If the half-life of a substance is $5$ years,then the total amount of the substance left after $15$ years,when the initial amount is $64$ gms,is: (in $gm$)

The equation of a curve whose normal at any point has a slope which is the same as the ordinate and which passes through $(1, -1)$ is $2x = k(3 - y^2)$. Then $k$ is:

Suppose a continuous function $f:(0, \infty) \rightarrow R$ satisfies $f(x)=2 \int_0^x t f(t) d t+1, \forall x \geq 0$. Then,$f(1)$ equals

The decay rate of radioactive material at any time $t$ is proportional to its mass at that time. The mass is $27 \text{ grams}$ when $t=0$. After $3 \text{ hours}$,it was found that $8 \text{ grams}$ are left. Then the substance left after one more hour is

In a certain culture of bacteria,the rate of increase is proportional to the number present. If there are $10^4$ at the end of $3$ hours and $4 \cdot 10^4$ at the end of $5$ hours,then there were $\qquad$ in the beginning.

At any point $(x, y)$ on a curve,if the length of the subnormal is $(x - 1)$ and the curve passes through $(1, 2)$,then the curve is a conic. $A$ vertex of the curve is: