If $A$ and $B$ are two events such that $P(A) = \frac{1}{3}$,$P(B) = \frac{1}{5}$,and $P(A \cup B) = \frac{1}{2}$,then $P(A \mid B') + P(B \mid A')$ is equal to

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:

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

$A$ die is thrown twice. Let $A$ be the event of getting a prime number when the die is thrown first time and $B$ be the event of getting an even number when the die is thrown second time. Then $P(A / \overline{B})=$

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

If $y' = \frac{x - y}{x + y}$,then its solution is