Logistic/bounded growth

13. A scientist is studying a population of insects. The number of insects, \(N\), in the population, \(t\) days after the start of the study is modelled by the equation $$N = \frac { 240 } { 1 + k \mathrm { e } ^ { - \frac { t } { 16 } } }$$ where \(k\) is a constant.
Given that there were 50 insects at the start of the study,

1. find the value of \(k\)
2. use the model to find the value of \(t\) when \(N = 100\)
3. Show that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { 1 } { p } N - \frac { 1 } { q } N ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
   

   END

11. A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 15 } P ( 5 - P ) , \quad t \geqslant 0$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that when \(t = 0 , P = 1\),

1. solve the differential equation, giving your answer in the form $$P = \frac { a } { b + c \mathrm { e } ^ { - \frac { 1 } { 3 } t } }$$ where \(a\), \(b\) and \(c\) are integers.
2. Hence show that the population cannot exceed 5000

1. (a) Express \(\frac { 1 } { P ( 5 - P ) }\) in partial fractions.

A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 15 } P ( 5 - P ) , \quad t \geqslant 0$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that when \(t = 0 , P = 1\),
(b) solve the differential equation, giving your answer in the form, $$P = \frac { a } { b + c \mathrm { e } ^ { - \frac { 1 } { 3 } t } }$$ where \(a\), \(b\) and \(c\) are integers.
(c) Hence show that the population cannot exceed 5000

8. When a plague of locusts attacks a wheat crop, the proportion of the crop destroyed after \(t\) hours is denoted by \(x\). In a model, it is assumed that the rate at which the crop is destroyed is proportional to \(x ( 1 - x )\). A plague of locusts is discovered in a wheat crop when one quarter of the crop has been destroyed. Given that the rate of destruction at this instant is such that if it remained constant, the crop would be completely destroyed in a further six hours,

1. show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 } x ( 1 - x )\),
2. find the percentage of the crop destroyed three hours after the plague of locusts is first discovered.

7 Fig. 7 illustrates the growth of a population with time. The proportion of the ultimate (long term) population is denoted by \(x\), and the time in years by \(t\). When \(t = 0 , x = 0.5\), and as \(t\) increases, \(x\) approaches 1 . \begin{figure}[h] \end{figure} One model for this situation is given by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = x ( 1 - x )$$

1. Verify that \(x = \frac { 1 } { 1 + \mathrm { e } ^ { - t } }\) satisfies this differential equation, including the initial condition.
2. Find how long it will take, according to this model, for the population to reach three-quarters of its ultimate value. An alternative model for this situation is given by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = x ^ { 2 } ( 1 - x ) ,$$ with \(x = 0.5\) when \(t = 0\) as before.
3. Find constants \(A , B\) and \(C\) such that \(\frac { 1 } { x ^ { 2 } ( 1 - x ) } = \frac { A } { x ^ { 2 } } + \frac { B } { x } + \frac { C } { 1 - x }\).
4. Hence show that \(t = 2 + \ln \left( \frac { x } { 1 - x } \right) - \frac { 1 } { x }\).
5. Find how long it will take, according to this model, for the population to reach three-quarters of its ultimate value.

7 A drug is administered by an intravenous drip. The concentration, \(x\), of the drug in the blood is measured as a fraction of its maximum level. The drug concentration after \(t\) hours is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k \left( 1 + x - 2 x ^ { 2 } \right) ,$$ where \(0 \leqslant x < 1\), and \(k\) is a positive constant. Initially, \(x = 0\).

1. Express \(\frac { 1 } { ( 1 + 2 x ) ( 1 - x ) }\) in partial fractions.
2. Hence solve the differential equation to show that \(\frac { 1 + 2 x } { 1 - x } = \mathrm { e } ^ { 3 k t }\).
3. After 1 hour the drug concentration reaches \(75 \%\) of its maximum value and so \(x = 0.75\). Find the value of \(k\), and the time taken for the drug concentration to reach \(90 \%\) of its maximum value.
4. Rearrange the equation in part (ii) to show that \(x = \frac { 1 - \mathrm { e } ^ { - 3 k t } } { 1 + 2 \mathrm { e } ^ { - 3 k t } }\). Verify that in the long term the drug concentration approaches its maximum value. \section*{END OF QUESTION PAPER} \section*{Tuesday 16 J une 2015 - Afternoon} \section*{A2 GCE MATHEMATICS (MEI)} 4754/01B Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{QUESTION PAPER} \section*{Candidates answer on the Question Paper.} \section*{OCR supplied materials:}
   • Insert (inserted)
   • MEI Examination Formulae and Tables (MF2)
   \section*{Other materials required:}
   • Scientific or graphical calculator
   • Rough paper
   Duration: Up to 1 hour \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-05_117_495_1014_1308} PLEASE DO NOT WRITE IN THIS SPACE 2 In line 79 it says "For most journeys, more than half the journey time is composed of load time and transfer time". For what percentage of the journey time for the round trip made by car A in Table 4 is the car stationary?
   \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_645_1746_388_164}
   3 Using the expression on line 51, work out the answer to the question on lines 39 and 40 for the case where there are 10 upper floors and 7 people. Give your answer to 2 decimal places.
   \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_488_1746_1233_164}
   4 In lines 89 and 90 it says "... on average there will be approximately 8 stops per trip. A round trip with 8 stops would take between 188 and 200 seconds". Explain how the figure of 188 seconds has been derived. 5
5. Referring to Strategy 3 and lines 99 to 101, complete the table below for car C .
6. Calculate the time car C will take to transport all the people who work on floors 7 and 8 , and return to the ground floor.

   5	\includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-08_1095_816_484_700}

   68 people make independent visits to any one of the upper floors of a building with 10 upper floors. What is the probability that at least one of the visitors goes to the top floor?

   6

   7 On lines 94 and 95 it says "Table 4 gives the timings for round trips in which the cars are required to stop at every floor they serve; Table 2 suggests this is a common occurrence in this case". Explain how Table 2 is used to make this claim. \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-09_1093_1740_1238_166} END OF QUESTION PAPER

14. A population of ants being studied on an island. The number of ants, \(P\), in the population, is modelled by the equation. $$P = \frac { 900 k e ^ { 0.2 t } } { 1 + k e ^ { 0.2 t } } , \text { where } k \text { is a constant. }$$ Given that there were 360 ants when the study started,
a. show that \(k = \frac { 2 } { 3 }\).
b. Show that \(P = \frac { 1800 } { 2 + 3 e ^ { - 0.2 t } }\). The model predicts an upper limit to the number of ants on the island.
c. State the value of this limit.
d. Find the value of \(t\) when \(P = 520\). Give your answer to one decimal place.
e. i. Show that the rate of growth, \(\frac { \mathrm { d } P } { d t } = \frac { P ( 900 - P ) } { 4500 }\) ii. Hence state the value of \(P\) at which the rate of growth is a maximum.

7. When a plague of locusts attacks a wheat crop, the proportion of the crop destroyed after \(t\) hours is denoted by \(x\). In a model, it is assumed that the rate at which the crop is destroyed is proportional to \(x ( 1 - x )\). A plague of locusts is discovered in a wheat crop when one quarter of the crop has been destroyed. Given that the rate of destruction at this instant is such that if it remained constant, the crop would be completely destroyed in a further six hours,

1. show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 } x ( 1 - x )\),
2. find the percentage of the crop destroyed three hours after the plague of locusts is first discovered.
   7. continued
   7. continued

1. A population of deer was introduced onto an island.

The number of deer, \(P\), on the island at time \(t\) years following their introduction is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { P } { 5000 } \left( 1000 - \frac { P ( t + 1 ) } { 6 t + 5 } \right) \quad t > 0$$ It was estimated that there were 540 deer on the island six months after they were introduced.
Use two applications of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }\) to estimate the number of deer on the island 10 months after they were introduced.

1. An area of woodland contains a mixture of blue and yellow flowers.

A study found that the proportion, \(x\), of blue flowers in the woodland area satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x t ( 0.8 - x ) } { x ^ { 2 } + 5 t } \quad t > 0$$ where \(t\) is the number of years since the start of the study.
Given that exactly 3 years after the start of the study half of the flowers in the woodland area were blue,

1. use one application of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }\) to estimate the proportion of blue flowers in the woodland area half a year later.
2. Deduce from the differential equation the proportion of flowers that will be blue in the long term.

