Differential Equations · JEE Main previous year questions

2023 14 questions

1. Q52023 · 13 Apr 2023 · Shift 1

   Let $y=y_1(x)$ and $y=y_2(x)$ be the solution curves of the differential equation $\frac{dy}{dx}=y+7$ with initial conditions $y_1(0)=0$ and $y_2(0)=1$ respectively. Then the curves $y=y_1(x)$ and $y=y_2(x)$ intersect at

   1. A.no point
   2. B.two points
   3. C.infinite number of points
   4. D.one point
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   Answer: A

2. Q272023 · 13 Apr 2023 · Shift 2· Numerical answer

   If $y=y(x)$ is the solution of the differential equation $$\frac{dy}{dx}+\frac{4x}{(x^2-1)}y=\frac{x+2}{(x^2-1{)}^{\frac{5}{2}}},x>1$$ such that $y(2)=\frac{2}{9}{\log }_e(2+\sqrt{3})$ and $y(\sqrt{2})=\alpha {\log }_e(\sqrt{\alpha }+\beta )+\beta -\sqrt{\gamma }$, $\alpha ,\beta ,\gamma \in \mathbb{N}$, then $\alpha \beta \gamma$ is equal to :

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   Answer: 6

3. Q32023 · 12 Apr 2023 · Shift 1

   Let $y=y(x)$, $y>0$, be a solution curve of the differential equation $(1+x^2)dy=y(x-y)dx$. If $y(0)=1$ and $y(2\sqrt{2})=\beta$, then

   1. A.$e^{{\beta }^{-1}}=e^{-2}(3+2\sqrt{2})$
   2. B.$e^{3{\beta }^{-1}}=e(5+\sqrt{2})$
   3. C.$e^{3{\beta }^{-1}}=e(3+2\sqrt{2})$
   4. D.$e^{{\beta }^{-1}}=e^{-2}(5+\sqrt{2})$
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   Answer: C

4. Q52023 · 11 Apr 2023 · Shift 1

   Let $y=y(x)$ be a solution curve of the differential equation $(1-x^2{y}^2)dx=ydx+xdy$. If the line $x=1$ intersects the curve $y=y(x)$ at $y=2$ and the line $x=2$ intersects the curve $y=y(x)$ at $y=\alpha$, then a value of $\alpha$ is :

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

5. Q122023 · 11 Apr 2023 · Shift 2

   Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{5}{x(x^5+1)}y=\frac{(x^5+1{)}^2}{x^7}$, $x>0$. If $y(1)=2$, then $y(2)$ is equal to :

   1. A.$\frac{693}{128}$
   2. B.$\frac{697}{128}$
   3. C.$\frac{637}{128}$
   4. D.$\frac{679}{128}$
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   Answer: A

6. Q212023 · 10 Apr 2023 · Shift 2· Numerical answer

   Let the tangent at any point P on a curve passing through the points $(1,1)$ and $\left(\frac{1}{10},100\right)$, intersect positive x-axis and y-axis at the points A and B respectively. If $\text{PA}:\text{PB}=1:k$ and $y=y(x)$ is the solution of the differential equation $e^{\frac{dy}{dx}}=kx+\frac{k}{2},y(0)=k$, then $4y(1)-6{\log }_{e}3$ is equal to _________.

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   Answer: 4

7. Q232023 · 8 Apr 2023 · Shift 1· Numerical answer

   If the solution curve of the differential equation $(y-2{\log }_{e}x)dx+(x{\log }_e{x}^2)dy=0,x>1$ passes through the points $\left(e,\frac{4}{3}\right)$ and $(e^4,\alpha )$, then $\alpha$ is equal to ___________.

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   Answer: 3

8. Q262023 · 8 Apr 2023 · Shift 2· Numerical answer

   Let the solution curve $x=x(y),0<y<\frac{\pi }{2}$, of the differential equation $({\log }_e(\cos y){)}^2\cos ydx-(1+3x{\log }_e(\cos y))\sin ydy=0$ satisfy $x\left(\frac{\pi }{3}\right)=\frac{1}{2{\log }_{e}2}$. If $x\left(\frac{\pi }{6}\right)=\frac{1}{{\log }_{e}m-{\log }_{e}n}$, where $m$ and $n$ are coprime, then $mn$ is equal to ___________.

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   Answer: 12

9. Q72023 · 6 Apr 2023 · Shift 2

   If the solution curve $f(x,y)=0$ of the differential equation $$(1+{\log }_{e}x)\frac{dx}{dy}-x{\log }_{e}x=e^y,x>0,$$ passes through the points $(1,0)$ and $(\alpha ,2)$, then ${\alpha }^{\alpha }$ is equal to :

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

   Answer: D

10. Q112023 · 1 Feb 2023 · Shift 1

    The area enclosed by the closed curve $C$ given by the differential equation $$\frac{dy}{dx}+\frac{x+a}{y-2}=0,y(1)=0\text{ is }4\pi .$$ Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is :

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

11. Q132023 · 1 Feb 2023 · Shift 1

    If $y=y(x)$ is the solution curve of the differential equation $$\frac{dy}{dx}+y\tan x=x\sec x,0\le x\le \frac{\pi }{3},y(0)=1$$ then $y\left(\frac{\pi }{6}\right)$ is equal to

    1. A.$\frac{\pi }{12}-\frac{\sqrt{3}}{2}{\log }_e\left(\frac{2\sqrt{3}}{e}\right)$
    2. B.$\frac{\pi }{12}+\frac{\sqrt{3}}{2}{\log }_e\left(\frac{2\sqrt{3}}{e}\right)$
    3. C.$\frac{\pi }{12}+\frac{\sqrt{3}}{2}{\log }_e\left(\frac{2}{e\sqrt{3}}\right)$
    4. D.$\frac{\pi }{12}-\frac{\sqrt{3}}{2}{\log }_e\left(\frac{2}{e\sqrt{3}}\right)$
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    Answer: D

12. Q12023 · 30 Jan 2023 · Shift 1

    Let the solution curve $y=y(x)$ of the differential equation $$\frac{dy}{dx}-\frac{3x^5{\tan }^{-1}(x^3)}{(1+x^6{)}^{3/2}}y=2x\exp \left\{\frac{x^3-{\tan }^{-1}{x}^3}{\sqrt{(1+x^6)}}\right\}$$ pass through the origin. Then $y(1)$ is equal to :

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

13. Q12023 · 29 Jan 2023 · Shift 2

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

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

14. Q142023 · 25 Jan 2023 · Shift 1

    Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}=\frac{y}{x}\left(1+x{y}^2(1+{\log }_{e}x)\right)$, $x>0,y(1)=3$. Then $\frac{y^2(x)}{9}$ is equal to :

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