If the half-life of radium is $77$ days,its decay constant in $day^{-1}$ will be:

$A$ radioactive isotope $X$ has a half-life of $20$ years and decays into a stable element $Y$. In a given rock sample, the ratio of the amounts of $X$ and $Y$ is found to be $1:7$. The estimated age of the rock is ......... years.

The atomic weight of a radioactive element is $M_w$ grams. The radioactivity of $m$ grams of this element is ......... ($N_A$ is Avogadro's number,$\lambda$ is the decay constant).

Element $X$ decays into element $Y$ with a half-life of $3$ days. On March $1$st,the mass of $X$ is $10 \, g$. What will be the masses of $X$ and $Y$ after $6$ days?

In the radioactive decay of an element,it is found that the count rate reduces from $1024$ to $128$ in $3$ minutes. Its half-life will be ...... minutes.

Half-lives of two radioactive substances $A$ and $B$ are,respectively,$20 \ min$ and $40 \ min$. Initially,the samples of $A$ and $B$ have an equal number of nuclei. After $80 \ min$,the ratio of the remaining number of $A$ and $B$ nuclei is