8 A disease is spreading through a colony of rabbits. There are 5000 rabbits in the colony. At time \(t\) hours, \(x\) is the number of rabbits infected. The rate of increase of the number of rabbits infected is proportional to the product of the number of rabbits infected and the number not yet infected.

1. 1. Formulate a differential equation for \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) in terms of the variables \(x\) and \(t\) and a constant of proportionality \(k\).
   2. Initially, 1000 rabbits are infected and the disease is spreading at a rate of 200 rabbits per hour. Find the value of the constant \(k\).
      (You are not required to solve your differential equation.)
2. The solution of the differential equation in this model is $$t = 4 \ln \left( \frac { 4 x } { 5000 - x } \right)$$
   1. Find the time after which 2500 rabbits will be infected, giving your answer in hours to one decimal place.
   2. Find, according to this model, the number of rabbits infected after 30 hours.

8 Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced. The population \(x\), in thousands, of red squirrels is modelled by the equation $$x = \frac { a } { 1 + k t } ,$$ where \(t\) is the time in years, and \(a\) and \(k\) are constants. When \(t = 0 , x = 2.5\).

1. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { k x ^ { 2 } } { a }\).
2. Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate \(a\) and \(k\).
3. What is the long-term population of red squirrels predicted by this model? The population \(y\), in thousands, of grey squirrels is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 2 y - y ^ { 2 } .$$ When \(t = 0 , y = 1\).
4. Express \(\frac { 1 } { 2 y - y ^ { 2 } }\) in partial fractions.
5. Hence show by integration that \(\ln \left( \frac { y } { 2 - y } \right) = 2 t\). Show that \(y = \frac { 2 } { 1 + \mathrm { e } ^ { - 2 t } }\).
6. What is the long-term population of grey squirrels predicted by this model? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4)
   Paper B: Comprehension
   Monday
   23 JANUARY 2006
   Afternooon U Additional materials:
   Rough paper
   MEI Examination Formulae and Tables (MF2)
   4754(B) \section*{Up to 1 hour $$\text { o to } 1 \text { hour }$$} \section*{TIME} \section*{Up to 1 hour}
   • Write your name, centre number and candidate number in the spaces at the top of this page.
   • Answer all the questions.
   • Write your answers in the spaces provided on the question paper.
   • You are permitted to use a graphical calculator in this paper.
   • Final answers should be given to a degree of accuracy appropriate to the context.
   • The number of marks is given in brackets [ ] at the end of each question or part question.
   • The insert contains the text for use with the questions.
   • You may find it helpful to make notes and do some calculations as you read the passage.
   • You are not required to hand in these notes with your question paper.
   • You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
   • The total number of marks for this section is 18.

   For Examiner's Use
   Qu.	Mark
   1
   2
   3
   4
   5
   Total

   1 Line 59 says "Again Party G just misses out; if there had been 7 seats G would have got the last one." Where is the evidence for this in the article? 26 parties, P, Q, R, S, T and U take part in an election for 7 seats. Their results are shown in the table below.

   Party	Votes (\%)
   P	30.2
   Q	11.4
   R	22.4
   S	14.8
   T	10.9
   U	10.3

7. Use the Trial-and-Improvement method, starting with values of \(10 \%\) and \(14 \%\), to find an acceptance percentage for 7 seats, and state the allocation of the seats.

   Acceptance percentage, \(\boldsymbol { a }\) \%		10\%	14\%
   Party	Votes (\%)	Seats	Seats	Seats	Seats	Seats
   P	30.2
   Q	11.4
   R	22.4
   S	14.8
   T	10.9
   U	10.3
   Total seats

   Seat Allocation \(\quad \mathrm { P } \ldots\)... \(\mathrm { Q } \ldots\) R ... S ... T ... \(\mathrm { U } \ldots\).
8. Now apply the d'Hondt Formula to the same figures to find the allocation of the seats.

