Questions (30808 questions)

4 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{2c30c315-c666-47d6-b05e-da71cb4decdd-05_519_604_1617_335} The angle \(A O B\) is \(\theta\) radians. The arc length \(A B\) is 15 cm and the area of the sector is \(45 \mathrm {~cm} ^ { 2 }\).

1. Find the values of \(r\) and \(\theta\).
2. Find the area of the segment bounded by the arc \(A B\) and the chord \(A B\).

5 In this question you must show detailed reasoning.
Use logarithms to solve the equation \(3 ^ { 2 x + 1 } = 4 ^ { 100 }\), giving your answer correct to 3 significant figures.

7 Business A made a \(\pounds 5000\) profit during its first year.
In each subsequent year, the profit increased by \(\pounds 1500\) so that the profit was \(\pounds 6500\) during the second year, £8000 during the third year and so on. Business B made a \(\pounds 5000\) profit during its first year.
In each subsequent year, the profit was 90\% of the previous year's profit.

1. Find an expression for the total profit made by business A during the first \(n\) years. Give your answer in its simplest form.
2. Find an expression for the total profit made by business B during the first \(n\) years. Give your answer in its simplest form.
3. Find how many years it will take for the total profit of business A to reach \(\pounds 385000\).
4. Comment on the profits made by each business in the long term.

8

1. Show that \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = \sin 2 \theta\).
2. In this question you must show detailed reasoning. Solve \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = 3 \cos 2 \theta\) for \(0 \leq \theta \leq \pi\).

10 A curve has equation \(x = ( y + 5 ) \ln ( 2 y - 7 )\).

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of y .
2. Find the gradient of the curve where it crosses the \(y\)-axis.

11 For all real values of \(x\), the functions f and g are defined by \(\mathrm { f } ( x ) = x ^ { 2 } + 8 a x + 4 a ^ { 2 }\) and \(\mathrm { g } ( x ) = 6 x - 2 a\), where \(a\) is a positive constant.

1. Find \(\mathrm { fg } ( x )\). Determine the range of \(\mathrm { fg } ( x )\) in terms of \(a\).
2. If \(\operatorname { fg } ( 2 ) = 144\), find the value of \(a\).
3. Determine whether the function fg has an inverse.

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 $$n = \frac { 50 A } { \mathrm { e } ^ { - 0.1 t } + A } , \text { where } A \text { is an arbitrary positive constant. }$$
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 Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{...day June 20XX - Morning/Afternoon} A Level Mathematics A
   H240/01 Pure Mathematics \section*{SAMPLE MARK SCHEME}
   \includegraphics[max width=\textwidth, alt={}]{2c30c315-c666-47d6-b05e-da71cb4decdd-09_268_1241_1256_777}
   \section*{Text Instructions} \section*{1. Annotations and abbreviations} \section*{2. Subject-specific Marking Instructions for A Level Mathematics 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. \section*{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. \section*{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.
   j If in any case the scheme operates with considerable unfairness consult your Team Leader. PS = Problem Solving
   M = Modelling \section*{Summary of Updates} \section*{Accredited} \section*{A Level Mathematics A} \section*{H240/01 Pure Mathematics} Printed Answer Booklet \section*{Date - Morning/Afternoon} \section*{Time allowed: \(\mathbf { 2 }\) hours} \section*{You must have:}
   • Question Paper H240/01 (inserted)
   \section*{You may use:}
   • a scientific or graphical calculator
     \includegraphics[max width=\textwidth, alt={}, center]{2c30c315-c666-47d6-b05e-da71cb4decdd-25_115_487_1052_1239}
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   \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.
   • Final answers should be given to a degree of accuracy appropriate to the context.
   • 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.
     \includegraphics[max width=\textwidth, alt={}, center]{2c30c315-c666-47d6-b05e-da71cb4decdd-26_2287_1632_234_238}
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   7(a)
   7(b)
   7(c)

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

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

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

   \includegraphics[max width=\textwidth, alt={}, center]{2c30c315-c666-47d6-b05e-da71cb4decdd-36_1552_1582_180_238}
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   14(b)

   \section*{DO NOT WRITE ON THIS PAGE} \section*{Copyright Information:} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answerrelated information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{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 Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge

1 Simplify fully.

1. \(\sqrt { a ^ { 3 } } \times \sqrt { 16 a }\)
2. \(\left( 4 b ^ { 6 } \right) ^ { \frac { 5 } { 2 } }\)

3 A publisher has to choose the price at which to sell a certain new book.
The total profit, \(\pounds t\), that the publisher will make depends on the price, \(\pounds p\).
He decides to use a model that includes the following assumptions.

