Questions H240/02 (151 questions)

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OCR H240/02 2018 September Q10
6 marks Easy -1.8
10 The table shows information, derived from the 2011 UK census, about the percentage of employees who used various methods of travel to work in four Local Authorities.
Local AuthorityUnderground, metro, light rail or tramTrainBusDriveWalk or cycle
A0.3\%4.5\%17\%52.8\%11\%
B0.2\%1.7\%1.7\%63.4\%11\%
C35.2\%3.0\%12\%11.7\%16\%
D8.9\%1.4\%9\%54.7\%10\%
One of the Local Authorities is a London borough and two are metropolitan boroughs, not in London.
  1. Which one of the Local Authorities is a London borough? Give a reason for your answer.
  2. Which two of the Local Authorities are metropolitan boroughs outside London? In each case give a reason for your answer.
  3. Describe one difference between the public transport available in the two metropolitan boroughs, as suggested by the table.
  4. Comment on the availability of public transport in Local Authority B as suggested by the table.
OCR H240/02 2018 September Q11
8 marks Moderate -0.3
11 In an experiment involving a bivariate distribution ( \(X , Y\) ) a random sample of 7 pairs of values was obtained and Pearson's product-moment correlation coefficient \(r\) was calculated for these values.
  1. The value of \(r\) was found to be 0.894 . Use the table below to test, at the \(5 \%\) significance level, whether there is positive linear correlation in the population, stating your hypotheses and conclusion clearly.
    1-tail test 2-tail test5\%2.5\%1\%0.5\%
    10\%5\%2\%1\%
    \(n\)
    1----
    2----
    30.98770.99690.99950.9999
    40.90000.95000.98000.9900
    50.80540.87830.93430.9587
    60.72930.81140.88220.9587
    70.66940.75450.83290.9745
    80.62150.70670.78870.8343
    90.58820.66640.74980.7977
    100.54940.63190.71550.7646
    Scatter diagrams for four sets of bivariate data, are shown. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_380_371_301_191} \captionsetup{labelformat=empty} \caption{Diagram A}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_373_373_301_628} \captionsetup{labelformat=empty} \caption{Diagram B}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1064} \captionsetup{labelformat=empty} \caption{Diagram C}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1503} \captionsetup{labelformat=empty} \caption{Diagram D}
    \end{figure} It is given that \(r = 0.894\) for one of these diagrams.
  2. For each of the other diagrams, state how you can tell that \(r \neq 0.894\).
OCR H240/02 2018 September Q12
8 marks Moderate -0.3
12 In the past, the time spent by customers in a certain shop had mean 10.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 12.0 minutes.
  1. Assuming that the standard deviation is unchanged, test at the \(1 \%\) significance level whether the mean time spent by customers in the shop has changed.
  2. Another random sample of 50 customers is chosen and a similar test at the \(1 \%\) significance level is carried out. Given that the population mean time has not changed, state the probability that the conclusion of the test will be that the population mean time has changed.
OCR H240/02 2018 September Q13
7 marks Challenging +1.2
13 Bag A contains 3 black discs and 2 white discs only. Initially Bag B is empty. Discs are removed at random from bag A, and are placed in bag B, one at a time, until all 5 discs are in bag B.
  1. Write down the probability that the last disc that is placed in bag B is black.
  2. Find the probability that the first disc and the last disc that are placed in bag B are both black.
  3. Find the probability that, starting from when the first disc is placed in bag B , the number of black discs in bag B is always greater than the number of white discs in bag B.
OCR H240/02 2018 September Q14
7 marks Standard +0.3
14 A counter is initially at point \(O\) on the \(x\)-axis. A fair coin is thrown 6 times. Each time the coin shows heads, the counter is moved one unit in the positive \(x\)-direction. Each time the coin shows tails, the counter is moved one unit in the negative \(x\)-direction. The final distance of the counter from \(O\), in either direction, is denoted by \(D\). Determine the most probable value of \(D\). \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}
OCR H240/02 2022 June Q1
8 marks Moderate -0.8
1 In this question you must show detailed reasoning. Solve the following equations.
  1. \(\frac { x } { x + 1 } - \frac { x - 1 } { x + 2 } = 0\)
  2. \(\frac { 8 } { x ^ { 6 } } - \frac { 7 } { x ^ { 3 } } - 1 = 0\)
  3. \(3 ^ { x ^ { 2 } - 7 } = \frac { 1 } { 243 }\)
OCR H240/02 2022 June Q2
5 marks Standard +0.8
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.
  1. 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.
