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 $cm$)?

$A$ spherical rain drop evaporates at a rate proportional to its surface area. The differential equation corresponding to the rate of change of the radius of the rain drop,if the constant of proportionality is $K > 0$,is

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t = 0$. The number of bacteria is increased by $20\%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{k}{\log_{e}\left(\frac{6}{5}\right)}$ hours,then $\left(\frac{k}{\log_{e} 2}\right)^{2}$ is equal to

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:

If the tangent at a point $P(x, y)$ of a curve is perpendicular to the line that joins the origin with the point $P$,then the curve is

$A$ family of curves is such that the length intercepted on the $y$-axis between the origin and the tangent at any point $(x, y)$ is three times the ordinate of the point of contact. The family of curves is: