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Let \( y=y_{1}(x) \) and \( y=y_{2}(x) \) be two distinct solutions of the differential equation \( \dfrac{dy}{dx}=x+y \) with \( y_{1}(0)=0 \) and \( y_{2}(0)=1 \) respectively. Then, the number of points of intersection of \( y=y_{1}(x) \) and \( y=y_{2}(x) \) is:

If \( y = y(x) \) is the solution of the differential equation \( x \dfrac{dy}{dx} + 2y = x e^{x} , \, y(1) = 0 \) then the local maximum value of the function \( z(x) = x^{2} y(x) - e^{x} , \, x \in \mathbb{R} \) is:

Let \( y = y(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{\sqrt{2}\,y}{2\cos^4 x - \cos 2x} = x e^{\tan^{-1}(\sqrt{2}\cot 2x)} \), \( 0 < x < \frac{\pi}{2} \) with \( y\!\left(\frac{\pi}{4}\right) = \frac{\pi^2}{32} \). If \( y\!\left(\frac{\pi}{3}\right) = \frac{\pi^2}{18} e^{-\tan^{-1}(\alpha)} \), then the value of \( 3\alpha^2 \) is equal to ________.

If \( \dfrac{dy}{dx} + 2y \tan x = \sin x, \, 0 < x < \dfrac{\pi}{2} \) and \( y\!\left(\dfrac{\pi}{3}\right) = 0 \), then the maximum value of \( y(x) \) is:

Let \( y = y(x),\; x>1 \), be the solution of the differential equation \( (x-1)\frac{dy}{dx} + 2xy = \frac{1}{x-1} \) with \( y(2) = \frac{1+e^{4}}{2e^{4}} \). If \( y(3) = \frac{e^{\alpha}+1}{\beta e^{\alpha}} \), then the value of \( \alpha + \beta \) is equal to ______.

Let the solution curve \( y=f(x) \) of the differential equation \( \dfrac{dy}{dx}+\dfrac{xy}{x^2-1}=\dfrac{x^4+2x}{\sqrt{1-x^2}}, \, x \in (-1,1) \) pass through the origin. Then \( \displaystyle \int\limits_{-\dfrac{\sqrt{3}}{2}}^{\dfrac{\sqrt{3}}{2}} f(x)\,dx \) is equal to:

If the solution of the differential equation \( \dfrac{dy}{dx} + e^{x}(x^{2} - 2)y = (x^{2} - 2x)(x^{2} - 2)e^{2x} \) satisfies \( y(0) = 0 \), then the value of \( y(2) \) is ________.

If \( y = y(x), \ x \in (0,\pi/2) \) be the solution curve of the differential equation \( (\sin^2 2x)\dfrac{dy}{dx} + (8\sin^2 2x + 2\sin 4x)y = 2e^{-4x}(2\sin 2x + \cos 2x), \) with \( y(\pi/4)=e^{-\pi} \), then \( y(\pi/6) \) is equal to:

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 differential equation

$\frac{dy}{dx} = \frac{1}{1 + \sin 2x}, y\left(\frac{\pi}{4}\right) = \frac{1}{2}.$

If the sum of abscissas of all the points of intersection of the curve $y = y(x)$ with the curve ($y = \sqrt{2} \sin x$) is $\frac{k\pi}{12}$, then $k$ is equal to ___________.

If the solution curve of the differential equation \( \big((\tan^{-1}y)-x\big)\,dy=(1+y^{2})\,dx \) passes through the point \( (1,0) \), then the abscissa of the point on the curve whose ordinate is \( \tan(1) \), is

Let \( y = y(x) \) be the solution of the differential equation \( x(1-x^{2})\dfrac{dy}{dx} + \left(3x^{2}y - y - 4x^{3}\right) = 0 \), \( x > 1 \), with \( y(2) = -2 \). Then \( y(3) \) is equal to

Let $y = y(x)$ be the solution of the differential equation $(1-x^2)dy = \left(xy + (x^3+2)\sqrt{1-x^2}\right) dx$, $-1 < x < 1$, and $y(0) = 0$. If $\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^2} \, y(x) \, dx = k$, then $k^{-1}$ is equal to _________.

