The population of a town increases at a rate proportional to the population at that time. If the population increases from $40,000$ to $80,000$ in $40$ years,then the population in another $40$ years will be (in $,000$)

$A$ spherical balloon expands at a rate proportional to its surface area. Initially, its radius is $2 \text{ cm}$ and $5$ minutes later it increases to $7 \text{ cm}$. What will be the surface area of the spherical balloon after $12$ minutes (in $\text{ cm}^2$)?

The number of solutions of $y' = \frac{y + 1}{x - 1}, y(1) = 2$ is

If the solution of the differential equation $\frac{dy}{dx} = \frac{1+x}{2y}$ is a conic passing through the point $(1, 1)$,then its eccentricity is:

$A$ tangent to the curve $y = f(x)$ at $P(x, y)$ meets the $x-$axis at $A$ and the $y-$axis at $B$. If $AP : BP = 1 : 3$ and $f(1) = 1$,then the curve also passes through the point:

Let $f$ be a differentiable function defined on $\left[0, \frac{\pi}{2}\right]$ such that $f(x) > 0$ and $f(x)+\int \limits_0^x f(t) \sqrt{1-\left(\log _e f(t)\right)^2} d t=e, \forall x \in\left[0, \frac{\pi}{2}\right]$. Then $\left(6 \log _{ e } f \left(\frac{\pi}{6}\right)\right)^2$ is equal to $.............$