Consider a radioactive nucleus $A$ which decays to a stable nucleus $C$ through the following sequence : $A \to B \to C$ Here $B$ is an intermediate nuclei which is also radioactive. Considering that there are $N_0$, atoms of $A$ initially, plot the graph showing the variation of number of atoms of $A$ and $B$ versus time. 

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Here at an instant of time when $\mathrm{N}_{\mathrm{B}}=\left(\mathrm{N}_{\mathrm{B}}\right)_{\max }^{\prime}$ growth rate and decay rate have become equal for element B. Before this time, its growth rate is more (than its decay rate) and after this time its decay rate is more (than its growth rate).

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The radioactivity of a certain radioactive elements drops to $\frac{1}{64}$ of its initial value in $30$ seconds. Its half life is ............. seconds

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Match List $I$ (Wavelength range of electromagnetic spectrum) with List $II$ (Method of production of these waves) and select the correct option from the options given below the lists

List $I$ List $II$
$(1)$ $700\, nm$ to $1\,mm$ $(i)$ Vibration of atoms and molecules
$(2)$ $1\,nm$ to $400\, nm$ $(ii)$ Inner shell electrons in atoms moving from one energy level to a lower level
$(3)$ $ < 10^{-3}\,nm$ $(iii)$ Radioactive decay of the nucleus
$(4)$ $1\,mm$ to $0.1\,m$ $(iv)$ Magnetron valve

  • [JEE MAIN 2014]