   	Round
   Party	1	2	3	4	5	6	7	Residual
   P	30.2
   Q	11.4
   R	22.4
   S	14.8
   T	10.9
   U	10.3
   Seat allocated to

   Seat Allocation \(\mathrm { P } \ldots\). Q ... \(\mathrm { R } \ldots\). S ... T ... \(\mathrm { U } \ldots\). 3 In this question, use the figures for the example used in Table 5 in the article, the notation described in the section "Equivalence of the two methods" and the value of 11 found for \(a\) in Table 4. Treating Party E as Party 5, verify that \(\frac { V _ { 5 } } { N _ { 5 } + 1 } < a \leqslant \frac { V _ { 5 } } { N _ { 5 } }\).
   4 Some of the intervals illustrated by the lines in the graph in Fig. 8 are given in this table.

   Seats	Interval	Seats	Interval
   1	\(22.2 < a \leqslant 27.0\)	5
   2	\(16.6 < a \leqslant 22.2\)	6	\(10.6 < a \leqslant 11.1\)
   3		7
   4

9. Describe briefly, giving an example, the relationship between the end-points of these intervals and the values in Table 5, which is reproduced below.
10. Complete the table above. \begin{table}[h]

    	Round
    Party	1	2	3	4	5	6	Residual
    A	22.2	22.2	11.1	11.1	11.1	11.1	7.4
    B	6.1	6.1	6.1	6.1	6.1	6.1	6.1
    C	27.0	13.5	13.5	13.5	9.0	9.0	9.0
    D	16.6	16.6	16.6	8.3	8.3	8.3	8.3
    E	11.2	11.2	11.2	11.2	11.2	5.6	5.6
    F	3.7	3.7	3.7	3.7	3.7	3.7	3.7
    G	10.6	10.6	10.6	10.6	10.6	10.6	10.6
    H	2.6	2.6	2.6	2.6	2.6	2.6	2.6
    Seat allocated to	C	A	D	C	E	A

    \captionsetup{labelformat=empty} \caption{Table 5}
    \end{table} 5 The ends of the vertical lines in Fig. 8 are marked with circles. Those at the tops of the lines are filled in, e.g. • whereas those at the bottom are not, e.g. o.
11. Relate this distinction to the use of inequality signs.
12. Show that the inequality on line 102 can be rearranged to give \(0 \leqslant V _ { k } - N _ { k } a < a\). [1]
13. Hence justify the use of the inequality signs in line 102.

14 John wants to encourage more birds to come into the park near his house. Each day, starting on day 1, he puts bird food out and then observes the birds for one hour. He records the maximum number of birds that he observes at any given moment in the park each day. He believes that his observations may be modelled by the following differential equation, where \(n\) is the maximum number of birds that he observed at any given moment on day \(t\). \(\frac { \mathrm { d } n } { \mathrm {~d} t } = 0.1 n \left( 1 - \frac { n } { 50 } \right)\)

