$A$ radioactive substance has a half-life of $h$ days. Its initial decay rate is given by (where the radioactive substance has an initial mass $m_0$).

The slope at any point of a curve $y=f(x)$ is given by $\frac{dy}{dx}=3x^2$ and it passes through $(-1,1)$. The equation of the curve is

The solution of the differential equation $x \frac{d^2y}{dx^2} = 1$,given that $y = 1$ and $\frac{dy}{dx} = 0$ when $x = 1$,is

The rate of change of volume of a spherical balloon at any instant is directly proportional to its surface area. If initially its radius is $3 \ cm$,and after $2 \ minutes$ its radius becomes $9 \ cm$,then what is the radius of the balloon after $4 \ minutes$ (in $cm$)?

If a body cools from $80^{\circ} C$ to $50^{\circ} C$ in a room at a temperature of $25^{\circ} C$ in $30$ minutes,then the temperature of the body after $1$ hour is: (in $^{\circ} C$)

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