Let $A$ and $B$ be two independent events. If $P(A) = \frac{1}{5}$ and $P(A \cup B) = \frac{7}{10}$,then what is the value of $P(\overline{B})$?

Suppose that $f(x, y)$ and $g(x, y)$ are homogeneous functions of the same order. If $x=Vy$ reduces the equation $\frac{dy}{dx}=\frac{f(x, y)}{g(x, y)}$ to the form $\frac{dV}{dy}=\frac{1}{y}(F(V))$,then $F(V)=$

If $\frac{dy}{dx} = f(x, y)$ is a homogeneous differential equation,then the general form of $f(x, y)$ is

The general solution of the differential equation $(x \sin \frac{y}{x}) dy = (y \sin \frac{y}{x} - x) dx$ is

Find the solution of the following differential equation: $\{x \cos (y/x) + y \sin (y/x)\} y dx = \{y \sin (y/x) - x \cos (y/x)\} x dy$.

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