$A$ $280\, \text{day}$ old radioactive substance shows an activity of $6000\, \text{dps}$. $140\, \text{days}$ later, its activity becomes $3000\, \text{dps}$. What was its initial activity? ......... $\text{dps}$

In a radioactive material,the activity at time $t_{1}$ is $R_{1}$ and at a later time $t_{2}$,it is $R_{2}$. If the decay constant of the material is $\lambda$,then:

$A$ sample of radioactive material $A$,which has an activity of $10\, mCi$ $(1\, Ci = 3.7 \times 10^{10}\, \text{decays/s})$,has twice the number of nuclei as another sample of different radioactive material $B$,which has an activity of $20\, mCi$. The correct choices for half-lives of $A$ and $B$ would then be respectively:

An alloy is composed of two radioactive materials $A$ and $B$ having equal weight. The half-lives of $A$ and $B$ are $10 \ yrs$ and $20 \ yrs$ respectively. After time $t$, the alloy was found to consist of $(1/e) \ kg$ of $A$ and $1 \ kg$ of $B$. If the atomic weights of $A$ and $B$ are the same, then the value of $t$ is (Assume $\ln 2 = 0.7$):

$A$ radioactive nucleus can decay by two different processes. The half-life for the first process is $3.0 \, hours$ while it is $4.5 \, hours$ for the second process. The effective half-life of the nucleus will be $......... \, hours.$

$A$ radioactive sample has an average life of $30 \, ms$ and is decaying. $A$ capacitor of capacitance $200 \, \mu F$ is first charged and later connected with a resistor $R$. If the ratio of charge on the capacitor to the activity of the radioactive sample is fixed with respect to time,then the value of $R$ should be $.... \, \Omega$.