# The mean lives of an radio active substance are 1620 and 405 years for $\;\alpha-\;$emission and $\;\beta-\;$ emission respee. Find out time during which three fourth of a sample will decay if it is decaying both the $\;\alpha-\;$emission and $\;\beta-\;$ emission simultaneously

$(a)\;512 years\qquad(b)\;268 years\qquad(c)\;449 years\qquad(d)\;352 years$

Explanation :
When a substance decays by $\;\alpha\;$ and $\;\beta\;$ emission simultaneously $\;\lambda_{ar}\;$ is given by
$\lambda_{ar} =\lambda_{\alpha}+\lambda_{\beta}$
Where $\;\lambda_{\alpha}\;$ = disintegration constant
for $\;\alpha-\;$ emission only
$\lambda \beta=\;$ disintegration constant for $\; \beta-\;$ emission only
Mean life is given by :
$T_{m}=\large\frac{1}{\lambda}$
$\lambda_{ar}=\lambda_{\alpha}+\lambda_{\beta}\;=>\;\large\frac{1}{T_{m}}+\large\frac{1}{T_{\alpha}}+\large\frac{1}{T_{\beta}}$
$=\large\frac{1}{1620}+\large\frac{1}{405}=3.08\times10^{-3}$
$\lambda \alpha v t=2.03\;log(\large\frac{100}{25})$
$t=2.303 \times\large\frac{1}{3.08\times10^{-3}}\;log 4=449.24 years\;$