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Nuclei of radioactive element A are produced at rate ‘ $t^{2}$ ’ at any time t. The element A has decay constant $λ$ . Let N be the number of nuclei of element A at any time t. At time $t = t_{o}, \frac{d N}{d t}$ is minimum. Then the number of nuclei of element A at time t=t₀ is:

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Correct option is A

The rate of formation of radioactive nuclei is t2 Rate of decay of radioactive nuclei is λN∴dNdt=t2−λN For dNdt to be minimum; d2Ndt2=0⇒d2Ndt2=2t−λdNdt=2t−λt2−λN=0⇒N=2t−λt2λ2⇒N=2t0−λt02λ2 at t=t0

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