1
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1

If $${{dy} \over {dx}} + 2y\tan x = \sin x,\,0 < x < {\pi \over 2}$$ and $$y\left( {{\pi \over 3}} \right) = 0$$, then the maximum value of $$y(x)$$ is :

A
$${1 \over 8}$$
B
$${3 \over 4}$$
C
$${1 \over 4}$$
D
$${3 \over 8}$$
2
JEE Main 2022 (Online) 25th July Evening Shift
+4
-1
Out of Syllabus

Let a smooth curve $$y=f(x)$$ be such that the slope of the tangent at any point $$(x, y)$$ on it is directly proportional to $$\left(\frac{-y}{x}\right)$$. If the curve passes through the points $$(1,2)$$ and $$(8,1)$$, then $$\left|y\left(\frac{1}{8}\right)\right|$$ is equal to

A
$$2 \log _{e} 2$$
B
4
C
1
D
$$4 \log _{e} 2$$
3
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1

The slope of the tangent to a curve $$C: y=y(x)$$ at any point $$(x, y)$$ on it is $$\frac{2 \mathrm{e}^{2 x}-6 \mathrm{e}^{-x}+9}{2+9 \mathrm{e}^{-2 x}}$$. If $$C$$ passes through the points $$\left(0, \frac{1}{2}+\frac{\pi}{2 \sqrt{2}}\right)$$ and $$\left(\alpha, \frac{1}{2} \mathrm{e}^{2 \alpha}\right)$$, then $$\mathrm{e}^{\alpha}$$ is equal to :

A
$$\frac{3+\sqrt{2}}{3-\sqrt{2}}$$
B
$$\frac{3}{\sqrt{2}}\left(\frac{3+\sqrt{2}}{3-\sqrt{2}}\right)$$
C
$$\frac{1}{\sqrt{2}}\left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)$$
D
$$\frac{\sqrt{2}+1}{\sqrt{2}-1}$$
4
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1

The general solution of the differential equation $$\left(x-y^{2}\right) \mathrm{d} x+y\left(5 x+y^{2}\right) \mathrm{d} y=0$$ is :

A
$$\left(y^{2}+x\right)^{4}=\mathrm{C}\left|\left(y^{2}+2 x\right)^{3}\right|$$
B
$$\left(y^{2}+2 x\right)^{4}=C\left|\left(y^{2}+x\right)^{3}\right|$$
C
$$\left|\left(y^{2}+x\right)^{3}\right|=\mathrm{C}\left(2 y^{2}+x\right)^{4}$$
D
$$\left|\left(y^{2}+2 x\right)^{3}\right|=C\left(2 y^{2}+x\right)^{4}$$
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