A radioactive sample consists of two distinct species having equal number of atoms initially. The mean life time of one species is $\tau$ and that of the other is $5\tau$. The decay products in both cases are stable. A plot is made of the total number of radioactive nuclei as a function of time. Which of the following figures best represents the form of this plot

Let the initial number of atoms of each species be $N_0$. Then the total number of radioactive nuclei at time $t$ is $$N(t)=N_0 e^{-t/\tau}+N_0 e^{-t/(5\tau)}.$$ At $t=0$, $$N(0)=2N_0.$$ As $t$ increases, both exponential terms decrease continuously, so $N(t)$ must decrease continuously. Also, $$\frac{dN}{dt}=-\frac{N_0}{\tau}e^{-t/\tau}-\frac{N_0}{5\tau}e^{-t/(5\tau)}<0,$$ so there cannot be any flat part or increase. Further, $$\frac{d^2N}{dt^2}=\frac{N_0}{\tau^2}e^{-t/\tau}+\frac{N_0}{25\tau^2}e^{-t/(5\tau)}>0,$$ so the curve is concave upward throughout. It falls rapidly at first and then more slowly, approaching zero asymptotically. Among the given figures, only option $D$ shows a monotonically decreasing, concave-up decay curve with no rise or plateau.