• If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small.
• If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small.

The graphs below show two possible models. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Explain how model A is inconsistent with one of the assumptions given above.
2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k \left( 12 p - p ^ { 2 } \right)\), and find the value of the constant \(k\).
3. The publisher needs to make a total profit of at least \(\pounds 6400\). Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie.
4. Comment briefly on how realistic model B may be in the following cases.
   • \(p = 0\)
   • \(p = 12.1\)

4

1. Express \(\frac { 1 } { ( x - 1 ) ( x + 2 ) }\) in partial fractions
2. In this question you must show detailed reasoning. Hence find \(\int _ { 2 } ^ { 3 } \frac { 1 } { ( x - 1 ) ( x + 2 ) } \mathrm { d } x\).
   Give your answer in its simplest form.

5 The diagram shows the circle with centre O and radius 2 , and the parabola \(y = \frac { 1 } { \sqrt { 3 } } \left( 4 - x ^ { 2 } \right)\).
\includegraphics[max width=\textwidth, alt={}, center]{cac31da4-f1ad-4c34-a47f-2bc68c2304f1-06_850_916_996_352} The circle meets the parabola at points \(P\) and \(Q\), as shown in the diagram.

1. Verify that the coordinates of \(Q\) are \(( 1 , \sqrt { 3 } )\).
2. Find the exact area of the shaded region enclosed by the arc \(P Q\) of the circle and the parabola.

6 Helga invests \(\pounds 4000\) in a savings account.
After \(t\) days, her investment is worth \(\pounds y\).
The rate of increase of \(y\) is \(k y\), where \(k\) is a constant.

1. Write down a differential equation in terms of \(t , y\) and \(k\).
2. Solve your differential equation to find the value of Helga's investment after \(t\) days. Give your answer in terms of \(k\) and \(t\). It is given that \(k = \frac { 1 } { 365 } \ln \left( 1 + \frac { r } { 100 } \right)\) where \(r \%\) is the rate of interest per annum.
   During the first year the rate of interest is \(6 \%\) per annum.
3. Find the value of Helga's investment after 90 days. After one year (365 days), the rate of interest drops to 5\% per annum.
4. Find the total time that it will take for Helga's investment to double in value.

7

1. The heights of English men aged 25 to 34 are normally distributed with mean 178 cm and standard deviation 8 cm .
   Three English men aged 25 to 34 are chosen at random.
   Find the probability that all three men have a height less than 194 cm .
2. The diagram shows the distribution of heights of Scottish women aged 25 to 34 .
   \includegraphics[max width=\textwidth, alt={}, center]{cac31da4-f1ad-4c34-a47f-2bc68c2304f1-08_559_1389_855_412} The distribution is approximately normal. Use the diagram in the Printed Answer Booklet to estimate the standard deviation of these heights, explaining your method.

9 The diagram below shows some "Cycle to work" data taken from the 2001 and 2011 UK censuses. The diagram shows the percentages, by age group, of male and female workers in England and Wales, excluding London, who cycled to work in 2001 and 2011.
\includegraphics[max width=\textwidth, alt={}, center]{cac31da4-f1ad-4c34-a47f-2bc68c2304f1-10_899_1537_524_283} The following questions refer to the workers represented by the graphs in the diagram.

1. A researcher is going to take a sample of men and a sample of women and ask them whether or not they cycle to work. Why would it be more important to stratify the sample of men? A research project followed a randomly chosen large sample of the group of male workers who were aged 30-34 in 2001.
2. Does the diagram suggest that the proportion of this group who cycled to work has increased or decreased from 2001 to 2011?
   Justify your answer.
3. Write down one assumption that you have to make about these workers in order to draw this conclusion.

12 The table shows information for England and Wales, taken from the UK 2011 census. A random sample of 10000 people in another country was chosen in 2011 , and the number, \(m\), of children aged 5-17 was noted.
It was found that there was evidence at the \(2.5 \%\) level that the proportion of children aged \(5 - 17\) in the same year was higher than in the UK.
Unfortunately, when the results were recorded the value of \(m\) was omitted. Use an appropriate normal distribution to find an estimate of the smallest possible value of \(m\).

