Integral Calculus & Area · JEE Main previous year questions

2023 45 questions

1. Q272023 · 31 Jan 2023 · Shift 1· Numerical answer

   Let for $x\in \mathbb{R}$, $f(x)=\frac{x+|x|}{2}$ and $g(x)=\{\begin{array}{ll}x,& x<0\\ x^2,& x\ge 0\end{array}$ Then area bounded by the curve $y=(f\circ g)(x)$ and the lines $y=0,2y-x=15$ is equal to ___________.

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   Answer: 72

2. Q52023 · 31 Jan 2023 · Shift 2

   Let $\alpha >0$. If ${\int }_0^{\alpha }\frac{x}{\sqrt{x+\alpha }-\sqrt{x}}\mathrm{d}x=\frac{16+20\sqrt{2}}{15}$, then $\alpha$ is equal to :

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

3. Q112023 · 31 Jan 2023 · Shift 2

   If $\varphi (x)=\frac{1}{\sqrt{x}}{\int }_{\frac{\pi }{4}}^x\left(4\sqrt{2}\sin t-3{\varphi }^{\prime }(t)\right)\mathrm{d}t,x>0$, then ${\varphi }^{\prime }\left(\frac{\pi }{4}\right)$ is equal to :

   1. A.$\frac{4}{6+\sqrt{\pi }}$
   2. B.$\frac{4}{6-\sqrt{\pi }}$
   3. C.$\frac{8}{\sqrt{\pi }}$
   4. D.$\frac{8}{6+\sqrt{\pi }}$
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   Answer: D

4. Q142023 · 31 Jan 2023 · Shift 2

   Let $y=y(x)$ be the solution of the differential equation $\left(3y^2-5x^2\right)y\mathrm{d}x+2x\left(x^2-y^2\right)\mathrm{d}y=0$ such that $y(1)=1$. Then $\left|(y(2){)}^3-12y(2)\right|$ is equal to :

   1. A.64
   2. B.$16\sqrt{2}$
   3. C.32
   4. D.$32\sqrt{2}$
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   Answer: D

5. Q212023 · 31 Jan 2023 · Shift 2· Numerical answer

   Let the area of the region $\left\{(x,y):|2x-1|\le y\le \left|x^2-x\right|,0\le x\le 1\right\}$ be A. Then $(6\mathrm{A}+11{)}^2$ is equal to

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   Answer: 125

6. Q242023 · 30 Jan 2023 · Shift 1· Numerical answer

   Let $\alpha$ be the area of the larger region bounded by the curve $y^2=8x$ and the lines $y=x$ and $x=2$, which lies in the first quadrant. Then the value of $3\alpha$ is equal to ____.

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   Answer: 22

7. Q282023 · 30 Jan 2023 · Shift 1· Numerical answer

   Let ${\sum }_{n=0}^{\infty }\frac{n^3((2n)!)+(2n-1)(n!)}{(n!)((2n)!)}=a\mathrm{e}+\frac{b}{\mathrm{e}}+c$, where $\mathrm{a},\mathrm{b},\mathrm{c}\in \mathbb{Z}$ and $\mathrm{e}={\sum }_{n=0}^{\infty }\frac{1}{n!}$ Then $a^2-b+c$ is equal to ____.

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   Answer: 26

8. Q122023 · 29 Jan 2023 · Shift 1

   Let $\Delta$ be the area of the region $\left\{(x,y)\in R^2:x^2+y^2\le 21,y^2\le 4x,x\ge 1\right\}$. Then $\frac{1}{2}\left(\Delta -21{\sin }^{-1}\frac{2}{\sqrt{7}}\right)$ is equal to

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

9. Q112023 · 29 Jan 2023 · Shift 2

   The value of the integral ${\int }_1^2\left(\frac{t^4+1}{t^6+1}\right)dt$ is

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

10. Q132023 · 29 Jan 2023 · Shift 2

    The value of the integral ${\int }_{1/2}^2\frac{{\tan }^{-1}x}{x}dx$ is equal to :

