Higher November 2021 Paper 2 Q14
14. At the end of 2020, the population, \(P_0\), of a particular type of insect on an island was estimated to be 25 000.
Anwar and Irina are scientists studying these insects.
Anwar uses this iterative formula to predict the population of the insects, \(P_n\), \(n\) years after the end of 2020.
\[\begin{aligned} &P_0 = 25\,000\\ &P_{n+1} = 1{\cdot}12P_n \quad \text{where } n \geqslant 0 \text{ and } n \text{ is an integer} \end{aligned}\](a) Use Anwar’s formula to predict the population at the end of 2021. [1]
(b) Use Anwar’s formula to predict the increase in population during the year 2025.
You must show all your working. [3]
You must show all your working. [3]
(c) The island can support a maximum of 50 000 of these insects.
When this number has been reached, the population stops increasing at the same rate.
Irina says:
When this number has been reached, the population stops increasing at the same rate.
Irina says:
“The first time that Anwar’s iterative formula cannot be used to predict the population is at the end of 2027.”
Is Irina correct?Yes ☐ No ☐
Show how you decide. [2]| Scheme | Marks | Comments |
|---|---|---|
| 28 000 | B1 |
| Scheme | Marks | Comments |
|---|---|---|
| Allow values to be consistently rounded or truncated | ||
| (end of 2024). \(1.12^3 \times 28\,000\) (= 39 337.984) | M1 | FT ‘\(1.12^3 \times\)(their 28 000)’; may be seen in stages (end of 2021) 1.12 × 28 000 ( = 31 360 ) (end of 2022) 1.12 × 31 360 ( = 35 123.2 ) (end of 2023) 1.12 × 35 123(.2) (= 39 337.984) |
| (end of 2025) 1.12 × 39 337(.984) = 44 058(.54…) or 44059 or 44 057(.44) | M1 | FT ‘their 39 337….’ Interpretation of the formula as e.g. \(25\,000(1.12)^5 - 25\,000(1.12)^4\) implies M1 M1 |
| (44 058.54… – 39 337.984 =) 4720 or 4721 | A1 |
| Scheme | Marks | Comments |
|---|---|---|
| Finds (end of 2026 = ) 49 345(.56….) or (end of 2027 = ) 55 267(.03….) | S1 | FT ‘their 44058(..)’ if a step-by-step method used Accept 49 343 to 49 346 Accept 55 264 to 55 268 Allow for appropriate use of \(25000 \times 1.12^n\) where \(n\) is a decimal between 6 and 7 |
| Yes indicated and sufficient work to show the population is both below and above 50 000 during 2027 | B1 | FT ‘their 44058(..)’ and the correct FT conclusion If no marks award SC1 for the correct conclusion following incorrect work in (b) |
| (6) |
Corrected by Westie's Workshop
- The board's mark scheme printed “during 2017”. Changed to “during 2027” because the population passes 50 000 between the end of 2026 (49 345) and the end of 2027 (55 267), so it happens during 2027.
Original: see the board's mark scheme PDF (link above).