Let \( y=y(x) \) be the solution curve of the differential equation \( \dfrac{dy}{dx}+\left(\dfrac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)y=\dfrac{(x+3)}{x+1},\, x>-1, \) which passes through the point \( (0,1) \). The \( y(1) \) is equal to

Let \( y=y(x) \) be the solution curve of the differential equation \( \sin(2x^{2})\log_{e}(\tan x^{2})\,dy+\left(4xy-4\sqrt{2}\,x\sin\!\left(x^{2}-\dfrac{\pi}{4}\right)\right)dx=0,\; 0<x<\sqrt{\dfrac{\pi}{2}}\), which passes through the point \(\left(\sqrt{\dfrac{\pi}{6}},1\right)\). Then \(\Big|y\left(\sqrt{\dfrac{\pi}{3}}\right)\Big|\) is equal to_______.

Let \( y = y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx} = \dfrac{4y^3 + 2y x^2}{3xy^2 + x^3} \), \( y(1) = 1 \). If for some \( n \in \mathbb{N} \), \( y(2) \in [n-1, n) \), then \( n \) is equal to ____________ .

If the solution curve \( y=y(x) \) of the differential equation \( y^{2}dx+(x^{2}-xy+y^{2})dy=0 \), which passes through the point \( (1,1) \) and intersects the line \( y=\sqrt{3}\,x \) at the point \( (\alpha,\sqrt{3}\alpha) \), then value of \( \log_{e}(\sqrt{3}\alpha) \) is equal to:

Let the slope of the tangent to a curve \( y=f(x) \) at \( (x,y) \) be given by \( 2 \tan x \left( \cos x - y \right) \). If the curve passes through the point \( \left( \frac{\pi}{4} , 0 \right) \), then the value of \( \int_{0}^{\pi/2} y \, dx \) is equal to:

If the solution curve of the differential equation \( \dfrac{dy}{dx}=\dfrac{x+y-2}{x-y} \) passes through the points \( (2,1) \) and \( (k+1,2) \), \( k>0 \), then

If \( y=y(x) \) is the solution of the differential equation \( 2x^{2}\frac{dy}{dx}-2xy+3y^{2}=0 \) such that \( y(e)=\frac{e}{3} \), then \( y(1) \) is equal to

Let the solutions curve of the differential equation: \( x \dfrac{dy}{dx}-y=\sqrt{y^{2}+16x^{2}}, \ y(1)=3 \) be \( y=y(x) \). Then \( y(2) \) is equal to:

The differential equation of the family of circles passing through the points \( (0,2) \) and \( (0,-2) \) is:

Let the solution curve \( y=y(x) \) of the differential equation \( (1+e^{2x})\!\left(\dfrac{dy}{dx}+y\right)=1 \) pass through the point \( \left(0,\dfrac{\pi}{2}\right) \). Then, \( \displaystyle\lim_{x\to \infty} e^{x}y(x) \) is equal to:

Let the solution curve \( y = y(x) \) of the differential equation
$\left[ \frac{x}{\sqrt{x^2 - y^2}} + e^{\frac{y}{x}} \right] x \frac{dy}{dx} = x + \left[ \frac{x}{\sqrt{x^2 - y^2}} + e^{\frac{y}{x}} \right] y$
pass through the points \( (1,0) \) and \( (2\alpha,\alpha) \), \( \alpha > 0 \). Then \( \alpha \) is equal to:

The general solution of the differential equation \( (x-y^{2})dx+y(5x+y^{2})dy=0 \) is :

Let \( f \) be a twice differentiable function on \( \mathbb{R} \). If \( f'(0) = 4 \) and \( f(x) + \displaystyle\int\limits_{0}^{x} (x-t) f'(t) \, dt = (e^{2x} + e^{-2x}) \cos 2x + \dfrac{2}{a} x \), then \( (2a+1)^5 a^2 \) is equal to ____________ .

