The rate of increase of the population of a country is proportional to the number present. If the population doubles in $50$ years,then the time taken by it to become four times itself is (in $years$)

The area bounded by a curve,the $x$-axis,and the ordinate of some point $(x, y)$ of the curve is equal to the length of the corresponding arc of the curve from a fixed point. If the curve passes through the point $P(0, 1)$,then the equation of this curve is:

If the population grows at the rate of $5 \%$ per year,then the time taken for the population to become double is (Given $\log 2=0.6912$ ) (in $years$)

The population of towns $A$ and $B$ increases at a rate proportional to their population present at that time. At the end of the year $1984$,the population of both the towns was $20,000$. At the end of the year $1989$,the population of town $A$ was $25,000$ and that of town $B$ was $28,000$. The difference of populations of towns $A$ and $B$ at the end of $1994$ was

The bacteria increase at a rate proportional to the number of bacteria present. If the original number $N$ doubles in $8$ hours,then the number of bacteria in $24$ hours will be (in $N$)

If the slope of the tangent on a curve at any point $(x, y)$ is equal to $\frac{y^2-x^2}{2xy}$,then the equation of the normal at the point $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ is