Limits, Continuity & Differentiability · JEE Main previous year questions

2025 17 questions

1. Q112025 · 8 Apr 2025 · Shift 2

   Given below are two statements: Statement I: ${\lim }_{x\to 0}\left(\frac{{\tan }^{-1}x+{\log }_e\sqrt{\frac{1+x}{1-x}}-2x}{x^5}\right)=\frac{2}{5}$ Statement II: ${\lim }_{x\to 1}\left(x^{\frac{2}{1-x}}\right)=\frac{1}{e^2}$ In the light of the above statements, choose the correct answer from the options given below:

   1. A.Statement I is false but Statement II is true
   2. B.Both Statement I and Statement II are false
   3. C.Both Statement I and Statement II are true
   4. D.Statement I is true but Statement II is false
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   Answer: C

2. Q12025 · 4 Apr 2025 · Shift 1

   Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2x)-f(x)=x$ for all $x\in \mathbb{R}$. If ${\lim }_{n\to \infty }\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)$, then ${\sum }_{r=1}^{10}G\left(r^2\right)$ is equal to

   1. A.$215$
   2. B.$420$
   3. C.$385$
   4. D.$540$
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   Answer: C

3. Q172025 · 4 Apr 2025 · Shift 1

   If ${\lim }_{x\to 1^{+}}\frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin (1-x)}{(x-1{)}^3}=-1$, where $\lambda ,\mu \in \mathbb{R}$, then $\lambda +\mu$ is equal to

   1. A.$20$
   2. B.$19$
   3. C.$18$
   4. D.$17$
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   Answer: C

4. Q222025 · 4 Apr 2025 · Shift 1· Numerical answer

   Let $m$ and $n$ be the number of points at which the function $f(x)=\max \left\{x,x^3,x^5,\dots x^{21}\right\},x\in \mathbb{R}$, is not differentiable and not continuous, respectively. Then $m+n$ is equal to _________.

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   Answer: 3

5. Q202025 · 4 Apr 2025 · Shift 2

   Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=1,f^{\prime }(2)=4$. Let ${\lim }_{x\to 0}(f(2+x){)}^{3/x}={\mathrm{e}}^{\alpha }$. Then the number of times the curve $y=4x^3-4x^2-4(\alpha -7)x-\alpha$ meets $x$-axis is :

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

6. Q112025 · 3 Apr 2025 · Shift 1

   Let $$f(x)=\{\begin{array}{ll}(1+ax{)}^{1/x}& ,x<0\\ 1+b,& x=0\\ \frac{(x+4{)}^{1/2}-2}{(x+c{)}^{1/3}-2},& x>0\end{array}$$ be continuous at $x=0$. Then $e^{a}bc$ is equal to:

   1. A.64
   2. B.48
   3. C.36
   4. D.72
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   Answer: B

7. Q212025 · 3 Apr 2025 · Shift 2· Numerical answer

   If ${\lim }_{x\to 0}{\left(\frac{\tan x}{x}\right)}^{\frac{1}{x^2}}=p$, then $96{\log }_{\mathrm{e}}p$ is equal to _________

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   Answer: 32

8. Q182025 · 2 Apr 2025 · Shift 1

   For $\alpha ,\beta ,\gamma \in \mathbf{R}$, if ${\lim }_{x\to 0}\frac{x^2\sin \alpha x+(\gamma -1)e^{x^2}}{\sin 2x-\beta x}=3$, then $\beta +\gamma -\alpha$ is equal to :

   1. A.-1
   2. B.4
   3. C.6
   4. D.7
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   Answer: D

9. Q72025 · 2 Apr 2025 · Shift 2

   If ${\lim }_{x\to 0}\frac{\cos (2x)+a\cos (4x)-b}{x^4}$ is finite, then $(a+b)$ is equal to:

   1. A.0
   2. B.$\frac{3}{4}$
   3. C.-1
   4. D.$\frac{1}{2}$
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   Answer: D

10. Q202025 · 29 Jan 2025 · Shift 2

    Let the function $f(x)=(x^2-1)|x^2-ax+2|+\cos |x|$ be not differentiable at the two points $x=\alpha =2$ and $x=\beta$. Then the distance of the point $(\alpha ,\beta )$ from the line $12x+5y+10=0$ is equal to :

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

11. Q242025 · 29 Jan 2025 · Shift 2· Numerical answer

    If $\underset{t\to 0}{\lim }{\left(\underset{0}{\overset{1}{\int }}(3x+5{)}^t\mathrm{dx}\right)}^{\frac{1}{t}}=\frac{\alpha }{5\mathrm{e}}{\left(\frac{8}{5}\right)}^{\frac{2}{3}}$, then $\alpha$ is equal to ____.

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    Answer: 64

12. Q222025 · 28 Jan 2025 · Shift 1· Numerical answer

    Let $f(x)=\{\begin{array}{ll}3x,& x<0\\ \min \{1+x+[x],x+2[x]\},& 0\le x\le 2\\ 5,& x>2\end{array}$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where $f$ is not continuous and is not differentiable, respectively, then $\alpha +\beta$ equals ____ .

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    Answer: 5

13. Q12025 · 24 Jan 2025 · Shift 1

    ${\lim }_{x\to 0}\mathrm{cosec}x\left(\sqrt{2{\cos }^{2}x+3\cos x}-\sqrt{{\cos }^{2}x+\sin x+4}\right)$ is:

    1. A.$\frac{1}{\sqrt{15}}$
    2. B.$\frac{1}{2\sqrt{5}}$
    3. C.$0$
    4. D.$-\frac{1}{2\sqrt{5}}$
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    Answer: D

14. Q22025 · 24 Jan 2025 · Shift 1

    Let $f:\mathbb{R}-\{0\}\to \mathbb{R}$ be a function such that $f(x)-6f\left(\frac{1}{x}\right)=\frac{35}{3x}-\frac{5}{2}$. If the ${\lim }_{x\to 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ;\alpha ,\beta \in \mathbb{R}$, then $\alpha +2\beta$ is equal to

    1. A.$6$
    2. B.$5$
    3. C.$3$
    4. D.$4$
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    Answer: D

15. Q62025 · 24 Jan 2025 · Shift 2

    For some $a,b$, let $f(x)=\left|\begin{array}{ccc}a+\frac{\sin x}{x}& 1& b\\ a& 1+\frac{\sin x}{x}& b\\ a& 1& b+\frac{\sin x}{x}\end{array}\right|$, $x\ne 0$, ${\lim }_{x\to 0}f(x)=\lambda +\mu a+\nu b$. Then $(\lambda +\mu +\nu {)}^2$ is equal to :

    1. A.25
    2. B.16
    3. C.9
    4. D.36
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    Answer: B

16. Q172025 · 24 Jan 2025 · Shift 2

    Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x)=[x]+|x-2|,-2<x<3$, is not continuous and not differentiable. Then $\mathrm{m}+\mathrm{n}$ is equal to :

    1. A.6
    2. B.9
    3. C.8
    4. D.7
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    Answer: C

17. Q122025 · 23 Jan 2025 · Shift 2

    ${\lim }_{x\to \infty }\frac{(2x^2-3x+5)(3x-1{)}^{\frac{x}{2}}}{(3x^2+5x+4)\sqrt{(3x+2{)}^x}}$ is equal to :

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