  2. Find the two possible values of \(p\).
OCR H240/02 2022 June Q3
10 marks Moderate -0.3
3
  1. 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.
  2. 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 \(a\).
  3. 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
5 marks Moderate -0.3
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
4 marks Moderate -0.8
5 In this question you must show detailed reasoning. A curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 4 x\).
  1. Show that the curve has no stationary points.
  2. Show that the curve has exactly one point of inflection.
OCR H240/02 2022 June Q6
6 marks Moderate -0.8
6
  1. 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.
  2. 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.
OCR H240/02 2022 June Q7
8 marks Moderate -0.3
7 It is given that any integer can be expressed in the form \(3 m + r\), where \(m\) is an integer and \(r\) is 0,1 or 2 . Use this fact to answer the following.
  1. By considering the different values of \(r\), prove that the square of any integer cannot be expressed in the form \(3 n + 2\), where \(n\) is an integer.
  2. Three integers are chosen at random from the integers 1 to 99 inclusive. The three integers are not necessarily different. By considering the different values of \(r\), determine the probability that the sum of these three integers is divisible by 3 .
OCR H240/02 2022 June Q8
7 marks Standard +0.8
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 \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
OCR H240/02 2022 June Q9
14 marks Standard +0.3
9 The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-08_842_1651_495_207} One of the 150 plants is chosen at random, and its height, \(X \mathrm {~cm}\), is noted.
  1. Show that \(\mathrm { P } ( 20 < X < 30 ) = 0.147\), correct to 3 significant figures. Sam suggests that the distribution of \(X\) can be well modelled by the distribution \(\mathrm { N } ( 40,100 )\).
    1. Give a brief justification for the use of the normal distribution in this context.
    2. Give a brief justification for the choice of the parameter values 40 and 100 .
  3. Use Sam's model to find \(\mathrm { P } ( 20 < X < 30 )\). Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, \(m\) and \(s\), for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution \(\mathrm { N } \left( m , s ^ { 2 } \right)\) as her model.
  4. Use Nina's model to find \(\mathrm { P } ( 20 < X < 30 )\).
    1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model.
    2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution.
OCR H240/02 2022 June Q10
10 marks Easy -1.8
10 The table shows the age structure of usual residents of 18 Local Authorities (LAs) in the North West region of the UK in 2011.
Local AuthorityAge 0 to 17Age 18 to 24Age 25 to 64Age 65 and over
A26.20\%9.06\%51.81\%12.92\%
B23.32\%8.99\%52.32\%15.37\%
C22.24\%8.96\%52.56\%16.23\%
D22.67\%8.10\%53.27\%15.96\%
E20.70\%7.77\%54.77\%16.76\%
F18.14\%6.51\%51.13\%24.21\%
G18.96\%14.20\%48.51\%18.33\%
H19.06\%14.79\%52.12\%14.04\%
I25.15\%9.04\%51.16\%14.65\%
J22.93\%8.81\%52.22\%16.04\%
K21.48\%13.98\%50.82\%13.73\%
L23.98\%9.20\%52.26\%14.56\%
M21.67\%11.19\%52.94\%14.19\%
N17.82\%6.01\%51.93\%24.23\%
O22.83\%7.30\%53.86\%16.01\%
P21.76\%8.28\%54.03\%15.93\%
Q21.42\%8.43\%53.90\%16.25\%
R18.61\%7.33\%49.35\%24.71\%
\section*{Percentage of residents}
  1. Without reference to any other columns, explain how you would use only the columns for the age ranges 0 to 17 and 18 to 24 to decide whether an LA might be one of the following.
    1. An LA that includes a university
    2. An LA that attracts young couples to live
    3. An LA that attracts retired people to live
  2. Using your answers to part (a), identify the following.
    1. Four LAs that might include a university
    2. Three LAs that might be attractive to retired people
  3. Explain why your answer to part (b)(ii), based only on the columns for the age ranges 0 to 17 and 18 to 24, may not be reliable.
  4. The lower quartile, median and upper quartile of the percentages in the column "Age 65 and over" are \(14.56 \% , 15.99 \%\) and \(16.76 \%\) respectively. Use this information to comment on your answers to part (b)(ii) and part (c). In a magazine article, a councillor plans to describe a typical LA in the North West region. He wants to quote the average percentage of residents aged 65 or over.
  5. The mean of the percentages in the column "Age 65 and over" is \(16.90 \%\). Use this information, and the information given in part (d), to explain whether the median or the mean better represents the data in the column "Age 65 and over".
