1
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 $$\left( {0,{\pi \over 2}} \right)$$
2
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
3
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
$${\pi \over 6}$$
4
JEE Main 2022 (Online) 24th June Morning Shift
+4
-1

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

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

A
$${e^3}({e^e} - 1)$$
B
$${e^e}({e^3} - 1)$$
C
$${e^2}({e^e} + 1)$$
D
$${e^e}({e^2} - 1)$$
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