1
JEE Main 2021 (Online) 27th July Morning Shift
+4
-1
Let y = y(x) be solution of the differential equation

$${\log _{}}\left( {{{dy} \over {dx}}} \right) = 3x + 4y$$, with y(0) = 0.

If $$y\left( { - {2 \over 3}{{\log }_e}2} \right) = \alpha {\log _e}2$$, then the value of $$\alpha$$ is equal to :
A
$$- {1 \over 4}$$
B
$${1 \over 4}$$
C
$$2$$
D
$$- {1 \over 2}$$
2
JEE Main 2021 (Online) 25th July Evening Shift
+4
-1
Let y = y(x) be the solution of the differential

equation xdy = (y + x3 cosx)dx with y($$\pi$$) = 0, then $$y\left( {{\pi \over 2}} \right)$$ is equal to :
A
$${{{\pi ^2}} \over 4} + {\pi \over 2}$$
B
$${{{\pi ^2}} \over 2} + {\pi \over 4}$$
C
$${{{\pi ^2}} \over 2} - {\pi \over 4}$$
D
$${{{\pi ^4}} \over 4} - {\pi \over 2}$$
3
JEE Main 2021 (Online) 25th July Morning Shift
+4
-1
Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x < \sqrt 2 ,y(0) = 0$$

then, the minimum value of $$y(x),x \in \left( { - \sqrt 2 ,\sqrt 2 } \right)$$ is equal to :
A
$$\left( {2 - \sqrt 3 } \right) - {\log _e}2$$
B
$$\left( {2 + \sqrt 3 } \right) + {\log _e}2$$
C
$$\left( {1 + \sqrt 3 } \right) - {\log _e}\left( {\sqrt 3 - 1} \right)$$
D
$$\left( {1 - \sqrt 3 } \right) - {\log _e}\left( {\sqrt 3 - 1} \right)$$
4
JEE Main 2021 (Online) 22th July Evening Shift
+4
-1
Let y = y(x) be the solution of the differential equation $$\cos e{c^2}xdy + 2dx = (1 + y\cos 2x)\cos e{c^2}xdx$$, with $$y\left( {{\pi \over 4}} \right) = 0$$. Then, the value of $${(y(0) + 1)^2}$$ is equal to :
A
e1/2
B
e$$-$$1/2
C
e$$-$$1
D
e
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