A particle is moving in the \( xy \)-plane along a curve \( C \) passing through the point \( (3,3) \). The tangent to the curve \( C \) at the point \( P \) meets the \( x \)-axis at \( Q \). If the \( y \)-axis bisects the segment \( PQ \), then \( C \) is a parabola with:

Let the solution curve of the differential equation \( xdy = \left( \sqrt{x^2 + y^2} + y \right) dx, \ x > 0, \) intersect the line \( x = 1 \) at \( y = 0 \) and the line \( x = 2 \) at \( y = \alpha \). Then the value of \( \alpha \) is:

For the curve \( C:\ (x^{2}+y^{2}-3)+(x^{2}-y^{2}-1)^{5}=0 \), the value of \( 3y' - y^{3}y'' \), at the point \( (\alpha,\alpha) \), \( \alpha>0 \), on \( C \), is equal to ________.

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

Let a smooth curve \( y=f(x) \) be such that the slope of the tangent at any point \( (x,y) \) on it is directly proportional to \( \left( \dfrac{-y}{x} \right) \). If the curve passes through the points \( (1,2) \) and \( (8,1) \) then \( \left| y\left(\dfrac{1}{8}\right) \right| \) is equal to:

Let \( x = x(y) \) be the solution of the differential equation \( 2 y e^{x / y^{2}} \, dx + \left( y^{2} - 4 x e^{x / y^{2}} \right) dy = 0 \) such that \( x(1) = 0 \). Then, \( x(e) \) is equal to:

The slope of normal at any point \( (x,y) \), \( x>0 \), \( y>0 \) on the curve \( y=y(x) \) is given by \( \frac{x^2}{xy-x^2y^2-1} \). If the curve passes through the point \( (1,1) \), then \( e \cdot y(e) \) is equal to:

If \( y=y(x) \) is the solution of the differential equation \( (1+e^{2x})\frac{dy}{dx}+2(1+y^{2})e^{x}=0 \) and \( y(0)=0 \), then \( 6\left(y'(0)+y(\log_{e}\sqrt{3})\right)^{2} \) is equal to:

Let \( y=y(x) \) be the solution curve of the differential equation \( \dfrac{dy}{dx}+\dfrac{1}{x^2-1}y=\left(\dfrac{x-1}{x+1}\right)^{1/2},\ x>1 \) passing through the point \( \left(2,\sqrt{\dfrac{1}{3}}\right) \). Then \( \sqrt{7}\,y(8) \) is equal to:

Let the solution curve \( y = y(x) \) of the differential equation \( (4 + x^2)\,\,dy - 2x(x^2 + 3y + 4)\,dx = 0 \) pass through the origin. Then \( y(2) \) is equal to ________________ .

Let \( y=y(x) \) be the solution of the differential equation \( (x+1)y'-y=e^{3x}(x+1)^{2} \), with \( y(0)=\dfrac{1}{3} \). Then, the point \( x=-\dfrac{4}{3} \) for the curve \( y=y(x) \) is:

Let \( \frac{dy}{dx} = \frac{ax - by + a}{bx + cy + a} \), where \( a, b, c \) are constants, represent a circle passing through the point \( (2,5) \). Then the shortest distance of the point \( (11,6) \) from this circle is:

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JEE Main 2022 27th July Evening Shift

For the curve \( C:\ (x^{2}+y^{2}-3)+(x^{2}-y^{2}-1)^{5}=0 \), the value of \( 3y' - y^{3}y'' \), at the point \( (\alpha,\alpha) \), \( \alpha>0 \), on \( C \), is equal to ________.