$A$ heavy nucleus $Q$ of half-life $20 \text{ minutes}$ undergoes alpha-decay with a probability of $60 \%$ and beta-decay with a probability of $40 \%$. Initially,the number of $Q$ nuclei is $1000$. The number of alpha-decays of $Q$ in the first one hour is:

At time $t=0$, a material is composed of two radioactive atoms $A$ and $B$, where $N_{A}(0)=2 N_{B}(0)$. The decay constant of both kinds of radioactive atoms is $\lambda$. However, $A$ disintegrates to $B$ and $B$ disintegrates to $C$. Which of the following figures represents the evolution of $N_{B}(t) / N_{B}(0)$ with respect to time $t$?
$N_{A}(0) = \text{Number of } A \text{ atoms at } t=0$
$N_{B}(0) = \text{Number of } B \text{ atoms at } t=0$

The half-life of a particle of mass $1.6 \times 10^{-26} \,kg$ is $6.9 \,s$. $A$ stream of such particles is travelling with a kinetic energy of $0.05 \,eV$ per particle. The fraction of particles that will decay when they travel a distance of $1 \,m$ is

The half-life of a stream of radioactive particles moving along a straight path with a constant kinetic energy of $4 \text{ eV}$ is $1 \text{ minute}$. The percentage of particles which decay before travelling a distance of $3.6 \text{ km}$ is (Mass of the radioactive particles $= 3.2 \times 10^{-21} \text{ kg}$ and charge of the electron $= 1.6 \times 10^{-19} \text{ C}$).

What is the half-life (in years) of a radioactive material if its activity drops to $1/16$th of its initial value in $30$ years (in $.5$)?

Two different radioactive elements with half-lives $T_1$ and $T_2$ have undecayed atoms $N_1$ and $N_2$ respectively present at a given instant. The ratio of their activities at that instant is