1
JEE Main 2022 (Online) 25th June Evening Shift
+4
-1 If $$y = y(x)$$ is the solution of the differential equation

$$2{x^2}{{dy} \over {dx}} - 2xy + 3{y^2} = 0$$ such that $$y(e) = {e \over 3}$$, then y(1) is equal to

A
$${1 \over 3}$$
B
$${2 \over 3}$$
C
$${3 \over 2}$$
D
3
2
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1 Let $$g:(0,\infty ) \to R$$ be a differentiable function such that

$$\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over {{{({e^x} + 1)}^2}}}} \right)dx = {{x\,g(x)} \over {{e^x} + 1}} + c}$$, for all x > 0, where c is an arbitrary constant. Then :

A
g is decreasing in $$\left( {0,{\pi \over 4}} \right)$$
B
g' is increasing in $$\left( {0,{\pi \over 4}} \right)$$
C
g + g' is increasing in $$\left( {0,{\pi \over 2}} \right)$$
D
g $$-$$ g' is increasing in $$\left( {0,{\pi \over 2}} \right)$$
3
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1 Let $$y = y(x)$$ be the solution of the differential equation $$(x + 1)y' - y = {e^{3x}}{(x + 1)^2}$$, with $$y(0) = {1 \over 3}$$. Then, the point $$x = - {4 \over 3}$$ for the curve $$y = y(x)$$ is :

A
not a critical point
B
a point of local minima
C
a point of local maxima
D
a point of inflection
4
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1 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 :

A
$${\pi \over 3}$$
B
$${\pi \over 2}$$
C
$${\pi \over 12}$$
D
$${\pi \over 6}$$
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