Questions — OCR MEI AS Paper 2 (98 questions)

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OCR MEI AS Paper 2 2024 June Q7
7 Determine the coefficient of \(x ^ { 5 }\) in the expansion of \(( 3 - 2 x ) ^ { 7 }\).
OCR MEI AS Paper 2 2024 June Q9
9 A fair six-sided die has its faces numbered 1, 3, 4, 5, 6 and 7. The die is rolled once.
\(A\) is the event that the die shows an even number.
\(B\) is the event that the die shows a prime number.
  1. Write down the value of \(\mathrm { p } ( A )\).
  2. Write down the value of \(\mathrm { p } ( B )\).
  3. Write down the value of \(\mathrm { p } ( A\) or \(B )\). The die is rolled again.
  4. Calculate the probability that the sum of the scores from the two rolls is even.
OCR MEI AS Paper 2 2024 June Q10
10 The pre-release material contains information about the birth rate per 1000 people in different countries of the world. These countries have been classified into different regions. The table shows some data for three of these regions: the mean and standard deviation (sd) of the birth rate per 1000, and the number of countries for which data was used, n. \section*{Birth rate per 1000 by region}
AfricaEuropeOceania
\(n\)554921
mean29.310.017.8
sd8.431.944.50
  1. Use the information in the table to compare and contrast the birth rate per 1000 in Africa with the birth rate per 1000 in Europe.
  2. The birth rate per 1000 in Mauritius, which is in Africa, is recorded as 9.86. Use the information in the table to show that this value is an outlier.
  3. Use your knowledge of the pre-release material to explain whether the value for Mauritius should be discarded.
  4. The pre-release material identifies 27 countries in Oceania. Suggest a reason why only 21 values were used to calculate the mean and standard deviation.
OCR MEI AS Paper 2 2024 June Q12
12 Data collected in the twentieth century showed that the probability of a randomly selected person having blue eyes was 0.08 . A medical researcher believes that the probability in 2024 is less than this so they decide to carry out a hypothesis test at the \(5 \%\) significance level.
  1. Write down suitable hypotheses for the test, defining the parameter used.
  2. Assuming that the probability that a person selected at random has blue eyes is still 0.08 , calculate the probability that 3 or fewer people in a random sample of 92 people have blue eyes. The researcher collects a random sample of 92 people and finds that 3 of them have blue eyes.
  3. Use your answer to part (b) to carry out the test, giving your conclusion in context.
OCR MEI AS Paper 2 2024 June Q13
13 Determine the range of values of \(x\) for which \(y = 4 x ^ { 3 } + 7 x ^ { 2 } - 6 x + 8\) is a decreasing function.
OCR MEI AS Paper 2 2024 June Q14
14 In this question you must show detailed reasoning.
Solve the equation \(5 - \cos \theta - 6 \sin ^ { 2 } \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\). Turn over for question 15
OCR MEI AS Paper 2 2024 June Q15
15 Ali and Sam are playing a game in which Ali tosses a coin 5 times. If there are 4 or 5 heads, Ali wins the game. Otherwise Sam wins the game. They decide to play the game 50 times.
  1. Initially Sam models the situation by assuming the coin is fair. Determine the number of games Ali is expected to win according to this model. Ali thinks the coin may be biased, with probability \(p\) of obtaining heads when the coin is tossed. Before playing the game, Ali and Sam decide to collect some data to estimate the value of \(p\). Sam tosses the coin 15 times and records the number of heads obtained. Ali tosses the coin 25 times and records the number of heads obtained.
  2. Explain why it is better to use the combined data rather than just Sam's data or just Ali's data to estimate the value of \(p\). Ali records 20 heads and Sam records 8 heads.
  3. Use the combined data to estimate the value of \(p\). Ali now models the situation using the value of \(p\) found in part (c) as the probability of obtaining heads when the coin is tossed.
  4. Determine how many games Ali expects to win using this value of \(p\) to model the situation.
  5. Ali wins 25 of the 50 games. Explain whether Sam's model or Ali's model is a better fit for the data. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its 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 answer-related 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.
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OCR MEI AS Paper 2 2020 November Q1
1 Solve the inequality \(2 x + 5 < 6 x - 3\).
OCR MEI AS Paper 2 2020 November Q2
2 A student measures the upper arm lengths of a sample of 97 women. The results are summarised in the frequency table in Fig. 2.1. \begin{table}[h]
Arm length in cm\(30 -\)\(31 -\)\(32 -\)\(33 -\)\(34 -\)\(35 -\)\(36 -\)\(37 -\)\(38 -\)\(39 -\)\(40 - 41\)
Frequency145913191717435
\captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{table} The student constructs two cumulative frequency diagrams to represent the data using different class intervals. These are shown in Fig. 2.2 opposite One of these diagrams is correct and the other is incorrect.
