JEE Main & Advanced Sample Paper JEE Main Sample Paper-47

  • question_answer
    The half lives of radioisotopes \[{{P}^{32}}\] and \[{{P}^{33}}\] are 20 days and 30 days respectively. The radioisotopes are mixed in the ratio of 4 :1 of their atoms. If the initial activity of the mixed sample is \[7\,mCi,\] then the activity of the mixed isotopes after 60 days.

    A)  \[2.5\,mCi\]                

    B)  \[1\,mCi\]

    C)  \[3.2\,mCi\]                

    D)  \[3\,mCi\]

    Correct Answer: B

    Solution :

     As,      \[\frac{{{N}_{32}}}{{{N}_{33}}}=\frac{4}{1}\]             \[[{{N}_{33}}={{N}_{0}}]\] \[\Rightarrow \]            \[\frac{dN}{N}={{\lambda }_{1}}{{N}_{32}}+{{\lambda }_{2}}{{N}_{33}}\] \[\Rightarrow \]            \[7\times {{10}^{-3}}={{\lambda }_{1}}4{{N}_{0}}+{{\lambda }_{2}}{{N}_{0}}\] \[\Rightarrow \]            \[7\times {{10}^{-3}}={{N}_{0}}\,\left[ \frac{In\,(2)}{5}+\frac{In\,(2)}{30} \right]\] \[\Rightarrow \]            \[{{N}_{0}}=\frac{30\times {{10}^{-3}}}{In\,(2)}\] After 60 days, \[\frac{dN}{dt}={{\lambda }_{1}}\frac{4{{N}_{0}}}{8}+\frac{{{\lambda }_{2}}{{N}_{0}}}{4}=1\,mCi\]

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