1.08l Interpret differential equation solutions: in context

6. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$

1. Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
2. Find the value which the population of the town will approach in the long term, according to the model.
   6. continued

9 In an experiment, at time \(t\) minutes there is \(Q\) grams of substance present.
It is known that the substance decays at a rate that is proportional to \(1 + Q ^ { 2 }\). Initially there are 100 grams of the substance present and after 100 minutes there are 50 grams present. Find the amount of the substance present after 400 minutes.

8

1. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$ to find \(t\) in terms of \(x\), given that \(x = 70\) when \(t = 0\).
2. Liquid fuel is stored in a tank. At time \(t\) minutes, the depth of fuel in the tank is \(x \mathrm {~cm}\). Initially there is a depth of 70 cm of fuel in the tank. There is a tap 6 cm above the bottom of the tank. The flow of fuel out of the tank is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$
   1. Explain what happens when \(x = 6\).
   2. Find how long it will take for the depth of fuel to fall from 70 cm to 22 cm .

8

1. 1. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} t } = y \sin t\) to obtain \(y\) in terms of \(t\).
   2. Given that \(y = 50\) when \(t = \pi\), show that \(y = 50 \mathrm { e } ^ { - ( 1 + \cos t ) }\).
2. A wave machine at a leisure pool produces waves. The height of the water, \(y \mathrm {~cm}\), above a fixed point at time \(t\) seconds is given by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = y \sin t$$
   1. Given that this height is 50 cm after \(\pi\) seconds, find, to the nearest centimetre, the height of the water after 6 seconds.
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\) and hence verify that the water reaches a maximum height after \(\pi\) seconds.

7

1. A differential equation is given by \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - k t \mathrm { e } ^ { \frac { 1 } { 2 } x }\), where \(k\) is a positive constant.
   1. Solve the differential equation.
   2. Hence, given that \(x = 6\) when \(t = 0\), show that \(x = - 2 \ln \left( \frac { k t ^ { 2 } } { 4 } + \mathrm { e } ^ { - 3 } \right)\).
      (3 marks)
2. The population of a colony of insects is decreasing according to the model \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - k t \mathrm { e } ^ { \frac { 1 } { 2 } x }\), where \(x\) thousands is the number of insects in the colony after time \(t\) minutes. Initially, there were 6000 insects in the colony. Given that \(k = 0.004\), find:
   1. the population of the colony after 10 minutes, giving your answer to the nearest hundred;
   2. the time after which there will be no insects left in the colony, giving your answer to the nearest 0.1 of a minute.

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.
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   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}
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   \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.
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   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.
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   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.
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   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)
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   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.
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8 \includegraphics[max width=\textwidth, alt={}, center]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-07_360_489_1027_788} The diagram shows a water tank which is shaped as an inverted cone with semi-vertical angle \(30 ^ { \circ }\) and height 50 cm . Initially the tank is full, and the depth of the water is 50 cm . Water flows out of a small hole at the bottom of the tank. The rate at which the water flows out is modelled by \(\frac { \mathrm { d } V } { \mathrm {~d} t } = - 2 h\), where \(V \mathrm {~cm} ^ { 3 }\) is the volume of water remaining and \(h \mathrm {~cm}\) is the depth of water in the tank \(t\) seconds after the water begins to flow out. Determine the time taken for the tank to become empty.
[0pt] [For a cone with base radius \(r\) and height \(h\) the volume \(V\) is given by \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]

10 A tank with vertical sides and rectangular cross-section is initially full of water. The water is leaking out of a hole in the base of the tank at a rate which is proportional to the square root of the depth of the water. \(V \mathrm {~m} ^ { 3 }\) is the volume of water in the tank at time \(t\) hours.

1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = a \sqrt { V }\), where \(a\) is a constant.
2. Given that the tank is half full after one hour, show that \(V = V _ { 0 } \left( \left( \frac { 1 } { \sqrt { 2 } } - 1 \right) t + 1 \right) ^ { 2 }\), where \(V _ { 0 } \mathrm {~m} ^ { 3 }\) is the initial volume of water in the tank.
3. Hence show that the tank will be empty after approximately 3 hours and 25 minutes.

A radioactive isotope decays in such a way that the rate of change of the number \(N\) of radioactive atoms present after \(t\) days, is proportional to \(N\).

1. Write down a differential equation relating \(N\) and \(t\). [2]
2. Show that the general solution may be written as \(N = Ae^{-kt}\), where \(A\) and \(k\) are positive constants. [5]

Initially the number of radioactive atoms present is \(7 \times 10^{18}\) and 8 days later the number present is \(3 \times 10^{17}\).

1. Find the value of \(k\). [3]
2. Find the number of radioactive atoms present after a further 8 days. [2]

A Pancho car has value \(£V\) at time \(t\) years. A model for \(V\) assumes that the rate of decrease of \(V\) at time \(t\) is proportional to \(V\).

1. By forming and solving an appropriate differential equation, show that \(V = Ae^{-kt}\), where \(A\) and \(k\) are positive constants. [3]

The value of a new Pancho car is \(£20\,000\), and when it is 3 years old its value is \(£11\,000\).

1. Find, to the nearest \(£100\), an estimate for the value of the Pancho when it is 10 years old. [5]

A Pancho car is regarded as 'scrap' when its value falls below \(£500\).

1. Find the approximate age of the Pancho when it becomes 'scrap'. [3]

Fluid flows out of a cylindrical tank with constant cross section. At time \(t\) minutes, \(t \geq 0\), the volume of fluid remaining in the tank is \(V\) m\(^3\). The rate at which the fluid flows, in m\(^3\) min\(^{-1}\), is proportional to the square root of \(V\).

