Questions - OCR - A-Level Maths

OCR H240/01 Q10

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

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

OCR H240/01 Q11

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.

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

OCR H240/01 Q12

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

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 \(( 2,6 )\).
Show that the values of \(\theta\) at the points P and Q satisfy the equation \(2 \sin \theta + \cos \theta = 1\).
Find the values of \(\theta\) at the points \(P\) and \(Q\).

OCR H240/01 Q13

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

OCR H240/01 Q14

82 marks

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 \(t\).
\(\frac { \mathrm { d } n } { \mathrm {~d} t } = 0.1 n \left( 1 - \frac { n } { 50 } \right)\)

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.
Using his model, determine the maximum number of birds that John would expect to observe at any given moment in the long term.
Write down one possible refinement of this model.
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.
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:} 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

OCR H240/02 2020 November Q1

1

Differentiate the following with respect to \(x\).
1. \(( 2 x + 3 ) ^ { 7 }\)
2. \(x ^ { 3 } \ln x\)
Find \(\int \cos 5 x \mathrm {~d} x\).
Find the equation of the curve through \(( 1,3 )\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x - 5\).

OCR H240/02 2020 November Q2

2 Simplify fully \(\frac { 2 x ^ { 3 } + x ^ { 2 } - 7 x - 6 } { x ^ { 2 } - x - 2 }\).

OCR H240/02 2020 November Q3

3 In this question you should assume that \(- 1 < x < 1\).

For the binomial expansion of \(( 1 - x ) ^ { - 2 }\)
1. find and simplify the first four terms,
2. write down the term in \(x ^ { n }\).
Write down the sum to infinity of the series \(1 + x + x ^ { 2 } + x ^ { 3 } + \ldots\).
Hence or otherwise find and simplify an expression for \(2 + 3 x + 4 x ^ { 2 } + 5 x ^ { 3 } + \ldots\) in the form \(\frac { a - x } { ( b - x ) ^ { 2 } }\) where \(a\) and \(b\) are constants to be determined.

OCR H240/02 2020 November Q4

4 In this question you must show detailed reasoning.
Solve the equation \(3 \sin ^ { 4 } \phi + \sin ^ { 2 } \phi = 4\), for \(0 \leqslant \phi < 2 \pi\), where \(\phi\) is measured in radians.

OCR H240/02 2020 November Q5

5

Determine the set of values of \(n\) for which \(\frac { n ^ { 2 } - 1 } { 2 }\) and \(\frac { n ^ { 2 } + 1 } { 2 }\) are positive integers. A 'Pythagorean triple' is a set of three positive integers \(a , b\) and \(c\) such that \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).
Prove that, for the set of values of \(n\) found in part (a), the numbers \(n , \frac { n ^ { 2 } - 1 } { 2 }\) and \(\frac { n ^ { 2 } + 1 } { 2 }\) form a Pythagorean triple.

OCR H240/02 2020 November Q6

6 Prove that \(\sqrt { 2 } \cos \left( 2 \theta + 45 ^ { \circ } \right) \equiv \cos ^ { 2 } \theta - 2 \sin \theta \cos \theta - \sin ^ { 2 } \theta\), where \(\theta\) is measured in degrees.
\(7 \quad A\) and \(B\) are fixed points in the \(x - y\) plane. The position vectors of \(A\) and \(B\) are \(\mathbf { a }\) and \(\mathbf { b }\) respectively. State, with reference to points \(A\) and \(B\), the geometrical significance of

the quantity \(| \mathbf { a } - \mathbf { b } |\),
the vector \(\frac { 1 } { 2 } ( \mathbf { a } + \mathbf { b } )\). The circle \(P\) is the set of points with position vector \(\mathbf { p }\) in the \(x - y\) plane which satisfy \(\left| \mathbf { p } - \frac { 1 } { 2 } ( \mathbf { a } + \mathbf { b } ) \right| = \frac { 1 } { 2 } | \mathbf { a } - \mathbf { b } |\).
State, in terms of \(\mathbf { a }\) and \(\mathbf { b }\),
1. the position vector of the centre of \(P\),
2. the radius of \(P\). It is now given that \(\mathbf { a } = \binom { 2 } { - 1 } , \mathbf { b } = \binom { 4 } { 5 }\) and \(\mathbf { p } = \binom { x } { y }\).
Find a cartesian equation of \(P\).

OCR H240/02 2020 November Q8

8 The rate of change of a certain population \(P\) at time \(t\) is modelled by the equation \(\frac { \mathrm { d } P } { \mathrm {~d} t } = ( 100 - P )\). Initially \(P = 2000\).

Determine an expression for \(P\) in terms of \(t\).
Describe how the population changes over time.

OCR H240/02 2020 November Q9

9 The histogram shows information about the numbers of cars in five different price ranges, sold in one year at a car showroom.
\includegraphics[max width=\textwidth, alt={}, center]{4ba60d6b-c987-4f6b-8c55-8b594c90c854-06_922_1413_495_244} It is given that 66 cars in the price range \(\pounds 10000\) to \(\pounds 20000\) were sold.

Find the number of cars sold in the price range \(\pounds 50000\) to \(\pounds 90000\).
State the units of the frequency density.
Suggest one change that the management could make to the diagram so that it would provide more information.
Estimate the number of cars sold in the price range \(\pounds 50000\) to \(\pounds 60000\).

OCR H240/02 2020 November Q10

10 Pierre is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5\% significance level.

OCR H240/02 2020 November Q11

11 As part of a research project, the masses, \(m\) grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.

Mass \(( \mathrm { g } )\)	\(50 \leqslant m < 150\)	\(150 \leqslant m < 200\)	\(200 \leqslant m < 250\)	\(250 \leqslant m < 350\)
Frequency	162	318	355	165

Calculate estimates of the mean and standard deviation of these masses. The masses, \(x\) grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable \(X \sim \mathrm {~N} ( 200,3600 )\).
Use the model to find \(\mathrm { P } ( 150 < X < 210 )\).
Use the model to determine \(x _ { 1 }\) such that \(\mathrm { P } \left( 160 < X < x _ { 1 } \right) = 0.6\), giving your answer correct to five significant figures. It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.
Use these results to show that the model may not be appropriate.
Suggest a different value of a parameter of the model in the light of these results.

OCR H240/02 2020 November Q12

12 In the past, the time for Jeff's journey to work had mean 45.7 minutes and standard deviation 5.6 minutes. This year he is trying a new route. In order to test whether the new route has reduced his journey time, Jeff finds the mean time for a random sample of 30 journeys using the new route. He carries out a hypothesis test at the 2.5\% significance level. Jeff assumes that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.

State appropriate null and alternative hypotheses for the test.
Determine the rejection region for the test.

OCR H240/02 2020 November Q13

13 Andy and Bev are playing a game.

The game consists of three points.
On each point, \(\mathrm { P } (\) Andy wins \() = 0.4\) and \(\mathrm { P } (\) Bev wins \() = 0.6\).
If one player wins two consecutive points, then they win the game, otherwise neither player wins.
1. Determine the probability of the following events.
  1. Andy wins the game.
  2. Neither player wins the game.

Andy and Bev now decide to play a match which consists of a series of games.

In each game, if a player wins the game then they win the match.
If neither player wins the game then the players play another game.
Determine the probability that Andy wins the match.

OCR H240/02 2020 November Q14

14 Table 1 shows the numbers of usual residents in the age range 0 to 4 in 15 Local Authorities (LAs) in 2001 and 2011. The table also shows the increase in the numbers in this age group, and the same increase as a percentage. \begin{table}[h]

	2001	2011	Increase	\% Increase
Bolton	16779	18765	1986	11.84\%
Bury	11117	12235	1118	10.06\%
Knowsley	9454	9121	-333	-3.52\%
Liverpool	24840	26099	1259	5.07\%
Manchester	24693	36413	11720	47.46\%
Oldham	15196	16491	1295	8.52\%
Rochdale	13771	14754	983	7.14\%
Salford	12529	16255	3726	29.74\%
Sefton	14896	14601	-295	-1.98\%
St. Helens	10083	10269	186	1.84\%
Stockport	16457	17342	885	5.38\%
Tameside	12803	14439	1636	12.78\%
Trafford	11971	14870	2899	24.22\%
Wigan	17561	19681	2120	12.07\%
Wirral	17475	18514	1039	5.95\%

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} Fig. 2 shows the increase in each LA in raw numbers, and Fig. 3 shows the percentage increase in each LA. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4ba60d6b-c987-4f6b-8c55-8b594c90c854-10_792_1691_1838_187} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4ba60d6b-c987-4f6b-8c55-8b594c90c854-11_707_1700_214_185} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

The Education Committees in these LAs need to plan for the provision of schools for pupils in their districts.
1. Explain why, in this context, the increase is more important than the actual numbers.
2. In which of the following LAs was there likely to have been the greatest need for extra teachers in the years following 2011: Bolton, Sefton, Tameside or Wigan? Give a reason for your answer.
3. State an assumption about the populations needed to make your answer in part (ii) valid.
In two of the 15 LAs the proportion of young families is greater than in the other 13 LAs. Suggest, using only data from Fig. 2 and Fig. 3 and/or Table 1, which two LAs these are most likely to be.

OCR H240/02 2020 November Q15

15 In this question you must show detailed reasoning. The random variable \(X\) has probability distribution defined as follows.
\(\mathrm { P } ( X = x ) = \begin{cases} \frac { 15 } { 64 } \times \frac { 2 ^ { x } } { x ! } & x = 2,3,4,5 ,
0 & \text { otherwise. } \end{cases}\)

Show that \(\mathrm { P } ( X = 2 ) = \frac { 15 } { 32 }\). The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 9\), determine the probability that at least one of these three values is equal to 2 . Freda chooses values of \(X\) at random until she has obtained \(X = 2\) exactly three times. She then stops.
Determine the probability that she chooses exactly 10 values of \(X\). \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA

OCR H240/02 2022 June Q1

1 In this question you must show detailed reasoning. Solve the following equations.

\(\frac { x } { x + 1 } - \frac { x - 1 } { x + 2 } = 0\)
\(\frac { 8 } { x ^ { 6 } } - \frac { 7 } { x ^ { 3 } } - 1 = 0\)
\(3 ^ { x ^ { 2 } - 7 } = \frac { 1 } { 243 }\)

OCR H240/02 2022 June Q2

2 The points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j }\) and \(4 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k }\) respectively.

Find the length of \(A B\). Point \(P\) has position vector \(p \mathbf { i } - 3 \mathbf { k }\), where \(p\) is a constant. \(P\) lies on the circumference of a circle of which \(A B\) is a diameter.
Find the two possible values of \(p\).

OCR H240/02 2022 June Q3

3

Amaya and Ben integrated \(( 1 + x ) ^ { 2 }\), with respect to \(x\), using different methods, as follows. Amaya: \(\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \frac { ( 1 + x ) ^ { 3 } } { 3 } + c \quad = \frac { 1 } { 3 } + x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c\) Ben: \(\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \int \left( 1 + 2 x + x ^ { 2 } \right) \mathrm { d } x = x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c\) Charlie said that, because these answers are different, at least one of them must be wrong. Explain whether you agree with Charlie's statement.
You are given that \(a\) is a constant greater than 1 .
1. Find \(\int _ { 1 } ^ { a } \frac { 1 } { ( 1 + x ) ^ { 2 } } \mathrm {~d} x\), giving your answer as a single fraction in terms of the constant \(a\).
2. You are given that the area enclosed by the curve \(y = \frac { 1 } { ( 1 + x ) ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = a\) is equal to \(\frac { 1 } { 3 }\). Determine the value of \(a\).
In this question you must show detailed reasoning. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } \frac { \cos 2 x } { \sin 2 x + 2 } \mathrm {~d} x\), giving your answer in its simplest form.

OCR H240/02 2022 June Q4

4 An artist is creating a design for a large painting. The design includes a set of steps of varying heights. In the painting the lowest step has height 20 cm and the height of each other step is \(5 \%\) less than the height of the step immediately below it. In the painting the total height of the steps is 205 cm , correct to the nearest centimetre. Determine the number of steps in the design.

OCR H240/02 2022 June Q5

5 In this question you must show detailed reasoning. A curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 4 x\).

Show that the curve has no stationary points.
Show that the curve has exactly one point of inflection.

OCR H240/02 2022 June Q6

6

The diagrams show five different graphs. In each case the whole of the graph is shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_376_382_310_306} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_376_378_310_842} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_378_378_310_1379} \captionsetup{labelformat=empty} \caption{Fig. 1.3}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_378_382_872_306} \captionsetup{labelformat=empty} \caption{Fig. 1.4}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_378_378_872_845} \captionsetup{labelformat=empty} \caption{Fig. 1.5}
\end{figure} Place ticks in the boxes in the table in the Printed Answer Booklet to indicate, for each graph, whether it represents a one-one function, a many-one function, a function that is its own inverse or it does not represent a function. There may be more than one tick in some rows or columns of the table.
A function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { x }\) for the domain \(\{ x : 0 < x \leqslant 2 \}\). State the range of f , giving your answer in set notation.

Questions — OCR (4619 questions)

Browse by module

Browse by board