13 The table and the four scatter diagrams below show data taken from the 2011 UK census for four regions. On the scatter diagrams the names have been replaced by letters.
The table shows, for each region, the mean and standard deviation of the proportion of workers in each Local Authority who travel to work by driving a car or van and the proportion of workers in each Local Authority who travel to work as a passenger in a car or van.
Each scatter diagram shows, for each of the Local Authorities in a particular region, the proportion of workers who travel to work by driving a car or van and the proportion of workers who travel to work as a passenger in a car or van. Region A
\includegraphics[max width=\textwidth, alt={}, center]{cac31da4-f1ad-4c34-a47f-2bc68c2304f1-14_607_1047_1228_369} Region B
\includegraphics[max width=\textwidth, alt={}, center]{cac31da4-f1ad-4c34-a47f-2bc68c2304f1-14_598_1043_1927_371}
\includegraphics[max width=\textwidth, alt={}, center]{cac31da4-f1ad-4c34-a47f-2bc68c2304f1-15_678_1104_227_317} Region D
\includegraphics[max width=\textwidth, alt={}, center]{cac31da4-f1ad-4c34-a47f-2bc68c2304f1-15_615_1058_1048_367}

1. Using the values given in the table, match each region to its corresponding scatter diagram, explaining your reasoning.
2. Steven claims that the outlier in the scatter diagram for Region C consists of a group of small islands. Explain whether or not the data given above support his claim.
3. One of the Local Authorities in Region B consists of a single large island. Explain whether or not you would expect this Local Authority to appear as an outlier in the scatter diagram for Region B.

8 In this question you must show detailed reasoning. The diagram shows triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-06_735_1383_456_342} The angles \(C A B\) and \(A B C\) are each \(45 ^ { \circ }\), and angle \(A C B = 90 ^ { \circ }\).
The points \(D\) and \(E\) lie on \(A C\) and \(A B\) respectively. \(A E = D E = 1 , D B = 2\).
Angle \(B E D = 90 ^ { \circ }\), angle \(E B D = 30 ^ { \circ }\) and angle \(D B C = 15 ^ { \circ }\).

1. Show that \(B C = \frac { \sqrt { 2 } + \sqrt { 6 } } { 2 }\).
2. By considering triangle \(B C D\), show that \(\sin 15 ^ { \circ } = \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }\).

9 Two forces, of magnitudes 2 N and 5 N , act on a particle in the directions shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-07_323_755_548_283} \section*{(a) Calculate the magnitude of the resultant force on the particle.
(b) Calculate the angle between this resultant force and the force of magnitude 5 N .}

10 A body of mass 20 kg is on a rough plane inclined at angle \(\alpha\) to the horizontal. The body is held at rest on the plane by the action of a force of magnitude \(P \mathrm {~N}\). The force is acting up the plane in a direction parallel to a line of greatest slope of the plane. The coefficient of friction between the body and the plane is \(\mu\).

1. When \(P = 100\), the body is on the point of sliding down the plane. Show that \(g \sin \alpha = g \mu \cos \alpha + 5\).
2. When \(P\) is increased to 150, the body is on the point of sliding up the plane. Use this, and your answer to part (a), to find an expression for \(\alpha\) in terms of \(g\).

12 A girl is practising netball.
She throws the ball from a height of 1.5 m above horizontal ground and aims to get the ball through a hoop.
The hoop is 2.5 m vertically above the ground and is 6 m horizontally from the point of projection. The situation is modelled as follows.

• The initial velocity of the ball has magnitude \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
• The angle of projection is \(40 ^ { \circ }\).
• The ball is modelled as a particle.
• The hoop is modelled as a point.

This is shown on the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-09_378_1210_1119_278}

1. For \(U = 10\), find
   1. the greatest height above the ground reached by the ball
   2. the distance between the ball and the hoop when the ball is vertically above the hoop.
2. Calculate the value of \(U\) which allows her to hit the hoop.
3. How appropriate is this model for predicting the path of the ball when it is thrown by the girl?
4. Suggest one improvement that might be made to this model.

