Nuclei of a radioactive element `X` are being produced at a constant rate `K` and this element decays to a stable nucleus `Y` with a decay constant `lambda` and half-life `T_(1//3)`. At the time `t=0`, there are `N_(0) `nuclei of the element `X`.
The number `N_(X)` of nuclei of `X` at time `t=T_(1//2)` is .

Options

A. `(q+lambdaN_(0))/(2 lambda)`
B. `(2lambdaN_(0)-1)(q)/(lambda)`
C. `[lambdaN_(0)+(q)/(2)](1)/(lambda)`
D. `[lambdaN_(0)-(q)/(2)](1)/(lambda)`

Answer

A

Explanation

`(dN_(X))/(dt)=q-lambdaN_(X)rArrN_(X)=(1)/(lambda)[q-(q-lambdaN_(0))e^(-lambda t)]`
Also,