Higher November 2023 Paper 1 Q16
16 At the start of year \(n\) the population of a species is \(P_n\)
At the start of the following year the population of the species is given by
\[P_{n + 1} = kP_n \quad \text{where } k \text{ is a positive constant.}\]The population of the species at the start of year 1 is 8 million.
The population of the species at the start of year 2 is 6 million.
(a) Work out the population of the species at the start of year 3 (3)
At the start of year 5 the value of \(k\) is increased by 0.3 to a new constant value.
Louise thinks that from the start of year 5 the population of the species would increase year on year.
(b) Is Louise correct?
You must give a reason for your answer. (1)
You must give a reason for your answer. (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| 4.5 | P1 | for initial use of formula, eg \(6 = 8k\) |
| P1 | for a full process to find \(P_3\) eg \(\text{``}\dfrac{6}{8}\text{''} \times 6\) | |
| A1 | oe |
| Answer | Mark | Mark scheme |
|---|---|---|
| Explanation | C1 | for explanation Acceptable examples Yes, the population will increase as \(k\) is over 1 She is correct because \(1.05 \gt 1\) Yes, each year the population will increase by 5% Yes, because \(0.75 + 0.3 = 1.05\) Yes, a 0.3 increase is greater than the current 0.25 decrease Not acceptable examples Yes, the population will increase each year Yes, because there is an increase in \(k\) No because 0.3 is less than 0.75 |
Additional guidance
If figures are given as part of the answer they must be correct, but can allow ft