The bacteria increases at the rate proportional to the number of bacteria present. If the original number '$$N$$' doubles in $$4 \mathrm{~h}$$, then the number of bacteria in $$12 \mathrm{~h}$$ will be

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are $$27 \mathrm{~gm}$$ of certain substance and $$3 \mathrm{~h}$$ later it is found that $$8 \mathrm{~gm}$$ are left, then the amount left after one more hour is

The integrating factor of the differential equation $$\left(1+x^2\right) d t=\left(\tan ^{-1} x-t\right) d x$$ is

The order of the differential equation of all circles which lie in the first quadrant and touch both the axes is......