IB AA HL May 2026 P1 TZA Q12

Rowan is investigating the spread of an infection through a population of rabbits.

He models \(N\), the number (in thousands) of infected rabbits in the population, at time \(t\) days.

Rowan models the spread of the infection by the differential equation \(\frac{\mathrm{d}N}{\mathrm{d}t}=k(-3+4N-N^2)\), where \(t\geq0\) and \(k\in\mathbb{R}^+\).

Initially, \(1500\) rabbits are infected.

1. Express \(\frac1{-3+4x-x^2}\) in the form \(\frac{A}{x-1}+\frac{B}{3-x}\), where \(A,B\in\mathbb{R}\).[3]
2. Hence, show that \(\ln\left(\frac{N-1}{3-N}\right)=2kt-\ln p\), where \(p\in\mathbb{R}^+\) is a constant to be determined.[8]
3. Find an expression for \(N\) in the form \(N=\frac{a+be^{-2kt}}{1+ce^{-2kt}}\), where \(a,b,c\in\mathbb{Z}\).[5]
4. According to Rowan’s model, the number of infected rabbits approaches a limit, \(L\), over the long term.
   Find the value of \(L\).[2]