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OCR MEI Paper 2 2021 November Q11
8 marks Standard +0.3
11 In 2010 the heights of adult women in the UK were found to have mean \(\mu = 161.6 \mathrm {~cm}\) and variance \(\sigma ^ { 2 } = 1.96 \mathrm {~cm} ^ { 2 }\). It is believed that the mean height of adult women in 2020 in the UK is greater than in 2010. In 2020 a researcher collected a random sample of the heights of 200 adult women in the UK. The researcher calculated the sample mean height and carried out a hypothesis test at the \(5 \%\) level to investigate whether there was any evidence to suggest that the mean height of adult women in the UK had increased. The researcher assumed that the variance was unaltered.
  1. - State suitable hypotheses for the test, defining any variables you use.
    • Explain whether the researcher conducted a 1-tail or a 2-tail test.
    • Determine the critical region for the test.
    The researcher found that the sample mean was 161.9 cm and made the following statements.
    • The sample mean is in the critical region.
    • The null hypothesis is accepted.
    • This proves that the mean height of adult women in the UK is unaltered at 161.6 cm .
    • Explain whether each of these statements is correct.
OCR MEI Paper 2 2021 November Q12
5 marks Moderate -0.5
12 Fig. 12.1 shows an excerpt from the pre-release material. \begin{table}[h]
ABCDEFGH
1SexAgeMaritalWeightHeightBMIWaistPulse
2Female34Married60.3173.420.0582.574
3Female85Widowed64.7161.224.9\#N/A\#N/A
4Female48Divorced100.6171.434.24105.692
5Male61Married70.9169.524.6892.270
6Male68Divorced96.8181.629.35112.968
\captionsetup{labelformat=empty} \caption{Fig. 12.1}
\end{table} There was no data available for cell H3.
  1. Explain why \#N/A is used when no data is available. Fig. 12.2 shows a scatter diagram of pulse rate against BMI (Body Mass Index) for females. All the available data was used. Pulse rate against BMI for females \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c9d14a4d-a1c8-42ad-9c0b-42cef6b3612f-08_659_1552_1363_233} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
    \end{figure} There are two outliers on the diagram.
  2. On the copy of Fig. 12.2 in the Printed Answer Booklet, ring these outliers.
  3. Use your knowledge of the pre-release material to explain whether either of these outliers should be removed.
  4. State whether the diagram suggests there is any correlation between pulse rate and BMI. The product moment correlation coefficient between waist measurement, \(w\), in cm and BMI, \(b\), for females was found to be 0.912 . All the available data was used.
  5. Explain why a model of the form \(\mathrm { w } = \mathrm { mb } + \mathrm { c }\) for the relationship between waist measurement and BMI is likely to be appropriate. The LINEST function on a spreadsheet gives \(m = 2.16\) and \(c = 33.0\).
  6. Calculate an estimate of the value for cell G3 in Fig. 12.1.
OCR MEI Paper 2 2021 November Q13
7 marks Moderate -0.3
13 At a certain factory Christmas tree decorations are packed in boxes of 10 . The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged. His results are displayed in Fig. 13.1. \begin{table}[h]
Number of damaged decorations012345 or more
Number of boxes1935281350
\captionsetup{labelformat=empty} \caption{Fig. 13.1}
\end{table}
  1. Calculate
    • the mean number of damaged decorations per box,
    • the standard deviation of the number of damaged decorations per box.
    It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )\).
  2. State suitable values for \(n\) and \(p\).
  3. Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to \(\mathbf { 1 }\) decimal place. \begin{table}[h]
    Number of damaged decorations012345 or more
    Observed number of boxes1935281350
    Expected number of boxes
    \captionsetup{labelformat=empty} \caption{Fig. 13.2}
    \end{table}
  4. Explain whether the model is a good fit for these data.
OCR MEI Paper 2 2021 November Q14
13 marks Moderate -0.3
14 The equation of a curve is
\(y = x ^ { 2 } ( x - 2 ) ^ { 3 }\).
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\), giving your answer in factorised form.
  2. Determine the coordinates of the stationary points on the curve. In part (c) you may use the result \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = 4 ( x - 2 ) \left( 5 x ^ { 2 } - 8 x + 2 \right)\).
  3. Determine the nature of the stationary points on the curve.
  4. Sketch the curve.
OCR MEI Paper 2 2021 November Q15
11 marks Moderate -0.8
15
  1. Show that \(\sum _ { r = 1 } ^ { \infty } 0.99 ^ { r - 1 } \times 0.01 = 1\). Kofi is a very good table tennis player. Layla is determined to beat him.
    Every week they play one match of table tennis against each other. They will stop playing when Layla wins the match for the first time.
    \(X\) is the discrete random variable "the number of matches they play in total". Kofi models the situation using the probability function
    \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = 0.99 ^ { \mathrm { r } - 1 } \times 0.01 \quad r = 1,2,3,4 , \ldots\) Kofi states that he is \(95 \%\) certain that Layla will not beat him within 6 years.
  2. Determine whether Kofi's statement is consistent with his model. In between matches, Layla practises, but Kofi does not.
  3. Explain why Layla might disagree with Kofi's model. Layla models the situation using the probability function
    \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 } \quad r = 1,2,3,4,5,6,7,8\).
  4. Explain how Layla's model takes into account the fact that she practises between matches, but Kofi’s does not. Layla states that she is \(95 \%\) certain that she will beat Kofi within the first 6 matches.
  5. Determine whether Layla’s statement is consistent with her model.
OCR MEI Paper 2 2021 November Q16
8 marks Standard +0.8
16 In this question you must show detailed reasoning.
Find \(\int \frac { x } { 1 + \sqrt { x } } d x\). END OF QUESTION PAPER
OCR MEI Paper 2 Specimen Q2
4 marks Moderate -0.8
2 Given that \(\mathrm { f } ( x ) = x ^ { 3 }\) and \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 1\), describe a sequence of two transformations which maps the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { g } ( x )\).
OCR MEI Paper 2 Specimen Q3
3 marks Moderate -0.8
3 Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 12 } } \cos 3 x \mathrm {~d} x\), giving your answer in exact form.
OCR MEI Paper 2 Specimen Q4
5 marks Moderate -0.8
4 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 4\) for \(- 1 \leq x \leq 2\).
For \(\mathrm { f } ^ { - 1 } ( x )\), determine
  • the domain
  • the range.
OCR MEI Paper 2 Specimen Q5
2 marks Moderate -0.8
5 In a particular country, \(8 \%\) of the population has blue eyes. A random sample of 20 people is selected from this population.
Find the probability that exactly two of these people have blue eyes.
OCR MEI Paper 2 Specimen Q6
4 marks Moderate -0.8
6 Each day, for many years, the maximum temperature in degrees Celsius at a particular location is recorded. The maximum temperatures for days in October can be modelled by a Normal distribution. The appropriate Normal curve is shown in Fig. 6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-05_761_1397_484_246} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
    1. Use the model to write down the mean of the maximum temperatures.
    2. Explain why the curve indicates that the standard deviation is approximately 3 degrees Celsius. Temperatures can be converted from Celsius to Fahrenheit using the formula \(F = 1.8 C + 32\), where \(F\) is the temperature in degrees Fahrenheit and \(C\) is the temperature in degrees Celsius.
  2. For maximum temperature in October in degrees Fahrenheit, estimate
    • the mean
    • the standard deviation.
    \begin{displayquote} Answer all the questions.
    Section B (77 marks) \end{displayquote} \(7 \quad\) Two events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.5\) and \(\mathrm { P } ( A \cup B ) = 0.85\). Find \(\mathrm { P } ( A \mid B )\).
OCR MEI Paper 2 Specimen Q8
3 marks Standard +0.3
8 Alison selects 10 of her male friends. For each one she measures the distance between his eyes. The distances, measured in mm , are as follows:
51575859616464656768
The mean of these data is 61.4 . The sample standard deviation is 5.232 , correct to 3 decimal places. One of the friends decides he does not want his measurement to be used. Alison replaces his measurement with the measurement from another male friend. This increases the mean to 62.0 and reduces the standard deviation. Give a possible value for the measurement which has been removed and find the measurement which has replaced it.
OCR MEI Paper 2 Specimen Q9
4 marks Moderate -0.8
9 A geyser is a hot spring which erupts from time to time. For two geysers, the duration of each eruption, \(x\) minutes, and the waiting time until the next eruption, \(y\) minutes, are recorded.
  1. For a random sample of 50 eruptions of the first geyser, the correlation coefficient between \(x\) and \(y\) is 0.758 .
    The critical value for a 2 -tailed hypothesis test for correlation at the \(5 \%\) level is 0.279 . Explain whether or not there is evidence of correlation in the population of eruptions. The scatter diagram in Fig. 9 shows the data from a random sample of 50 eruptions of the second geyser. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-07_794_1298_383_251} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure}
  2. Stella claims the scatter diagram shows evidence of correlation between duration of eruption and waiting time. Make two comments about Stella's claim.
OCR MEI Paper 2 Specimen Q10
3 marks Easy -1.2
10 A researcher wants to find out how many adults in a large town use the internet at least once a week. The researcher has formulated a suitable question to ask. For each of the following methods of taking a sample of the adults in the town, give a reason why the method may be biased. Method A: Ask people walking along a particular street between 9 am and 5 pm on one Monday.
Method B: Put the question through every letter box in the town and ask people to send back answers.
Method C: Put the question on the local council website for people to answer online.
OCR MEI Paper 2 Specimen Q11
4 marks Moderate -0.5
11 Suppose \(x\) is an irrational number, and \(y\) is a rational number, so that \(y = \frac { m } { n }\), where \(m\) and \(n\) are integers and \(n \neq 0\).
Prove by contradiction that \(x + y\) is not rational.
OCR MEI Paper 2 Specimen Q12
6 marks Standard +0.8
12 Fig. 12 shows the curve \(2 x ^ { 3 } + y ^ { 3 } = 5 y\).
\includegraphics[max width=\textwidth, alt={}, center]{e9f3a5f3-210b-453d-9ff5-8518340f5689-08_841_606_900_212}
  1. Find the gradient of the curve \(2 x ^ { 3 } + y ^ { 3 } = 5 y\) at the point \(( 1,2 )\), giving your answer in exact form.
  2. Show that all the stationary points of the curve lie on the \(y\)-axis.
OCR MEI Paper 2 Specimen Q13
6 marks Standard +0.8
13 Evaluate \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + \sqrt { x } } \mathrm {~d} x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.
OCR MEI Paper 2 Specimen Q14
12 marks Standard +0.3
14 In a chemical reaction, the mass \(m\) grams of a chemical at time \(t\) minutes is modelled by the differential equation
\(\frac { \mathrm { d } m } { \mathrm {~d} t } = \frac { m } { t ( 1 + 2 t ) }\).
At time 1 minute, the mass of the chemical is 1 gram.
  1. Solve the differential equation to show that \(m = \frac { 3 t } { ( 1 + 2 t ) }\).
  2. Hence
    1. find the time when the mass is 1.25 grams,
    2. show what happens to the mass of the chemical as \(t\) becomes large.
OCR MEI Paper 2 Specimen Q15
15 marks Standard +0.3
15 A quality control department checks the lifetimes of batteries produced by a company. The lifetimes, \(x\) minutes, for a random sample of 80 'Superstrength' batteries are shown in the table below.
Lifetime\(160 \leq x < 165\)\(165 \leq x < 168\)\(168 \leq x < 170\)\(170 \leq x < 172\)\(172 \leq x < 175\)\(175 \leq x < 180\)
Frequency5142021164
  1. Estimate the proportion of these batteries which have a lifetime of at least 174.0 minutes.
  2. Use the data in the table to estimate
    • the sample mean,
    • the sample standard deviation.
    The data in the table on the previous page are represented in the following histogram, Fig 15: \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-10_728_1577_312_315} \captionsetup{labelformat=empty} \caption{Fig. 15}
    \end{figure} A quality control manager models the data by a Normal distribution with the mean and standard deviation you calculated in part (b).
  3. Comment briefly on whether the histogram supports this choice of model.
    1. Use this model to estimate the probability that a randomly selected battery will have a lifetime of more than 174.0 minutes.
    2. Compare your answer with your answer to part (a). The company also manufactures 'Ultrapower' batteries, which are stated to have a mean lifetime of 210 minutes.
  5. A random sample of 8 Ultrapower batteries is selected. The mean lifetime of these batteries is 207.3 minutes. Carry out a hypothesis test at the \(5 \%\) level to investigate whether the mean lifetime is as high as stated. You should use the following hypotheses \(\mathrm { H } _ { 0 } : \mu = 210 , \mathrm { H } _ { 1 } : \mu < 210\), where \(\mu\) represents the population mean for Ultrapower batteries. You should assume that the population is Normally distributed with standard deviation 3.4.
    [0pt] [5]
OCR MEI Paper 2 Specimen Q16
19 marks Easy -1.2
16 Fig. 16.1, Fig. 16.2 and Fig. 16.3 show some data about life expectancy, including some from the pre-release data set. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Life expectancy at birth 1974 for 193 countries} \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-11_520_1671_479_182}
\end{figure} Life expectancy at birth 2014 for 222 countries \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 16.1} \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-11_589_1477_1324_349}
\end{figure} Fig. 16.2
\includegraphics[max width=\textwidth, alt={}, center]{e9f3a5f3-210b-453d-9ff5-8518340f5689-11_207_828_2213_310} Increase in life expectancy from 1974 to 2014 (years) \begin{table}[h]
Increase in life expectancy for
193 countries from 1974 to 2014
Number of values193
Minimum- 4.618
Lower quartile6.9576
Median9.986
Upper quartile15.873
Maximum30.742
\captionsetup{labelformat=empty} \caption{Fig. 16.3}
\end{table} Source: CIA World
Factbook and
  1. Comment on the shapes of the distributions of life expectancy at birth in 2014 and 1974.
    1. The minimum value shown in the box plot is negative. What does a negative value indicate?
    2. What feature of Fig 16.3 suggests that a Normal distribution would not be an appropriate model for increase in life expectancy from one year to another year?
    3. Software has been used to obtain the values in the table in Fig. 16.3. Decide whether the level of accuracy is appropriate. Justify your answer.
    4. John claims that for half the people in the world their life expectancy has improved by 10 years or more.
      Explain why Fig. 16.3 does not provide conclusive evidence for John's claim.
  3. Decide whether the maximum increase in life expectancy from 1974 to 2014 is an outlier. Justify your answer. Here is some further information from the pre-release data set.
    Country
    Life expectancy
    at birth in 2014
    Ethiopia60.8
    Sweden81.9
    1. Estimate the change in life expectancy at birth for Ethiopia between 1974 and 2014.
    2. Estimate the change in life expectancy at birth for Sweden between 1974 and 2014.
    3. Give one possible reason why the answers to parts (i) and (ii) are so different. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Fig. 16.4 shows the relationship between life expectancy at birth in 2014 and 1974.} \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-13_981_1520_333_292}
      \end{figure} Fig. 16.4 A spreadsheet gives the following linear model for all the data in Fig 16.4.
      (Life expectancy at birth 2014) \(= 30.98 + 0.67 \times\) (Life expectancy at birth 1974) The life expectancy at birth in 1974 for the region that now constitutes the country of South Sudan was 37.4 years. The value for this country in 2014 is not available.
    1. Use the linear model to estimate the life expectancy at birth in 2014 for South Sudan.
    2. Give two reasons why your answer to part (i) is not likely to be an accurate estimate for the life expectancy at birth in 2014 for South Sudan.
      You should refer to both information from Fig 16.4 and your knowledge of the large data set.
  6. In how many of the countries represented in Fig. 16.4 did life expectancy drop between 1974 and 2014? Justify your answer.
OCR MEI Paper 3 2018 June Q1
3 marks Easy -1.2
1 Triangle ABC is shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-4_451_565_520_744} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Find the perimeter of triangle ABC .
OCR MEI Paper 3 2018 June Q2
2 marks Easy -1.2
2 The curve \(y = x ^ { 3 } - 2 x\) is translated by the vector \(\binom { 1 } { - 4 }\). Write down the equation of the translated curve. [2]
OCR MEI Paper 3 2018 June Q3
2 marks Challenging +1.2
3 Fig. 3 shows a circle with centre O and radius 1 unit. Points A and B lie on the circle with angle \(\mathrm { AOB } = \theta\) radians. C lies on AO , and BC is perpendicular to AO . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-4_648_627_1507_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Show that, when \(\theta\) is small, \(\mathrm { AC } \approx \frac { 1 } { 2 } \theta ^ { 2 }\).
OCR MEI Paper 3 2018 June Q4
10 marks Standard +0.3
4 In this question you must show detailed reasoning.
A curve has equation \(y = x - 5 + \frac { 1 } { x - 2 }\). The curve is shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-5_723_844_424_612} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Determine the coordinates of the stationary points on the curve.
  2. Determine the nature of each stationary point.
  3. Write down the equation of the vertical asymptote.
  4. Deduce the set of values of \(x\) for which the curve is concave upwards.
OCR MEI Paper 3 2018 June Q5
11 marks Moderate -0.3
5 A social media website launched on 1 January 2017. The owners of the website report the number of users the site has at the start of each month. They believe that the relationship between the number of users, \(n\), and the number of months after launch, \(t\), can be modelled by \(n = a \times 2 ^ { k t }\) where \(a\) and \(k\) are constants.
  1. Show that, according to the model, the graph of \(\log _ { 10 } n\) against \(t\) is a straight line.
  2. Fig. 5 shows a plot of the values of \(t\) and \(\log _ { 10 } n\) for the first seven months. The point at \(t = 1\) is for 1 February 2017, and so on. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-6_831_1442_609_388} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} Find estimates of the values of \(a\) and \(k\).
  3. The owners of the website wanted to know the date on which they would report that the website had half a million users. Use the model to estimate this date.
  4. Give a reason why the model may not be appropriate for large values of \(t\).