Integral Calculus & Area · JEE Main previous year questions

2019 29 questions

1. Q12019 · 12 Apr 2019 · Shift 1

   The integral $\int \frac{2x^3-1}{x^4+x}dx$ is equal to : (Here C is a constant of integration)

   1. A.${\log }_e\left|\frac{x^3+1}{x^2}\right|+C$
   2. B.$\frac{1}{2}{\log }_e\left|\frac{x^3+1}{x^2}\right|+C$
   3. C.${\log }_e\left|\frac{x^3+1}{x}\right|+C$
   4. D.$\frac{1}{2}{\log }_e\frac{(x^3+1{)}^2}{|x^3|}+C$
   Show answer

   Answer: C

2. Q32019 · 12 Apr 2019 · Shift 1

   If ${\int }_0^{\frac{\pi }{2}}\frac{\cot x}{\cot x+\mathrm{cosec}x}dx=m(\pi +n)$, then m.n is equal to

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

   Answer: A

3. Q112019 · 12 Apr 2019 · Shift 1

   Consider the differential equation, $y^{2}dx+\left(x-\frac{1}{y}\right)dy=0$. If value of y is 1 when x = 1, then the value of x for which y = 2, is :

   1. A.$\frac{3}{2}-\frac{1}{\sqrt{e}}$
   2. B.$\frac{1}{2}+\frac{1}{\sqrt{e}}$
   3. C.$\frac{5}{2}+\frac{1}{\sqrt{e}}$
   4. D.$\frac{3}{2}-\sqrt{e}$
   Show answer

   Answer: A

4. Q122019 · 12 Apr 2019 · Shift 1

   If the area (in sq. units) of the region $\{(x,y):y^2\le 4x,x+y\le 1,x\ge 0,y\ge 0\}$ is $a\sqrt{2}+b$, then $a-b$ is equal to :

   1. A.$\frac{8}{3}$
   2. B.$-\frac{2}{3}$
   3. C.6
   4. D.$\frac{10}{3}$
   Show answer

   Answer: C

5. Q12019 · 12 Apr 2019 · Shift 2

   Let $\alpha \in \left(0,\frac{\pi }{2}\right)$ be fixed. If the integral $\int \frac{\tan x+\tan \alpha }{\tan x-\tan \alpha }dx=\text{A}(x)\cos 2\alpha +\text{B}(x)\sin 2\alpha +\text{C}$, where C is a constant of integration, then the functions A(x) and B(x) are respectively :

   1. A.$x-\alpha$ and ${\log }_e|\cos (x-\alpha )|$
   2. B.$x+\alpha$ and ${\log }_e|\sin (x-\alpha )|$
   3. C.$x+\alpha$ and ${\log }_e|\sin (x+\alpha )|$
   4. D.$x-\alpha$ and ${\log }_e|\sin (x-\alpha )|$
   Show answer

   Answer: D

6. Q22019 · 12 Apr 2019 · Shift 2

   The general solution of the differential equation $(y^2-x^3)dx-xydy=0$ ($x\ne 0$) is : (where c is a constant of integration)

   1. A.$y^2+2x^3+c{x}^2=0$
   2. B.$y^2+2x^2+c{x}^3=0$
   3. C.$y^2-2x+c{x}^3=0$
   4. D.$y^2-2x^3+c{x}^2=0$
   Show answer

   Answer: A

7. Q12019 · 10 Apr 2019 · Shift 1

   The region represented by $|x-y|\le 2$ and $|x+y|\le 2$ is bounded by a :

   1. A.rhombus of area $8\sqrt{2}sq.\mathrm{units}$
   2. B.square of side length $2\sqrt{2}\mathrm{units}$
   3. C.square of area 16 sq. units
   4. D.rhombus of side length 2 units
   Show answer

   Answer: B

8. Q142019 · 10 Apr 2019 · Shift 1

   If $\int \frac{dx}{(x^2-2x+10{)}^2}=A\left({\tan }^{-1}\left(\frac{x-1}{3}\right)+\frac{f(x)}{x^2-2x+10}\right)+C$ where C is a constant of integration then :

   1. A.$A=\frac{1}{54}$ and $f(x)=9(x-1{)}^2$
   2. B.$A=\frac{1}{54}$ and $f(x)=3(x-1)$
   3. C.$A=\frac{1}{81}$ and $f(x)=3(x-1)$
   4. D.$A=\frac{1}{27}$ and $f(x)=9(x-1{)}^2$
   Show answer

   Answer: B

9. Q152019 · 10 Apr 2019 · Shift 1

   The value of ${\int }_0^{2\pi }[\sin 2x(1+\cos 3x)]dx$, where [t] denotes the greatest integer function is :

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

   Answer: D

10. Q112019 · 10 Apr 2019 · Shift 2

    The area (in sq.units) of the region bounded by the curves $y=2^x$ and $y=|x+1|$, in the first quadrant is :

    1. A.$\frac{1}{2}$
    2. B.$\frac{3}{2}$
    3. C.$\frac{3}{2}-\frac{1}{{\log }_{e}2}$
    4. D.${\log }_{e}2+\frac{3}{2}$
    Show answer

    Answer: C

11. Q122019 · 10 Apr 2019 · Shift 2

    The integral ${\int }_{\pi /6}^{\pi /3}{\sec }^{2/3}x{\mathrm{cosec}}^{4/3}xdx$ is equal to :

    1. A.$3^{\frac{5}{3}}-3^{\frac{1}{3}}$
    2. B.$3^{\frac{5}{6}}-3^{\frac{2}{3}}$
    3. C.$3^{\frac{4}{3}}-3^{\frac{1}{3}}$
    4. D.$3^{\frac{7}{6}}-3^{\frac{5}{6}}$
    Show answer

    Answer: D

12. Q152019 · 10 Apr 2019 · Shift 2

    If $\int x^5{e}^{-x^2}dx=g(x)e^{-x^2}+c$, where c is a constant of integration, then $g(-1)$ is equal to :

    1. A.1
    2. B.- 1
    3. C.$-\frac{5}{2}$
    4. D.$-\frac{1}{2}$
    Show answer

    Answer: C

13. Q22019 · 9 Apr 2019 · Shift 1

    The value of ${\int }_0^{\pi /2}\frac{{\sin }^{3}x}{\sin x+\cos x}dx$ is

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

    Answer: D

14. Q122019 · 9 Apr 2019 · Shift 1

    The integral $\int {\sec }^{2/3}x{\mathrm{cosec}}^{4/3}xdx$ is equal to (Hence C is a constant of integration)

    1. A.$-\frac{3}{4}{\tan }^{-4/3}x+C$
    2. B.$3{\tan }^{-1/3}x+C$
    3. C.$-3{\cot }^{-1/3}x+C$
    4. D.$-3{\tan }^{-1/3}x+C$
    Show answer

    Answer: D

15. Q142019 · 9 Apr 2019 · Shift 1

    The area (in sq. units) of the region $A=\{(x,y):x^2\le y\le x+2\}$ is

    1. A.$\frac{31}{6}$
    2. B.$\frac{10}{3}$
    3. C.$\frac{13}{6}$
    4. D.$\frac{9}{2}$
    Show answer

    Answer: D

16. Q42019 · 9 Apr 2019 · Shift 2

    The value of the integral ${\int }_0^{1}x{\cot }^{-1}(1-x^2+x^4)dx$ is :-

    1. A.$\frac{\pi }{2}-\frac{1}{2}{\log }_{e}2$
    2. B.$\frac{\pi }{4}-{\log }_{e}2$
    3. C.$\frac{\pi }{4}-\frac{1}{2}{\log }_{e}2$
    4. D.$\frac{\pi }{2}-{\log }_{e}2$
    Show answer

    Answer: C

17. Q72019 · 9 Apr 2019 · Shift 2

    If $\cos x\frac{dy}{dx}-y\sin x=6x$, $(0<x<\frac{\pi }{2})$ and $y\left(\frac{\pi }{3}\right)=0$ then $y\left(\frac{\pi }{6}\right)$ is equal to :-