  1. State which diagram is incorrect, justifying your answer.
  2. Use the correct diagram in Fig. 2.2 to find an estimate of the median. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-05_2256_1230_191_148} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure}
OCR MEI AS Paper 2 2020 November Q3
3 A researcher is conducting an investigation into the number of portions of fruit adults consume each day. The researcher decides to ask 50 men and 50 women to complete a simple questionnaire.
  1. State the type of sampling procedure the researcher is using.
  2. Write down one disadvantage of this sampling procedure. The researcher represents the data in Fig. 3.1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Number of portions of fruit consumed by adults} \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-06_531_991_701_248}
    \end{figure} Fig. 3.1
  3. Describe the shape of the distribution. The data are summarised in the frequency table in Fig. 3.2. \begin{table}[h]
    Number of portions of fruit012345
    Number of adults1834261174
    \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{table}
  4. For the data in Fig. 3.2, use your calculator to find
    • the mean,
    • the standard deviation.
    Give your answers correct to 2 decimal places. A second researcher chooses a proportional stratified sample of 100 children from years 5 and 6 in a certain primary school. There are 220 children to choose from. In year 5 there are 125 children, of whom 81 are boys.
  5. How many year 5 girls should be included in the sample? The second researcher found that the mean number of portions of fruit consumed per day by the children in this sample was 1.61 and the standard deviation was 0.53 .
  6. Comment on the amount of fruit consumed per day by the children compared to the amount of fruit consumed per day by the adults.
OCR MEI AS Paper 2 2020 November Q4
4 In a certain country it is known that 11\% of people are left-handed.
  1. Calculate the probability that, in a random sample of 98 people from this country, 5 or fewer are found to be left-handed, giving your answer correct to 3 significant figures. An anthropologist believes that the proportion of left-handed people is lower in a particular ethnic group. The anthropologist collects a random sample of 98 people from this particular ethnic group in order to test the hypothesis that the proportion of left-handed people is less than \(11 \%\). The anthropologist carries out the test at the \(1 \%\) level.
  2. Determine the critical region for this test.
OCR MEI AS Paper 2 2020 November Q5
5 A company needs to appoint 3 new assistants. 8 candidates are invited for interview; each candidate has a different surname. The candidates are to be interviewed one after another. The personnel officer randomly selects the order in which the candidates are to be interviewed by drawing their names out of a hat. One of the candidates is called Mr Browne and another is called Mrs Green.
  1. Calculate the probability that Mr Browne is interviewed first and Mrs Green is interviewed last. 5 of the 8 candidates invited for interview are women and the other 3 are men. The chief executive can't make up his mind who to appoint so he randomly selects 3 candidates by drawing their names out of a hat.
  2. Determine the probability that more women than men are selected.
OCR MEI AS Paper 2 2020 November Q6
6 Use integration to show that the area bounded by the \(x\)-axis and the curve with equation \(y = ( x - 1 ) ^ { 2 } ( x - 3 )\) is \(\frac { 4 } { 3 }\) square units.
OCR MEI AS Paper 2 2020 November Q8
8 In this question you must show detailed reasoning.
Solve the equation \(3 \cos \theta + 8 \tan \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your answers correct to the nearest degree.
OCR MEI AS Paper 2 2020 November Q9
9 The equation of a curve is \(y = 24 \sqrt { x } - 8 x ^ { \frac { 3 } { 2 } } + 16\).
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
  2. Find the coordinates of the turning point.
  3. Determine the nature of the turning point.
OCR MEI AS Paper 2 2020 November Q10
10 Fig. 10.1 shows a sample collected from the large data set. BMI is defined as \(\frac { \text { mass of person in kilograms } } { \text { square of person's height in metres } }\). \begin{table}[h]
SexAge in yearsMass in kgHeight in cmBMI
Male3877.6164.828.57
Male1763.5170.321.89
Male1868.0172.322.91
Male1857.2172.219.29
Male1977.6191.221.23
Male2472.7177.023.21
Male2592.5177.929.23
Male2670.4159.427.71
Male3177.5174.025.60
Male34132.4182.239.88
Male38115.0186.433.10
Male40112.1171.738.02
\captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{table}
  1. Calculate the mass in kg of a person with a BMI of 23.56 and a height of 181.6 cm , giving your answer correct to 1 decimal place. Fig. 10.2 shows a scatter diagram of BMI against age for the data in the table. A line of best fit has also been drawn. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-09_682_1212_351_248} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{figure}
  2. Describe the correlation between age and BMI.
  3. Use the line of best fit to estimate the BMI of a 30-year-old man.
  4. Explain why it would not be sensible to use the line of best fit to estimate the BMI of a 60-year-old man.
  5. Use your knowledge of the large data set to suggest two reasons why the sample data in the table may not be representative of the population.
