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 $1.5 \text{ g}$ will disintegrate into its mass of $0.5 \text{ g}$ is proportional to

$A$ spherical raindrop evaporates at a rate proportional to its surface area. The differential equation involving the rate of change of its radius $r$ with time $t$ is ... (where $k$ is a positive constant)

The temperature $T(t)$ of a body at time $t=0$ is $160^{\circ} F$ and it decreases continuously as per the differential equation $\frac{dT}{dt}=-K(T-80)$,where $K$ is a positive constant. If $T(15)=120^{\circ} F$,then $T(45)$ is equal to . . . . . . . . (in $^{\circ} F$)

If the equation of the curve which passes through the point $(1,1)$ satisfies the differential equation $\frac{dy}{dx} = \frac{2x-5y+3}{5x+2y-3}$, then the equation of that curve is:

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 $.............$

If a body cools from $80^{\circ} C$ to $50^{\circ} C$ in a room at a temperature of $25^{\circ} C$ in $30$ minutes,then the temperature of the body after $1$ hour is: (in $^{\circ} C$)