The solution of the differential equation $x^{2} \frac{dy}{dx} = y^{2} + xy$ is

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

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

Which of the following is not a homogeneous function?

$A$ curve passes through the point $\left(1, \frac{\pi}{6}\right)$. Let the slope of the curve at each point $(x, y)$ be given by $\frac{dy}{dx} = \frac{y}{x} + \sec \left(\frac{y}{x}\right)$,where $x > 0$. Then the equation of the curve is:

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{4y^3+2yx^2}{3xy^2+x^3}$ with $y(1)=1$. If for some $n \in N$,$y(2) \in [n-1, n)$,then $n$ is equal to...