If the tangent at a point $P(x, y)$ of a curve is perpendicular to the line that joins the origin with the point $P$,then the curve is

The rate of disintegration of a radioactive element at any time is proportional to its mass at that time, where $k (k > 0)$ is the constant of proportionality. The time during which an original mass of $1.5 \text{ gm}$ will disintegrate to a mass of $0.5 \text{ gm}$ is ...

$A$ spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is $3 \text{ mm}$ and $1 \text{ hour}$ later has been reduced to $2 \text{ mm}$,then the expression of radius $r$ of the raindrop at any time $t$ is (where $0 \leq t < 3$):

If a body cools from $80^{\circ} C$ to $60^{\circ} C$ in a room temperature of $30^{\circ} C$ in $30 \text{ min}$,then the temperature of the body after one hour is (in $^{\circ} C$)

Let a curve $y=f(x), x \in(0, \infty)$ pass through the points $P\left(1, \frac{3}{2}\right)$ and $Q\left(a, \frac{1}{2}\right)$. If the tangent at any point $R(b, f(b))$ to the given curve cuts the $y$-axis at the point $S(0, c)$ such that $bc=3$,then $(PQ)^2$ is equal to $.........$.

The rate of disintegration of a radioactive element at any time $t$ is proportional to its mass $M$ at that time. If $k (k > 0)$ is the constant of proportionality, find the time required for an initial mass of $1.5 \text{ g}$ to disintegrate to a mass of $0.5 \text{ g}$.