The activity of a radioactive sample is measured as $9750$ counts per minute at $t=0$ and as $975$ counts per minute at $t=5$ minutes. The decay constant is approximately,

In which sequence the radioactive radiations are emitted in the following nuclear reaction? 

 ${}_{\mathrm{Z}}{}^{}\mathrm{X}^{\mathrm{A}}\to {}_{\mathrm{Z}+1}{}^{}\mathrm{Y}^{\mathrm{A}}\to {}_{\mathrm{Z}-1}{}^{}\mathrm{K}^{\mathrm{A}-4}\to {}_{\mathrm{Z}-1}{}^{}\mathrm{K}^{\mathrm{A}-4}$

Let ${\lambda }_{\alpha }, {\lambda }_{\beta }$ and ${\lambda }_{\alpha }^{\text{'}}$ denote the wavelengths of the $X$-rays of the $K_{\alpha }, K_{\beta }$ and $L_{\alpha }$ lines in the characteristic $X$-rays for a metal. Which of the following statements must hold true?