For a substance the average life for $\alpha$-emission is $1620$ years and for $\beta$ emission is $405$ years. After how much time the $1/4$ of the material remains after $\alpha$ and $\beta$ emission .......$years$

Define the average life of a radioactive substance. 

If $10\%$ of a radioactive material decays in $5$ days, then the amount of original material left after $20$ days is approximately ...............$\%$

$10\, gm$ of radioactive material of half-life $15$ year is kept in store for $20$ years. The disintegrated material is ............$gm$

A radioactive nucleus $A$ has a single decay mode with half-life $\tau_A$. Another radioactive nucleus $B$ has two decay modes $1$ and $2$. If decay mode $2$ were absent, the half-life of $B$ would have been $\tau_A / 2$. If decay mode $1$ were absent, the half-life of $B$ would have been $3 \tau_A$. If the actual half life of $B$ is $\tau_B$, then the ratio $\tau_B / \tau_A$ is

What is radioactivity ?