The decay rate of radium is proportional to the amount present at any time $t$. If initially $60 \text{ gms}$ was present and half-life period of radium is $1600 \text{ years}$, then the amount of radium present after $3200 \text{ years}$ is (in $\text{ grams}$)

The normal to a curve at $P(x, y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is

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

Radium decomposes at a rate proportional to the amount present at any time. If $P \%$ of the amount disappears in one year,then the amount of radium left after $2$ years is

$A$ spherical rain drop evaporates at a rate proportional to its surface area. If initially its radius is $3 \ mm$ and after $1 \ second$ it is reduced to $2 \ mm$,then at any time $t$ its radius is (where $0 \leq t < 3$)

In a certain culture of bacteria,the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are $10,000$ bacteria at the end of $3$ hours and $40,000$ bacteria at the end of $5$ hours. Find the number of bacteria present in the beginning.