Q JEE MAIN 2022

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\frac{4 y^3+2 y x^2}{3 x y^2+x^3}, y(1)=1$. If for...

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Q JEE-Main_2026

If the curve $y=f(x)$ passes through the point ( $1, \mathrm{e}$ ) and satisfies the differential equation $...

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution of the differential equation $...

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Q JEE MAIN 2026

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be such that $f(x y)=f(x) f(y)$, for all $x, y \in \mathbf{R}$ and $...

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution of the differential equation $...

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Q JEE MAIN_2026

Let $y=y(x)$ be the solution of the differential equation $\left(x^2-x \sqrt{x^2-1}\right) d y+\left(y\left(x-\sqrt{x^2-1}\right)-x\right) d x=0, x \geq 1$. If $y(1)=1$,...

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Q JEE MAIN 2026

The area of the region $\left\{(x, y): 0 \leq y \leq 6-x, y^2 \geq 4 x-3, x \geq 0\right\}$ is:

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Q JEE MAIN 2026

Let $f$ be a twice differentiable function such that $...

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution of the differential equation $x \sin \left(\frac{y}{x}\right) d y=\left(y \sin \left(\frac{y}{x}\right)-x\right) d x, y(1)=\frac{\pi}{2}$ and...

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution of the differential equation: $$ \frac{d y}{d x}+\left(\frac{6 x^2+\left(3 x^2+2 x^3+4\right) e^{-2 x}}{\left(x^3+2\right)\left(2+e^{-2 x}\right)}\right) y=2+e^{-2...

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Q JEE Main 2026

Let $x=x(y)$ be the solution of the differential equation $2 y^2 \frac{\mathrm{~d} x}{\mathrm{~d} y}-2 x y+x^2=0, y>1, x(\mathrm{e})=\mathrm{e}$. Then $x\left(\mathrm{e}^2\right)$...

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Q Jee Main 2026

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\left(1+x+x^2\right)\left(1-y+y^2\right), y(0)=\frac{1}{2}$. Then $(2 y(1)-1)$ is equal to Options...

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Q JEE Main 2024

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+\frac{2 x}{\left(1+x^2\right)^2} y=x e^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0$. Then the area...

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Q JEE Main 2024

The differential equation of the family of circles passing the origin and having center at the line $\mathrm{y}=\mathrm{x}$ is :

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Q JEE Main 2024

For a differentiable function $\mathrm{f}: I R \rightarrow I R$, suppose $f^{\prime}(\mathrm{x})=3 f(\mathrm{x})+\alpha$, where $\alpha \in \mathrm{IR}, f(0)=1$ and $...

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Q JEE Main 2024

Let $\int_0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} \mathrm{dt}=\int_0^{\mathrm{x}} \mathrm{y}(\mathrm{t}) \mathrm{dt}, 0 \leq \mathrm{x} \leq 3, \mathrm{y} \geq 0, y(0)=0$. Then at $x=2, y^{\prime \prime}+y+1$...

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Q JEE Main 2024

Let y=y(x) be the solution of the differential equation $(x+y+2)^2 d x=d y, y(0)=-2$. Let the maximum and minimum values...

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Q JEE Main 2024

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x y-2\right) d x=0$. If $y(0)=$...

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Q JEE MAIN 2024

If $\frac{d x}{d y}=\frac{1+x-y^2}{y}, x(1)=1$, then $5 x(2)$ is equal to :

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Q Jee Main 2024

Let $f(x)=\sqrt{\lim _{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}$ be differentiable in $(-\infty, 0) \cup(0, \infty)$ and $f(1)=1$. Then the value...

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Q JEE MAIN 2024

Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $\mathrm{Y}-\mathrm{y}=\mathrm{Y}^{\prime}(\mathrm{x})(\mathrm{X}- \mathrm{x})$...

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Q JEE-Main 2024

If the solution curve, of the differential equation $\frac{d y}{d x}=\frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $...

JEE Main Mathematics Medium

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Q Jee Main 2024

If $\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}$ is the solution of the differential equation $x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$ and...

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Q Jee Main 2024

Let $y=y(x)$ be the solution of the differential equation $$ \begin{gathered} \sec ^2 x d x+\left(e^{2 y} \tan ^2 x+\tan...

JEE Main Mathematics Hard

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Q JEE-Main 2024

If $y=y(x)$ is the solution curve of the differential equation $\left(x^2-4\right) d y-\left(y^2-3 y\right) d x=0, x>2, y(4)=\frac{3}{2}$ and the...

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Q Jee Main 2024

The temperature $\mathrm{T}(\mathrm{t})$ of a body at time $\mathrm{t}=0$ is $160^{\circ} \mathrm{F}$ and it decreases continuously as per the differential...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution of the differential equation $$ x \frac{d y}{d x}-\sin 2 y=x^3\left(2-x^3\right) \cos ^2 y, x...

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Q JEE MAIN 2026

If the solution curve $y=f(x)$ of the differential equation $$ \left(x^2-4\right) y^{\prime}-2 x y+2 x\left(4-x^2\right)^2=0, x>2, $$ passes through the...

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Q JEE MAIN 2026

Let a differentiable function f satisfy the equation $\int_0^{36} f\left(\frac{t x}{36}\right) d t=4 \alpha f(x)$. If $y=f(x)$ is a standard...

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Q JEE MAIN_2026_

Let $y=y(x)$ be the solution of the differential equation $x^4 \mathrm{~d} y+\left(4 x^3 y+2 \sin x\right) \mathrm{d} x=0$, $x>0, y\left(\frac{\pi}{2}\right)=0$....

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Q JEE MAIN 2026

If $y=y(x)$ satisfies the differential equation $16(\sqrt{x+9 \sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos y \mathrm{~d} y=(1+2 \sin y) \mathrm{d} x, x>0$ and $y(256)=\frac{\pi}{2}, y(49)=\alpha$,...

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Q JEE MAIN_2026_

Let the solution curve of the differential equation $x d y-y d x=\sqrt{x^2+y^2} d x, x>0, y(1)=0$, be $y=y(x)$. Then...

JEE Main Mathematics Medium

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution of the differential equation $...

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Q JEE MAIN 2026

Let $y=y(x)$ be the solution curve of the differential equation $\left(1+x^2\right) d y+\left(y-\tan ^{-1} x\right) d x=0, y(0)=1$. Then the...

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Q JEE MAIN_2019_

If $y=y(x)$ is the solution of the differential equation $\frac{\mathrm{dt}}{\mathrm{dx}}=(\tan \mathrm{x}-\mathrm{y}) \sec ^2 \mathrm{x}, \mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $...

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Q JEE MAIN 2019

The solution of the differential equation $x \frac{d y}{d x}+2 y=x^2(x \neq 0)$ with $y(1)=1$, is :

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Q JEE MAIN 2019

The general solution of the differential equation $\left(y^2-x^3\right) d x-x y d y=0(x \neq 0)$ is : (where c is...

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Q JEE MAIN 2019

Let $y=y(x)$ be the solution of the differential equation, $x \frac{d y}{d x}+y=x \log _e x x,(x>1)$. If $...

JEE Main Mathematics Easy

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Q JEE Main 2020

Let $y=y(x)$ be the solution of the differential equation, $\frac{2+\sin x}{y+1} \cdot \frac{d y}{d x}-\cos x, y>0, y(0)=1$. If $...

JEE Main Mathematics Medium

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Q JEE MAIN 2020

If for $x \geq 0, y=y(x)$ is the solution of the differential equation, $(x+1) d y=\left((x+1)^2+y-3\right) d x, y(2)=0$, then...

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Q JEE MAIN 2019

Let $f:[0,1] \rightarrow R$ be such that $f(x y)=f(x) \cdot f(y)$, for all $x, y \in[0,1]$, and $f(0) \neq 0$....

JEE Main Mathematics Hard

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Q JEE MAIN 2020

If $y=y(x)$ is the solution of the differential equation $\frac{5+\mathrm{e}^{\mathrm{x}}}{2+\mathrm{y}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{e}^{\mathrm{x}}=0$ satisfying $\mathrm{y}(0)=1$, then a value of $...

JEE Main Mathematics Medium

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Q JEE MAIN 2020

The general solution of the differential equation $\sqrt{1+x^2+y^2+x^2 y^2}+x y \frac{d y}{d x}=0$ is : (where C is a constant...

JEE Main Mathematics Hard

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Q JEE MAIN 2020

JEE Main Mathematics Medium

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Q JEE MAIN 2020

If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^2-3 x+2$ at the points where the curve intersects the $x$-axis,...

JEE Main Mathematics Medium

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Q JEE MAIN 2020

Let the normal at a point $P$ on the curve $y^2-3 x^2+y+10=0$ intersect the $y$-axis at $\left(0, \frac{3}{2}\right)$. If $m$...

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Q JEE MAIN 2019

Let $y=y(x)$ be the solution of the differential equation, $\frac{d y}{d x}+y \tan 2 x=x^2+x^2 \tan x$, $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,...

JEE Main Mathematics Hard

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Q JEE-MAIN 2020

Let $y=y(x)$ be the solution of the differential equation $...

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Q JEE MAIN_2021_

If the solution curve of the differential equation $\left(2 x-10 y^3\right) d y+y d x=0$, passes through the points $(0,1)$...

JEE Main Mathematics Easy

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Q JEE MAIN_2021_

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is...

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Q JEE Main 2019

If $\frac{d y}{d x}+\frac{3}{\cos ^2 x} y=\frac{1}{\cos ^2 x}, x \in\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)$ and $y\left(\frac{\pi}{4}\right)=\frac{4}{3}$, they $y\left(-\frac{\pi}{4}\right)$ equals

JEE Main Mathematics Hard

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Q JEE MAIN 2020

Let $y=y(x)$ be the solution of the differential equation, $x y^{\prime}-y=x^2(x \cos x+\sin x), x>0$. If $y(\pi)=\pi$, then $y^{\prime \prime}\left(\frac{\pi}{2}\right)+y\left(\frac{\pi}{2}\right)$...

JEE Main Mathematics Medium

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Q JEE MAIN 2020

The solution of the dif ferential equation $\frac{d y}{d x}-\frac{y+3 x}{\log _e(y+3 x)}+3=0$ is (where C is a constant of...

JEE Main Mathematics Easy

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Q JEE MAIN 2020

If $\frac{d y}{d x}=\frac{x y}{x^2+y^2} ; y(1)=1 ; \quad$ then a value of $x$ satisfying $y(x)=e$ is :

JEE Main Mathematics Easy

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Q JEE-MAIN 2020

If $x^3 d y+x y d x=x^2 d y+2 y d x ; x(2)=e$ and $x>1$, then $x(4)$ is equal...

JEE Main Mathematics Easy

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Q JEE MAIN 2019

Let $f:[0,1] \rightarrow R$ be such that $f(x y)=f(x) \cdot f(y)$, for all $x, y \in[0,1]$, and $f(0) \neq 0$....

JEE Main Mathematics Medium

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Q JEE MAIN 2020

The differential equation of the family of curves, $x^2=4 b(y+b), b \in R$, is

JEE Main Mathematics Easy

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Q JEE MAIN 2019

JEE Main Mathematics Medium

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Q JEE MAIN 2020

If $y=y(x)$ is the solution of the differential equation, $e^y\left(\frac{d y}{d x}-1\right)=e^x$ such that $y(0)=0$, then $y(1)$ is equal to

JEE Main Mathematics Easy

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Q JEE Main 2021

If $y \frac{d y}{d x}=x\left|\frac{y^2}{x^2}+\frac{\phi\left(\frac{y^2}{x^2}\right)}{\phi^{\prime}\left(\frac{y^2}{x^2}\right)}\right|, x>0, \phi>0$, and $y(1)=-1$, then $\phi\left(\frac{y^2}{4}\right)$ is equal to :

JEE Main Mathematics Hard

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Q JEE MAIN 2019

The curve amongst the family of curves represented by the differential equation, $\left(x^2-y^2\right) d x+2 x y d y=0$ which...

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Q JEE MAIN 2021

Let us consider a curve, $y=f(x)$ passing through the point $(-2,2)$ and the slope of the tangent to the curve...

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Q JEE Main 2021

Let $y=y(x)$ be a solution curve of the differential equation $...

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Q JEE Main 2019

If $y=y(x)$ is the solution of the differential equation, $x \frac{d y}{d x}+2 y=x^2$ satisfying $y(1)=1$, then $y\left(\frac{1}{2}\right)$ is equal...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

Let $y=y(x)$ be the solution of the differential equation, $\left(x^2+1\right)^2 \frac{d y}{d x}+2 x\left(x^2+1\right) y=1$ such that $y(0)=0$. If $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $y(x)$ be the solution of the differential equation $2 x^2 d y+\left(e^y-2 x\right) d x=0, x>0$. If $y(e)=1$, then...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

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

JEE Main Mathematics Hard

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Q _JEE MAIN_2021

Let $y=y(x)$ be solution of the following differential equation $...

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Q JEE MAIN 2021 S2

Let $y=y(x)$ be the solution of the differential equation $d y=e^{\alpha x+y} d x ; \alpha \in N$. If $...

JEE Main Mathematics Easy

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Q JEE MAIN_2021

Let $y=y(x)$ be the solution of the differential equation $\frac{d}{d x}=1+x e^{y-x},-\sqrt{2}

JEE Main Mathematics Medium

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Q JEE MAIN 2021 S2

Let $y=y(x)$ be the solution of the differential equation $\left(x-x^3\right) d y=\left(y+y x^2-3 x^4\right) d x, x>2$. If $y(3)=3$, then...

JEE Main Mathematics Hard

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Q JEE MAIN 2019

Given that the slope of the tangent to a curve $y=y(x)$ at any point ( $x, y$ ) is $...

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Q JEE MAIN 2021 S2

Let a curve $y=f(x)$ pass through the point $\left(2,\left(\log _e 2\right)^2\right)$ and have slope $\frac{2 y}{x \log _e x}$ for...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $y=y(x)$ be the solution of the differential equation $x d y=\left(y+x^3 \cos x\right) d x$ with $y(\pi)=0$, then $y\left(\frac{\pi}{2}\right)$...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If $y=y(x), y \in\left[0, \frac{\pi}{2}\right)$ is the solution of the dirrential equation $\sec y \frac{d y}{d x}-\sin x(x+y)-\sin (x-y)=0$, with...

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Q JEE MAIN 2021

Let $F:[3,5] \rightarrow R$ be a twice differentiable function on $(3,5)$ such that $F(x)=e^{-x} \int_3^x\left(3 t^2+2 t+4 F^{\prime}(t)\right) d t$...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

Let $y=y(x)$ be solution of the differential equation $\log _e\left(\frac{d y}{d x}\right)=3 x+4 y$, with $y(0)=0$. If $...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $y=y(x)$ be the solution of the differential equation $e^x \sqrt{1-y^2} d x+\left(\frac{y}{x}\right) d y=0, y(1)=-1$. Then the value of...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $y=y(x)$ be the solution of the differential equation $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If the curve, $y=y(x)$ represented by the solution of the differential equation $\left(2 x y^2-y\right) d x+x d y=0$, passes...

JEE Main Mathematics Medium

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Q JEE MAIN_2021_

Let $y=y(x)$ be the solution of the differential equation $...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $y=y(x)$ be the solution of the differential equation $x d y-y d x=\sqrt{\left(x^2-y^2\right)} d x, x \geq 1$, with...

JEE Main Mathematics Hard

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Q JEE-MAIN 2021

In the curve y = y (x) is the solution of the differential equation $...

JEE Main Mathematics Hard

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Q JEE-MAIN 2021

Let y = y (x) be the solution of the differential equation

JEE Main Mathematics Hard

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Q JEE MAIN_2021_

If $f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)$ and its firs t derivative with respect to $x$ is $-\frac{b}{a} \log _e 2$...

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Q JEE MAIN 2021

Let a curve $y=y(x)$ be given by the solution of the differential equation $$ \cos \left(\frac{1}{2} \cos ^{-1}\left(e^{-x}\right)\right) d x=\sqrt{e^{2...

JEE Main Mathematics Hard

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Q JEE MAIN_2021_

Which of the following is true for y (x) that satisfies the differential equation

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let C 1 be the curve obtained by the solution of differential equation $2 x y \frac{d y}{d x}=y^2-x^2, x>0$....

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Q JEE MAIN 2021

Let $y=y(x)$ satisfies the equation $\frac{d y}{d x}-|A|=0$, for all $x>0$ where $...