OCR H240/02 2022 June Q11
7 marks Standard +0.3
11 In the past the masses of new-born babies in a certain country were normally distributed with mean 3300 g . Last year a publicity campaign was held to encourage pregnant women to improve their diet. Following this campaign, it is required to test whether the mean mass of new-born babies has increased. A random sample of 200 new-born babies is chosen, and it is found that their mean mass is 3360 g . It is given that the standard deviation of the masses of new-born babies is 450 g . Carry out the test at the 2.5\% significance level.
OCR H240/02 2022 June Q12
6 marks Standard +0.3
12 A firm claims that no more than \(2 \%\) of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than \(2 \%\). In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the \(5 \%\) significance level, using the null hypothesis \(p = 0.02\).
  1. Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect.
  2. The researcher finds that 18 out of the 600 packets are underweight. A colleague says
    " 18 out of 600 is \(3 \%\), so there is evidence that the actual proportion of underweight bags is greater than \(2 \%\)." Criticise this statement.
OCR H240/02 2022 June Q13
10 marks Moderate -0.8
13 There are 25 students in a class.
  • The number of students who study both History and English is 3.
  • The number of students who study neither History nor English is 14 .
  • The number of students who study History but not English is three times the number who study English but not History.
    1. - Show this information on a Venn diagram.
    2. Determine the probability that a student selected at random studies English.
Two different students from the class are chosen at random.
  • Given that exactly one of the two students studies English, determine the probability that exactly one of the two students studies History. \section*{END OF QUESTION PAPER}
    •
    OCR H240/02 2020 November Q1
    9 marks Easy -1.3
    1. Differentiate the following with respect to \(x\).
      1. \((2x + 3)^7\) [2]
      2. \(x^3 \ln x\) [3]
    2. Find \(\int \cos 5x \, dx\). [2]
    3. Find the equation of the curve through \((1, 3)\) for which \(\frac{dy}{dx} = 6x - 5\). [2]
    OCR H240/02 2020 November Q2
    4 marks Moderate -0.8
    Simplify fully \(\frac{2x^3 + x^2 - 7x - 6}{x^2 - x - 2}\). [4]
    OCR H240/02 2020 November Q3
    7 marks Moderate -0.3
    In this question you should assume that \(-1 < x < 1\).
    1. For the binomial expansion of \((1 - x)^{-2}\)
      1. find and simplify the first four terms, [2]
      2. write down the term in \(x^n\). [1]
    2. Write down the sum to infinity of the series \(1 + x + x^2 + x^3 + \ldots\). [1]
    3. Hence or otherwise find and simplify an expression for \(2 + 3x + 4x^2 + 5x^3 + \ldots\) in the form \(\frac{a - x}{(b - x)^2}\) where \(a\) and \(b\) are constants to be determined. [3]
    OCR H240/02 2020 November Q4
    5 marks Standard +0.3
    In this question you must show detailed reasoning. Solve the equation \(3\sin^4 \phi + \sin^2 \phi = 4\), for \(0 \leq \phi < 2\pi\), where \(\phi\) is measured in radians. [5]
    OCR H240/02 2020 November Q5
    5 marks Standard +0.3
    1. Determine the set of values of \(n\) for which \(\frac{n^2 - 1}{2}\) and \(\frac{n^2 + 1}{2}\) are positive integers. [3]
    A 'Pythagorean triple' is a set of three positive integers \(a\), \(b\) and \(c\) such that \(a^2 + b^2 = c^2\).
    1. 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. [2]
    OCR H240/02 2020 November Q6
    3 marks Standard +0.3
    Prove that \(\sqrt{2} \cos(2\theta + 45°) = \cos^2 \theta - 2\sin \theta \cos \theta - \sin^2 \theta\), where \(\theta\) is measured in degrees. [3]
    OCR H240/02 2020 November Q7
    8 marks Moderate -0.8
    \(A\) 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
    1. the quantity \(|\mathbf{a} - \mathbf{b}|\), [1]
    2. the vector \(\frac{1}{2}(\mathbf{a} + \mathbf{b})\). [1]
    The circle \(P\) is the set of points with position vector \(\mathbf{p}\) in the \(x\)-\(y\) plane which satisfy $$\left|\mathbf{p} - \frac{1}{2}(\mathbf{a} + \mathbf{b})\right| = \frac{1}{2}|\mathbf{a} - \mathbf{b}|.$$
    1. State, in terms of \(\mathbf{a}\) and \(\mathbf{b}\),
      1. the position vector of the centre of \(P\), [1]
      2. the radius of \(P\). [1]
    It is now given that \(\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and \(\mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix}\).
    1. Find a cartesian equation of \(P\). [4]