Integral Calculus & Area · JEE Main previous year questions

2023 45 questions

1. Q252023 · 13 Apr 2023 · Shift 1· Numerical answer

   Let for $x\in \mathbb{R}$, $S_0(x)=x$, $S_k(x)=C_{k}x+k\underset{0}{\overset{x}{\int }}{S}_{k-1}(t)dt$, where $C_0=1$, $C_k=1-\underset{0}{\overset{1}{\int }}{S}_{k-1}(x)dx$, $k=1,2,3,\dots$ Then $S_2(3)+6C_3$ is equal to ___________.

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   Answer: 18

2. Q112023 · 13 Apr 2023 · Shift 2

   The area of the region $\left\{(x,y):x^2\le y\le |x^2-4|,y\ge 1\right\}$ is

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

3. Q122023 · 13 Apr 2023 · Shift 2

   The value of $\frac{e^{-\frac{\pi }{4}}+\underset{0}{\overset{\frac{\pi }{4}}{\int }}{e}^{-x}{\tan }^{50}xdx}{\underset{0}{\overset{\frac{\pi }{4}}{\int }}{e}^{-x}({\tan }^{49}x+{\tan }^{51}x)dx}$ is

   1. A.$51$
   2. B.$50$
   3. C.$25$
   4. D.$49$
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   Answer: B

4. Q42023 · 12 Apr 2023 · Shift 1

   Let the lines $l_1:\frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha }{-2}$ and $l_2:3x+2y+z-2=0=x-3y+2z-13$ be coplanar. If the point $\mathrm{P}(a,b,c)$ on $l_1$ is nearest to the point $\mathrm{Q}(-4,-3,2)$, then $|a|+|b|+|c|$ is equal to

   1. A.$12$
   2. B.$14$
   3. C.$10$
   4. D.$8$
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   Answer: C

5. Q212023 · 12 Apr 2023 · Shift 1· Numerical answer

   Let ${\mathrm{D}}_k=\left|\begin{array}{ccc}1& 2k& 2k-1\\ n& n^2+n+2& n^2\\ n& n^2+n& n^2+n+2\end{array}\right|$. If ${\sum }_{k=1}^n{\mathrm{D}}_k=96$, then $n$ is equal to _________.

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   Answer: 6

6. Q252023 · 12 Apr 2023 · Shift 1· Numerical answer

   If ${\int }_{-0.15}^{0.15}|100x^2-1|dx=\frac{k}{3000}$, then $k$ is equal to _________.

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   Answer: 575

7. Q292023 · 12 Apr 2023 · Shift 1· Numerical answer

   Let $I(x)=\int \sqrt{\frac{x+7}{x}}dx$ and $I(9)=12+7{\log }_{e}7$. If $I(1)=\alpha +7{\log }_e(1+2\sqrt{2})$, then ${\alpha }^4$ is equal to _________.

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   Answer: 64

8. Q62023 · 11 Apr 2023 · Shift 1

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

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

9. Q122023 · 11 Apr 2023 · Shift 1

   Area of the region $\{(x,y):x^2+(y-2{)}^2\le 4,x^2\ge 2y\}$ is

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

10. Q152023 · 11 Apr 2023 · Shift 1

    The number of integral solutions $x$ of ${\log }_{(x+\frac{7}{2})}{\left(\frac{x-7}{2x-3}\right)}^2\ge 0$ is :

    1. A.8
    2. B.7
    3. C.5
    4. D.6
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    Answer: D

11. Q132023 · 11 Apr 2023 · Shift 2

    Let the function $f:[0,2]\to \mathbb{R}$ be defined as $$f(x)=\{\begin{array}{ll}{e}^{\min \{x^2,x-[x]\}},& x\in [0,1)\\ e^{[x-{\log }_{e}x]},& x\in [1,2]\end{array}$$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral ${\int }_0^{2}xf(x)dx$ is :

    1. A.$2e-1$
    2. B.$2e-\frac{1}{2}$
    3. C.$1+\frac{3e}{2}$
    4. D.$(e-1)\left(e^2+\frac{1}{2}\right)$
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    Answer: B

12. Q12023 · 10 Apr 2023 · Shift 1

    If $I(x)=\int e^{{\sin }^{2}x}(\cos x\sin 2x-\sin x)dx$ and $I(0)=1$, then $I\left(\frac{\pi }{3}\right)$ is equal to :

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

13. Q132023 · 10 Apr 2023 · Shift 1

    Let $f$ be a differentiable function such that $x^{2}f(x)-x=4{\int }_0^{x}tf(t)dt$, $f(1)=\frac{2}{3}$. Then $18f(3)$ is equal to :

    1. A.160
    2. B.210
    3. C.150
    4. D.180
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    Answer: A

14. Q72023 · 10 Apr 2023 · Shift 2

    For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if $\int \left({\left(\frac{x}{e}\right)}^{2x}+{\left(\frac{e}{x}\right)}^{2x}\right){\log }_{e}xdx=\frac{1}{\alpha }{\left(\frac{x}{e}\right)}^{\beta x}-\frac{1}{\gamma }{\left(\frac{e}{x}\right)}^{\delta x}+C$, where $e={\sum }_{n=0}^{\infty }\frac{1}{n!}$ and C is constant of integration, then $\alpha +2\beta +3\gamma -4\delta$ is equal to :

    1. A.$-8$
    2. B.$-4$
    3. C.1
    4. D.4
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    Answer: D

15. Q222023 · 10 Apr 2023 · Shift 2· Numerical answer

    If the area of the region $\{(x,y):|x^2-2|\le y\le x\}$ is A, then $6A+16\sqrt{2}$ is equal to _________.

