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 rate of increase of the population of a city is proportional to the population present at that instant. In the period of $40$ years,the population increased from $30,000$ to $40,000$. At any time $t$,the population is given by $P(t) = (a)(b)^{\frac{t}{40}}$. Then the values of $a$ and $b$ are respectively:

The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx} + \frac{x+a}{y-2} = 0, y(1) = 0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect the $x$-axis at points $R$ and $S$ respectively,then the length of the line segment $RS$ is

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 $^{\circ} C$)

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:

$A$ 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: