Integral Calculus & Area · JEE Main previous year questions

2024 43 questions

1. Q52024 · 9 Apr 2024 · Shift 1

   The solution of the differential equation $(x^2+y^2)\mathrm{d}x-5xy\mathrm{d}y=0$, $y(1)=0$, is :

   1. A.$|x^2-4y^2{|}^5=x^2$
   2. B.$|x^2-2y^2{|}^6=x$
   3. C.$|x^2-2y^2{|}^5=x^2$
   4. D.$|x^2-4y^2{|}^6=x$
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   Answer: A

2. Q112024 · 9 Apr 2024 · Shift 1

   Let $\int \frac{2-\tan x}{3+\tan x}\mathrm{d}x=\frac{1}{2}(\alpha x+{\log }_e|\beta \sin x+\gamma \cos x|)+C$, where $C$ is the constant of integration. Then $\alpha +\frac{\gamma }{\beta }$ is equal to :

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

3. Q42024 · 9 Apr 2024 · Shift 2

   Let $\underset{0}{\overset{x}{\int }}\sqrt{1-(y^{\prime }(t){)}^2}dt={\int }_0^{x}y(t)dt$, $0\le x\le 3$, $y\ge 0$, $y(0)=0$. Then at $x=2$, $y^{\prime \prime }+y+1$ is equal to

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

4. Q62024 · 9 Apr 2024 · Shift 2

   The integral $\underset{1/4}{\overset{3/4}{\int }}\cos \left(2{\cot }^{-1}\sqrt{\frac{1-x}{1+x}}\right)dx$ is equal to

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

5. Q112024 · 9 Apr 2024 · Shift 2

   The value of the integral $\underset{-1}{\overset{2}{\int }}{\log }_e\left(x+\sqrt{x^2+1}\right)dx$ is

   1. A.$\sqrt{5}-\sqrt{2}+{\log }_e\left(\frac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$
   2. B.$\sqrt{2}-\sqrt{5}+{\log }_e\left(\frac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$
   3. C.$\sqrt{5}-\sqrt{2}+{\log }_e\left(\frac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$
   4. D.$\sqrt{2}-\sqrt{5}+{\log }_e\left(\frac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$
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   Answer: D

6. Q22024 · 8 Apr 2024 · Shift 1

   The value of $k\in \mathbb{N}$ for which the integral $I_n={\int }_0^1(1-x^k{)}^{n}dx,n\in \mathbb{N}$, satisfies $147I_{20}=148I_{21}$ is

   1. A.$8$
   2. B.$14$
   3. C.$7$
   4. D.$10$
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   Answer: C

7. Q202024 · 8 Apr 2024 · Shift 1

   Let $I(x)=\int \frac{6}{{\sin }^{2}x(1-\cot x{)}^2}dx$. If $I(0)=3$, then $I\left(\frac{\pi }{12}\right)$ is equal to

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

8. Q232024 · 8 Apr 2024 · Shift 2· Numerical answer

   If $\int \frac{1}{\sqrt[5]{(x-1{)}^4(x+3{)}^6}}\mathrm{d}x=\mathrm{A}{\left(\frac{\alpha x-1}{\beta x+3}\right)}^{\mathrm{B}}+\mathrm{C}$, where $\mathrm{C}$ is the constant of integration, then the value of $\alpha +\beta +20\mathrm{AB}$ is ____.

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   Answer: 7

9. Q272024 · 8 Apr 2024 · Shift 2· Numerical answer

   Let $\alpha |x|=|y|e^{xy-\beta },\alpha ,\beta \in \mathbf{N}$ be the solution of the differential equation $x\mathrm{d}y-y\mathrm{d}x+xy(x\mathrm{d}y+y\mathrm{d}x)=0,y(1)=2$. Then $\alpha +\beta$ is equal to ____

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   Answer: 4

10. Q102024 · 6 Apr 2024 · Shift 2

    If the area of the region $\{(x,y):\frac{a}{x^2}\le y\le \frac{1}{x},1\le x\le 2,0<a<1\}$ is $({\log }_{\mathrm{e}}2)-\frac{1}{7}$ then the value of $7a-3$ is equal to :

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

11. Q192024 · 6 Apr 2024 · Shift 2

    If $\int \frac{1}{a^2{\sin }^{2}x+b^2{\cos }^{2}x}dx=\frac{1}{12}{\tan }^{-1}(3\tan x)+\text{constant}$, then the maximum value of $a\sin x+b\cos x$, is :

    1. A.$\sqrt{41}$
    2. B.$\sqrt{39}$
    3. C.$\sqrt{40}$
    4. D.$\sqrt{42}$
    Show answer

    Answer: C

12. Q202024 · 6 Apr 2024 · Shift 2

    If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5:4$, is $a{x}^2+b{y}^2+cxy+dx+ey+170=0$, then the value of $a^2+2b+3c+4d+e$ is equal to :

    1. A.37
    2. B.$-27$
    3. C.437
    4. D.5
    Show answer

    Answer: A

13. Q242024 · 6 Apr 2024 · Shift 2· Numerical answer

    If the solution $y(x)$ of the given differential equation $(e^y+1)\cos xdx+e^y\sin xdy=0$ passes through the point $\left(\frac{\pi }{2},0\right)$, then the value of $e^{y\left(\frac{\pi }{6}\right)}$ is equal to ____.

