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OCR MEI Paper 2 2024 June Q13
9 marks Standard +0.8
13 Determine the coordinates of the turning points on the curve with equation $$y ^ { 2 } + x y + x ^ { 2 } - x = 1 .$$
OCR MEI Paper 2 2024 June Q14
8 marks Moderate -0.8
14 The pre-release material contains medical data for 103 women and 97 men.
The boxplot represents the weights in kg of 101 of the women from the pre-release material.
\includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-09_421_1232_735_244}
  1. Use your knowledge of the pre-release material to give a reason why the weights of all 103 women were not included in the diagram.
  2. Determine the range of values in which any outliers lie.
  3. Use your knowledge of the pre-release material to explain whether these outliers should be removed from any further analysis of the data.
  4. The median weight of men in the sample was found to be 79.9 kg . Explain what may be inferred by comparing the median weight of men with the median weight of women. Further analysis of the weights of both men and women is carried out. The table shows some of the results.
    meanstandard deviation
    men82.69 kg19.98 kg
    women72.5 kg19.95 kg
  5. Use the information in the table to make two inferences about the distribution of the weights of men compared with the distribution of the weights of women.
OCR MEI Paper 2 2024 June Q15
17 marks Standard +0.3
15 Bottles of Fizzipop nominally contain 330 ml of drink. A consumer affairs researcher collects a random sample of 55 bottles of Fizzipop and records the volume of drink in each bottle. Summary statistics for the researcher's sample are shown in the table.
\(n\)55
\(\sum x\)18535
\(\sum x ^ { 2 }\)6247066.6
    1. Calculate the mean volume of drink in a bottle of Fizzipop.
    2. Show that the standard deviation of the volume of drink in a bottle of Fizzipop is 3.78 ml . The researcher uses software to produce a histogram with equal class intervals, which is shown below.
      \includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-10_533_759_1181_251}
  2. Explain why the researcher decides that the Normal distribution is a suitable model for the volume of drink in a bottle of Fizzipop.
  3. Use your answers to parts (a) and (b) to determine the expected number of bottles which contain less than 330 ml in a random sample of 100 bottles. In order to comply with new regulations, no more than 1\% of bottles of Fizzipop should contain less than 330 ml . The manufacturer decides to meet the new regulations by adjusting the manufacturing process so that the mean volume of drink in a bottle of Fizzipop is increased. The standard deviation is unaltered.
  4. Determine the minimum mean volume of drink in a bottle of Fizzipop which should ensure that the new regulations are met. Give your answer to \(\mathbf { 3 }\) significant figures. The mean volume of drink in a bottle of Fizzipop is set to 340 ml . After several weeks the quality control manager suspects the mean volume may have reduced. She collects a random sample of 100 bottles of Fizzipop. The mean volume of drink in a bottle in the sample is found to be 339.37 ml .
  5. Assuming the standard deviation is unaltered, conduct a hypothesis test at the \(5 \%\) level to determine whether there is any evidence to suggest that the mean volume of drink in a bottle of Fizzipop is less than 340 ml .
OCR MEI Paper 2 2020 November Q1
2 marks Moderate -0.8
1 Fig. 1 shows triangle \(A B C\). Fig. 1 Calculate the area of triangle \(A B C\), giving your answer correct to 3 significant figures.
OCR MEI Paper 2 2020 November Q2
3 marks Easy -1.3
2 Fig. 2 shows a sector of a circle of radius 8 cm . The angle of the sector is 2.1 radians. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-04_423_296_1366_246} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Calculate the length of the arc \(L\).
  2. Calculate the area of the sector.
OCR MEI Paper 2 2020 November Q3
4 marks Moderate -0.8
3 You are given that \(y = 4 x + \sin 8 x\).
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
  2. Find the smallest positive value of \(x\) for which \(\frac { \mathrm { dy } } { \mathrm { dx } } = 0\), giving your answer in an exact form.
OCR MEI Paper 2 2020 November Q4
2 marks Easy -1.8
4 Fig. 4 shows a cumulative frequency diagram for the time spent revising mathematics by year 11 students at a certain school during a week in the summer term. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-05_554_1070_737_242} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Use the diagram to estimate the median time spent revising mathematics by these students. [1] A teacher comments that \(90 \%\) of the students spent less than an hour revising mathematics during this week.
  2. Determine whether the information in the diagram supports this comment.
OCR MEI Paper 2 2020 November Q5
3 marks Moderate -0.8
5 The first \(n\) terms of an arithmetic series are
\(17 + 28 + 39 + \ldots + 281 + 292\).
  1. Find the value of \(n\).
  2. Find the sum of these \(n\) terms.
OCR MEI Paper 2 2020 November Q6
4 marks Moderate -0.8
6
  1. Find the first three terms in ascending powers of \(x\) of the binomial expansion of \(( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\).
  2. State the range of values of \(x\) for which this expansion is valid.
OCR MEI Paper 2 2020 November Q7
5 marks Moderate -0.5
7 You are given that \(P ( A ) = 0.6 , P ( B ) = 0.5\) and \(P ( A \cup B ) ^ { \prime } = 0.2\).
  1. Find \(\mathrm { P } ( \mathrm { A } \cap \mathrm { B } )\).
  2. Find \(\mathrm { P } ( \mathrm { A } \mid \mathrm { B } )\).
  3. State, with a reason, whether \(A\) and \(B\) are independent.
OCR MEI Paper 2 2020 November Q8
12 marks Moderate -0.8
8 Rosella is carrying out an investigation into the age at which adults retire from work in the city where she lives. She collects a sample of size 50 , ensuring this comprises of 25 randomly selected retired men and 25 randomly selected retired women.
  1. State the name of the sampling method she uses. Fig. 8.1 shows the data she obtains in a frequency table and Fig. 8.2 shows these data displayed in a histogram. \begin{table}[h]
    Age in years at retirement\(45 -\)\(50 -\)\(55 -\)\(60 -\)\(65 -\)\(70 -\)\(75 - 80\)
    Frequency density0.41.82.42.21.81.20.2
    \captionsetup{labelformat=empty} \caption{Fig. 8.1}
    \end{table} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-08_805_1006_1160_244} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure}
  2. How many people in the sample are aged between 50 and 55? Rosella obtains a list of the names of all 4960 people who have retired in the city during the previous month.
  3. Describe how Rosella could collect a sample of size 200 from her list using
    • systematic sampling such that every item on the list could be selected,
    • simple random sampling.
    Rosella collects two simple random samples, one of size 200 and one of size 500, from her list. The histograms in Fig. 8.3 show the data from the sample of size 200 on the left and the data from the sample of 500 on the right. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-09_659_1909_388_77} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
    \end{figure}
  4. With reference to the histograms shown in Fig. 8.2 and Fig. 8.3, explain why it appears reasonable to model the age of retirement in this city using the Normal distribution. Summary statistics for the sample of 500 are shown in Fig. 8.4. \begin{table}[h]
    Statistics
    n500
    Mean60.0515
    \(\sigma\)6.5717
    s6.5783
    \(\Sigma x\)30025.7601
    \(\Sigma \mathrm { x } ^ { 2 }\)1824686.322
    Min36.0793
    Q155.2573
    Median59.9202
    Q364.4239
    Max81.742
    \captionsetup{labelformat=empty} \caption{Fig. 8.4}
    \end{table}
  5. Use an appropriate Normal model based on the information in Fig. 8.4 to estimate the number of people aged over 65 who retired in the city in the previous month.
  6. Identify a limitation in using this model to predict the number of people aged over 65 retiring in the following month.
OCR MEI Paper 2 2020 November Q9
9 marks Standard +0.3
9 A company supplies computers to businesses. In the past the company has found that computers are kept by businesses for a mean time of 5 years before being replaced. Claud, the manager of the company, thinks that the mean time before replacing computers is now different.
  1. Describe how Claud could obtain a cluster sample of 120 computers used by businesses the company supplies. Claud decides to conduct a hypothesis test at the \(5 \%\) level to test whether there is evidence to suggest that the mean time that businesses keep computers is not 5 years. He takes a random sample of 120 computers. Summary statistics for the length of time computers in this sample are kept are shown in Fig. 9. \begin{table}[h]
    Statistics
    \(n\)120
    Mean4.8855
    \(\sigma\)2.6941
    \(s\)2.7054
    \(\Sigma x\)586.2566
    \(\Sigma x ^ { 2 }\)3735.1475
    Min0.1213
    Q12.5472
    Median4.8692
    Q37.0349
    Max9.9856
    \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{table} \section*{(b) In this question you must show detailed reasoning.}
    • State the hypotheses for this test, explaining why the alternative hypothesis takes the form it does.
    • Use a suitable distribution to carry out the test.
OCR MEI Paper 2 2020 November Q11
10 marks Moderate -0.8
11 The pre-release material contains information concerning median house prices over the period 2004-2015. A spreadsheet has been used to generate a time series graph for two areas: the London borough of "Barking and Dagenham" and "North West". This is shown together with the raw data in Fig. 11.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-12_572_1751_447_159} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
\end{figure} Dr Procter suggests that it is unusual for median house prices in a London borough to be consistently higher than those in other parts of the country.
