Three fair coins are tossed. If both heads and tails appear,then the probability that exactly one head appears is:

If $x \sin \left(\frac{y}{x}\right) dy = \left[y \sin \left(\frac{y}{x}\right) - x\right] dx$,$x > 0$ and $y(1) = \frac{\pi}{2}$,then the value of $\cos \left(\frac{y}{x}\right)$ is

The general solution of the differential equation $\frac{d y}{d x}=\frac{3 x+y}{x-y}$ is (where $C$ is a constant of integration.)

Show that the differential equation $(x^{2}-y^{2}) dx + 2xy dy = 0$ is a homogeneous equation and find its solution.

Which one of the following is a homogeneous differential equation?

One card is drawn at random from a well-shuffled deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent?
$E:$ 'The card drawn is a king or a queen'
$F:$ 'The card drawn is a queen or a jack'