In a certain culture of bacteria,the rate of increase is proportional to the number present. It is found that the number doubles in $4$ hours. Then the number of times the bacteria are increased in $12$ hours is

The latus rectum of the conic satisfying the differential equation $x dy + y dx = 0$ and passing through the point $(2, 8)$ is:

The solution of the differential equation $x \frac{d^2 y}{d x^2} = 1$ at $x = y = 1$ with $\frac{dy}{d x} = 0$ at $x = 1$ is:

The rate at which the population of a city increases varies as the population present. Within the period of $30$ years,the population grew from $20$ lakhs to $40$ lakhs. Then,the population after a further period of $15$ years will be (Take $\sqrt{2} = 1.41$) (in $lakhs$)

The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is $290 \ K$ and the metal temperature drops from $370 \ K$ to $330 \ K$ in $10 \ \text{minutes}$,then the time required to drop the temperature up to $295 \ K$ is

$A$ body cools according to Newton's law of cooling from $100^{\circ} C$ to $60^{\circ} C$ in $15$ minutes. If the temperature of the surrounding is $20^{\circ} C$,then the temperature of the body after cooling down for one hour is (in $^{\circ} C$)