The probability that exactly $3$ heads appear in six tosses of an unbiased coin,given that the first three tosses resulted in $2$ or more heads is

In a single throw of two dice,what is the probability of obtaining a sum of numbers greater than $7$,given that $4$ appears on the first die?

Show that the given differential equation is homogeneous and solve it:
$y \, dx + x \log \left(\frac{y}{x}\right) dy - 2x \, dy = 0$

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

Consider the differential equation $\frac{dy}{dx} = \frac{y^3}{2(xy^2 - x^2)}$.
Statement $-1:$ The substitution $z = y^2$ transforms the above equation into a first-order homogeneous differential equation.
Statement $-2:$ The solution of this differential equation is $y^2 e^{-y^2/x} = C$.

Let $f(x) = \sqrt{\lim_{r \rightarrow x} \left\{ \frac{2r^2 \left[(f(r))^2 - f(x)f(r)\right]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}} \right\}}$ be differentiable in $(-\infty, 0) \cup (0, \infty)$ and $f(1) = 1$. Then the value of $ea$,such that $f(a) = 0$,is equal to: