The microorganisms double themselves in $3$ hours. Assuming that the quantity increases at a rate proportional to itself,then the number of times it multiplies itself in $18$ hours is:

Let $f$ be a differentiable function defined on $\left[0, \frac{\pi}{2}\right]$ such that $f(x) > 0$ and $f(x)+\int \limits_0^x f(t) \sqrt{1-\left(\log _e f(t)\right)^2} d t=e, \forall x \in\left[0, \frac{\pi}{2}\right]$. Then $\left(6 \log _{ e } f \left(\frac{\pi}{6}\right)\right)^2$ is equal to $.............$

$A$ wet substance in the open air loses its moisture at a rate proportional to the moisture content. If a sheet,hung in the open air,loses half its moisture during the first hour,then $90 \%$ of the moisture will be lost in ...... hours.

Let the curve $y = y(x)$ be the solution of the differential equation,$\frac{dy}{dx} = 2(x + 1)$. If the numerical value of the area bounded by the curve $y = y(x)$ and the $x-$axis is $\frac{4\sqrt{8}}{3}$,then the value of $y(1)$ is equal to

The number of solutions of $y' = \frac{y + 1}{x - 1}, y(1) = 2$ is

$A$ spherical rain drop evaporates at a rate proportional to its surface area. If initially its radius is $3 \ mm$ and after $1 \ second$ it is reduced to $2 \ mm$,then at any time $t$ its radius is (where $0 \leq t < 3$)