• # question_answer SOLUTION A bacterial infection in an internal wound grows as $N'(t)={{N}_{0}}$ exp(t), where the time t is in hours. A dose of antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there, the bacterial population goes down as $\frac{dN}{dt}=-5{{N}^{2}}.$What will be the plot of $\frac{{{N}_{0}}}{N}$vs. t after 1 hour ? [JEE Main 10-4-2019 Morning] A) B) C)                 D)

From 0 to 1 hour, $N'={{N}_{0}}{{e}^{t}}$ From 1 hour onwards$\frac{dN}{dt}=-5{{N}^{2}}$ So at t = 1 hour, $N'=e{{N}_{0}}$ $\frac{dN}{dt}=-5{{N}^{2}}$ $\int\limits_{e{{N}_{0}}}^{N}{{{N}^{-2}}}dN=-5\int\limits_{1}^{t}{dt}$ $\frac{1}{N}-\frac{1}{e{{N}_{0}}}=5(t-1)$ $\frac{{{N}_{0}}}{N}-\frac{1}{e}=5{{N}_{0}}(t-1)$ $\frac{{{N}_{0}}}{N}=5{{N}_{0}}(t-1)+\frac{1}{e}$ $\frac{{{N}_{0}}}{N}=5{{N}_{0}}t+\left( \frac{1}{e}-5{{N}_{0}} \right)$    which is following $y=mx+C$