OCR H240/01 (Pure Mathematics)

1 Solve the simultaneous equations. $$\begin{array} { r } x ^ { 2 } + 8 x + y ^ { 2 } = 84
x - y = 10 \end{array}$$

2 The points \(A\), \(B\) and \(C\) have position vectors \(3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } , - \mathbf { i } + 6 \mathbf { k }\) and \(7 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k }\) respectively. M is the midpoint of BC .

1. Show that the magnitude of \(\overrightarrow { O M }\) is equal to \(\sqrt { 17 }\). Point D is such that \(\overrightarrow { B C } = \overrightarrow { A D }\).
2. Show that position vector of the point D is \(11 \mathbf { i } - 8 \mathbf { j } - 6 \mathbf { k }\).

3 The diagram below shows the graph of \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-05_801_1483_413_251}

1. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } ( x - 2 ) + 1\).

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]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-05_510_606_1745_274} 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.

6 Prove by contradiction that there is no greatest even positive integer.

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, \(\pounds 8000\) 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 \(0 \leq \theta \leq \pi\).

9 The equation \(x ^ { 3 } - x ^ { 2 } - 5 x + 10 = 0\) has exactly one real root \(\alpha\).

1. Show that the Newton-Raphson iterative formula for finding this root can be written as $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } - x _ { n } ^ { 2 } - 10 } { 3 x _ { n } ^ { 2 } - 2 x _ { n } - 5 }$$
2. Apply the iterative formula in part (a) with initial value \(x _ { 1 } = - 3\) to find \(x _ { 2 } , x _ { 3 } , x _ { 4 }\) correct to 4 significant figures.
3. Use a change of sign method to show that \(\alpha = - 2.533\) is correct to 4 significant figures.
4. Explain why the Newton-Raphson method with initial value \(x _ { 1 } = - 1\) would not converge to \(\alpha\).

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 \(g ( x ) = 6 x - 2 a\), where \(a\) is a positive constant.

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

12 The parametric equations of a curve are given by \(x = 2 \cos \theta\) and \(y = 3 \sin \theta\) for \(0 \leq \theta < 2 \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). The tangents to the curve at the points P and Q pass through the point \(( 2,6 )\).
2. Show that the values of \(\theta\) at the points P and Q satisfy the equation \(2 \sin \theta + \cos \theta = 1\).
3. Find the values of \(\theta\) at the points \(P\) and \(Q\).

13 In this question you must show detailed reasoning. Find the exact values of the \(x\)-coordinates of the stationary points of the curve \(x ^ { 3 } + y ^ { 3 } = 3 x y + 35\).

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 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} 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.

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   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