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KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    A sample of radioactive material decays simultaneously by two processes A & B with half lives \[\frac{1}{2}\] & \[\frac{1}{4}hr\] respectively. For first half hr it decays with the process A, next one \[hr\] with the process B & for further half an hour with both A & B. If originally there were \[{{N}_{0}}\] nuclei the number of nuclei after 2 hr of such decay \[{{N}_{0}}{{\left( \frac{1}{2} \right)}^{a}}\] find the value of a -

    A) 2                                 

    B) 4    

    C) 8         

    D) 1

    Correct Answer: C

    Solution :

    [C]
    For first half sample decay with process A of half lives \[\frac{1}{2}hr.\]
    After first half hrs \[N={{N}_{0}}\frac{1}{2}\]
    For next one hour decay process B of half lives \[\frac{1}{4}\]occur
    So for \[t=\frac{1}{2}\] to \[t=1\frac{1}{2}\]
    Total 4 half lives occur \[N={{N}_{0}}\frac{1}{2}{{\left[ \frac{1}{2} \right]}^{4}}={{N}_{0}}{{\left( \frac{1}{2} \right)}^{5}}\]
    For last \[\frac{1}{2}hr\] both A and B process occur simultaneously
    For \[t=1\frac{1}{2}\] to t = 2hr [For both A & B \[\frac{1}{{{t}_{1/2}}}=\frac{1}{1/2}+\frac{1}{1/4}=6]\]
    \[{{t}_{1/2}}=\frac{1}{6}hr\]
    Number of half-life in\[\frac{1}{2}hr\] is 3
    \[\therefore \,\,\,N={{N}_{0}}{{\left[ \frac{1}{2} \right]}^{5}}\,\,{{\left[ \frac{1}{2} \right]}^{3}}\]\[={{N}_{0}}{{\left[ \frac{1}{2} \right]}^{8}}\]
    a = 8               


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