  6. Once the data in the large data set had been cleaned there were 196 values available for selection. Describe how a sample of size 12 could be generated using systematic sampling so that each of the 196 values could be selected in the sample.
OCR MEI AS Paper 2 2020 November Q11
11 A car is travelling along a stretch of road at a steady speed of \(11 \mathrm {~ms} ^ { - 1 }\).
The driver accelerates, and \(t\) seconds after starting to accelerate the speed of the car, \(V\), is modelled by the formula
\(\mathrm { V } = \mathrm { A } + \mathrm { B } \left( 1 - \mathrm { e } ^ { - 0.17 \mathrm { t } } \right)\).
When \(t = 3 , V = 13.8\).
  1. Find the values of \(A\) and \(B\), giving your answers correct to 2 significant figures. When \(t = 4 , V = 14.5\) and when \(t = 5 , V = 14.9\).
  2. Determine whether the model is a good fit for these data.
  3. Determine the acceleration of the car according to the model when \(t = 5\), giving your answer correct to 3 decimal places. The car continues to accelerate until it reaches its maximum speed.
    The speed limit on this road is \(60 \mathrm { kmh } ^ { - 1 }\). All drivers who exceed this speed limit are recorded by a speed camera and automatically fined \(\pounds 100\).
  4. Determine whether, according to the model, the driver of this car is fined \(\pounds 100\).
OCR MEI AS Paper 2 2021 November Q1
1 Find the coefficient of \(x ^ { 4 }\) in the expansion of \(( 1 + 3 x ) ^ { 6 }\).
OCR MEI AS Paper 2 2021 November Q2
2 Mia rolls a six-sided die 24 times and records the scores. She displays her results in a vertical line chart. This is shown in Fig. 2.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-03_534_1168_648_242} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure}
  1. Describe the shape of the distribution. She repeats the experiment, but this time she rolls the die 50 times. Her results are displayed in Fig. 2.2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Scores on a six-sided die} \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-03_476_1161_1617_242}
    \end{figure} Fig. 2.2 Her brother Kai rolls the same die 1000 times and displays his results in a similar diagram.
  2. Assuming the die is fair, describe the distribution you would expect to see in Kai's diagram.
OCR MEI AS Paper 2 2021 November Q4
4 Find \(\int \left( 9 x ^ { 2 } + \frac { 6 } { \sqrt { x } } \right) \mathrm { d } x\).
OCR MEI AS Paper 2 2021 November Q5
5 In 2019 scientists developed a model for comparing the ages of humans and dogs.
According to the model,
\(Y = A \ln X + B\)
where \(X =\) dog age in years and \(Y =\) human age in years.
For the model, it is known that when \(X = 1 , Y = 31\) and when \(X = 12 , Y = 71\).
  1. Find the value of \(B\).
  2. Determine the value of \(A\), correct to the nearest whole number. Use the model, with the exact value of \(B\) and the value of \(A\) correct to the nearest whole number, to answer parts (c) and (d).
  3. Find the human age corresponding to a dog age of 20 years.
  4. Determine the dog age corresponding to a human age of 120 years.
OCR MEI AS Paper 2 2021 November Q6
6 The probability distribution for the discrete random variable \(X\) is shown below.
\(x\)0123
\(\mathrm { P } ( X = x )\)\(3 p ^ { 2 }\)\(0.5 p ^ { 2 } + 2 p\)\(1.5 p\)\(1.5 p ^ { 2 } + 0.5 p\)
  1. Determine the value of \(p\).
  2. Determine the modal value of \(X\).
OCR MEI AS Paper 2 2021 November Q7
7 The pre-release material contains information about health expenditure. Fig. 7.1 shows an extract from the data. \begin{table}[h]
CountryHealth expenditure (\% of GDP)
Algeria7.2
Egypt5.6
Libya5
Morocco5.9
Sudan8.4
Tunisia7
Western Sahara\#N/A
Angola3.3
Benin4.6
Botswana5.4
Burkina Faso5
\captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{table}
  1. Explain how the data should be cleaned before any analysis takes place. Kareem uses all the available data to conduct an investigation into health expenditure as a percentage of GDP in different countries. He calculates the mean to be 6.79 and the standard deviation to be 2.78 . Fig. 7.2 shows the smallest values and the largest values of health expenditure as a percentage of GDP. \begin{table}[h]
    Smallest values of Health expenditure (\% of GDP)Largest values of Health expenditure (\% of GDP)
    1.511.7
    1.911.9
    2.113.7
    13.7
    16.5
    17.1
    17.1
    \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{table}
  2. Determine which of these values are outliers. Kareem removes the outliers from the data and finds that there are 187 values left. He decides to collect a sample of size 30 . He uses the following sampling procedure.
    Assign each value a number from 1 to 187. Generate a random number, \(n\), between 1 and 13 . Starting with the \(n\)th value, choose every 6th value after that until 30 values have been chosen.
  3. Explain whether Kareem is using simple random sampling.
OCR MEI AS Paper 2 2021 November Q8
8 With respect to an origin O , the position vectors of the points A and B are
\(\overrightarrow { \mathrm { OA } } = \binom { - 3 } { 20 }\) and \(\overrightarrow { \mathrm { OB } } = \binom { 6 } { 8 }\).
  1. Determine whether \(| \overrightarrow { \mathrm { AB } } | > 200\). The point C is such that \(\overrightarrow { \mathrm { AC } } = \binom { 18 } { - 24 }\).
  2. Determine whether \(\mathrm { A } , \mathrm { B }\) and C are collinear.
OCR MEI AS Paper 2 2021 November Q9
9 Arun, Beth and Charlie are investigating whether there is any association between death rate per 1000 and physician density per 1000. They each collect a random sample of size 10. Arun’s sample is shown in Fig.9.1. \begin{table}[h]
death rate per 1000physician density per 1000
Canberra7.23.62
Dhaka5.30.49
Brasilia6.82.23
Yaounde9.30.08
Zagreb12.53.08
Tehran5.41.16
Rome10.74.14
Tripoli3.82.09
Oslo7.94.51
Abuja9.70.35
\captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{table}
  1. Explain whether or not Arun collected his data from the pre-release material, or whether it is not possible to say. Beth and Charlie collected their samples from the pre-release material. Each of them drew a scatter diagram for their samples. The samples and scatter diagrams are shown in Figs. 9.2 and 9.3.
    Beth's sampledeath rate per 1000physician density per 1000
    Sudan6.70.41
    Cambodia7.40.17
    Gabon6.20.36
    Seychelles70.95
    Mexico5.42.25
    Kuwait2.32.58
    Haiti7.50.23
    Maldives41.04
    Nauru5.91.24
    Jordan3.42.34
    \includegraphics[max width=\textwidth, alt={}]{2b9ce212-84e2-4817-be94-98e2adff12a3-08_545_1024_340_918}
    \begin{table}[h]
    Charlie's sampledeath rate per 1000physician density per 1000
    Vanuata40.17
    Solomon Islands3.80.2
    N. Mariana Islands4.90.36
    Nauru5.91.24
    United Kingdom9.42.81
    Portugal10.63.34
    North Macedonia9.62.87
    Faroe Islands8.82.62
    Bulgaria14.53.99
    St. Kitts and Nevis7.22.52
    \captionsetup{labelformat=empty} \caption{Fig. 9.3}
    \end{table} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 9.2} \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-08_572_899_1400_1041}
    \end{figure} Arun states that Charlie's sample and Beth's sample cannot both be random for the following reasons.
    • Both samples include Nauru - there should not be any common values.
    • Beth's diagram suggests a negative association between death rate and physician density, whereas Charlie's diagram suggests a positive association. If both samples are random the same relationship would be suggested.
    • - Explain whether Arun’s reasons are valid.
    • State whether or not Arun is correct, or whether it is not possible to say.
    Kofi collects a sample of 10 African countries and 10 European countries. The scatter diagram for his results is shown in Fig. 9.4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-09_485_903_902_260} \captionsetup{labelformat=empty} \caption{Fig. 9.4}
    \end{figure}
  2. On the copy of Fig. 9.4 in the Printed Answer Booklet, use your knowledge of the pre-release material to identify the points representing the 10 European countries, justifying your choice.