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

$A$ cube having an edge of length $5 \ cm$ is painted on all faces and then it is cut into equal cubes of unit volume. $A$ small cube is selected at random and it is found that at least one of its faces is painted. What is the probability that $2$ more faces of it are also painted?

The particular solution of the differential equation $(x+y) dy + (x-y) dx = 0$ at $x=1, y=1$ is

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

Show that the differential equation $y^{\prime} = \frac{x+y}{x}$ is a homogeneous equation and find its general solution.