  1. Use your knowledge of the large data set to comment on Dr Procter's suggestion. Dr Procter wishes to predict the median house price in Barking and Dagenham in 2016. She uses the spreadsheet function LINEST to find the equation of the line of best fit for the given data. She obtains the equation
    \(P = 4897 Y - 9657847\), where \(P\) is the median house price in pounds and \(Y\) is the calendar year, for example 2015.
  2. Use Dr Procter's equation to predict the median house price in Barking and Dagenham in
    • 2016
    • 2017.
    Professor Jackson uses a simpler model by using the data from 2014 and 2015 only to form a straight-line model.
  3. Find the equation Professor Jackson uses in her model.
  4. Use Professor Jackson’s equation to predict the median house price in Barking and Dagenham in
    • 2016
    • 2017.
    Professor Jackson carries out some research online. She finds some information about median house prices in Barking and Dagenham, which is shown in Fig. 11.2. \begin{table}[h]
    20162017
    \(\pounds 290000\)\(\pounds 300000\)
    \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{table}
  5. Comment on how well
    • Dr Procter’s model fits the data,
    • Professor Jackson’s model fits the data.
    • Explain which, if any, of the models is likely to be more reliable for predicting median house prices in Barking and Dagenham in 2020.
OCR MEI Paper 2 2020 November Q13
7 marks Moderate -0.5
13 The pre-release material contains information concerning median house prices, recycling rates and employment rates. Fig. 13.1 shows a scatter diagram of recycling rate against employment rate for a random sample of 33 regions. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-14_629_1424_397_242} \captionsetup{labelformat=empty} \caption{Fig. 13.1}
\end{figure} The product moment correlation coefficient for this sample is 0.37154 and the associated \(p\)-value is 0.033. Lee conducts a hypothesis test at the \(5 \%\) level to test whether there is any evidence to suggest there is positive correlation between recycling rate and employment rate. He concludes that there is no evidence to suggest positive correlation because \(0.033 \approx 0\) and \(0.37154 > 0.05\).
  1. Explain whether Lee's reasoning is correct. Fig. 13.2 shows a scatter diagram of recycling rate against median house price for a random sample of 33 regions. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-14_648_1474_1758_242} \captionsetup{labelformat=empty} \caption{Fig. 13.2}
    \end{figure} The product moment correlation coefficient for this sample is - 0.33278 and the associated \(p\)-value is 0.058 . Fig. 13.3 shows summary statistics for the median house prices for the data in this sample. \begin{table}[h]
    Statistics
    \(n\)33
    Mean465467.9697
    \(\sigma\)201236.1345
    \(s\)204356.2606
    \(\Sigma x\)15360443
    \(\Sigma x ^ { 2 }\)8486161617387
    Min243500
    Q1342500
    Median410000
    Q3521000
    Max1200000
    \captionsetup{labelformat=empty} \caption{Fig. 13.3}
    \end{table}
  2. Use the information in Fig. 13.3 and Fig. 13.2 to show that there are at least two outliers.
  3. Describe the effect of removing the outliers on
    • the product moment correlation coefficient between recycling rate and median house price,
    • the \(p\)-value associated with this correlation coefficient,
      in each case explaining your answer.
      [0pt] [2]
      All 33 items in the sample are areas in London. A student suggests that it is very unlikely that only areas in London would be selected in a random sample.
    • Use your knowledge of the pre-release material to explain whether you think the student's suggestion is reasonable.
OCR MEI Paper 2 2020 November Q15
7 marks Moderate -0.3
15 Functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined as follows.
\(\mathrm { f } ( x ) = \sqrt { x }\) for \(x > 0\) and \(\mathrm { g } ( x ) = x ^ { 3 } - x - 6\) for \(x > 2\). The function \(\mathrm { h } ( x )\) is defined as
\(\mathrm { h } ( x ) = \mathrm { fg } ( x )\).
  1. Find \(\mathrm { h } ( x )\) in terms of \(x\) and state its domain.
  2. Find \(\mathrm { h } ( 3 )\). Fig. 15 shows \(\mathrm { h } ( x )\) and \(\mathrm { h } ^ { - 1 } ( x )\), together with the straight line \(y = x\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-17_780_796_895_242} \captionsetup{labelformat=empty} \caption{Fig. 15}
    \end{figure}
  3. Determine the gradient of \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) at the point where \(y = 3\).
OCR MEI Paper 2 2021 November Q1
2 marks Easy -1.2
1 The equation of a curve is \(y = 4 x ^ { 2 } + 8 x + 1\).
The curve is stretched parallel to the \(x\)-axis with scale factor 2 .
Find the equation of the new curve, giving your answer in the form \(\mathrm { y } = a \mathrm { x } ^ { 2 } + b \mathrm { x } + c\), where \(a , b\) and \(c\) are integers to be determined.
OCR MEI Paper 2 2021 November Q2
3 marks Easy -1.8
2
  1. Write \(65 ^ { \circ }\) in radians, giving your answer in the form \(k \pi\), where \(k\) is a fraction in its lowest terms.
  2. Write 0.211 radians in degrees, giving your answer correct to \(\mathbf { 1 }\) decimal place.
OCR MEI Paper 2 2021 November Q3
3 marks Easy -2.0
3 Draw a number line to show the values of \(x\) which belong to the set \(\{ x : x \geqslant 2 \} \cap \{ x : x < 7 \}\).
OCR MEI Paper 2 2021 November Q4
3 marks Easy -1.8
4 Sketch the graph of \(y = | 2 x - 3 |\).
OCR MEI Paper 2 2021 November Q5
3 marks Moderate -0.8
5 It is known that 40\% of people in Britain carry a certain gene.
A random sample of 32 people is collected.
  1. Calculate the probability that exactly 12 people carry the gene.
  2. Calculate the probability that at least 8 people carry the gene, giving your answer correct to \(\mathbf { 3 }\) decimal places.
OCR MEI Paper 2 2021 November Q6
5 marks Moderate -0.8
6 You are given that \(\mathbf { v } = 2 \mathbf { a } + 3 \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are the position vectors
\(\mathbf { a } = \binom { 5 } { 3 }\) and \(\mathbf { b } = \binom { - 1 } { 6 }\).
  1. Determine the magnitude of \(\mathbf { v }\).
  2. Determine the angle between \(\mathbf { v }\) and the vector \(\binom { 1 } { 0 }\).
OCR MEI Paper 2 2021 November Q7
4 marks Easy -1.2
7 The parametric equations of a circle are
\(x = 7 + 5 \cos \theta , \quad y = 5 \sin \theta - 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).
  1. Find a cartesian equation of the circle.
  2. State the coordinates of the centre of the circle. Answer all the questions.
    Section B (77 marks)
OCR MEI Paper 2 2021 November Q8
4 marks Standard +0.3
8 The Normal variable \(X\) is transformed to the Normal variable \(Y\).
The transformation is \(\mathrm { y } = \mathrm { a } + \mathrm { bx }\), where \(a\) and \(b\) are positive constants.
You are given that \(X \sim N ( 42,6.8 )\) and \(Y \sim N ( 57.2,11.492 )\).
Determine the values of \(a\) and \(b\).
OCR MEI Paper 2 2021 November Q9
10 marks Standard +0.8
9 Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table.
malefemale
black13
yellow21
chocolate11
Four puppies are chosen at random to train as guide dogs.
  1. Determine the probability that exactly 3 females are chosen.
  2. Determine the probability that at least 3 black puppies are chosen.
  3. Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen.
  4. Explain whether the 2 events ‘choosing exactly 3 females’ and ‘choosing at least 3 black puppies’ are independent events.
OCR MEI Paper 2 2021 November Q10
9 marks Moderate -0.8
10 Ben has an interest in birdwatching. For many years he has identified, at the start of the year, 32 days on which he will spend an hour counting the number of birds he sees in his garden. He divides the year into four using the Meteorological Office definition of seasons. Each year he uses stratified sampling to identify the 32 days on which he will count the birds in his garden, drawn equally from the four seasons. Ben’s data for 2019 are shown in the stem and leaf diagram in Fig. 10.1. \begin{table}[h]
035999
100112456789
20146789
30023
4036
51
60
\captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{table}
  1. Suggest a reason why Ben chose to use stratified sampling instead of simple random sampling.
  2. Describe the shape of the distribution.
  3. Explain why the mode is not a useful measure of central tendency in this case.
  4. For Ben's sample, determine
    • the median,
    • the interquartile range.
    Ben found a boxplot for the sample of size 32 he collected using stratified sampling in 2015. The boxplot is shown in Fig. 10.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c9d14a4d-a1c8-42ad-9c0b-42cef6b3612f-06_483_1163_1982_242} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{figure} In 2016 Ben replaced his hedge with a garden fence.
    Ben now believes that
    • he sees fewer birds in his garden,
    • the number of birds he sees in his garden is more variable.
    • With reference to Fig. 10.2 and your answer to part (d), comment on whether there is any evidence to support Ben’s belief.
    Jane says she can tell that the data for 2015 is definitely uniformly distributed by looking at the boxplot.
  5. Explain why Jane is wrong.