Questions H240/02 (94 questions)

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OCR H240/02 2022 June Q8
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
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 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
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
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
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 2023 June Q1
    1
      1. Express \(x ^ { 2 } - 8 x + 11\) in the form \(( x - a ) ^ { 2 } + b\) where \(a\) and \(b\) are constants.
      2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 11\).
    2. Determine the value of the constant \(k\) for which the equation \(x ^ { 2 } - 8 x + 11 = k\) has two equal roots.
      \(2 \xrightarrow { \text { The points } } O\) and \(A\) have position vectors \(\left( \begin{array} { l } 0
      0
      0 \end{array} \right)\) and \(\left( \begin{array} { l } 6
      0
      8 \end{array} \right)\) respectively. The point \(P\) is such that \(\overrightarrow { O P } = k \overrightarrow { O A }\), where \(k\) is a non-zero constant.
    3. Find, in terms of \(k\), the length of \(O P\). Point \(B\) has position vector \(\left( \begin{array} { l } 1
      2
      3 \end{array} \right)\) and angle \(O P B\) is a right angle.
    4. Determine the value of \(k\).
    OCR H240/02 2023 June Q3
    3 In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve \(y = \frac { 1 } { x + 2 }\), the two axes and the line \(x = 2.5\).
    OCR H240/02 2023 June Q4
    4 The diagram shows part of the graph of \(y = x ^ { 2 }\). The normal to the curve at the point \(A ( 1,1 )\) meets the curve again at \(B\). Angle \(A O B\) is denoted by \(\alpha\).
    \includegraphics[max width=\textwidth, alt={}, center]{40d40a0b-5b33-4940-b15b-ee03e1291f61-04_522_766_1978_246}
    1. Determine the coordinates of \(B\).
    2. Hence determine the exact value of \(\tan \alpha\).
    OCR H240/02 2023 June Q5
    5 In this question you must show detailed reasoning. The function f is defined by \(\mathrm { f } ( x ) = \cos x + \sqrt { 3 } \sin x\) with domain \(0 \leqslant x \leqslant 2 \pi\).
    1. Solve the following equations.
      1. \(\mathrm { f } ^ { \prime } ( x ) = 0\)
      2. \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) The diagram shows the graph of the gradient function \(y = \mathrm { f } ^ { \prime } ( x )\) for the domain \(0 \leqslant x \leqslant 2 \pi\).
        \includegraphics[max width=\textwidth, alt={}, center]{40d40a0b-5b33-4940-b15b-ee03e1291f61-05_583_741_781_242}
    2. Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points \(A , B , C\) and \(D\).
      1. Explain how to use the graph of the gradient function to find the values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
      2. Using set notation, write down the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing in the domain \(0 \leqslant x \leqslant 2 \pi\).
    OCR H240/02 2023 June Q6
    6 A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(A B\) is \(M\).
    \includegraphics[max width=\textwidth, alt={}, center]{40d40a0b-5b33-4940-b15b-ee03e1291f61-06_794_753_351_239} The equation of the circle is \(x ^ { 2 } - 6 x + y ^ { 2 } + a = 0\), where \(a\) is a constant.
    1. In this question you must show detailed reasoning. Show that the area of triangle \(A B C\) is \(\frac { 3 } { 2 } \sqrt { 9 - 2 a }\).
      1. Find the value of \(a\) when the area of triangle \(A B C\) is zero.
      2. Give a geometrical interpretation of the case in part (b)(i).
    3. Give a geometrical interpretation of the case where \(a = 5\).
    OCR H240/02 2023 June Q7
    7 A student wishes to prove that, for all positive integers \(a\) and \(b , a ^ { 2 } - 4 b \neq 2\).
    1. Prove that \(a ^ { 2 } - 4 b = 2 \Rightarrow a\) is even.
    2. Hence or otherwise prove that, for all positive integers \(a\) and \(b , a ^ { 2 } - 4 b \neq 2\).
    OCR H240/02 2023 June Q8
    8 The stem-and-leaf diagram shows the heights, in centimetres, of 15 plants. \(|\)
    \(\mid\)02
    \(\mid\)10
    \(\mid\)24
    \(\mid\)30249
    \(\mid\)412479
    \(\mid\)537
    \(\mid\)62
    Key: | 2 | 5 means 25 cm .
    1. Draw a box-and-whisker plot to illustrate the data. A statistician intends to analyse the data, but wants to ignore any outliers before doing so.
    2. Discuss briefly whether there are any heights in the diagram which the statistician should ignore.
    OCR H240/02 2023 June Q9
    9 A school contains 500 students in years 7 to 11 and 250 students in years 12 and 13. A random sample of 20 students is selected to represent the school at a parents' evening. The number of students in the sample who are from years 12 and 13 is denoted by \(X\).
