Limits, Continuity & Differentiability · JEE Main previous year questions

2023 20 questions

1. Q182023 · 13 Apr 2023 · Shift 1

   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
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   Answer: B

2. Q52023 · 13 Apr 2023 · Shift 2

   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.$76$
   Show answer

   Answer: B

3. Q272023 · 12 Apr 2023 · Shift 1· Numerical answer

   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 _________.

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   Answer: 2

4. Q102023 · 11 Apr 2023 · Shift 2

   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.$\sqrt{3}$
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   Answer: C

5. Q212023 · 10 Apr 2023 · Shift 1· Numerical answer

   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 _________.

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   Answer: 4

6. Q62023 · 10 Apr 2023 · Shift 2

   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.${\pi }^2\left(1-\frac{{\pi }^2}{16}\right)$
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   Answer: B

7. Q42023 · 8 Apr 2023 · Shift 1

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

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

8. Q112023 · 8 Apr 2023 · Shift 1

   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.$\frac{e+2}{e+1}+{\log }_e(e+1)$
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   Answer: C

9. Q112023 · 8 Apr 2023 · Shift 2

   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.$2\alpha$
   Show answer

   Answer: D

10. Q92023 · 6 Apr 2023 · Shift 2

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

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

    Answer: D

11. Q302023 · 6 Apr 2023 · Shift 2· Numerical answer

    Let $f(x)=\frac{x}{(1+x^n{)}^{\frac{1}{n}}},x\in \mathbb{R}-\{-1\},n\in \mathbb{N},n>2$. If $f^n(x)=(f\circ f\circ f\dots \text{ upto }n\text{ times})(x)$, then ${\lim }_{n\to \infty }{\int }_0^1{x}^{n-2}\left(f^n(x)\right)\mathrm{d}x$ is equal to ____.

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    Answer: 0

12. Q132023 · 31 Jan 2023 · Shift 2

    ${\lim }_{x\to \infty }\frac{(\sqrt{3x+1}+\sqrt{3x-1}{)}^6+(\sqrt{3x+1}-\sqrt{3x-1}{)}^6}{{\left(x+\sqrt{x^2-1}\right)}^6+{\left(x-\sqrt{x^2-1}\right)}^6}{x}^3$

    1. A.is equal to 9
    2. B.is equal to $\frac{27}{2}$
    3. C.does not exist
    4. D.is equal to 27
    Show answer

    Answer: D

13. Q212023 · 30 Jan 2023 · Shift 1· Numerical answer

    ${\lim }_{x\to 0}\frac{48}{x^4}{\int }_0^x\frac{t^3}{t^6+1}\mathrm{d}t$ is equal to ____.

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    Answer: 12

14. Q112023 · 29 Jan 2023 · Shift 1

    Let $y=f(x)$ be the solution of the differential equation $y(x+1)dx-x^{2}dy=0,y(1)=e$. Then $\underset{x\to 0^{+}}{\lim }f(x)$ is equal to

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

    Answer: B

15. Q192023 · 29 Jan 2023 · Shift 1

    Let $x=2$ be a root of the equation $x^2+px+q=0$ and $f(x)=\{\begin{array}{ll}\frac{1-\cos (x^2-4px+q^2+8q+16)}{(x-2p{)}^4},& x\ne 2p\\ 0,& x=2p\end{array}$ Then $\underset{x\to 2p^{+}}{\lim }[f(x)]$, where $[.]$ denotes greatest integer function, is

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

    Answer: C

16. Q52023 · 25 Jan 2023 · Shift 1

    The value of $\underset{n\to \infty }{\lim }\frac{1+2-3+4+5-6+.....+(3n-2)+(3n-1)-3n}{\sqrt{2n^4+4n+3}-\sqrt{n^4+5n+4}}$ is :

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

17. Q52023 · 25 Jan 2023 · Shift 2

    If the function $f(x)=\{\begin{array}{ll}(1+|\cos x|{)}^{\frac{\lambda }{|\cos x|}},& 0<x<\frac{\pi }{2}\\ \mu ,& x=\frac{\pi }{2}\\ e^{\frac{\cot 6x}{\cot 4x}},& \frac{\pi }{2}<x<\pi \end{array}$ is continuous at $x=\frac{\pi }{2}$, then $9\lambda +6{\log }_e\mu +{\mu }^6-e^{6\lambda }$ is equal to

    1. A.11
    2. B.10
    3. C.8
    4. D.$2e^4+8$
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    Answer: B

18. Q142023 · 25 Jan 2023 · Shift 2

    Let $y=y(t)$ be a solution of the differential equation $\frac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$ where, $\alpha >0,\beta >0$ and $\gamma >0$. Then $\underset{t\to \infty }{\lim }y(t)$

    1. A.is 0
    2. B.is 1
    3. C.is $-1$
    4. D.does not exist
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    Answer: A

19. Q42023 · 24 Jan 2023 · Shift 1

    ${\lim }_{t\to 0}{\left(1^{\frac{1}{{\sin }^{2}t}}+2^{\frac{1}{{\sin }^{2}t}}+\dots +n^{\frac{1}{{\sin }^{2}t}}\right)}^{{\sin }^{2}t}$ is equal to

    1. A.$\frac{n(n+1)}{2}$
    2. B.$n$
    3. C.$n^2+n$
    4. D.$n^2$
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    Answer: B

20. Q62023 · 24 Jan 2023 · Shift 2

    The set of all values of $a$ for which $\underset{x\to a}{\lim }([x-5]-[2x+2])=0$, where $[\alpha ]$ denotes the greatest integer less than or equal to $\alpha$ is equal to

    1. A.$[-7.5,-6.5]$
    2. B.$(-7.5,-6.5]$
    3. C.$[-7.5,-6.5)$
    4. D.$(-7.5,-6.5)$
    Show answer

    Answer: D