The number, b, of bacteria in a sample is given by b = P+Qe^(2t), where P and Q are constants and t is time in weeks. Initially there are 500 bacteria which increase to 600 after 1 week.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

1. [B1] (a) P + Q = 500 and P + Qe² = 600
2. [M1] (a) Q = 100 / (e² - 1) for attempt to solve by removing P from two equations both containing 3 terms
3. [A1] (a) Q = 15.7 or 15.6 awrt
4. [A1] (a) P = 484 or 485 awrt
5. [B1] (b) B = 484.3 + 15.65e⁴ = 1338 Integer value rounded down from 1338... if seen.
6. [M1] (c) e^(2t) = (1000000 - 484.3) / 15.65 Make e^(2t) the subject
7. [M1] (c) 2t = ln((1000000 - 484.3) / 15.65) Take logs correctly where e^(2t) > 0 or e^t > 0
8. [A1] (c) [t = 5.5(3) or t = 5.5....] → 6th week. nfww