1
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
2
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
$$\frac{3}{1+2 \sqrt{2}}$$
3
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
Let $x=x(\mathrm{t})$ and $y=y(\mathrm{t})$ be solutions of the differential equations $\frac{\mathrm{d} x}{\mathrm{dt}}+\mathrm{a} x=0$ and $\frac{\mathrm{d} y}{\mathrm{dt}}+\mathrm{by}=0$ respectively, $\mathrm{a}, \mathrm{b} \in \mathbf{R}$. Given that $x(0)=2 ; y(0)=1$ and $3 y(1)=2 x(1)$, the value of $\mathrm{t}$, for which $x(\mathrm{t})=y(\mathrm{t})$, is :
A
$\log _{\frac{2}{3}} 2$
B
$\log _{\frac{4}{3}} 2$
C
$\log _4 3$
D
$\log _3 4$
4
JEE Main 2023 (Online) 15th April Morning Shift
+4
-1
Let $x=x(y)$ be the solution of the differential equation

$2(y+2) \log _{e}(y+2) d x+\left(x+4-2 \log _{e}(y+2)\right) d y=0, y>-1$

with $x\left(e^{4}-2\right)=1$. Then $x\left(e^{9}-2\right)$ is equal to :
A
$\frac{4}{9}$
B
$\frac{32}{9}$
C
$\frac{10}{3}$
D
3
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