Integral Calculus & Area · JEE Main previous year questions

2021 36 questions

1. Q42021 · 26 Aug 2021 · Shift 2

   If the value of the integral $\underset{0}{\overset{5}{\int }}\frac{x+[x]}{e^{x-[x]}}dx=\alpha e^{-1}+\beta$, where $\alpha ,\beta \in \mathbb{R}$, $5\alpha +6\beta =0$, and $[x]$ denotes the greatest integer less than or equal to x; then the value of $(\alpha +\beta {)}^2$ is equal to :

   1. A.100
   2. B.25
   3. C.16
   4. D.36
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   Answer: B

2. Q52021 · 26 Aug 2021 · Shift 2

   Let y(x) be the solution of the differential equation $2x^{2}dy+(e^y-2x)dx=0$, $x>0$. If $y(e)=1$, then $y(1)$ is equal to :

   1. A.0
   2. B.2
   3. C.${\log }_{e}2$
   4. D.${\log }_e(2e)$
   Show answer

   Answer: C

3. Q72021 · 27 Jul 2021 · Shift 2

   The area of the region bounded by $y-x=2$ and $x^2=y$ is equal to :

   1. A.$\frac{16}{3}$
   2. B.$\frac{2}{3}$
   3. C.$\frac{9}{2}$
   4. D.$\frac{4}{3}$
   Show answer

   Answer: C

4. Q82021 · 27 Jul 2021 · Shift 2

   Let y = y(x) be the solution of the differential equation $(x-x^3)dy=(y+y{x}^2-3x^4)dx$, x > 2. If y(3) = 3, then y(4) is equal to :

   1. A.4
   2. B.12
   3. C.8
   4. D.16
   Show answer

   Answer: B

5. Q152021 · 27 Jul 2021 · Shift 2

   Let f : (a, b) $\to$ R be twice differentiable function such that $f(x)={\int }_a^{x}g(t)dt$ for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

   1. A.twelve roots in (a, b)
   2. B.five roots in (a, b)
   3. C.seven roots in (a, b)
   4. D.three roots in (a, b)
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   Answer: C

6. Q222021 · 27 Jul 2021 · Shift 2· Numerical answer

   If ${\int }_0^{\pi }({\sin }^{3}x)e^{-{\sin }^{2}x}dx=\alpha -\frac{\beta }{e}{\int }_0^1\sqrt{t}{e}^{t}dt$, then $\alpha +\beta$ is equal to ____.

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   Answer: 5

7. Q242021 · 27 Jul 2021 · Shift 2· Numerical answer

   Let y = y(x) be the solution of the differential equation $dy=e^{\alpha x+y}dx;\alpha \in N$. If $y({\log }_{e}2)={\log }_{e}2$ and $y(0)={\log }_e\left(\frac{1}{2}\right)$, then the value of $\alpha$ is equal to ____.

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   Answer: 2

8. Q52021 · 25 Jul 2021 · Shift 1

   The value of the definite integral ${\int }_{\pi /24}^{5\pi /24}\frac{dx}{1+\sqrt[3]{\tan 2x}}$ is :

   1. A.$\frac{\pi }{3}$
   2. B.$\frac{\pi }{6}$
   3. C.$\frac{\pi }{12}$
   4. D.$\frac{\pi }{18}$
   Show answer

   Answer: C

9. Q32021 · 25 Jul 2021 · Shift 2

   If $f(x)=\{\begin{array}{ll}{\int }_0^x(5+|1-t|)dt,& x>2\\ 5x+1,& x\le 2\end{array}$, then

   1. A.f(x) is not continuous at x = 2
   2. B.f(x) is everywhere differentiable
   3. C.f(x) is continuous but not differentiable at x = 2
   4. D.f(x) is not differentiable at x = 1
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   Answer: C

10. Q72021 · 25 Jul 2021 · Shift 2

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

    1. A.2
    2. B.0
    3. C.$-1$
    4. D.1
    Show answer

    Answer: B

11. Q172021 · 25 Jul 2021 · Shift 2

    Let y = y(x) be the solution of the differential equation $xdy=(y+x^3\cos x)dx$ with $y(\pi )=0$, then $y\left(\frac{\pi }{2}\right)$ is equal to :

    1. A.$\frac{{\pi }^2}{4}+\frac{\pi }{2}$
    2. B.$\frac{{\pi }^2}{2}+\frac{\pi }{4}$
    3. C.$\frac{{\pi }^2}{2}-\frac{\pi }{4}$
    4. D.$\frac{{\pi }^4}{4}-\frac{\pi }{2}$
    Show answer

    Answer: A

12. Q242021 · 25 Jul 2021 · Shift 2· Numerical answer

    If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of $a^4+b^4+c^4$ is equal to ____.

