$A$ body cools according to Newton's law of cooling from $100^{\circ} C$ to $60^{\circ} C$ in $20 \text{ minutes}$. If the temperature of the surrounding is $20^{\circ} C$,then the temperature of the body after one hour is: (in $^{\circ} C$)

The equation of the curve passing through $\left(2, \frac{9}{2}\right)$ and having the slope $\left(1-\frac{1}{x^2}\right)$ at $(x, y)$ is

Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth $y$,where the constant of proportionality $k > 0$ depends on the acceleration due to gravity and the geometry of the hole. If $t$ is measured in minutes and $k = \frac{1}{15}$,then the time required to drain the tank if the water is $4 \text{ m}$ deep to start with is .......... $\text{min}$.

The rate of disintegration of a radioactive element at time $t$ is proportional to its mass at that time. Then the time during which the original mass of $6 \text{ gm}$ will disintegrate into its mass of $3 \text{ gm}$ is proportional to

$A$ tangent is drawn at any point $P(x, y)$ on a curve,which passes through $(1, 1)$. The tangent cuts the $X$-axis and $Y$-axis at $A$ and $B$ respectively. If $AP:BP = 3:1$,then:

Let $f:[0,2] \rightarrow R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let $F(x)=\int_0^{x^2} f(\sqrt{t}) dt$ for $x \in [0,2]$. If $F'(x)=f'(x)$ for all $x \in (0,2)$,then $F(2)$ equals