in a circuit there are two resistances each R in series Across any one R there is resistance 2R in parallel The arrangement is connected to a source The charge supplied by source varies with time t as Q=atbt^2 what is the total heat produced in resistor 2R

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2 Answers
More details are needed in this quesiton because you have not specified the starting and end time.
Even if the starting time is 0. the ending time is not specified.
I will give an outline on how ths can be done assuming that u have taken Q
Q=atbt^2
=> I= a2bt
so the time when I becomes zero is t=a/2b
If we try to solve this problem for this condition, look into the next section of this question.
Continuing,
We get I (t) in the previous part as the derivative of Q.
if current I passes thru the wire or Resistance '2R' will be R*I/(2R+R)
so the current passing thru 2R will be I/3
so heat generation rate I^R
but we will have to integrate this value from initial to end point.
This method should be used to reach the final answer.
I hope this helps you solve the problem :)