1
JEE Main 2024 (Online) 30th January Morning Shift
+4
-1

Let $$y=y(x)$$ be the solution of the differential equation $$\sec x \mathrm{~d} y+\{2(1-x) \tan x+x(2-x)\} \mathrm{d} x=0$$ such that $$y(0)=2$$. Then $$y(2)$$ is equal to:

A
$$2\{\sin (2)+1\}$$
B
2
C
1
D
$$2\{1-\sin (2)\}$$
2
JEE Main 2024 (Online) 29th January Evening Shift
+4
-1

If $$\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}$$ is the solution of the differential equation $$x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$$ and $$y(1)=\frac{\pi}{3}$$, then $$\alpha^2$$ is equal to

A
12
B
9
C
4
D
3
3
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

A function $$y=f(x)$$ satisfies $$f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0$$ with condition $$f(0)=0$$. Then, $$f\left(\frac{\pi}{2}\right)$$ is equal to

A
2
B
1
C
$$-$$1
D
0
4
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

If $$y=y(x)$$ is the solution curve of the differential equation $$\left(x^2-4\right) \mathrm{d} y-\left(y^2-3 y\right) \mathrm{d} x=0, x>2, y(4)=\frac{3}{2}$$ and the slope of the curve is never zero, then the value of $$y(10)$$ equals :

A
$$\frac{3}{1+(8)^{1 / 4}}$$
B
$$\frac{3}{1-(8)^{1 / 4}}$$
C
$$\frac{3}{1-2 \sqrt{2}}$$
D
$$\frac{3}{1+2 \sqrt{2}}$$
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