Answer :

(a)We know that in radioactive decay


... (1)


where N(t)=no. of particles after t time


N0=no. of particles in starting of radioactive decay


λ= decay constant


Half-life of a radioactive substance is defined as the time taken for the substance to reduce to /2.



Taking log on both sides,




And theory of average life is,



where, S is sum of all particle lives which were there during the period t to t+dt


Therefore, sum of lives of


As,








As,



(b)


Initially for A no. Of particles is No and for B 2No


And half-life of A is 60 years, 30 years for B.


Decay constant of A =


And Decay constant of B =


Using the eq. Of radioactive decay


i.e.


According to question for A after some time no. Of particles of A becomes as twice of no. Of particles of B




And



Dividing both equations, we get




Taking log both sides, we get



Substituting the value of decay constant of A and B respectively we get




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