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

11. Q72023 · 25 Jan 2023 · Shift 1

    Let $f(x)=\int \frac{2x}{(x^2+1)(x^2+3)}dx$. If $f(3)=\frac{1}{2}({\log }_{e}5-{\log }_{e}6)$, then $f(4)$ is equal to

    1. A.${\log }_{e}19-{\log }_{e}20$
    2. B.${\log }_{e}17-{\log }_{e}18$
    3. C.$\frac{1}{2}({\log }_{e}19-{\log }_{e}17)$
    4. D.$\frac{1}{2}({\log }_{e}17-{\log }_{e}19)$
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    Answer: D

12. Q112023 · 25 Jan 2023 · Shift 2

    The integral $16{\int }_1^2\frac{dx}{x^3(x^2+2{)}^2}$ is equal to

    1. A.$\frac{11}{12}+{\log }_{e}4$
    2. B.$\frac{11}{6}+{\log }_{e}4$
    3. C.$\frac{11}{12}-{\log }_{e}4$
    4. D.$\frac{11}{6}-{\log }_{e}4$
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    Answer: D

13. Q242023 · 25 Jan 2023 · Shift 2· Numerical answer

    If ${\int }_{\frac{1}{3}}^3|{\log }_{e}x|dx=\frac{m}{n}{\log }_e\left(\frac{n^2}{e}\right)$, where m and n are coprime natural numbers, then $m^2+n^2-5$ is equal to ____.

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    Answer: 20

14. Q62023 · 24 Jan 2023 · Shift 1

    Let $y=y(x)$ be the solution of the differential equation $x^{3}dy+(xy-1)dx=0$, $x>0$, $y\left(\frac{1}{2}\right)=3-e$. Then $y(1)$ is equal to

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

15. Q82023 · 24 Jan 2023 · Shift 1

    The area enclosed by the curves $y^2+4x=4$ and $y-2x=2$ is :

    1. A.$\frac{22}{3}$
    2. B.9
    3. C.$\frac{23}{3}$
    4. D.$\frac{25}{3}$
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    Answer: B

16. Q272023 · 24 Jan 2023 · Shift 1· Numerical answer

    The value of $\frac{8}{\pi }{\int }_0^{\frac{\pi }{2}}\frac{(\cos x{)}^{2023}}{(\sin x{)}^{2023}+(\cos x{)}^{2023}}dx$ is ____

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    Answer: 2

17. Q82023 · 24 Jan 2023 · Shift 2

    Let $y=y(x)$ be the solution of the differential equation $(x^2-3y^2)dx+3xydy=0,y(1)=1$. Then $6y^2(e)$ is equal to

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

18. Q142023 · 24 Jan 2023 · Shift 2

    ${\int }_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}}\frac{48}{\sqrt{9-4x^2}}dx$ is equal to :

    1. A.$\frac{\pi }{2}$
    2. B.$\frac{\pi }{3}$
    3. C.$\frac{\pi }{6}$
    4. D.$2\pi$
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    Answer: D

19. Q262023 · 24 Jan 2023 · Shift 2· Numerical answer

    Let $f$ be a differentiable function defined on $\left[0,\frac{\pi }{2}\right]$ such that $f(x)>0$ and $f(x)+{\int }_0^{x}f(t)\sqrt{1-({\log }_{e}f(t){)}^2}dt=e,\forall x\in \left[0,\frac{\pi }{2}\right]$. Then ${\left(6{\log }_{e}f\left(\frac{\pi }{6}\right)\right)}^2$ is equal to _______.

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    Answer: 27

20. Q272023 · 24 Jan 2023 · Shift 2· Numerical answer

    If the area of the region bounded by the curves $y^2-2y=-x,x+y=0$ is A, then 8 A is equal to _______

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    Answer: 36