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    Answer: 3

14. Q52024 · 5 Apr 2024 · Shift 2

    Let $S_1=\{z\in \mathbf{C}:|z|\le 5\}$, $S_2=\left\{z\in \mathbf{C}:\mathrm{Im}\left(\frac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right)\ge 0\right\}$ and $S_3=\{z\in \mathbf{C}:\mathrm{Re}(z)\ge 0\}$. Then the area of the region $S_1\cap S_2\cap S_3$ is :

    1. A.$\frac{125\pi }{24}$
    2. B.$\frac{125\pi }{6}$
    3. C.$\frac{125\pi }{12}$
    4. D.$\frac{125\pi }{4}$
    Show answer

    Answer: C

15. Q112024 · 5 Apr 2024 · Shift 2

    Let $\beta (m,n)=\underset{0}{\overset{1}{\int }}{x}^{m-1}(1-x{)}^{n-1}dx,m,n>0$. If $\underset{0}{\overset{1}{\int }}(1-x^{10}{)}^{20}dx=a\times \beta (b,c)$, then $100(a+b+c)$ equals _________.

    1. A.2012
    2. B.1021
    3. C.1120
    4. D.2120
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    Answer: D

16. Q172024 · 5 Apr 2024 · Shift 2

    The area enclosed between the curves $y=x|x|$ and $y=x-|x|$ is :

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

17. Q22024 · 4 Apr 2024 · Shift 1

    One of the points of intersection of the curves $y=1+3x-2x^2$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2},2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(l\sqrt{5}+m)-n{\log }_e(1+\sqrt{5})$, where $l,m,n\in \mathbf{N}$. Then $l+m+n$ is equal to

    1. A.30
    2. B.29
    3. C.31
    4. D.32
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    Answer: A

18. Q222024 · 4 Apr 2024 · Shift 1· Numerical answer

    If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3\sqrt{5}}k$, and $\underset{0}{\overset{k}{\int }}[x^2]\mathrm{d}x=\alpha -\sqrt{\alpha }$, where $[x]$ denotes the greatest integer function, then $6{\alpha }^3$ is equal to ____.

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    Answer: 48

19. Q262024 · 4 Apr 2024 · Shift 1· Numerical answer

    If $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\frac{{\sin }^{2}x}{1+\sin x\cos x}\mathrm{d}x=\frac{1}{\mathrm{a}}{\log }_e\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi }{\mathrm{b}\sqrt{3}}$, where $\mathrm{a},\mathrm{b}\in \mathrm{N}$, then $\mathrm{a}+\mathrm{b}$ is equal to ____.

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    Answer: 8

20. Q302024 · 4 Apr 2024 · Shift 1· Numerical answer

    Let $a=1+\frac{{}^2{\mathrm{C}}_2}{3!}+\frac{{}^3{\mathrm{C}}_2}{4!}+\frac{{}^4{\mathrm{C}}_2}{5!}+\dots$, $\mathrm{b}=1+\frac{{}^1{\mathrm{C}}_0+{}^1{\mathrm{C}}_1}{1!}+\frac{{}^2{\mathrm{C}}_0+{}^2{\mathrm{C}}_1+{}^2{\mathrm{C}}_2}{2!}+\frac{{}^3{\mathrm{C}}_0+{}^3{\mathrm{C}}_1+{}^3{\mathrm{C}}_2+{}^3{\mathrm{C}}_3}{3!}+\dots$. Then $\frac{2b}{a^2}$ is equal to ____.

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    Answer: 8

21. Q82024 · 4 Apr 2024 · Shift 2

    The area (in sq. units) of the region described by $\left\{(x,y):y^2\le 2x\text{, and }y\ge 4x-1\right\}$ is

    1. A.$\frac{9}{32}$
    2. B.$\frac{11}{12}$
    3. C.$\frac{8}{9}$
    4. D.$\frac{11}{32}$
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    Answer: A

22. Q162024 · 4 Apr 2024 · Shift 2

    If the value of the integral ${\int }_{-1}^1\frac{\cos \alpha x}{1+3^x}dx$ is $\frac{2}{\pi }$, then a value of $\alpha$ is

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

23. Q222024 · 4 Apr 2024 · Shift 2· Numerical answer

    If $\int {\mathrm{cosec}}^{5}xdx=\alpha \cot x\mathrm{cosec}x\left({\mathrm{cosec}}^{2}x+\frac{3}{2}\right)+\beta {\log }_e\left|\tan \frac{x}{2}\right|+C$ where $\alpha ,\beta \in \mathbb{R}$ and $C$ is the constant of integration, then the value of $8(\alpha +\beta )$ equals _________.

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    Answer: 1

24. Q22024 · 1 Feb 2024 · Shift 1

    The value of the integral ${\int }_0^{\pi /4}\frac{xdx}{{\sin }^4(2x)+{\cos }^4(2x)}$ equals :

    1. A.$\frac{\sqrt{2}{\pi }^2}{8}$
    2. B.$\frac{\sqrt{2}{\pi }^2}{16}$
    3. C.$\frac{\sqrt{2}{\pi }^2}{32}$
    4. D.$\frac{\sqrt{2}{\pi }^2}{64}$
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    Answer: C

25. Q102024 · 1 Feb 2024 · Shift 1

    The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to :

    1. A.$28-30{\log }_{e}2$
    2. B.$30-28{\log }_{e}2$
    3. C.$30-32{\log }_{e}2$
    4. D.$32-30{\log }_{e}2$
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    Answer: C