Differential Equations · JEE Main previous year questions

2021 9 questions

1. Q52021 · 1 Sept 2021 · Shift 2

   If $y=y(x)$ is the solution curve of the differential equation $x^{2}dy+\left(y-\frac{1}{x}\right)dx=0$ ; $x>0$ and $y(1)=1$, then $y\left(\frac{1}{2}\right)$ is equal to :

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

2. Q72021 · 25 Jul 2021 · Shift 1

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

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

3. Q212021 · 25 Jul 2021 · Shift 1· Numerical answer

   Let y = y(x) be solution of the following differential equation $e^y\frac{dy}{dx}-2e^y\sin x+\sin x{\cos }^{2}x=0,y\left(\frac{\pi }{2}\right)=0$ If $y(0)={\log }_e(\alpha +\beta e^{-2})$, then $4(\alpha +\beta )$ is equal to ____.

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   Answer: 4

4. Q162021 · 18 Mar 2021 · Shift 1

   The differential equation satisfied by the system of parabolas $y^2=4a(x+a)$ is :

   1. A.$y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)-y=0$
   2. B.$y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)+y=0$
   3. C.$y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0$
   4. D.$y\left(\frac{dy}{dx}\right)+2x\left(\frac{dy}{dx}\right)-y=0$
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   Answer: C

5. Q12021 · 18 Mar 2021 · Shift 2

   Let y = y(x) be the solution of the differential equation $\frac{dy}{dx}=(y+1)\left((y+1)e^{x^2/2}-x\right),0<x<2.1$, with $y(2)=0$. Then the value of $\frac{dy}{dx}$ at $x=1$ is equal to :

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

6. Q142021 · 16 Mar 2021 · Shift 1

   If $y=y(x)$ is the solution of the differential equation, $\frac{dy}{dx}+2y\tan x=\sin x$, $y\left(\frac{\pi }{3}\right)=0$, then the maximum value of the function $y(x)$ over $\mathbb{R}$ is equal to:

   1. A.$\frac{1}{8}$
   2. B.8
   3. C.$-\frac{15}{4}$
   4. D.$\frac{1}{2}$
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   Answer: A

7. Q212021 · 16 Mar 2021 · Shift 1· Numerical answer

   Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx}=2(x+1)$. If the numerical value of area bounded by the curve $y=y(x)$ and x-axis is $\frac{4\sqrt{8}}{3}$, then the value of y(1) is equal to _________.

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   Answer: 2

8. Q252021 · 26 Feb 2021 · Shift 1· Numerical answer

   The difference between degree and order of a differential equation that represents the family of curves given by $y^2=a\left(x+\frac{\sqrt{a}}{2}\right),a>0$ is _________.

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   Answer: 2

9. Q262021 · 26 Feb 2021 · Shift 1· Numerical answer

   If $y=y(x)$ is the solution of the equation $e^{\sin y}\cos y\frac{dy}{dx}+e^{\sin y}\cos x=\cos x$, $y(0)=0$; then $1+y\left(\frac{\pi }{6}\right)+\frac{\sqrt{3}}{2}y\left(\frac{\pi }{3}\right)+\frac{1}{\sqrt{2}}y\left(\frac{\pi }{4}\right)$ is equal to _________.

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   Answer: 1