Water at $100^{\circ} C$ cools in $10 \text{ minutes}$ to $80^{\circ} C$ in a room temperature of $25^{\circ} C$. What will be the temperature of water after $20 \text{ minutes}$ (in $^{\circ} C$)?

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 rate of change of volume of a spherical balloon at any instant is directly proportional to its surface area. If initially its radius is $3 \ cm$,and after $2 \ minutes$ its radius becomes $9 \ cm$,then what is the radius of the balloon after $4 \ minutes$ (in $cm$)?

Let $f$ be a non-negative function defined in $[0, \pi / 2]$, $f^{\prime}$ exists and is continuous for all $x$, and $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} dt = \int_0^x f(t) dt$ with $f(0) = 0$. Then

The normal to a curve at $P(x, y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is

The rate of disintegration of a radioactive element at any time is proportional to its mass at that time, where $k (k > 0)$ is the constant of proportionality. The time during which an original mass of $1.5 \text{ gm}$ will disintegrate to a mass of $0.5 \text{ gm}$ is ...