Answer :

(a)We know that in radioactive decay

... (1)

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

N_{0}=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 N_{o} and for B 2N_{o}

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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