Let $y$ be the number of people in a village at time $t$. Assume that the rate of change of the population is proportional to the number of people in the village at any time and further assume that the population never increase in time. Then, the population of the village at any fixed $t$ is given by

The rate of growth of bacteria is proportional to the number present. If initially there were $1000$ bacteria and the number doubles in $1$ hour. Then, the number of bacteria after $2\frac{1 }{2}$ hours are $($Given $\left.\sqrt{2}=1.414\right)$

If the body cools from $135°\mathrm{C}$ to $80°\mathrm{C}$ at room temperature of $25°\mathrm{C}$ in $60$ minutes, then the temperature of body after $2$ hours is

Let $f:\left[\frac{1}{2},1\right]\to R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{\text{'}}\left(x\right)<2f\left(x\right)$ and $f\left(\frac{1}{2}\right)=1$. Then the value of $\underset{1/2}{\overset{1}{\int }}f\left(x\right)dx$ lies in the interval: