$3 P(A) = P(B) = \frac{5}{13}$ and $P(A \mid B) = \frac{3}{5}$,then $P(A \cup B) = $ . . . . . . .

An unbiased coin is tossed $3$ times. If the third toss results in a head,what is the probability of getting at least one more head in the first two tosses?

The differential equation $y^{\prime} = \frac{y}{x + \sqrt{xy}}$ has a general solution given by (where $C$ is a constant of integration):

$I: y^{\prime}=\frac{y+x}{x} ; \quad II: y^{\prime}=\frac{x^2+y}{x^3} ; \quad III: y^{\prime}=\frac{2xy}{y^2-x^2}$
$S1$: 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.

$(x^2 + y^2)dy = xy dx$. If $y(x_0) = e$ and $y(1) = 1$,then the value of $x_0$ is:

If the gradient of the tangent at any point $(x, y)$ of a curve which passes through the point $\left( 1, \frac{\pi}{4} \right)$ is $\left\{ \frac{y}{x} - \sin^2\left( \frac{y}{x} \right) \right\}$,then the equation of the curve is: