Questions — OCR MEI Paper 2 (127 questions)

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OCR MEI Paper 2 2023 June Q6
6 The parametric equations of a circle are
\(x = 2 \cos \theta - 3\) and \(y = 2 \sin \theta + 1\).
Determine the cartesian equation of the circle in the form \(( \mathrm { x } - \mathrm { a } ) ^ { 2 } + ( \mathrm { y } - \mathrm { b } ) ^ { 2 } = \mathrm { k }\), where \(a , b\) and \(k\) are integers.
OCR MEI Paper 2 2023 June Q7
7 The coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 x + k ) ^ { 12 }\), where \(k\) is a positive integer, is 79200000.
Determine the value of \(k\).
OCR MEI Paper 2 2023 June Q8
8 A garden centre stocks coniferous hedging plants. These are displayed in 10 rows, each of 120 plants. An employee collects a sample of the heights of these plants by recording the height of each plant on the front row of the display.
  1. Explain whether the data collected by the employee is a simple random sample. The data are shown in the cumulative frequency curve below.
    \includegraphics[max width=\textwidth, alt={}, center]{11788aaf-98fb-4a78-8a40-a40743b1fe15-06_1376_1344_680_233} The owner states that at least \(75 \%\) of the plants are between 40 cm and 80 cm tall.
  2. Show that the data collected by the employee supports this statement.
  3. Explain whether all samples of 120 plants would necessarily support the owner's statement.
OCR MEI Paper 2 2023 June Q9
9 The pre-release material contains information concerning the median income of taxpayers in different areas of London. Some of the data for Camden is shown in the table below. The years quoted in this question refer to the end of the financial years used in the pre-release material. For example, the year 2004 in the table refers to the year 2003/04 in the pre-release material.
Year20042005200620072008200920102011
Median
Income in \(\pounds\)
2130023200242002590026900\#N/A2840029400
  1. Explain whether these data are a sample or a population of Camden taxpayers. A time series for the data is shown below. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Median income of taxpayers in Camden 2004-2011} \includegraphics[alt={},max width=\textwidth]{11788aaf-98fb-4a78-8a40-a40743b1fe15-07_624_1469_950_242}
    \end{figure} The LINEST function on a spreadsheet is used to formulate the following model for the data:
    \(I = 1115 Y - 2212950\), where \(I =\) median income of taxpayers in \(\pounds\) and \(Y =\) year.
  2. Use this model to find an estimate of the median income of taxpayers in Camden in 2009.
  3. Give two reasons why this estimate is likely to be close to the true value. The median income of taxpayers in Croydon in 2009 is also not available.
  4. Use your knowledge of the pre-release material to explain whether the model used in part (b) would give a reasonable estimate of the missing value for Croydon.
OCR MEI Paper 2 2023 June Q10
10 Determine the exact value of \(\int _ { 0 } ^ { \frac { \pi } { 4 } } 4 x \cos 2 x d x\).
OCR MEI Paper 2 2023 June Q11
11 In this question you must show detailed reasoning.
The variables \(x\) and \(y\) are such that \(\frac { \mathrm { dy } } { \mathrm { dx } }\) is directly proportional to the square root of \(x\).
When \(x = 4 , \frac { d y } { d x } = 3\).
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) in terms of \(x\). When \(\mathrm { x } = 4 , \mathrm { y } = 10\).
  2. Find \(y\) in terms of \(x\).
OCR MEI Paper 2 2023 June Q12
12 It is given that
  • \(\mathrm { f } ( x ) = \pm \frac { 1 } { \sqrt { x } } , x > 0\)
  • \(\mathrm { g } ( x ) = \frac { x } { x - 3 } , x > 3\)
  • \(\mathrm { h } ( x ) = x ^ { 2 } + 2 , x \in \mathbb { R }\).
    1. Explain why \(\mathrm { f } ( x )\) is not a function.
    2. Find \(\mathrm { gh } ( x )\).
    3. State the domain of \(\mathrm { gh } ( x )\).
OCR MEI Paper 2 2023 June Q13
13 A large supermarket chain advertises that the mean mass of apples of a certain variety on sale in their stores is 0.14 kg . Following a poor growing season, the head of quality control believes that the mean mass of these apples is less than 0.14 kg and she decides to carry out a hypothesis test at the \(5 \%\) level of significance. She collects a random sample of this variety of apple from the supermarket chain and records the mass, in kg, of each apple. She uses software to analyse the data. The results are summarised in the output below.
\(n\)80
Mean0.1316
\(\sigma\)0.0198
\(s\)0.0199
\(\Sigma x\)10.525
\(\Sigma x ^ { 2 }\)1.4161
Min0.1
Q10.12
Median0.132
Q30.1435
Max0.19
  1. State the null hypothesis and the alternative hypothesis for the test, defining the parameter used.
  2. Write down the distribution of the sample mean for this hypothesis test.
  3. Determine the critical region for the test.
  4. Carry out the test, giving your conclusion in context.
OCR MEI Paper 2 2023 June Q14
14 The pre-release material contains information concerning the median income of taxpayers in \(\pounds\) and the percentage of all pupils at the end of KS4 achieving 5 or more GCSEs at grade A*-C, including English and Maths, for different areas of London. Some of the data for 2014/15 is shown in Fig. 14.1. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Fig. 14.1}
Median Income of Taxpayers in £Percentage of Pupils Achieving 5 or more A*-C, including English and Maths
City of London61100\#N/A
Barking and Dagenham2180054.0
Barnet2710070.1
Bexley2440055.0
Brent2270060.0
Bromley2810068.0
\end{table} A student investigated whether there is any relationship between median income of taxpayers and percentage of pupils achieving 5 or more GCSEs at grade A*-C, including English and Maths.
  1. With reference to Fig. 14.1, explain how the data should be cleaned before any analysis can take place. After the data was cleaned, the student used software to draw the scatter diagram shown in Fig. 14.2. Scatter diagram to show percentage of pupils achieving 5 A*-C grades against median income of taxpayers \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 14.2} \includegraphics[alt={},max width=\textwidth]{11788aaf-98fb-4a78-8a40-a40743b1fe15-10_574_1481_1900_241}
    \end{figure} The student calculated that the product moment correlation coefficient for these data is 0.3743 .
  2. Give two reasons why it may not be appropriate to use a linear model for the relationship between median income of taxpayers in \(\pounds\) and the percentage of all pupils at the end of KS4 achieving 5 or more GCSEs at grade A*-C. The student carried out some further analysis. The results are shown in Fig. 14.3. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 14.3}
    median income of
    taxpayers in \(\pounds\)
    percentage of pupils
    achieving \(5 + \mathrm { A } ^ { * } - \mathrm { C }\)
    mean2721661.0
    standard deviation4177.55.32
    \end{table} The student identified three outliers in total.
    • Use the information in Fig. 14.3 to determine the range of values of the median income of taxpayers in \(\pounds\) which are outliers.
    • Use the information in Fig. 14.3 to determine the range of values of the percentage of all pupils at the end of KS4 achieving 5 or more GCSEs at grade A*-C which are outliers.
    • On the copy of Fig. 14.2 in the Printed Answer Booklet, circle the three outliers identified by the student.
    The student decided to remove these outliers and recalculate the product moment correlation coefficient.
  4. Explain whether the new value of the product moment correlation coefficient would be between 0.3743 and 1 or between 0 and 0.3743 .
OCR MEI Paper 2 2023 June Q16
16 Research conducted by social scientists has shown that \(16 \%\) of young adults smoke cigarettes. Two young adults are selected at random.
  1. Determine the probability that one smokes cigarettes and the other doesn't. The same research has also shown that
    • 75\% of young adults drink alcohol.
    • \(66 \%\) of young adults drink alcohol, but do not smoke cigarettes.
    • Determine the probability that a young adult selected at random does smoke cigarettes, but does not drink alcohol.
    • A young adult who drinks alcohol is selected at random. Determine the probability that this young adult smokes cigarettes.
    • Using your answer to part (c), explain whether the event that a young adult selected at random smokes cigarettes is independent of the event that a young adult selected at random drinks alcohol.
OCR MEI Paper 2 2023 June Q18
18 Riley is investigating the daily water consumption, in litres, of his household.
He records the amount used for a random sample of 120 days from the previous twelve-month period. The daily water consumption, in litres, is denoted by \(x\). Summary statistics for Riley's sample are given below.
\(\sum \mathrm { x } = 31164.7 \sum \mathrm { x } ^ { 2 } = 8101050.91 \mathrm { n } = 120\)
  1. Calculate the sample mean giving your answer correct to \(\mathbf { 3 }\) significant figures. Riley displays the data in a histogram.
    \includegraphics[max width=\textwidth, alt={}, center]{11788aaf-98fb-4a78-8a40-a40743b1fe15-13_832_1383_934_242}
  2. Find the number of days on which between 255 and 260 litres were used.
  3. Give two reasons why a Normal distribution may be an appropriate model for the daily consumption of water. Riley uses the sample mean and the sample variance, both correct to \(\mathbf { 3 }\) significant figures, as parameters of a Normal distribution to model the daily consumption of water.
  4. Use Riley's model to calculate the probability that on a randomly chosen day the household uses less than 255 litres of water.
  5. Calculate the probability that the household uses less than 255 litres of water on at least 5 days out of a random sample of 28 days. The company which supplies the water makes charges relating to water consumption which are shown in the table below.
    Standing charge per day in pence7.8
    Charge per litre in pence0.18
  6. Adapt Riley's model for daily water consumption to model the daily charges for water consumption. \section*{END OF QUESTION PAPER}
OCR MEI Paper 2 2024 June Q1
1 Calculate the exact distance between the points ( \(2 , - 1\) ) and ( 6,1 ). Give your answer in the form \(\mathrm { a } \sqrt { \mathrm { b } }\), where \(a\) and \(b\) are prime numbers.
OCR MEI Paper 2 2024 June Q2
2 The equation of a curve is \(y = e ^ { x }\). The curve is subject to a translation \(\binom { 3 } { 0 }\) and a stretch scale factor 2 parallel to the \(y\)-axis. Write down the equation of the new curve.
OCR MEI Paper 2 2024 June Q3
3 The histogram shows the amount spent on electricity in pounds in a sample of households in March 2023.
\includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-04_542_1276_1133_244}
  1. Describe the shape of the distribution. A total of 16 households each spent between \(\pounds 60\) and \(\pounds 65\) on electricity.
  2. Determine how many households were in the sample altogether.
OCR MEI Paper 2 2024 June Q4
4
  1. On the axes in the Printed Answer Booklet, sketch the graph of \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  2. Solve the equation \(\sin 2 \theta = - \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
    \(5 M\) is the event that an A-level student selected at random studies mathematics.
    \(C\) is the event that an A-level student selected at random studies chemistry.
    You are given that \(\mathrm { P } ( M ) = 0.42 , \mathrm { P } ( C ) = 0.36\) and \(\mathrm { P } ( \mathrm { M }\) and \(\mathrm { C } ) = 0.24\). These probabilities are shown in the two-way table below.
    \cline { 2 - 4 } \multicolumn{1}{c|}{}\(M\)\(M ^ { \prime }\)Total
    \(C\)0.240.36
    \(C ^ { \prime }\)
    Total0.421
OCR MEI Paper 2 2024 June Q6
6 The probability distribution of the discrete random variable \(X\) is shown in the table.
\(x\)0123
\(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)0.2\(a\)\(3 a\)0.4
  1. Calculate the value of the constant \(a\).
  2. A single value of \(X\) is chosen at random. Find the probability that the value is an odd number.
  3. Two independent values of \(X\) are chosen at random. Calculate the probability that the total of the two values is 3 .
OCR MEI Paper 2 2024 June Q7
7 A sequence is defined by the recurrence relation \(\mathrm { u } _ { \mathrm { k } + 1 } = \mathrm { u } _ { \mathrm { k } } + 5\) with \(\mathrm { u } _ { 1 } = - 2\).
  1. Write down the values of \(u _ { 2 } , u _ { 3 }\), and \(u _ { 4 }\).
  2. Explain whether this sequence is divergent or convergent.
  3. Determine the value of \(u _ { 30 }\).
  4. Determine the value of \(\sum _ { \mathrm { k } = 1 } ^ { 30 } \mathrm { u } _ { \mathrm { k } }\).
OCR MEI Paper 2 2024 June Q8
8 The equation of a curve is \(y = 2 x ^ { 3 } + 3 m x ^ { 2 } - 9 m x + 4\). Determine the range of values of \(m\) for which the curve has no stationary values.
OCR MEI Paper 2 2024 June Q9
9 A teacher is investigating how pupils travel to and from school each day. Pupils can either travel by bus, train, car, bicycle or walk. The teacher decides to collect a sample of size 60 for the investigation.
  1. The teacher lives in a village 10 miles away from the school. Explain how collecting a sample which just consists of pupils who live in the same village as the teacher might introduce bias. The table below shows how many students there are in each year.
    Year 7Year 8Year 9Year 10Year 11
    86105107101101
  2. The teacher decides to use the method of proportional stratified sampling. Calculate the number of pupils in the sample who are in Year 9. The teacher generates a sample of 10 pupils from the 86 in Year 7 by listing them in alphabetical order and selecting the first name on the list and every ninth name thereafter.
  3. Explain whether this method will generate a simple random sample of the pupils who travel in Year 7.
OCR MEI Paper 2 2024 June Q10
10
  1. Determine the first three terms in ascending powers of \(x\) of the binomial expansion of \(( 8 + 3 x ) ^ { \frac { 1 } { 3 } }\).
  2. State the range of values of \(x\) for which this expansion is valid.
OCR MEI Paper 2 2024 June Q11
11 A householder is investigating whether there is any relationship between his monthly cost of gas and his monthly cost of electricity, both measured in pounds ( \(\pounds\) ). The householder collects a random sample of monthly costs and presents them in the scatter diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-08_604_1452_392_244} One of the points on the diagram represents the energy costs in a month when the householder was away on holiday for three weeks. The other points represent the energy costs in months when the householder did not go away on holiday.
  1. On the copy of the diagram in the Printed Answer Booklet, circle the point which represents the month when the householder was most likely to have been away on holiday for three weeks.
  2. With reference to the diagram, describe the relationship between the cost of gas and the cost of electricity. The householder decides to test whether there is evidence to suggest that there is any association between the monthly cost of gas and the monthly cost of electricity. The value of Spearman's rank correlation coefficient for this sample is 0.4359 and the associated \(p\)-value is 0.09195 .
  3. Determine whether there is any evidence to suggest, at the \(5 \%\) level, that there is any association between the monthly cost of gas and the monthly cost of electricity.
OCR MEI Paper 2 2024 June Q12
12 A survey conducted in 2021 showed that 10\% of British adults were vegetarians. A dietitian believes that the proportion of British adults who are vegetarians may have changed, so decides to conduct a hypothesis test at the \(5 \%\) level of significance. In a random sample of 112 adults, the dietitian finds that there are 19 vegetarians. Carry out the hypothesis test to determine whether there is any evidence to support the dietitian’s belief.
OCR MEI Paper 2 2024 June Q13
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
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
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 .