Integral Calculus & Area · JEE Main previous year questions

2022 34 questions

1. Q262022 · 27 Jun 2022 · Shift 2· Numerical answer

   If the area of the region $\left\{(x,y):x^{\frac{2}{3}}+y^{\frac{2}{3}}\le 1,x+y\ge 0,y\ge 0\right\}$ is A, then $\frac{256A}{\pi }$ is equal to ________.

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   Answer: 36

2. Q272022 · 26 Jun 2022 · Shift 1· Numerical answer

   The value of the integral $$\frac{48}{{\pi }^4}{\int }_0^{\pi }\left(\frac{3\pi x^2}{2}-x^3\right)\frac{\sin x}{1+{\cos }^{2}x}dx$$ is equal to ____.

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   Answer: 6

3. Q62022 · 26 Jun 2022 · Shift 2

   The area of the region bounded by $y^2=8x$ and $y^2=16(3-x)$ is equal to:

   1. A.$\frac{32}{3}$
   2. B.$\frac{40}{3}$
   3. C.16
   4. D.19
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   Answer: C

4. Q72022 · 26 Jun 2022 · Shift 2

   If $\int \frac{1}{x}\sqrt{\frac{1-x}{1+x}}dx=g(x)+c$, $g(1)=0$, then $g\left(\frac{1}{2}\right)$ is equal to :

   1. A.${\log }_e\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)+\frac{\pi }{3}$
   2. B.${\log }_e\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)+\frac{\pi }{3}$
   3. C.${\log }_e\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)-\frac{\pi }{3}$
   4. D.$\frac{1}{2}{\log }_e\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)-\frac{\pi }{6}$
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   Answer: A

5. Q272022 · 26 Jun 2022 · Shift 2· Numerical answer

   The integral $\frac{24}{\pi }{\int }_0^{\sqrt{2}}\frac{(2-x^2)dx}{(2+x^2)\sqrt{4+x^4}}$ is equal to ________.

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   Answer: 3

6. Q12022 · 25 Jun 2022 · Shift 1

   The value of $\underset{0}{\overset{\pi }{\int }}\frac{e^{\cos x}\sin x}{(1+{\cos }^{2}x)(e^{\cos x}+e^{-\cos x})}dx$ is equal to:

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

7. Q92022 · 25 Jun 2022 · Shift 1

   Let $g:(0,\infty )\to \mathbb{R}$ be a differentiable function such that $\int \left(\frac{x(\cos x-\sin x)}{e^x+1}+\frac{g(x)(e^x+1-x{e}^x)}{(e^x+1{)}^2}\right)dx=\frac{xg(x)}{e^x+1}+c$, for all $x>0$, where $c$ is an arbitrary constant. Then :

   1. A.g is decreasing in $\left(0,\frac{\pi }{4}\right)$
   2. B.g' is increasing in $\left(0,\frac{\pi }{4}\right)$
   3. C.g + g' is increasing in $\left(0,\frac{\pi }{2}\right)$
   4. D.g - g' is increasing in $\left(0,\frac{\pi }{2}\right)$
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   Answer: D

8. Q92022 · 25 Jun 2022 · Shift 2

   If $b_n={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{2}nx}{\sin x}dx$, $n\in \mathbb{N}$, then

   1. A.$b_3-b_2$, $b_4-b_3$, $b_5-b_4$ are in A.P. with common difference $-2$
   2. B.$\frac{1}{b_3-b_2}$, $\frac{1}{b_4-b_3}$, $\frac{1}{b_5-b_4}$ are in an A.P. with common difference $2$
   3. C.$b_3-b_2$, $b_4-b_3$, $b_5-b_4$ are in a G.P.
   4. D.$\frac{1}{b_3-b_2}$, $\frac{1}{b_4-b_3}$, $\frac{1}{b_5-b_4}$ are in an A.P. with common difference $-2$
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   Answer: D

9. Q62022 · 24 Jun 2022 · Shift 2

   The value of the integral $${\int }_{-\pi /2}^{\pi /2}\frac{dx}{(1+e^x)({\sin }^{6}x+{\cos }^{6}x)}$$ is equal to

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