The graph between number of decayed atoms $N'$ of a radioactive element and time $t$ is

The half life period of radioactive element ${x}$ is same as the mean life time of another radioactive element $y.$ Initially they have the same number of atoms. Then:

A radioactive reaction is $_{92}{U^{238}}{ \to _{82}}P{b^{206}}$. How many $\alpha $ and $\beta $  particles are emitted

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 :

At time $t=0$ some radioactive gas is injected into a sealed vessel. At time $T$ some more of the gas is injected into the vessel. Which one of the following graphs best represents the logarithm of the activity $A$ of the gas with time $t$ ?

A radioactive sample is $\alpha$-emitter with half life $138.6$ days is observed by a student to have $2000$ disintegration/sec. The number of radioactive nuclei for given activity are