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:

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

$|\frac{t_{o} - λ t_{o}^{2}}{λ^{2}}|$

$|\frac{t_{o} - λ t_{o}^{2}}{λ}|$

$|\frac{2 t_{o} - λ t_{o}^{2}}{λ^{2}}|$

$|\frac{2 t_{o} - λ t_{o}^{2}}{λ}|$

answer is A.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

The rate of formation of radioactive nuclei is $t^{2}$
Rate of decay of radioactive nuclei is $λ N$
$∴ \frac{d N}{d t} = t^{2} - λ N$

$\begin{array}{l} For \frac{d N}{d t} to be minimum; \frac{d^{2} N}{d t^{2}} = 0 \\ \Rightarrow \frac{d^{2} N}{d t^{2}} = 2 t - λ \frac{d N}{d t} = 2 t - λ (t^{2} - λ N) = 0 \\ \Rightarrow N = \frac{2 t - λ t^{2}}{λ^{2}} \\ \Rightarrow N = \frac{2 t_{0} - λ t_{0}^{2}}{λ^{2}} (at t = t_{0}) \end{array}$