The general solution of the differential equation $(3xy+y^2) dx + (x^2+xy) dy = 0$ is

If a curve $y=f(x),$ passing through the point $(1,2),$ is the solution of the differential equation $2 x^{2} dy=\left(2 xy+y^{2}\right) dx,$ then $f\left(\frac{1}{2}\right)$ is equal to

Two cards are drawn from a pack of $52$ playing cards one after the other without replacement. If the first card drawn is a queen,then the probability of getting a face card from a black suit in the second draw is

The substitution $x=vy$ converts which one of the following differential equations into an equation solvable by the variable separable method?

$I: y^{\prime}=\frac{y+x}{x} ; \quad II: y^{\prime}=\frac{x^2+y}{x^3} ; \quad III: y^{\prime}=\frac{2xy}{y^2-x^2}$
$S1$: Differential equations given by $I$ and $II$ are homogeneous differential equations.
$S2$: Differential equations given by $II$ and $III$ are homogeneous differential equations.
$S3$: Differential equations given by $I$ and $III$ are homogeneous differential equations.

The solution of the differential equation $\frac{dy}{dx} = \tan \left(\frac{y}{x}\right) + \frac{y}{x}$ is