13 Particle \(A\), of mass \(m \mathrm {~kg}\), lies on the plane \(\Pi _ { 1 }\) inclined at an angle of \(\tan ^ { - 1 } \frac { 3 } { 4 }\) to the horizontal.
Particle \(B\), of \(4 m \mathrm {~kg}\), lies on the plane \(\Pi _ { 2 }\) inclined at an angle of \(\tan ^ { - 1 } \frac { 4 } { 3 }\) to the horizontal.
The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at \(P\).
The coefficient of friction between particle \(A\) and \(\Pi _ { 1 }\) is \(\frac { 1 } { 3 }\) and plane \(\Pi _ { 2 }\) is smooth.
Particle \(A\) is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane. This is shown on the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-10_403_851_865_303}

1. Show that when \(A\) is released it accelerates towards the pulley at \(\frac { 7 g } { 15 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Assuming that \(A\) does not reach the pulley, show that it has moved a distance of \(\frac { 1 } { 4 } \mathrm {~m}\) when its speed is \(\sqrt { \frac { 7 g } { 30 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   [0pt] [2]

14 A uniform ladder \(A B\) of mass 35 kg and length 7 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a rough vertical wall.
The ladder is inclined at an angle of \(45 ^ { \circ }\) to the horizontal.
A man of mass 70 kg is standing on the ladder at a point \(C\), which is \(x\) metres from \(A\).
The coefficient of friction between the ladder and the wall is \(\frac { 1 } { 3 }\) and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 2 }\).
The system is in limiting equilibrium. Find \(x\).
[0pt] [8] \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 Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{...day June 20XX - Morning/Afternoon} A Level Mathematics A
H240/03 Pure Mathematics and Mechanics SAMPLE MARK SCHEME Duration: 2 hours MAXIMUM MARK 100
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-13_259_1320_1242_826} This document consists of 20 pages \section*{Text Instructions} \section*{1. Annotations and abbreviations} \section*{2. Subject-specific Marking Instructions for A Level Mathematics 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. \section*{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. \section*{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.
g 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.
j If in any case the scheme operates with considerable unfairness consult your Team Leader. \end{table} PS = Problem Solving
M = Modelling \section*{Summary of Updates} \section*{A Level Mathematics A} \section*{H240/03 Pure Mathematics and Mechanics
Printed Answer Booklet} \section*{Date - Morning/Afternoon} \section*{Time allowed: \(\mathbf { 2 }\) hours} \section*{You must have:}

• Question Paper H240/03 (inserted)

\section*{You may use:}

• a scientific or graphical calculator
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\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.
• Final answers should be given to a degree of accuracy appropriate to the context.
• 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 \(\mathbf { 1 2 }\) pages.

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\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-38_2276_1630_180_240} \begin{table}[h] \end{table} DO NOT WRITE IN THIS SPACE
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-42_2282_1636_178_236} \section*{DO NOT WRITE ON THIS PAGE} \section*{Copyright Information:} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{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 Builiding, Shaftesbury Road, Cambridge CB2 8EA.
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2 In this question you must show detailed reasoning.
Given that \(z _ { 1 } = 3 + 2 \mathrm { i }\) and \(z _ { 2 } = - 1 - \mathrm { i }\), find the following, giving each in the form \(a + b \mathrm { i }\).

1. \(\mathrm { z } _ { 1 } { } ^ { * } \mathrm { z } _ { 2 }\)
2. \(\frac { z _ { 1 } + 2 z _ { 2 } } { z _ { 2 } }\)

3

1. You are given two matrices, \(\mathbf { A }\) and \(\mathbf { B }\), where $$\mathbf { A } = \left( \begin{array} { l l } 1 & 2
   2 & 1 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } - 1 & 2
   2 & - 1 \end{array} \right)$$ Show that \(\mathbf { A B } = m \mathbf { I }\), where \(m\) is a constant to be determined.
2. You are given two matrices, \(\mathbf { C }\) and \(\mathbf { D }\), where $$\mathbf { C } = \left( \begin{array} { r r r } 2 & 1 & 5
   1 & 1 & 3
   - 1 & 2 & 2 \end{array} \right) \text { and } \mathbf { D } = \left( \begin{array} { r r r } - 4 & 8 & - 2
   - 5 & 9 & - 1
   3 & - 5 & 1 \end{array} \right)$$ Show that \(\mathbf { C } ^ { - 1 } = k \mathbf { D }\) where \(k\) is a constant to be determined.
3. The matrices \(\mathbf { E }\) and \(\mathbf { F }\) are given by \(\mathbf { E } = \left( \begin{array} { c c } k & k ^ { 2 }
   3 & 0 \end{array} \right)\) and \(\mathbf { F } = \binom { 2 } { k }\) where \(k\) is a constant. Determine any matrix \(\mathbf { F }\) for which \(\mathbf { E F } = \binom { - 2 k } { 6 }\).

7 In this question you must show detailed reasoning. It is given that \(\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 65 z - 125\).
The points representing the three roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) are plotted on an Argand diagram.
Show that these points lie on the circle \(| z | = k\), where \(k\) is a real number to be determined.