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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