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1
JEE Main 2022 (Online) 29th June Evening Shift
+4
-1 English
Hindi

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

A
2
B
$$-$$2
C
$$-$$4
D
$$-$$1
2
JEE Main 2022 (Online) 29th June Morning Shift
+4
-1 English
Hindi

Let the solution curve of the differential equation

$$x{{dy} \over {dx}} - y = \sqrt {{y^2} + 16{x^2}}$$, $$y(1) = 3$$ be $$y = y(x)$$. Then y(2) is equal to:

A
15
B
11
C
13
D
17
3
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1 English
Hindi

Let x = x(y) be the solution of the differential equation

$$2y\,{e^{x/{y^2}}}dx + \left( {{y^2} - 4x{e^{x/{y^2}}}} \right)dy = 0$$ such that x(1) = 0. Then, x(e) is equal to :

A
$$e{\log _e}(2)$$
B
$$- e{\log _e}(2)$$
C
$${e^2}{\log _e}(2)$$
D
$$- {e^2}{\log _e}(2)$$
4
JEE Main 2022 (Online) 28th June Morning Shift
+4
-1 English
Hindi

Let the solution curve $$y = y(x)$$ of the differential equation

$$\left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]x{{dy} \over {dx}} = x + \left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]y$$

pass through the points (1, 0) and (2$$\alpha$$, $$\alpha$$), $$\alpha$$ > 0. Then $$\alpha$$ is equal to

A
$${1 \over 2}\exp \left( {{\pi \over 6} + \sqrt e - 1} \right)$$
B
$${1 \over 2}\exp \left( {{\pi \over 6} + e - 1} \right)$$
C
$$\exp \left( {{\pi \over 6} + \sqrt e + 1} \right)$$
D
$$2\exp \left( {{\pi \over 3} + \sqrt e - 1} \right)$$
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