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Q JEE MAIN 2021

If $y=y(x)$ is the solution of the differential equation $\frac{d y}{d x}+(\tan x) y=\sin x, 0 \leq x \leq \frac{\pi}{3}$,...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x \frac{d y}{d x}+y=b x^4$, then for what value...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $f$ be a twice differentiable function defined on $R$ such that $f(0)=1, f^{\prime}(0)=2$ and $f^{\prime}(x) \neq 0$ for all...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

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

JEE Main Mathematics Easy

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Q JEE-MAIN 2021

Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{d y}{d x}=2(x+1)$. If the numerical value of area...

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Q JEE MAIN 2021

If $y=y(x)$ is the solution of the differential equation, $\frac{d y}{d x}+2 y \tan x=\sin x, y\left(\frac{\pi}{3}\right)=0$, then the maximum...

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Q JEE MAIN 2021

If $y=y(x)$ is the solution of the equation $...

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Q JEE MAIN 2021

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

JEE Main Mathematics Easy

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Q JEE MAIN 2021

If the tangent to the curve $y=x^3$ at the point $P\left(t, t^3\right)$ meets the curve again at $Q$, then the...

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Q JEE MAIN 2021

The rate of growth of bacteria in a culture is proportional to the number of bacteris present and the bacteria...

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Q JEE MAIN 2021

The population $P=P(t)$ at time ' $t$ ' of a certain species follows the differential equation $\frac{d P}{d t}=0.5 P-450$....

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If a curve passes through the origin and the slope of the tangent to it at any point $(x, y)$...

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Q JEE MAIN_2022

Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the...

JEE Main Mathematics Hard

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Q JEE MAIN_2022

If $\frac{d y}{d x}+2 y \tan x=\sin x, 0

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Q JEE MAIN 2022

The general solution of the differential equation $\left(x-y^2\right) d x+y\left(5 x+y^2\right) d y=0$ is :

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The slope of the tangent to a curve $C: y=y(x)$ at any point $[x, y)$ on it is $...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $y=y(x), x>1$, be the solution of the differential equation $(x-1) \frac{d y}{d x}+2 x y=\frac{1}{x-1}$, with $y(2)=\frac{1+e}{2 e^4}$. If...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

If $y=y(x)$ is the solution of the differential equation $\left(1+e^{2 x}\right) \frac{d y}{d x}+2\left(1+y^2\right) e^x=0$ and $y(0)=0$, then $...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

Let the slope of the tangent to a curve $y=f(x)$ at $(x, y)$ be given by $2 \tan (\cos x-y)$....

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $\mathrm{x}=\mathrm{x}(\mathrm{y})$ be the solution of the differential equation $2 \mathrm{ye}^{\mathrm{x} / \mathrm{y}^2} \mathrm{dx}+\left(\mathrm{y}^2-4 \mathrm{xe}^{\mathrm{x} / \mathrm{y}^2}\right) \mathrm{dy}=0$ such that...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The slope of normal at any point (x, y), x > 0, y > 0 on the curve y =...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If the solution of the differential equation $\frac{d y}{d x}+e^x\left(x^2-2\right) y=\left(x^2-2 x\right)\left(x^2-2\right) e^{2 x}$ satisfies y(0) = 0, then the...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If the solution curve of the differential equation $\left(\left(\tan ^{-1} y\right)-x\right) d y=\left(1+y^2\right) d x$ passes through the point $(1,0)$...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If $y=y(x)$ is the solution of the differential equation $x \frac{d y}{d x}+2 y=x e^x, y(1)=0$, then the local maximum...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

If $y=y(x)$ is the solution of the differential equation $2 x^2 \frac{d y}{d x}-2 x y+3 y^2=0$ such that $y(e)=\frac{e}{3}$,...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $S=(0,2 \pi)-\left\{\frac{\pi}{2}, \frac{3 \pi}{4}, \frac{3 \pi}{2}, \frac{7 \pi}{4}\right\}$. Let $y=y(x), x \in S$, be the solution curve of the...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $y=y(x)$ be the solution of the differential equation $x\left(1-x^2\right) \frac{d y}{d x}+\left(3 x^2 y-y-4 x^3\right)=0, x>1$, with $y(2)=-2$. Then...