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    Answer: 27

16. Q92023 · 8 Apr 2023 · Shift 1

    The area of the region $\left\{(x,y):x^2\le y\le 8-x^2,y\le 7\right\}$ is :

    1. A.$18$
    2. B.$24$
    3. C.$20$
    4. D.$21$
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    Answer: C

17. Q212023 · 8 Apr 2023 · Shift 1· Numerical answer

    Let $[t]$ denote the greatest integer $\le t$. Then $\frac{2}{\pi }{\int }_{\pi /6}^{5\pi /6}(8[\mathrm{cosec}x]-5[\cot x])dx$ is equal to ___________.

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    Answer: 14

18. Q32023 · 8 Apr 2023 · Shift 2

    The integral $\int \left[{\left(\frac{x}{e}\right)}^x+{\left(\frac{e}{x}\right)}^x\right]\ln \left(\frac{ex}{2}\right)dx$ is equal to :

    1. A.${\left(\frac{x}{2}\right)}^x+{\left(\frac{2}{x}\right)}^x+C$
    2. B.${\left(\frac{x}{2}\right)}^x-{\left(\frac{2}{x}\right)}^x+C$
    3. C.${\left(\frac{x}{2}\right)}^x{\log }_2\left(\frac{2}{x}\right)+C$
    4. D.None
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    Answer: B

19. Q222023 · 8 Apr 2023 · Shift 2· Numerical answer

    Let the area enclosed by the lines $x+y=2,y=0,x=0$ and the curve $f(x)=\min \left\{x^2+\frac{3}{4},1+[x]\right\}$ where $[x]$ denotes the greatest integer $\le x$, be A. Then the value of $12\mathrm{A}$ is ___________.

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    Answer: 17

20. Q212023 · 6 Apr 2023 · Shift 1· Numerical answer

    Let $y=y(x)$ be a solution of the differential equation $(x\cos x)dy+(xy\sin x+y\cos x-1)dx=0,0<x<\frac{\pi }{2}$. If $\frac{\pi }{3}y\left(\frac{\pi }{3}\right)=\sqrt{3}$, then $\left|\frac{\pi }{6}{y}^{\prime \prime }\left(\frac{\pi }{6}\right)+2y^{\prime }\left(\frac{\pi }{6}\right)\right|$ is equal to ________.

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    Answer: 2

21. Q222023 · 6 Apr 2023 · Shift 1· Numerical answer

    If the area of the region $S=\{(x,y):2y-y^2\le x^2\le 2y,x\ge y\}$ is equal to $\frac{n+2}{n+1}-\frac{\pi }{n-1}$, then the natural number $n$ is equal to ________.

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    Answer: 5

22. Q22023 · 6 Apr 2023 · Shift 2

    The area bounded by the curves $y=|x-1|+|x-2|$ and $y=3$ is equal to :

    1. A.5
    2. B.4
    3. C.6
    4. D.3
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    Answer: B

23. Q272023 · 1 Feb 2023 · Shift 1· Numerical answer

    Let $A$ be the area bounded by the curve $y=x|x-3|$, the $x$-axis and the ordinates $x=-1$ and $x=2$. Then $12A$ is equal to ________.

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    Answer: 62

24. Q92023 · 31 Jan 2023 · Shift 1

    Let $\alpha \in (0,1)$ and $\beta ={\log }_{\mathrm{e}}(1-\alpha )$. Let $P_n(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\dots +\frac{x^n}{n},x\in (0,1)$. Then the integral ${\int }_0^{\alpha }\frac{t^{50}}{1-t}\mathrm{d}t$ is equal to

    1. A.$-(\beta +P_{50}(\alpha ))$
    2. B.$\beta -P_{50}(\alpha )$
    3. C.$P_{50}(\alpha )-\beta$
    4. D.$\beta +P_{50}(\alpha )$
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    Answer: A

25. Q142023 · 31 Jan 2023 · Shift 1

    The value of ${\int }_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{(2+3\sin x)}{\sin x(1+\cos x)}\mathrm{d}x$ is equal to :

    1. A.$\frac{10}{3}-\sqrt{3}+{\log }_{\mathrm{e}}\sqrt{3}$
    2. B.$\frac{7}{2}-\sqrt{3}-{\log }_{\mathrm{e}}\sqrt{3}$
    3. C.$\frac{10}{3}-\sqrt{3}-{\log }_{\mathrm{e}}\sqrt{3}$
    4. D.$-2+3\sqrt{3}+{\log }_{\mathrm{e}}\sqrt{3}$
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    Answer: A