1
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1 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 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

A
$$\frac{1+3 e^{2}}{2\left(3 e^{2}-1\right)}$$
B
$$\frac{3 e^{2}}{2\left(3 e^{2}-1\right)}$$
C
$$\frac{1-3 e^{2}}{2\left(3 e^{2}+1\right)}$$
D
$$\frac{3 e^{2}}{2\left(3 e^{2}+1\right)}$$
2
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1 Let $$f$$ be a differentiable function such that $${x^2}f(x) - x = 4\int_0^x {tf(t)dt}$$, $$f(1) = {2 \over 3}$$. Then $$18f(3)$$ is equal to :

A
160
B
210
C
150
D
180
3
JEE Main 2023 (Online) 6th April Evening Shift
+4
-1 If the solution curve $$f(x, y)=0$$ of the differential equation $$\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0$$, passes through the points $$(1,0)$$ and $$(\alpha, 2)$$, then $$\alpha^{\alpha}$$ is equal to

A
$$e^{\sqrt{2} e^{2}}$$
B
$$e^{2 e^{\sqrt{2}}}$$
C
$$e^{e^{2}}$$
D
$$e^{2 e^{2}}$$
4
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1 Let $$\alpha x=\exp \left(x^{\beta} y^{\gamma}\right)$$ be the solution of the differential equation $$2 x^{2} y \mathrm{~d} y-\left(1-x y^{2}\right) \mathrm{d} x=0, x > 0,y(2)=\sqrt{\log _{e} 2}$$. Then $$\alpha+\beta-\gamma$$ equals :

A
1
B
0
C
3
D
$$-1$$
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