1
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

If $$y=y(x), x \in(0, \pi / 2)$$ be the solution curve of the differential equation

$$\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=2 \mathrm{e}^{-4 x}(2 \sin 2 x+\cos 2 x)$$,

with $$y(\pi / 4)=\mathrm{e}^{-\pi}$$, then $$y(\pi / 6)$$ is equal to :

A
$$\frac{2}{\sqrt{3}} e^{-2 \pi / 3}$$
B
$$\frac{2}{\sqrt{3}} \mathrm{e}^{2 \pi / 3}$$
C
$$\frac{1}{\sqrt{3}} e^{-2 \pi / 3}$$
D
$$\frac{1}{\sqrt{3}} e^{2 \pi / 3}$$
2
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1

Let $$y=y_{1}(x)$$ and $$y=y_{2}(x)$$ be two distinct solutions of the differential equation $$\frac{d y}{d x}=x+y$$, with $$y_{1}(0)=0$$ and $$y_{2}(0)=1$$ respectively. Then, the number of points of intersection of $$y=y_{1}(x)$$ and $$y=y_{2}(x)$$ is

A
0
B
1
C
2
D
3
3
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1

Let the solution curve $$y=f(x)$$ of the differential equation $$\frac{d y}{d x}+\frac{x y}{x^{2}-1}=\frac{x^{4}+2 x}{\sqrt{1-x^{2}}}$$, $$x\in(-1,1)$$ pass through the origin. Then $$\int\limits_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) d x$$ is equal to

A
$$\frac{\pi}{3}-\frac{1}{4}$$
B
$$\frac{\pi}{3}-\frac{\sqrt{3}}{4}$$
C
$$\frac{\pi}{6}-\frac{\sqrt{3}}{4}$$
D
$$\frac{\pi}{6}-\frac{\sqrt{3}}{2}$$
4
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
$${3 \over 8}$$
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