A sample contains $16\, gm$ of a radioactive material, the half life of which is two days. After $32\, days,$ the amount of radioactive material left in the sample is

In a radioactive decay process, the activity is defined as $A=-\frac{\mathrm{d} N}{\mathrm{~d} t}$, where $N(t)$ is the number of radioactive nuclei at time $t$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t=0$. At a later time, the activities of $S_1$ and $S_2$ are $A_1$ and $A_2$, respectively. When $S_1$ and $S_2$ have just completed their $3^{\text {rd }}$ and $7^{\text {th }}$ half-lives, respectively, the ratio $A_1 / A_2$ is. . . . . . .

Two radioactive elements $R$ and $S$ disintegrate as
$R \rightarrow P + \alpha; \lambda_R = 4.5 × 10^{-3} \,\, years^{-1}$
$S \rightarrow P + \beta; \lambda_S = 3 × 10^{-3} \,\, years^{-1}$
Starting with number of atoms of $R$ and $S$ in the ratio of $2 : 1,$ this ratio after the lapse of three half lives of $R$ will be :

A nucleus has a half-life of $30\; min$. At $3 \;PM$ its decay rate was measured as $120000 \,cps$. the decay rate ……….. $\,cps$ at $5 \,PM$ ?

Starting with a sample of pure ${}^{66}Cu$, $7/8$ of it decays into $Zn$ in $15\  minutes$ . The it decays into $Zn$ in $15\  minutes$ . The corresponding half-life is ……………. $minutes$