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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 19

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

    A) 5                        

    B)        4

    C) 3                        

    D)        8

    Correct Answer: D

    Solution :

    After first half hrs \[N={{N}_{0}}\frac{1}{2}\] for \[t=\frac{1}{2}\] to \[t=1\frac{1}{2}\] \[N=\left( {{N}_{0}}\frac{1}{2} \right){{\left[ \frac{1}{2} \right]}^{4}}={{N}_{0}}{{\left( \frac{1}{2} \right)}^{5}}\] for \[t=1\frac{1}{2}\]to \[t=2\] hrs. [for both A and B \[\frac{1}{{{t}_{1/2}}}=\frac{1}{1/2}+\frac{1}{1/4}=2+4=6;\]   \[{{t}_{1/2}}=1/6hrs]\] \[N=\left[ {{N}_{0}}{{\left( \frac{1}{2} \right)}^{5}} \right]{{\left( \frac{1}{2} \right)}^{3}}={{N}_{0}}{{\left( \frac{1}{2} \right)}^{8}}\]


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