$(a)\;469 years\qquad(b)\;512 years \qquad(c)\;449 years \qquad(d)\;412 years$

Answer : 449 years

The decay constant $\;\lambda\;$ is reciprocal of mean life

$\Rightarrow \lambda_{\alpha}=\large\frac{1}{1620}\;$ per year and $ \lambda_{\beta }=\large\frac{1}{405}\;$ per year.

Total decay constant , $\; \lambda=\lambda_{\alpha}+\lambda_{\beta}\; =\large\frac{1}{1620}+\large\frac{1}{405}=\large\frac{1}{324}$ per year

When $\;(\large\frac{3}{4})$$^{th}\;$ part of sample has disintegrated $\;N=\large\frac{N_{0}}{4}$

$\Rightarrow \large\frac{N_{0}}{4}$$=N_{0}e^{-\lambda t}$$ \quad or \quad e^{\lambda t}=4 \rightarrow \lambda t =ln 4$

$\Rightarrow t=\large\frac{1}{\lambda}$$\;\ln 2^2 = \large\frac{2}{\lambda}$$\;\ln 2=2 \times 324 \times 0.693 = 449$ years.

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