+1{)}^{1/3}}{n^{4/3}}+\frac{(n+2{)}^{1/3}}{n^{4/3}}+\cdots +\frac{(2n{)}^{1/3}}{n^{4/3}}\right)$ is equal to :

   1. A.$\frac{4}{3}(2{)}^{3/4}$
   2. B.$\frac{3}{4}(2{)}^{4/3}-\frac{3}{4}$
   3. C.$\frac{4}{3}(2{)}^{4/3}$
   4. D.$\frac{3}{4}(2{)}^{4/3}-\frac{4}{3}$
   Show answer

   Answer: B

8. Q12019 · 10 Apr 2019 · Shift 2

   If ${\lim }_{x\to 1}\frac{x^2-ax+b}{x-1}=5$, then a + b is equal to :

   1. A.1
   2. B.- 4
   3. C.- 7
   4. D.5
   Show answer

   Answer: C

9. Q192019 · 9 Apr 2019 · Shift 1

   If the function f defined on, $\left(\frac{\pi }{6},\frac{\pi }{3}\right)$ by $$f(x)=\{\begin{array}{ll}\frac{\sqrt{2}\cos x-1}{\cot x-1},& x\ne \frac{\pi }{4}\\ k,& x=\frac{\pi }{4}\end{array}$$ is continuous, then k is equal to

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

   Answer: C

10. Q12019 · 9 Apr 2019 · Shift 2

    If $f:R\to R$ is a differentiable function and f(2) = 6, then $\underset{x\to 2}{\lim }\frac{{\int }_6^{f(x)}2tdt}{(x-2)}$ is :-

    1. A.2f'(2)
    2. B.24f'(2)
    3. C.0
    4. D.12f'(2)
    Show answer

    Answer: D

11. Q122019 · 9 Apr 2019 · Shift 2

    If the function $f(x)=\{\begin{array}{l}a|\pi -x|+1,x\le 5\\ b|x-\pi |+3,x>5\end{array}$ is continuous at $x=5$, then the value of $a-b$ is :-

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

    Answer: B

12. Q82019 · 8 Apr 2019 · Shift 2

    Let $f:R\to R$ be a differentiable function satisfying $f^{\prime }(3)+f^{\prime }(2)=0$. Then $\underset{x\to 0}{\lim }{\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)}^{\frac{1}{x}}$ is equal to

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

    Answer: D

13. Q192019 · 8 Apr 2019 · Shift 2

    Let $f:[-1,3]\to R$ be defined as $$f(x)=\{\begin{array}{ll}|x|+[x],& -1\le x<1\\ x+|x|,& 1\le x<2\\ x+[x],& 2\le x\le 3\end{array}$$ where [t] denotes the greatest integer less than or equal to t. Then, f is discontinuous at:

    1. A.only three points
    2. B.four or more points
    3. C.only two points
    4. D.only one point
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    Answer: A

14. Q52019 · 12 Jan 2019 · Shift 1

    Let f and g be continuous functions on $[0,a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then ${\int }_0^{a}f(x)g(x)dx$ is equal to :

    1. A.$4{\int }_0^{a}f(x)dx$
    2. B.$-3{\int }_0^{a}f(x)dx$
    3. C.${\int }_0^{a}f(x)dx$
    4. D.$2{\int }_0^{a}f(x)dx$
    Show answer

    Answer: D

15. Q152019 · 12 Jan 2019 · Shift 1

    ${\lim }_{x\to \pi /4}\frac{{\cot }^{3}x-\tan x}{\cos (x+\frac{\pi }{4})}$ is :

    1. A.$8\sqrt{2}$
    2. B.4
    3. C.$4\sqrt{2}$
    4. D.8
    Show answer

    Answer: D

16. Q12019 · 12 Jan 2019 · Shift 2

    ${\lim }_{x\to 1^{-}}\frac{\sqrt{\pi }-\sqrt{2{\sin }^{-1}x}}{\sqrt{1-x}}$ is equal to :

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

    Answer: A

17. Q262019 · 12 Jan 2019 · Shift 2

    ${\lim }_{n\to \infty }\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\cdots +\frac{1}{5n}\right)$ is equal to :

    1. A.${\tan }^{-1}(2)$
    2. B.${\tan }^{-1}(3)$
    3. C.$\frac{\pi }{4}$
    4. D.$\frac{\pi }{2}$
    Show answer

    Answer: A

18. Q82019 · 11 Jan 2019 · Shift 2

    ${\lim }_{x\to 0}\frac{x\cot (4x)}{{\sin }^{2}x{\cot }^2(2x)}$ is equal to :

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

    Answer: C

19. Q92019 · 9 Jan 2019 · Shift 1

    $$\underset{y\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}$$

    1. A.exists and equals $\frac{1}{2\sqrt{2}}$
    2. B.exists and equals $\frac{1}{4\sqrt{2}}$
    3. C.exists and equals $\frac{1}{2\sqrt{2}(1+\sqrt{2})}$
    4. D.does not exist
    Show answer

    Answer: B