1. Show that the depth \(h\) metres of fluid in the tank satisfies the differential equation $$\frac{dh}{dt} = -k\sqrt{h}, \quad \text{where } k \text{ is a positive constant.}$$ [3]
2. Show that the general solution of the differential equation may be written as $$h = (A - Bt)^2, \quad \text{where } A \text{ and } B \text{ are constants.}$$ [4] Given that at time \(t = 0\) the depth of fluid in the tank is 1 m, and that 5 minutes later the depth of fluid has reduced to 0.5 m,
3. find the time, \(T\) minutes, which it takes for the tank to empty. [3]
4. Find the depth of water in the tank at time \(0.5T\) minutes. [2]

Water is leaking from a container. After \(t\) seconds, the depth of water in the container is \(x\) cm, and the volume of water is \(V\) cm\(^3\), where \(V = \frac{1}{3}x^3\). The rate at which water is lost is proportional to \(x\), so that \(\frac{dV}{dt} = -kx\), where \(k\) is a constant.

1. Show that \(x \frac{dx}{dt} = -k\). [3]

Initially, the depth of water in the container is 10 cm.

1. Show by integration that \(x = \sqrt{100 - 2kt}\). [4]
2. Given that the container empties after 50 seconds, find \(k\). [2]

Once the container is empty, water is poured into it at a constant rate of 1 cm\(^3\) per second. The container continues to lose water as before.

1. Show that, \(t\) seconds after starting to pour the water in, \(\frac{dx}{dt} = \frac{1-x}{x^2}\). [2]
2. Show that \(\frac{1}{1-x} - x - 1 = \frac{x^2}{1-x}\). Hence solve the differential equation in part (iv) to show that $$t = \ln\left(\frac{1}{1-x}\right) - \frac{1}{2}x^2 - x.$$ [6]
3. Show that the depth cannot reach 1 cm. [1]

The total value of the sales made by a new company in the first \(t\) years of its existence is denoted by \(£V\). A model is proposed in which the rate of increase of \(V\) is proportional to the square root of \(V\). The constant of proportionality is \(k\).

1. Express the model as a differential equation. Verify by differentiation that \(V = (\frac{1}{2}kt + c)^2\), where \(c\) is an arbitrary constant, satisfies this differential equation. [4]
2. The value of the company's sales in its first year is £10000, and the total value of the sales in the first two years is £40000. Find \(V\) in terms of \(t\). [4]

An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac{dP}{dt} = kP,$$ where \(k\) is a constant.

1. Find an expression for \(P\) in terms of \(k\) and \(t\). [5]

Given that after one week there are 360 insects in the colony,

1. find the value of \(k\) to 3 significant figures. [2]

Given also that after two and three weeks there are 440 and 600 insects respectively,

1. comment on suitability of the model. [2]

An alternative model assumes that $$\frac{dP}{dt} = P(0.4 - 0.25\cos 0.5t).$$

1. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation. [4]
2. Compare the suitability of the two models. [3]

The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac{dn}{dt} = e^{0.5t} - 5, \quad t \geq 0.$$

1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue. [3]
2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation. [4]
3. Explain why this model would not be appropriate for large values of \(t\). [1]

At time \(t = 0\), a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, \(y\) metres, after \(t\) hours satisfies the differential equation $$\frac{dy}{dt} = -ke^{-0.2t},$$ where \(k\) is a positive constant.

1. Find an expression for \(y\) in terms of \(k\) and \(t\). [4]

Given that two hours after being filled the depth of water in the tank is 1.6 metres,

1. find the value of \(k\) to 4 significant figures. [2]

Given also that the hole in the tank is \(h\) cm above the base of the tank,

1. show that \(h = 79\) to 2 significant figures. [3]

The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).

1. Write down a differential equation connecting \(N\) and \(t\). [1]

Given that initially there are \(N_0\) bacteria present in a culture,

1. Show that \(N = N_0 e^{kt}\), where \(k\) is a positive constant. [6]

Given also that the number of bacteria present doubles every six hours,

1. find the value of \(k\), [3]
2. Find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. [2]

A car starts from rest at time \(t = 0\) and moves along a straight road with constant acceleration 4 ms\(^{-2}\) for 10 seconds. It then travels at a constant speed for 50 seconds before decelerating to rest over a further distance of 240 m.

1. Sketch a graph of velocity against time for the total period of the car's motion. [3 marks]
2. Find the car's average speed for the whole journey. [6 marks]

In reality the car's acceleration \(a\) ms\(^{-2}\) in the first 10 seconds is not constant, but increases from 0 to 4 ms\(^{-2}\) in the first 5 seconds and then decreases to 0 again. A refined model designed to take account of this uses the formula \(a = k(mt - t^2)\) for \(0 \leq t \leq 10\).

1. Calculate the values of the constants \(k\) and \(m\). [5 marks]
2. Find the acceleration of the car when \(t = 2\) according to this model. [2 marks]

In a chemical reaction, the mass \(m\) grams of a chemical at time \(t\) minutes is modelled by the differential equation $$\frac{dm}{dt} = \frac{m}{t(1 + 2t)}.$$ At time 1 minute, the mass of the chemical is 1 gram.

1. Solve the differential equation to show that \(m = \frac{3t}{(1 + 2t)}\). [8]
2. Hence
   1. find the time when the mass is 1.25 grams, [2]
   2. show what happens to the mass of the chemical as \(t\) becomes large. [2]

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]