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

    Answer: D

18. Q142019 · 9 Apr 2019 · Shift 2

    $\int e^{\sec x}(\sec x\tan xf(x)+\sec x\tan x+{\sec }^{2}x)dx={\mathrm{e}}^{\sec x}f(x)+C$ then a possible choice of f(x) is :-

    1. A.x sec x + tan x + 1/2
    2. B.sec x + xtan x - 1/2
    3. C.sec x - tan x - 1/2
    4. D.sec x + tan x + 1/2
    Show answer

    Answer: D

19. Q192019 · 9 Apr 2019 · Shift 2

    The area (in sq. units) of the region $\mathrm{A}=\{(x,y):\frac{y^2}{2}\le x\le y+4\}$ is :-

    1. A.30
    2. B.18
    3. C.$\frac{53}{3}$
    4. D.16
    Show answer

    Answer: B

20. Q72019 · 8 Apr 2019 · Shift 2

    Let $S(\alpha )=\{(x,y):y^2\le x,0\le x\le \alpha \}$ and $A(\alpha )$ is area of the region $S(\alpha )$. If for a $\lambda$, $0<\lambda <4$, $A(\lambda ):A(4)=2:5$, then $\lambda$ equals

    1. A.$2{\left(\frac{4}{25}\right)}^{\frac{1}{3}}$
    2. B.$2{\left(\frac{2}{5}\right)}^{\frac{1}{3}}$
    3. C.$4{\left(\frac{4}{25}\right)}^{\frac{1}{3}}$
    4. D.$4{\left(\frac{2}{5}\right)}^{\frac{1}{3}}$
    Show answer

    Answer: C

21. Q112019 · 8 Apr 2019 · Shift 2

    The number of integral values of m for which the equation $(1+m^2)x^2-2(1+3m)x+(1+8m)=0$ has no real root is :

    1. A.2
    2. B.infinitely many
    3. C.1
    4. D.3
    Show answer

    Answer: B

22. Q172019 · 8 Apr 2019 · Shift 2

    If $\int \frac{dx}{x^3(1+x^6{)}^{2/3}}=xf(x)(1+x^6{)}^{\frac{1}{3}}+C$ where C is a constant of integration, then the function f(x) is equal to

    1. A.$\frac{3}{x^2}$
    2. B.$-\frac{1}{6x^3}$
    3. C.$-\frac{1}{2x^3}$
    4. D.$-\frac{1}{2x^2}$
    Show answer

    Answer: C

23. Q172019 · 12 Jan 2019 · Shift 1

    The integral $\int \cos ({\log }_{e}x)dx$ is equal to : (where C is a constant of integration)

    1. A.$\frac{x}{2}[\sin ({\log }_{e}x)-\cos ({\log }_{e}x)]+C$
    2. B.$x[\cos ({\log }_{e}x)+\sin ({\log }_{e}x)]+C$
    3. C.$\frac{x}{2}[\cos ({\log }_{e}x)+\sin ({\log }_{e}x)]+C$
    4. D.$x[\cos ({\log }_{e}x)-\sin ({\log }_{e}x)]+C$
    Show answer

    Answer: C

24. Q52019 · 12 Jan 2019 · Shift 2

    The integral ${\int }_1^e\left\{{\left(\frac{x}{e}\right)}^{2x}-{\left(\frac{e}{x}\right)}^x\right\}{\log }_{\mathrm{e}}xdx$ is equal to :

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

    Answer: B

25. Q82019 · 12 Jan 2019 · Shift 2

    The integral $\int \frac{3x^{13}+2x^{11}}{(2x^4+3x^2+1{)}^4}dx$ is equal to : (where C is a constant of integration)

    1. A.$\frac{x^{12}}{6(2x^4+3x^2+1{)}^3}+C$
    2. B.$\frac{x^4}{6(2x^4+3x^2+1{)}^3}+C$
    3. C.$\frac{x^{12}}{(2x^4+3x^2+1{)}^3}+C$
    4. D.$\frac{x^4}{(2x^4+3x^2+1{)}^3}+C$
    Show answer

    Answer: A