If it is mentioned that for a curve passing through $(3,4)$,the slope of the curve at any point is the reciprocal of twice the ordinate of that point,then that curve is a

The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx} + \frac{x+a}{y-2} = 0, y(1) = 0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect the $x$-axis at points $R$ and $S$ respectively,then the length of the line segment $RS$ is

The rate of increase of the population of a country is proportional to the number present. If the population doubles in $50$ years,then the time taken by it to become four times itself is (in $years$)

If a curve passes through the origin,such that the length of the subnormal is equal to one more than the square of the ordinate,then:

The solution of $\frac{d^2 y}{d x^2}=0$ represents

Radium decomposes at a rate proportional to the amount present at any time. If $P \%$ of the amount disappears in one year,then the amount of radium left after $2$ years is