    1. State a suitable binomial model for \(X\). Use your model to answer the following.
      1. Write down an expression for \(\mathrm { P } ( X = x )\).
      2. State, in set notation, the values of \(x\) for which your expression is valid.
    3. Find \(\mathrm { P } ( 5 \leqslant X \leqslant 9 )\).
    4. State one disadvantage of using a random sample in this context.
    OCR H240/02 2023 June Q10
    10 The mass, in kilograms, of a species of fish in the UK has population mean 4.2 and standard deviation 0.25. An environmentalist believes that the fish in a particular river are smaller, on average, than those in other rivers in the UK. A random sample of 100 fish of this species, taken from the river, has sample mean 4.16 kg . Stating a necessary assumption, test at the \(5 \%\) significance level whether the environmentalist is correct.
    OCR H240/02 2023 June Q11
    11 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\).
    1. Find \(\mathrm { P } ( Y > \mu - \sigma )\).
    2. Given that \(\mathrm { P } ( Y > 45 ) = 0.2\) and \(\mathrm { P } ( Y < 25 ) = 0.3\), determine the values of \(\mu\) and \(\sigma\). The random variables \(U\) and \(V\) have the distributions \(\mathrm { N } ( 10,4 )\) and \(\mathrm { N } ( 12,9 )\) respectively.
    3. It is given that \(\mathrm { P } ( U < b ) = \mathrm { P } ( V > c )\), where \(b > 10\) and \(c < 12\). Determine \(b\) in terms of \(c\).
    OCR H240/02 2023 June Q12
    12 A student has an ordinary six-sided dice. The student suspects that it is biased against six, so that when it is thrown, it is less likely to show a six than if it were fair. In order to test this suspicion, the student plans to carry out a hypothesis test at the 5\% significance level. The student throws the dice 100 times and notes the number of times, \(X\), that it shows a six.
    1. Determine the largest value of \(X\) that would provide evidence at the \(5 \%\) significance level that the dice is biased against six. Later another student carries out a similar test, at the 5\% significance level. This student also throws the dice 100 times.
    2. It is given that the dice is fair. Find the probability that the conclusion of the test is that there is significant evidence that the dice is biased against six.
    OCR H240/02 2023 June Q13
    13 The scatter diagram uses information about all the Local Authorities (LAs) in the UK, taken from the 2011 census. For each LA it shows the percentage ( \(x\) ) of employees who used public transport to travel to work and the percentage \(( y )\) who used motorised private transport.
    "Public transport" includes train, bus, minibus, coach, underground, metro and light rail.
    "Motorised private transport" includes car, van, motorcycle, scooter, moped, taxi and passenger in a car or van.
    \includegraphics[max width=\textwidth, alt={}, center]{40d40a0b-5b33-4940-b15b-ee03e1291f61-10_1038_1122_683_242}
    1. Most of the points in the diagram lie on or near the line with equation \(x + y = k\), where \(k\) is a constant.
      1. Give a possible value for \(k\).
      2. Hence give an approximate value for the percentage of employees who either worked from home or walked or cycled to work.
    2. The average amount of fuel used per person per day for travelling to work in any LA is denoted by F. Consider the two groups of LAs where the percentages using motorised private transport are highest and lowest.
      1. Using only the information in the diagram, suggest, with a reason, which of these two groups will have greater values of F than the other group. A student says that it is not possible to give a reliable answer to part (b)(i) without some further information.
      2. Suggest two kinds of further information which would enable a more reliable answer to be given.
    3. Points \(A\) and \(B\) in the diagram are the most extreme outliers. Use their positions on the diagram to answer the following questions about the two LAs represented by these two points.
      1. The two LAs share a certain characteristic. Describe, with a justification, this characteristic.
      2. The environments in these two LAs are very different. Describe, with a justification, this difference.
    4. A student says that it is difficult to extract detailed information from the scatter diagram. Explain whether you agree with this criticism.
    OCR H240/02 2023 June Q14
    14 In this question you must show detailed reasoning. A disease that affects trees shows no visible evidence for the first few years after the tree is infected. A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not \(100 \%\) reliable, and a researcher uses the following model.
    • If the tree has the disease, the probability of a positive result is 0.95 .
    • If the tree does not have the disease, the probability of a positive result is 0.1 .
      1. It is known that in a certain county, \(A , 35 \%\) of the trees have the disease. A tree in county \(A\) is chosen at random and is tested.
    Given that the result is positive, determine the probability that this tree has the disease. A forestry company wants to determine what proportion of trees in another county, \(B\), have the disease. They choose a large random sample of trees in county \(B\). Each tree in the sample is tested and it is found that the result is positive for \(43 \%\) of these trees.
  • By carrying out a calculation, determine an estimate of the proportion of trees in county \(B\) that have the disease.
    •