1. Show that the general solution to the differential equation can be written in the form \(n = \frac { 50 A } { \mathrm { e } ^ { - 0.1 t } + A }\), where \(A\) is an arbitrary positive constant.
2. Using his model, determine the maximum number of birds that John would expect to observe at any given moment in the long term.
3. Write down one possible refinement of this model.
4. Write down one way in which John's model is not appropriate. \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
   OCR is part of the }\section*{...day June 20XX - Morning/Afternoon} A Level Mathematics A
   H240/01 Pure Mathematics \section*{SAMPLE MARK SCHEME} MAXIMUM MARK 100 \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-09_259_1320_1242_826} \section*{Text Instructions}
   1. Annotations and abbreviations
   \section*{2. Subject-specific Marking Instructions for \(\mathbf { A }\) Level Mathematics \(\mathbf { A }\)} Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded. An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner. If you are in any doubt whatsoever you should contact your Team Leader. The following types of marks are available.
   M
   A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. B
   Mark for a correct result or statement independent of Method marks. \section*{E} Mark for explaining a result or establishing a given result. This usually requires more working or explanation than the establishment of an unknown result.
   Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
   d When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
   e The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. If this is not the case please, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
   Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question.
   f Unless units are specifically requested, there is no penalty for wrong or missing units as long as the answer is numerically correct and expressed either in SI or in the units of the question. (e.g. lengths will be assumed to be in metres unless in a particular question all the lengths are in km , when this would be assumed to be the unspecified unit.) We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so. When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value. This rule should be applied to each case. When a value is not given in the paper accept any answer that agrees with the correct value to 2 s.f. Follow through should be used so that only one mark is lost for each distinct accuracy error, except for errors due to premature approximation which should be penalised only once in the examination. There is no penalty for using a wrong value for \(g\). E marks will be lost except when results agree to the accuracy required in the question. Rules for replaced work: if a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests; if there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these maths-specific instructions rather than those in the assessor handbook.
   h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A mark in the question. Marks designated as cao may be awarded as long as there are no other errors. E marks are lost unless, by chance, the given results are established by equivalent working. 'Fresh starts' will not affect an earlier decision about a misread. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
   i If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers (provided, of course, that there is nothing in the wording of the question specifying that analytical methods are required). Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. If in doubt, consult your Team Leader. If in any case the scheme operates with considerable unfairness consult your Team Leader. PS = Problem Solving
   M = Modelling \section*{Summary of Updates} \section*{A Level Mathematics A} \section*{H240/01 Pure Mathematics} Printed Answer Booklet \section*{Date - Morning/Afternoon} \section*{Time allowed: \(\mathbf { 2 }\) hours} You must have:
   • Question Paper H240/01 (inserted)
   You may use:
   • a scientific or graphical calculator \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-25_113_517_1123_1215} \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-25_318_1548_1375_248}
   \section*{INSTRUCTIONS}
   • The Question Paper will be found inside the Printed Answer Booklet.
   • Use black ink. HB pencil may be used for graphs and diagrams only.
   • Complete the boxes provided on the Printed Answer Booklet with your name, centre number and candidate number.
   • Answer all the questions.
   • Write your answer to each question in the space provided in the Printed Answer Booklet.
   Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s).
   • Do not write in the bar codes.
   • You are permitted to use a scientific or graphical calculator in this paper.
   • Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question.
   • The acceleration due to gravity is denoted by \(g \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Unless otherwise instructed, when a numerical value is needed, use \(g = 9.8\).
   \section*{INFORMATION}
   • You are reminded of the need for clear presentation in your answers.
   • The Printed Answer Booklet consists of \(\mathbf { 1 6 }\) pages. The Question Paper consists of \(\mathbf { 8 }\) pages.

   1
   2(a)
   2(b)

   3(a) \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-27_801_1479_214_360} 3(b) \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-27_1150_1504_1452_338}

   4(a)

   7(d)
   8(a)
   8(b)

   9(d)
   10(a)
   10(b)

   11(a)
   11(b)

   11(c)
   12(a)
   12(b)

   12(c)

   14(a)

   14(b)
   \multirow[t]{6}{*}{14(c)}
   14(d)

   DO NOT WRITE IN THIS SPACE \section*{DO NOT WRITE ON THIS PAGE} \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
   OCR is part of the

10 A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects.
Initially there are 25 damaged tomato plants.
The number of tomato plants damaged by insects is increasing by \(32 \%\) each day.
10

1. The total number of plants damaged by insects, \(x\), is modelled by $$x = A \times B ^ { t }$$ where \(A\) and \(B\) are constants and \(t\) is the number of days after the gardener first noticed the damaged plants. 10
   1. (i) Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants.
      10
   2. (ii) Explain why this model is not realistic in the long term.
      10
   3. A refined model assumes the rate of increase of the number of plants damaged by insects is given by $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x ( 900 - x ) } { 2700 }$$ 10
   4. 1. Show that $$\int \left( \frac { A } { x } + \frac { B } { 900 - x } \right) \mathrm { d } x = \int \mathrm { d } t$$ where \(A\) and \(B\) are positive integers to be found.
         

         10 (iii) Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects. [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

The population of a country at time \(t\) years is \(N\) millions. At any time, \(N\) is assumed to increase at a rate proportional to the product of \(N\) and \((1 - 0.01N)\). When \(t = 0\), \(N = 20\) and \(\frac{dN}{dt} = 0.32\).

