The rate of increase of bacteria in a culture is proportional to the number of bacteria present at that instant. It is found that the number doubles in $6 \text{ hours}$. The number of bacteria becomes $n$ times the initial number at the end of $18 \text{ hours}$. Find $n$.

If the general solution of the differential equation $y' = \frac{y}{x} + \phi \left( \frac{x}{y} \right)$,for some function $\phi$,is given by $y \ln |cx| = x$,where $c$ is an arbitrary constant,then $\phi(2)$ is equal to:

$A$ body at an unknown temperature is placed in a room which is held at a constant temperature of $30^{\circ} F$. If after $10 \text{ minutes}$ the temperature of the body is $0^{\circ} F$ and after $20 \text{ minutes}$ the temperature of the body is $15^{\circ} F$,then the expression for the temperature of the body at any time $t$ is

If a curve passes through the origin,such that the length of the subnormal is equal to one more than the square of the ordinate,then:

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 $^{\circ} C$)?

If surrounding air is kept at $20^{\circ} C$ and a body cools from $80^{\circ} C$ to $70^{\circ} C$ in $5$ minutes,then the temperature of the body after $15$ minutes will be: (in $^{\circ} C$)