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

Answer : (c) 449 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 :

$T_{m}=\large\frac{1}{\lambda}$

$\lambda_{ar}=\lambda_{\alpha}+\lambda_{\beta}\;=>\;\large\frac{1}{T_{m}}+\large\frac{1}{T_{\alpha}}+\large\frac{1}{T_{\beta}}$

$=\large\frac{1}{1620}+\large\frac{1}{405}=3.08\times10^{-3}$

$\lambda \alpha v t=2.03\;log(\large\frac{100}{25})$

$t=2.303 \times\large\frac{1}{3.08\times10^{-3}}\;log 4=449.24 years\;$

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