1. Treating \(N\) and \(t\) as continuous variables, show that they satisfy the differential equation $$\frac{dN}{dt} = 0.02N(1 - 0.01N).$$ [1]
2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(N\). [8]
3. Find the time at which the population will be double its value at \(t = 0\). [1]

1. Express \(\frac{2}{P(P-2)}\) in partial fractions. [3]

A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation $$\frac{dP}{dt} = \frac{1}{2}P(P-2)\cos 2t, \quad t \geqslant 0$$ where \(P\) is the population in thousands, and \(t\) is the time measured in years since the start of the study. Given that \(P = 3\) when \(t = 0\),

1. solve this differential equation to show that $$P = \frac{6}{3 - e^{\frac{1}{2}\sin 2t}}$$ [7]
2. find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures. [3]

Fig. 7 illustrates the growth of a population with time. The proportion of the ultimate (long term) population is denoted by \(x\), and the time in years by \(t\). When \(t = 0\), \(x = 0.5\), and as \(t\) increases, \(x\) approaches 1. \includegraphics{figure_7} One model for this situation is given by the differential equation $$\frac{dx}{dt} = x(1-x).$$

1. Verify that \(x = \frac{1}{1+e^{-t}}\) satisfies this differential equation, including the initial condition. [6]
2. Find how long it will take, according to this model, for the population to reach three-quarters of its ultimate value. [3]

An alternative model for this situation is given by the differential equation $$\frac{dx}{dt} = x^2(1-x),$$ with \(x = 0.5\) when \(t = 0\) as before.

1. Find constants \(A\), \(B\) and \(C\) such that \(\frac{1}{x^2(1-x)} = \frac{A}{x^2} + \frac{B}{x} + \frac{C}{1-x}\). [4]
2. Hence show that \(t = 2 + \ln\left(\frac{x}{1-x}\right) - \frac{1}{x}\). [5]
3. Find how long it will take, according to this model, for the population to reach three-quarters of its ultimate value. [2]

Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced. The population \(x\), in thousands, of red squirrels is modelled by the equation $$x = \frac{a}{1 + kt},$$ where \(t\) is the time in years, and \(a\) and \(k\) are constants. When \(t = 0\), \(x = 2.5\).

1. Show that \(\frac{dx}{dt} = -\frac{kx^2}{a}\). [3]
2. Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate \(a\) and \(k\). [3]
3. What is the long-term population of red squirrels predicted by this model? [1]

The population \(y\), in thousands, of grey squirrels is modelled by the differential equation $$\frac{dy}{dt} = 2y - y^2.$$ When \(t = 0\), \(y = 1\).

1. Express \(\frac{1}{2y - y^2}\) in partial fractions. [4]
2. Hence show by integration that \(\ln\left(\frac{y}{2-y}\right) = 2t\). Show that \(y = \frac{2}{1 + e^{-2t}}\). [7]
3. What is the long-term population of grey squirrels predicted by this model? [1]

A population of meerkats is being studied. The population is modelled by the differential equation $$\frac{\mathrm{d}P}{\mathrm{d}t} = \frac{1}{22}P(11 - 2P), \quad t \geq 0, \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double. [7]

A population of meerkats is being studied. The population is modelled by the differential equation $$\frac{dP}{dt} = \frac{1}{22}P(11 - 2P), \quad t \geq 0, \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double. [7]

John wants to encourage more birds to come into the park near his house. Each day, starting on day 1, he puts bird food out and then observes the birds for one hour. He records the maximum number of birds that he observes at any given moment in the park each day. He believes that his observations may be modelled by the following differential equation, where \(n\) is the maximum number of birds that he observed at any given moment on day \(t\). $$\frac{dn}{dt} = 0.1n\left(1 - \frac{n}{50}\right)$$

1. Show that the general solution to the differential equation can be written in the form $$n = \frac{50A}{e^{-0.1t} + A},$$ where \(A\) is an arbitrary positive constant. [9]
2. Using his model, determine the maximum number of birds that John would expect to observe at any given moment in the long term. [1]
3. Write down one possible refinement of this model. [1]
4. Write down one way in which John's model is not appropriate. [1]