Limits, Continuity & Differentiability · JEE Main previous year questions

2024 33 questions

1. Q262024 · 31 Jan 2024 · Shift 2· Numerical answer

   If ${\lim }_{x\to 0}\frac{a{x}^2{e}^x-b{\log }_e(1+x)+cx{e}^{-x}}{x^2\sin x}=1$, then $16\left(a^2+b^2+c^2\right)$ is equal to ____.

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   Answer: 81

2. Q22024 · 30 Jan 2024 · Shift 1

   The value of ${\lim }_{n\to \infty }{\sum }_{k=1}^n\frac{n^3}{(n^2+k^2)(n^2+3k^2)}$ is :

   1. A.$\frac{\pi }{8(2\sqrt{3}+3)}$
   2. B.$\frac{(2\sqrt{3}+3)\pi }{24}$
   3. C.$\frac{13\pi }{8(4\sqrt{3}+3)}$
   4. D.$\frac{13(2\sqrt{3}-3)\pi }{8}$
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   Answer: C

3. Q102024 · 30 Jan 2024 · Shift 1

   Let $f:\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\to \mathbf{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If the ${\lim }_{x\to 0}\frac{x{\int }_0^{x}f(t)dt}{e^{x^2}-1}=\alpha$, then $8{\alpha }^2$ is equal to :

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

4. Q182024 · 29 Jan 2024 · Shift 1

   ${\lim }_{x\to \frac{\pi }{2}}\left(\frac{1}{{\left(x-\frac{\pi }{2}\right)}^2}{\int }_{x^3}^{{\left(\frac{\pi }{2}\right)}^3}\cos \left(t^{\frac{1}{3}}\right)\mathrm{d}t\right)$ is equal to

   1. A.$\frac{3{\pi }^2}{4}$
   2. B.$\frac{3{\pi }^2}{8}$
   3. C.$\frac{3\pi }{4}$
   4. D.$\frac{3\pi }{8}$
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   Answer: B

5. Q242024 · 29 Jan 2024 · Shift 2· Numerical answer

   Let the slope of the line $45x+5y+3=0$ be $27r_1+\frac{9r_2}{2}$ for some $r_1,r_2\in \mathbb{R}$. Then ${\lim }_{x\to 3}\left({\int }_3^x\frac{8t^2}{\frac{3r_{2}x}{2}-r_2{x}^2-r_1{x}^3-3x}dt\right)$ is equal to _________.

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   Answer: 12

6. Q292024 · 29 Jan 2024 · Shift 2· Numerical answer

   Let $f(x)=\sqrt{{\lim }_{r\to x}\left\{\frac{2r^2[(f(r){)}^2-f(x)f(r)]}{r^2-x^2}-r^3{e}^{\frac{f(r)}{r}}\right\}}$ be differentiable in $(-\infty ,0)\cup (0,\infty )$ and $f(1)=1$. Then the value of $ea$, such that $f(a)=0$, is equal to _________.

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   Answer: 2

7. Q102024 · 27 Jan 2024 · Shift 1

   If $\mathrm{a}=\underset{x\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $\mathrm{b}=\underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $a{b}^3$ is :

   1. A.36
   2. B.25
   3. C.32
   4. D.30
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   Answer: C

8. Q112024 · 27 Jan 2024 · Shift 2

   If ${\lim }_{x\to 0}\frac{3+\alpha \sin x+\beta \cos x+{\log }_e(1-x)}{3{\tan }^{2}x}=\frac{1}{3}$, then $2\alpha -\beta$ is equal to :

   1. A.$2$
   2. B.$1$
   3. C.$5$
   4. D.$7$
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   Answer: C