If $y = e^{-mx}$ is a solution of the differential equation $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 3y = 0$, then the values of $m$ are

The growth of population is proportional to the number present. If the population of a colony doubles in $50$ years,then the population will become triple in . . . . . . years.

$A$ spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is $3 \text{ mm}$ and $1 \text{ hour}$ later has been reduced to $2 \text{ mm}$,then the expression of radius $r$ of the raindrop at any time $t$ is (where $0 \leq t < 3$):

$A$ 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 the time $t$,in which $99 \%$ of the moisture will be lost,is

Given $\frac{d^2 y}{d x^2}+\cot x \frac{d y}{d x}+4 y \operatorname{cosec}^2 x=0$. Changing the independent variable $x$ to $z$ by the substitution $z=\log \tan \frac{x}{2}$, the equation is changed to

The slope of a curve passing through the point $(2, 0)$ at $(x, y)$ is $x^2 - 2x$. Find the value of $y$ at $x = 0$.