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Q JEE MAIN 2022

Let the solution curve $y=y(x)$ of the differential equation, $\left[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}\right] x \frac{d y}{d x}=x+\left[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}\right] y$ pass through the points $(1,0)$...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let the solution curve $y=y(x)$ of the differential equation $\left(4+x^2\right) d y-2 x\left(x^2+3 y+4\right) d x=0$ pass through the origin....

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let the solution curve of the differential equation $x \frac{d y}{d x}-y=\sqrt{y^2+16 x^2}, y(1)=3$ be $y=y(x)$. Then $y(2)$ is equal...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

If $\frac{d y}{d x}+\frac{2^{x-y}\left(2^y-1\right)}{2^x-1}=0, x, y>0, y(1)=1$, then $y(2)$ is equal to :

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If the solution curve $y=y(x)$ of the differential equation $y^2 d x+\left(x^2-x y+y^2\right) d y=0$, which passes through the point...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $y=y(x)$ be the solution of the differential equation $(x+1) y^{\prime}-y=e^{3 x}(x+1)^2$, with $y(0)=\frac{1}{3}$. Then, the point $x =-\frac{4}{3}$ for...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If $x=x(y)$ is the solution of the differential equation $y \frac{d x}{d y}=2 x+y^3(y+1) e^y, x(1)=0$; then $x(e)$ is equal...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

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...

JEE Main Mathematics Hard

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Q JEE MAIN 2023

If $y=y(x)$ is the solution of the differential equation $\frac{d y}{d x}+\frac{4 x}{\left(x^2-1\right)} y=\frac{x+2}{\left(x^2-1\right)^{5 / 2}}, x>1$ such that $...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $f(x)=\frac{x}{\left(1+x^n\right)^{\frac{1}{n}}}, x \in R-\{-1\}, n \in N, n>2$. If $f^n(x)=$ (fofof ..... upto $n$ times) ( $x$ ), then...

JEE Main Mathematics Hard

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Q JEE MAIN 2023

For the system of equations $$ \begin{aligned} & x+y+z=6 \\ & x+2 y+\alpha z=10 \end{aligned} $$ $x+3 y+5 z=\beta$, which...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $f(x)=x-1$ and $g(x)=e^x$ for $x \in \mathbb{R}$. If $\frac{d y}{d x}=\left(e^{-2 \sqrt{x}} g(f(f(x)))-\frac{y}{\sqrt{x}}\right), y(0)=0$, then $y(1)$ is

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $y=y(x)$ be the solution of the differential equation $...

JEE Main Mathematics Easy

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Q JEE MAIN_2025

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+1\right) y^2-2 x y=\left(x^4+2 x^2+1\right) \cos x$. $y(0)=1$. Then $...

JEE Main Physics Medium

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Q JEE MAIN 2025

If a curve $y=y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+3\left(\tan ^2 x\right) y+3 y=\sec ^2 x, y(0)=\frac{1}{3}+e^3$. Then...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

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

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let $x=x(y)$ be the solution of the differential equation $2(y+2) \log _e(y+2) d x+\left(x+4-2 \log _e(y+2)\right) d y=0, y>-1$ with...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

If the solution curve of the differential equation $\left(y-2 \log _e x\right) d x+\left(x \log _e x^2\right) d y=0, x>1$...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $y=y(x)$ be a solution curve of the differential equation, $\left(1-x^2 y^2\right) d x=y d x+x d y$. If the...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $f:[2,4] \rightarrow \mathbb{R}$ be a differentiable function such that $...

JEE Main Mathematics Hard

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Q JEE MAIN 2023

Let $y=y(x), y>0$, be a solution curve of the differential equation $\left(1+x^2\right) d y=y(x-y) d x$. If $y(0)=1$ and $...

JEE Main Mathematics Medium

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Q JEE-Main 2023

Let y = y(x) be a solution of the differential equation (xcos x)dy + (xysinx + ycos x – l)dx...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

The slope of tangent at any point $(x, y)$ on a curve $y=y(x)$ is $\frac{x^2+y^2}{2 x y}, x>0$. If $y(2)=0$,...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $f$ be a differentiable function such that $x^2 f(x)-x=4 \int_0^x t f(t) d t, f(1)=\frac{2}{3}$. Then $18 f(3)$ is...

JEE Main Mathematics Medium

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Q JEE MAIN

The solution of the differential equation $\left(x^2+y^2\right) d x-5 x y d y=0, y(1)=0$, is :

JEE Main Mathematics Easy

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Q JEE-Main 2024

Let a conic C pass through the point (4,-2) and P(x,y),x≥3, be any point on C. Let the slope of...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $...

JEE Main Mathematics Medium

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Q JEE-Main 2024

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $...

JEE Main Mathematics Medium

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Q JEE-Main 2024

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(1+x^2\right) \frac{d y}{d x}+y=e^{\tan ^{-1} x}, \mathrm{y}(1)=0$. Then $\mathrm{y}(0)$ is $\_\_\_\_$...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let the solution $y=y(x)$ of the differential equation $\frac{d y}{d x}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+$ 10 is equal...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $y=y(x)$ be the solution of the differential equation $\left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0$,...

JEE Main Mathematics Easy

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Q JEE MAIN

The solution curve, of the differential equation $2 y \frac{d y}{d x}+3=5 \frac{d y}{d x}$ , passing through the point...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

If y=y(x) is the solution of the differential equation dydx+2y=sin⁡(2x),y(0)=34, then y8 is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If the solution $y=y(x)$ of the differential equation $\left(x^4+2 x^3+3 x^2+2 x+2\right) d y-\left(2 x^2+2 x+3\right) d x=0$ satisfies $y(-1)=-\frac{\pi}{4}$,...

JEE Main Mathematics Hard

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Q JEE MAIN 2024

Let $y=y(x)$ be the solution of the differential equation $\left(1-x^2\right) d y=\left[x y+\left(x^3+2\right) \sqrt{3\left(1-x^2\right)}\right] d x,-1< x

JEE Main Mathematics Easy

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Q JEE-Main 2024

If $x=x(t)$ is the solution of the differential equation $(t+1) d x=\left(2 x+(t+1)^4\right) d t, x(0)=2$, then, $x(1)$ equals

JEE Main Mathematics Easy

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Q JEE MAIN

If the solution curve y=y(x) of the differential equation $\left(1+y^2\right)\left(1+\log _e x\right) d x+x d y=0, x>0$ passes through the...

JEE Main Mathematics Medium

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Q JEE-Main 2024

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=2 x(x+y)^3-x(x+y)-1, y(0)=1$. Then, $\left(\frac{1}{\sqrt{2}}+y\left(\frac{1}{\sqrt{2}}\right)\right)^2$ equals :

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let $y=y(x)$ be the solution of the differential equation $...

JEE Main Mathematics Hard

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Q JEE MAIN 2024

The solution curve of the differential equation $y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0$ passing through the...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let y=y(x) be the solution of the differential equation sec⁡xdy+{2(1-x)tan⁡x+x(2-x)} dx=0 such that y(0)=2. Then y(2) is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN

A function y=f(x) satisfies $f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0$ with condition $f(0)=0$. Then $f\left(\frac{\pi}{2}\right)$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If the solution of the differential equation $(2 x+3 y-2) d x+(4 x+6 y-7) d y=0, y(0)=3$, is $...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0$ and $\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{by}=0$ respectively, $\mathrm{a}, \mathrm{b} \in \mathrm{R}$. Given that...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If for the solution curve $y=f(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+(\tan x) y=\frac{2+\sec x}{(1+2 \sec x)^{2}}$, $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If $y=y(x)$ is the solution of the differential equation, $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $y = y(x)$ be the solution curve of the differential equation $...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let $f:[0,\infty)\to$ $\mathbb{R}$ be a differentiable function such that $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let g be a differentiable function such that $\int_0^x g (t)dt = x - \int_0^x t g(t)dt,x \ge 0$ and...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $f:R \to R$ be a thrice differentiable odd function satisfying $...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let $y=y(x)$ be the solution of the differential equation $...

JEE Main Mathematics Easy

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Q

Let $f:(0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right), y>0$ and $x(1)=\frac{\pi}{2}$ Then $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $y = y(x)$ be the solution of the differential equation $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let y = y(x) be the solution of the differential equation $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let a curve $y = f(x)$ pass through the points (0,5) and $\left( {{{\log }_e}2,k} \right)$. If the curve satisfies...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If $x=f(y)$ is the solution of the differential equation $$ \left(1+y^2\right)+\left(x-2 \mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)...

JEE Main Mathematics Easy

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