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    Answer: 13

13. Q32021 · 22 Jul 2021 · Shift 2

    Let $y=y(x)$ be the solution of the differential equation ${\mathrm{cosec}}^{2}xdy+2dx=(1+y\cos 2x){\mathrm{cosec}}^{2}xdx$, with $y\left(\frac{\pi }{4}\right)=0$. Then, the value of $(y(0)+1{)}^2$ is equal to :

    1. A.$e^{1/2}$
    2. B.$e^{-1/2}$
    3. C.$e^{-1}$
    4. D.$e$
    Show answer

    Answer: C

14. Q52021 · 22 Jul 2021 · Shift 2

    If ${\int }_0^{100\pi }\frac{{\sin }^{2}x}{e^{\left(\frac{x}{\pi }-\left[\frac{x}{\pi }\right]\right)}}dx=\frac{\alpha {\pi }^3}{1+4{\pi }^2}$, $\alpha \in \mathbb{R}$ where $[x]$ is the greatest integer less than or equal to x, then the value of $\alpha$ is :

    1. A.$200(1-e^{-1})$
    2. B.$100(1-e)$
    3. C.$50(e-1)$
    4. D.$150(e^{-1}-1)$
    Show answer

    Answer: A

15. Q72021 · 22 Jul 2021 · Shift 2

    If the shortest distance between the straight lines $3(x-1)=6(y-2)=2(z-1)$ and $4(x-2)=2(y-\lambda )=(z-3)$, $\lambda \in \mathbb{R}$ is $\frac{1}{\sqrt{38}}$, then the integral value of $\lambda$ is equal to :

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

    Answer: A

16. Q292021 · 22 Jul 2021 · Shift 2· Numerical answer

    Let $y=y(x)$ be the solution of the differential equation $\left((x+2)e^{\left(\frac{y+1}{x+2}\right)}+(y+1)\right)dx=(x+2)dy$, $y(1)=1$. If the domain of $y=y(x)$ is an open interval $(\alpha ,\beta )$, then $|\alpha +\beta |$ is equal to ________.

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    Answer: 4

17. Q12021 · 20 Jul 2021 · Shift 1

    Let a be a positive real number such that ${\int }_0^a{e}^{x-[x]}dx=10e-9$ where $[x]$ is the greatest integer less than or equal to x. Then a is equal to:

    1. A.$10-{\log }_e(1+e)$
    2. B.$10+{\log }_{e}2$
    3. C.$10+{\log }_{e}3$
    4. D.$10+{\log }_e(1+e)$
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    Answer: B

18. Q32021 · 20 Jul 2021 · Shift 1

    The value of the integral ${\int }_{-1}^1{\log }_e(\sqrt{1-x}+\sqrt{1+x})dx$ is equal to:

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

19. Q82021 · 20 Jul 2021 · Shift 1

    Let y = y(x) be the solution of the differential equation $x\tan \left(\frac{y}{x}\right)dy=\left(y\tan \left(\frac{y}{x}\right)-x\right)dx$, $-1\le x\le 1$, $y\left(\frac{1}{2}\right)=\frac{\pi }{6}$. Then the area of the region bounded by the curves $x=0$, $x=\frac{1}{\sqrt{2}}$ and y = y(x) in the upper half plane is :

    1. A.$\frac{1}{8}(\pi -1)$
    2. B.$\frac{1}{12}(\pi -3)$
    3. C.$\frac{1}{4}(\pi -2)$
    4. D.$\frac{1}{6}(\pi -1)$
    Show answer

    Answer: A

20. Q112021 · 20 Jul 2021 · Shift 1

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

    1. A.$1-4e^3$
    2. B.$1-4e^6$
    3. C.$1+4e^3$
    4. D.$1+4e^6$
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    Answer: B

21. Q12021 · 18 Mar 2021 · Shift 1

    The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :

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

22. Q42021 · 18 Mar 2021 · Shift 1

    The integral $\int \frac{(2x-1)\cos \sqrt{(2x-1{)}^2+5}}{\sqrt{4x^2-4x+6}}dx$ is equal to (where $c$ is a constant of integration)

    1. A.$\frac{1}{2}\sin \sqrt{(2x-1{)}^2+5}+c$
    2. B.$\frac{1}{2}\cos \sqrt{(2x+1{)}^2+5}+c$
    3. C.$\frac{1}{2}\cos \sqrt{(2x-1{)}^2+5}+c$
    4. D.$\frac{1}{2}\sin \sqrt{(2x+1{)}^2+5}+c$
    Show answer

    Answer: A

23. Q222021 · 18 Mar 2021 · Shift 1· Numerical answer

    Let $f(x)$ and $g(x)$ be two functions satisfying $f(x^2)+g(4-x)=4x^3$ and $g(4-x)+g(x)=0$, then the value of ${\int }_{-4}^{4}f(x^2)dx$ is

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    Answer: 512

24. Q32021 · 18 Mar 2021 · Shift 2

    The area bounded by the curve $4y^2=x^2(4-x)(x-2)$ is equal to :

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

25. Q242021 · 18 Mar 2021 · Shift 2· Numerical answer

    Let y = y(x) be the solution of the differential equation $xdy-ydx=\sqrt{(x^2-y^2)}dx,x\ge 1$, with $y(1)=0$. If the area bounded by the line $x=1,x=e^{\pi },y=0$ and $y=y(x)$ is $\alpha e^{2\pi }+\beta$, then the value of $10(\alpha +\beta )$ is equal to ____.

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    Answer: 4