The orthogonal trajectory of $x^2-y^2=a^2$ , where a is an arbitrary constant, is

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: